# Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution

Authors: Eunju Sohn; Charles Knessl
Journal: Advances in Operations Research
 (2016)

---

## Abstract

We consider a storage allocation model with a finite number of storage spaces. There arem primary spaces and R secondary spaces. All of them are numbered and ranked. Customers arrive according to a Poisson process and occupy a space for an exponentially distributed time period, and a new arrival takes the lowest ranked available space. We let N1 and N2 denote the numbers of occupied primary and secondary spaces and study the joint distribution Prob[N1=k,N2=r] in the steady state. The joint process (N1,N2) behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rate λ becomes large, and the storage capacities m and R are scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.


---

## Body

## 1. Introduction

We consider the following storage allocation model. There arem primary and R secondary storage spaces. The primary spaces are numbered {1,2,…,m} and the secondary ones are numbered {m+1,m+2,…,m+R}. Customers arrive according to a Poisson process of rate λ, and each customer occupies a storage space for an exponentially distributed amount of time, with the mean occupation time 1/μ. A new arrival takes the lowest ranked available space. If all m+R spaces are filled, then a new arrival is turned away and lost. The policy of taking the lowest ranked space is called “first-fit allocation.”We can consider the storage spaces as parking spaces of a restaurant. The primary spaces are in a lot right next to the restaurant, and the secondary spaces are located somewhere further away from the restaurant. Lower ranked spaces will be closer to the restaurant so it is natural for a customer to use the first-fit policy. Since spaces are occupied and emptied at random times, this model is called a dynamic storage allocation model. Design and analysis of algorithms for dynamic storage allocation are a fundamental part of computer science [1]. In such applications we can consider the customers as records, files, or lists and the storage device as a memory device. As time evolves, items are inserted and deleted, and the storage device, which is a linear array of “cells,” will have regions of occupied cells alternating with interior holes. This is referred to as memory fragmentation in computers, and collapsing the holes corresponds to running a defragmentation program.In the language of queueing theory, the model with finite secondary storage spaces can be called theM/M/(m+R)/(m+R) queue (the Erlang loss model) with ranked servers. The main contribution here is to study the effects of the finite storage capacity, for systems with a large number of both primary and secondary storage spaces and a commensurably large traffic intensity, which we denote by ρ=λ/μ. Thus we study the model asymptotically for ρ→∞ with m,R=Θ(ρ).We letN1 and N2 be the numbers of occupied primary and secondary spaces, and we will focus on the joint distribution of N1 and N2, in the steady state. The distributions of both N1 and N1+N2 are readily computed, as these processes behave as Erlang loss models, with m and m+R servers, respectively. Thus their steady state distributions are truncated Poisson distributions. However, the distribution of the number N2 of occupied secondary spaces is much more complicated, as is the joint distribution Prob[N1=k,N2=r].We focus here on only the steady state distribution but comment that the transient behavior of the standard Erlang loss model can be analyzed by singular perturbation methods of the type employed here (see [2]). Thus we believe that, with significant additional effort, the transient behavior of the joint process (N1, N2) could also be ultimately analyzed.There has been much past work on the model with an infinite (secondary) storage capacity (R=∞) since Kosten [3]. Various aspects of the solution were also studied in [4–7], but the solutions are in a complicated form, which is difficult to evaluate asymptotically for ρ→∞, due to the presence of an alternating sum. We derived the joint steady state distribution of the (N1,N2) process in [8] using a discrete version of the classic method of separation of variables. We obtained the solution as a contour integral that involves certain polynomials related to hypergeometric functions. Such representations enabled us to obtain a complete set of asymptotic results including the joint distribution Prob[N1=k,N2=r], for R=∞ [9–12].The solution of the finite capacity model withR<∞ seems more complicated than the solution of the model with R=∞. But we will show here that a singular perturbation analysis is again fruitful, and we will obtain a complete set of asymptotic results for π(k,r), which depends also parametrically on ρ, and the numbers m and R of primary and secondary storage spaces. Most of the time we will scale all of k, r, m, and R to be of the same order as the traffic intensity ρ. We will focus on understanding the effects of the finiteness of the secondary storage capacity R.The remainder of the paper is organized as follows. In Section2 we state the basic equations and briefly describe their forthcoming analysis. In Section 3 we summarize all of the main results, and the joint distribution will have different asymptotic expansions in three main regions of the state space, which is the lattice rectangle {(k,r):0⩽k⩽m,0⩽r⩽R}. Moreover, there are also various boundary, corner, and transition curves where different expansions will be needed. In Section 4 we derive the asymptotics of the joint distribution in the three main regions, while in Sections 5–7 we treat the boundary, corner, and transition ranges. In Section 8 we will do some numerical comparisons to test the accuracy and robustness of our asymptotic results. Some discussion of our results also appears in Section 8. Since the analysis is quite technical, we have written this paper so that the derivations in Sections 4–7 can be omitted upon a first (and perhaps even later) reading(s).


## 2. Statement of the Problem

We consider a system withm primary and R secondary storage spaces (or servers). The primary spaces are ranked and numbered 1,2,…,m while the secondary spaces are numbered m+1,m+2,…,m+R. Customers arrive according to a Poisson process with rate parameter λ and a new arrival takes the lowest ranked available space, if possible a primary one. If all m+R spaces are occupied further arrivals are turned away and lost. All of the storage spaces are identical and a customer occupies a space for an exponentially distributed amount of time, with the mean occupation (or service) time being 1/μ. We then let N1 and N2 be the numbers of occupied primary and secondary spaces, respectively. We also introduce a dimensionless parameter(1)ρ=λμto denote the traffic intensity.The joint process(N1,N2) corresponds to a continuous time random walk in a lattice rectangle. Figure 1 indicates transition rates. The steady state distribution(2)πk,r=πk,r;ρ,m,R=limt→∞⁡Prob⁡N1t=k,N2t=r∣N10=k0,N20=r0is independent of the initial values N1(0) and N2(0) and satisfies the following balance equations:(3)ρ+k+rπk,r=r+1πk,r+1+k+1πk+1,r+ρπk-1,r,1⩽k⩽m-1,0⩽r⩽R-1,(4)ρ+m+rπm,r=r+1πm,r+1+ρπm-1,r+ρπm,r-1,1⩽r⩽R-1,(5)ρ+k+Rπk,R=k+1πk+1,R+ρπk-1,R,1⩽k⩽m-1,(6)ρ+rπ0,r=r+1π0,r+1+π1,r,0⩽r⩽R-1,(7)ρ+Rπ0,R=π1,R,(8)ρ+mπm,0=ρπm-1,0+πm,1,(9)m+Rπm,R=ρπm,R-1+ρπm-1,R.The main balance equation (3) applies in the interior of the lattice rectangle and along the boundary r=0, (4)–(6) correspond to boundary conditions along three of the four boundaries of the rectangle, and (7)–(9) are corner conditions. Also, (6) applies at r=0 so the corner condition at (0,0) is ρπ(0,0)=π(0,1)+π(1,0). We also have the normalization condition(10)∑k=0m∑r=0Rπk,r=1.Figure 1
Steady state transition rates.The processN1 by itself behaves precisely as the Erlang loss model (M/M/m/m queue with m servers). This is well known to have, in the steady state, a truncated Poisson distribution; hence(11)∑r=0Rπk,r=ρke-ρ/k!∑k=0mρke-ρ/k!,0⩽k⩽m.The total number,N1+N2, of occupied servers also follows a truncated Poisson distribution. Therefore,(12)∑k+r=Lπk,r=∑k=max0,L-Rminm,Lπk,L-k=ρLe-ρ/L!∑l=0m+Rρle-ρ/l!,0⩽L⩽m+R.We recently obtained in [13] explicit expressions for the joint distribution π(k,r), but they are not very insightful due to their complexity. Thus we study the problem asymptotically, for ρ→∞ with m,R=O(ρ). This means that there are many arrivals but the numbers of storage spaces, both primary and secondary ones, are commensurately large. Note that if ρ→∞ with m,R=O(1) then the probability distribution π(k,r) would concentrate on a single lattice point, with π(k,r)→δk,mδr,R as ρ→∞. Here δk,m=0, k≠m, and δk,m=1, k=m. Thus this limit would not be particularly interesting. There are, however, certain cases where either m or R is large but o(ρ), that should lead to interesting results, but we do not analyze them here.We next introduce the parameters(13)X0=mρ,Y0=Rρand in the present limit we have 0<X0, Y0<∞, where we view X0 and Y0 as fixed as ρ→∞. We may then view the process (N1,N2) on a “coarse” spatial scale, with(14)X=kρ,0⩽X⩽X0,Y=rρ,0⩽Y⩽Y0.On the (X,Y) scale the random walk takes small steps (=1/ρ) and the state space may be approximately viewed as the continuous rectangle(15)X,Y:0⩽X⩽X0,0⩽Y⩽Y0. Setting(16)πk,r=πk,r;ρ,m,R=PX,Y=PX,Y;ρ,X0,Y0 the main balance equation (3) becomes(17)1+X+YPX,Y=Y+ρ-1PX,Y+ρ-1+X+ρ-1PX+ρ-1,Y+PX-ρ-1,Ywhich is a difference equation with small differences, of order O(ρ-1). The boundary condition along k=m in (4) may be replaced by the “artificial boundary condition”(18)m+1πm+1,r=ρπm,r-1,1⩽r⩽R-1.The above is obtained by requiring that (3) holds also at k=m and comparing this to (4). Introducing π(m+1,r) simplifies some of the calculations, but this quantity has no physical meaning.The asymptotic structure of the joint distribution will be very different for four main regions in the(X0,Y0) parameter space. We call these regions R1–R4 and they are sketched in Figure 2. They are defined by the inequalities(19)R1=X0,Y0:X0>1,Y0>0,(20)R2=X0,Y0:0<X0<1,Y0+X0>1,(21)R3=X0,Y0:0<X0<1,X0-X0<Y0<1-X0,(22)R4=X0,Y0:0<X0<1,0<Y0<X0-X0.It will also prove useful to define as follows the curves that separate these four regions:(23)R1∩R2=X0,Y0:X0=1,Y0>0,(24)R2∩R3=X0,Y0:X0+Y0=1,0<X0<1,(25)R3∩R4=X0,Y0:X0+Y0=X0,0<X0<1.Note that the union of all the sets in (19)–(25) is the entire open quarter plane in parameter space. We purposefully exclude the coordinate axes X0=0 and Y0=0, as they would require entirely different asymptotic analyses. The separating curves in (23)–(25) will also require separate analyses, and we will obtain results that apply not only along the curves but also in small neighborhoods of these curves, which will be defined precisely later. This will produce results that asymptotically match to those in the main regions.Figure 2
Four regions of the(X0,Y0) parameter space.The presence of the different regions can be explained intuitively. IfX0>1(m>ρ) there are enough primary spaces to service all storage requests and the secondary spaces will generally not be needed. If X0<1 but X0+Y0>1(m+R>ρ) the primary spaces are insufficient but the total number of spaces is adequate. Then we might expect that typically all m primary spaces and about ρ-m(<R) secondary spaces will be occupied. If X0+Y0<1 then typically all primary and secondary spaces will be occupied. Then we might expect π(k,r) to be concentrated near k=m, r=R. The further splitting of X0+Y0<1 into the regions R3 and R4 is difficult to explain intuitively in terms of the basic model, but we will explain this dichotomy via our asymptotic analysis. We also note that the asymptotic behavior of the distribution in (11) undergoes a transition when m/ρ passes through 1, while (12) undergoes an analogous transition when (m+R)/ρ passes through 1. However, neither (11) nor (12) undergoes a transition along R3∩R4. In the analysis that follows we will also need to, for each region of parameter space, separately analyze several different regions of the state space, which corresponds to the rectangle 0⩽X⩽X0, 0⩽Y⩽Y0 on the coarse spatial scale. It will sometimes prove necessary to analyze boundary and corner regions where the discrete nature of the model must be considered.


## 3. Summary of Results

In the analysis it proves sometimes useful to use the variables(n,p) where(26)n=m-k,p=R-r,so that n (resp., p) measures the number ofunoccupied primary (resp., secondary) spaces. Then we also let(27)πk,r=πm-n,R-p=Qn,pso that Q(0,0) corresponds to the probability that all of the storage spaces are full. In (27) we did not indicate the dependence of π and Q on the parameters ρ, m, and R.We begin by giving asymptotic results forQ(0,0).Proposition 1.
Forρ→∞ and fixed X0,Y0 one has(28)πm,R=Q0,0~1-X0-Y0,X0,Y0∈R3∪R4,(29)Q0,0~1ρe-γ2/2∫-∞γe-u2/2du,X0+Y0=1+γρ,γ=O1R2∩R3,(30)Q0,0~1ρ12πX0+Y0eρX0+Y0-1-X0+Y0logX0+Y0,X0,Y0∈R1∪R2.Here and throughout the paper, we use the convention thatR1∪R2 corresponds to the union of the open sets R1 and R2 and also the separating curve R1∩R2 (cf. (23)). Similar comments apply for R3∪R4 and R2∪R3. We refer to the asymptotic limit in (29) as corresponding to R2∩R3, where we now give the precise scaling, X0+Y0-1=O(ρ-1/2), that applies near the separating curve in (24). The results in (28)–(30) will follow from our asymptotic analysis of the joint distribution, but we note that these also follow easily from (12), by setting N=m+R and expanding the result for ρ→∞ and different ranges of Z0=X0+Y0 (thus Z0=N/ρ).It will prove convenient to express some of our results in terms of the three constantsC, C0, and C1; these depend only on the parameters ρ, X0, and Y0. We summarize below the leading order asymptotics of these constants.Proposition 2.
Define the constantsC, C1, and C0 by the relations(31)C~ρ-1/2,X0,Y0∈R1∪R2,(32)C~2πρ∫-∞γe-u2/2du-1,X0,Y0∈R2∩R3,(33)C~2πX0+Y01-X0-Y0eρ1-X0-Y0+X0+Y0log⁡X0+Y0,X0,Y0∈R3,(34)C1~ρ-2/3,X0,Y0∈R2,(35)C1~ρ-2/32π∫-∞γe-u2/2du,X0,Y0∈R2∩R3,(36)C1~ρ1/62πX0+Y01-X0-Y0eρ1-X0-Y0+X0+Y0logX0+Y0,X0,Y0∈R3,(37)C1~ρ-1/61-X0-Y0e-ρΦX0,Y02πX01/4∫r0∞eδ1uAiudu,(38)X0+Y0-X0=ρ-1/3δ∗=Oρ-1/3R3∩R4,(39)δ1=δ∗X01/61-X01/3,ΦX0,Y0=Y0+W0log⁡Y0+W0-Y0log⁡Y0-W02log⁡W0-X02log⁡X0+X0-1,W0=1-X02,where Ai(·) is the Airy function and r0 its maximal root(40)r0=max⁡z:Aiz=0=-2.3381⋯.(41)C1~ρ-5/62π1-X0-Y0Ai′r0e-ρΦX0,Y0e-ρ1/3Φ∗Y0Y0X01/31-X02/3X0-X0-Y02W0+Y0Y0+W01-X01/2X0,(42)Φ∗Y0=-r01-X02/3X0-1/6log⁡Y0+W01-X0,X0,Y0∈R4,(43)C0~ρ-1/21-X0-Y0,X0,Y0∈R3∪R4,(44)C0~ρ-1e-γ2/2∫-∞γe-u2/2du,X0,Y0∈R2∩R3,(45)C0~ρ-12πX0+Y0-1/2eρX0+Y0-1-X0+Y0logX0+Y0,X0,Y0∈R1∪R2.We note that the relationC0~ρ-1/2Q(0,0) holds forall cases of the parameters. In (38) we have thus defined the precise scaling near the separating curve R3∩R4, as Y0=X0-X0+O(ρ-1/3). Note that C is not defined for region R4 while C1 is not defined for R1, as then the corresponding constant will play no role in the analysis. We conclude by giving the precise scaling for R1∩R2 (near X0=1), which will be(46)X0=1+βρ=1+Oρ-1/2.Note that (46) can be predicted from the marginal distribution in (11), as the sum in the denominator undergoes a transition for m=ρ+O(ρ), which is the same scaling as in (46).


### 3.1. Joint Distribution and Its Limits

Now we consider the joint distribution;π(k,r)=π(k,r;ρ,m,R)=P(X,Y)=P(X,Y;ρ,X0,Y0) for ρ→∞. We recall that X0=m/ρ and Y0=R/ρ are the scaled numbers of primary and secondary spaces. The state space of the random walk is the lattice rectangle in Figure 1, and on the coarse (X,Y) spatial scale this can be viewed as the continuous rectangle {(X,Y):0⩽X⩽X0,0⩽Y⩽Y0}. Our goal is to give a complete asymptotic description of the joint distribution for ρ→∞, including ranges of the state space where there is appreciable mass and also ranges where π(k,r) is asymptotically small. This corresponds to the tails of the distribution and in such ranges π(k,r)=P(X,Y) is typically exponentially small for large ρ. We first discuss the ranges where there is significant mass, and this will lead to certain limiting distributions, which will be very different for regions R1–R4 of parameter space in (19)–(25).Proposition 3.
Forρ→∞ one has the following limiting distributions:(i)
X0=m/ρ>1 (thus (X0,Y0)∈R1)(47)πk,r≈δ0re-ρρkk!,kρ=X<X0,which can be recast as the limit(48)ρπρ+ρX′,r⟶δ0r12πe-X′2/2,ρ⟶∞.(ii)
0<X0<1,X0+Y0>1 (thus (X0,Y0)∈R2)(49)πk,r~1-X02πρX0nexp⁡-ρ2Y-1-X02and this applies forn=m-k=O(1) and Y-(1-X0)=O(ρ-1/2) (i.e., r=ρ-m+O(ρ)).(iii)
0<X0+Y0<1 (thus (X0,Y0)∈R3∪R4)(50)πk,r~1-X0-Y0X012πi∮1-wz-w-wz+w-n-1w-p-1dw,(51)z±w=12X01+X0+Y0-Y0w±1+X0+Y0-Y0w2-4X0,which holds forn=m-k=O(1) and p=R-r=O(1).WhenX0>1 we have m>ρ so the secondary storage spaces will be rarely needed, and then π(k,r) approximately follows the Poisson distribution in (47), which has also the Gaussian limit in (48). The results in (47) and (48) provide no information on π(k,r) for r⩾1, but later we will estimate precisely these probabilities. We also note that when r=0, (47) ceases to be valid for k=ρX0-O(1)=m=O(1), for then if almost all primary spaces are full there may well be some secondary spaces also occupied, and thus π(k,0) may become comparable to π(k,r) for r⩾1, for this range of k. If X0<1 and X0+Y0>1 the primary storage spaces are insufficient to meet the demand, but the total number of spaces does suffice. Then (49) shows that m-O(1) primary spaces and ρ(1-X0)+O(ρ)=ρ-m+O(ρ) secondary spaces will tend to be occupied, with the joint distribution being a product of a geometric and a Gaussian. This also shows that, to leading order for large ρ, the processes N1 and N2 decouple. When X0+Y0<1 we have m+R<ρ and the totality of storage spaces is not enough to meet the demand. Then typically all but a few spaces, both primary and secondary, will tend to be occupied, with the numbers n and p of available spaces following the discrete joint distribution in (50). From (50) we can easily show that(52)∑k+r=Lπk,r~1-X0-Y0X0+Y0L so that the total number of empty spaces is geometrically distributed; this result also follows easily from the exact expression in (12). We will later see that the tail behavior of (50), for n and/or p→∞, is quite different according as (X0,Y0)∈R3 or (X0,Y0)∈R4, which again will indicate that the triangle 0<X0+Y0<1 in parameter space needs to be split into the two regions R3 and R4.We next study the transitions between the three limiting results in Proposition4.Proposition 4.
Forρ→∞ one has the limiting distributions: (i)
X0-1=β/ρ=O(ρ-1/2), X-1=α/ρ, and Y=Ω/ρ (thus k=m+αρ and r=ρΩ)(53)πk,r~ρ-1e-α2/412πi∫-i∞i∞Ωz-1Dz-αDz-βDz-1-βdz,-∞<α<β,Ω>0,whereDz(-β) is the parabolic cylinder function of order Z and argument -β. When β=0(X0=1) the above simplifies to(54)πk,r~ρ-12πΩ-αe-Ω-α2/2,α<0,Ω>0.(ii)
X0+Y0-1=γ/ρ=O(ρ-1/2), n=m-k=O(1), and Y0-Y=ρ-1/2y~=O(ρ-1/2)(55)πk,r~ρ-1/21-X0X0n∫-∞γe-u2/2du-1e-y~-γ2/2,y~>0.Asγ→+∞ the truncated Gaussian distribution in (55) approaches the free space Gaussian in (49), which applies for 1-Y0<X0<1. For γ→-∞, (55) asymptotically matches to (50), when the latter is expanded for X0+Y0↑1 and simultaneously p→∞, with the product (1-X0-Y0)p=-γy~ held fixed.The complicated distribution in (53) is a necessary intermediate result since (47) and (49) do not asymptotically match. The right-hand side of (53) is of the form ρ-1× (density in (α,Ω)), with the density having support in the quarter plane α<β, Ω>0. Thus if m=ρ+O(ρ) there will tend to be O(ρ) empty primary spaces and O(ρ) full secondary spaces, with now an intricate coupling between the processes N1 and N2. Finally, we note that the results in items (i) and (ii) in Proposition 3 and in item (i) of Proposition 4 are independent of the secondary storage capacity Y0=R/ρ, while item (iii) in Proposition 3 and item (ii) in Proposition 4 do depend upon Y0.


### 3.2. Joint Distribution: Main Regions of State Space

The asymptotic expansion ofπ(k,r)=P(X,Y) will be different for the four parameter ranges indicated in Figure 2 and also for three main regions of the state space, which we call D0, D+, and D-, and we define/discuss these below.First consider regionR1 of parameter space, so that X0>1, and define the curve(56)Y=Y~X=Y~X;X0,Y0=Y0+Y02X0+Y02X0+Y0-1X0-X0+Y02+X0+Y0X-X0+X0+Y02+X0+Y0X0-X-X02-4X0+Y02X0+Y0-1X0-X.This curve depends on both X0 and Y0 and thus on both of the total numbers, m and R, of primary and secondary storage spaces. The curve is defined for 0⩽X⩽X0 and we have(57)Y~0=Y01-X0X0+Y02,Y~X0=Y0.For region R1 (and indeed also for R2 and R3) we have Y~(0)>0 so that (56) connects the point 0,Y01-X0/X0+Y02 to the corner point (X0,Y0) in the scaled state space. The curve Y~(X) divides the state space into the two regions D0 and D+, with(58)D+=X,Y:0<X⩽X0,0<Y<Y~X,(59)D0=X,Y:0<X<X0,Y~X<Y<Y0.Here we defined D0 as an open set, while D+ is bounded by the four curves X=0, Y=0, X=X0, and Y=Y~(X) and we include only the third of these (X=X0) as a part of D+. This is because the asymptotic expansion that will apply in the interior of D+ will remain valid near X=X0, but not near the other three bounding curves. The expansion valid in D0 will break down if either X≈0, Y≈Y0, or Y≈Y~(X). We sketch in Figure 3 the curve Y~(X) and we recall that if X0>1 most of the mass in π(k,r) is concentrated in the range r=0 and k=ρ+O(ρ)(X=1+O(ρ-1/2)) (see Proposition 3), and this corresponds to the lower bounding curve for D+.Figure 3
RegionR1.Proposition 5.
For(X0,Y0)∈R1 (X0>1) the asymptotic expansions of π(k,r)=π(ρX,ρY) are as follows: (i)
(X,Y)∈D+(0<Y<Y~(X))(60)πk,r~CρK+X,YeρΨ+X,Y,whereCρ~ρ-1/2,(61)Ψ+X,Y=X0log⁡X0+X0-1+s-2X0log⁡s+X0+ss+X01-et-ss+X0s+X0-1+e-tlog⁡1+s+X0-1et+e-t-ss+X01-X0-s-etlog⁡1-ss+X0et,where(s,t) are related to (X,Y) via the mapping, for 0<s<Y0,(62)X=1-ss+X0et1+s+X0-1e-t,Y=ss+X0e-t-1+X0+s,(63)K+X,Y=e-t+s+X0-1-1/2e-t-ss+X0-1/2Δ+s,t-1/2X02πs+X02-X03/2s+X03,whereΔ+ is the Jacobian associated with (62); that is,(64)Δ+s,t=XtYs-XsYt=11-EDE-Det-2D1-D-D-E2e-t+E+D-2EDe-2t,D=ss+X0=1-X0s+X0,E=1-X0-s.(ii)
(X,Y)∈D0(Y~(X)<Y<Y0)(65)πk,r~C0ρKX,YeρΨX,Y,whereC0(ρ) is given by (45) for region R1(66)ΨX,Y=A1-eτ+X0log⁡1-A+Y0log⁡1-B-Xlog⁡1-Aeτ-Ylog⁡1-Beτ=-Aeτ+Aeτ-1+e-τ-A1-X01-Alog⁡1-Aeτ+Y0e-τ-BB-1log⁡1-Beτ+A+X0log⁡1-A+Y0log⁡1-B,(67)B=A1-A-X01-AA+X0+Y0-X0,where(A,τ) are related to (X,Y) by(68)X=e-τ-AX01-A-1+eτ,Y=Y0-1-e-τA+X0+Y0-X01-A=Y0B-e-τB-1,whereA∈(Amin,Amax) with(69)Amin=1-X0,Amax=121-X0-Y0+X0+Y0+12-4X0,withA∈(Y0/(X0+Y0),Amax) corresponding to D0, and(70)KX,Y=e-τ-Ae-τ-B-1/2Δ-1/2A1-AY01-A2-X02πY0-AX0+Y0Y0+1-X0-Y0A-A2,whereΔ is the Jacobian associated with (68), so that(71)Δ=XτYA-XAYτ=e-τ-11-X01-A2-Aeτ+1-X01-Ae-τ+Y01-B1+X01-A2e-τ=e-τ-1A+X0+Y0-X01-A+AeτX01-A2-1+e-τ1-2X01-A+X0X0+Y01-A2.We can view (68) as representing a family of curves in the (X,Y) plane, with A indexing the family and τ increasing along a curve. When τ=0 the curves in (68) meet at the corner point (X,Y)=(X0,Y0) and we also note that the Jacobian in (71) vanishes when τ=0, indicating a singularity in the transformation in (68). When A=Amax the curve becomes the horizontal segment Y=Y0. But then (69) shows that (1-Amax)(Amax+X0+Y0)=X0 so that B=∞ in (67). Thus near Y=Y0 the expansion in (68) becomes invalid. When A=Y0/(X0+Y0) the curve in (68) becomes(72)Y=Y01-1-e-τX0+Y0,X=X0+Y0-1+eτe-τ-Y0X0+Y0and eliminating τ we see that (72) is precisely the curve Y=Y~(X) in (56) that separates D0 from D+. For A∈(Amin,Y0/(X0+Y0)) the curves in (68) fill a portion of D+, but then the leading term for π(k,r) is given by (60), and (65) corresponds to only an exponentially small correction to (60). When A=Amin=1-X0 the curve in (68) is tangent to the line X=X0 at the point Y=Y0, which will have significance for the parameter region R4. We also note that for A∈(Y0/(X0+Y0),Amax) when τ=-log⁡A the curves in (68) hit the y-axis (then X=0 by (68)) and then the first factor in (70) becomes singular, which indicates that the asymptotics become invalid. Along A=Y0/(X0+Y0) corresponding to Y=Y~(X), K(X,Y) in (70) is again singular. Thus (70) is singular when A=Y0/(X0+Y0), A=Amax(B=∞), and e-τ=A, corresponding to the three curves (Y=Y~(X), Y=Y0, and X=0) that bound the region D0. We will give the appropriate expansions near these bounding curves in Section 3.3.For0<s<Y0 the curves in (62) fill the entire region D+, with s=0 corresponding to the line segment Y=0, 0<X<X0, and s=Y0 corresponding to the curve Y=Y~(X) (then (62) coincides with (72)). When X=X0 we have t=0 and the curves in (62) hit the line X=X0 at finite and nonzero slopes, for all 0⩽s⩽Y0. As t increases each curve will hit first either the x-axis or the y-axis. When s=0 the x-axis is hit first. For 1<X<X0 this occurs at a finite value of t, when t=log[(X0-1)/(X-1)], but if 0<X<1 in order to approach the x-axis we must let s→0 and t→∞ in such a way that set is held fixed. In this limit (62) may be approximated by X~1-set/X0 and Y~(X0-1)s/X0. We discuss in more detail the behavior of (60) as Y→0 later, when we give the asymptotic expansion(s) for π(k,r) that apply for X0>1 and r=O(1). When s>0 the curves in (62) hit the y-axis when t=log(1+X0/s), for then X=0. In particular if s=Y0 the corresponding curve hits the y-axis at Y~(0) in (57). Near both the X- and Y-axes, (60) will have singular behaviors and other expansions must be constructed. Note, however, that (60) is not singular along the curve Y=Y~(X), whereas (65) is singular. We can simplify (60) near X=X0, and then we obtain the more explicit form(73)πk,r~ρ-1/2X0X0+Ym-kX0+Y2-X0X0+Y5/22πeρΨ+X0,Y,(74)Ψ+X0,Y=Y+X0-1-X0+Ylog⁡X0+Y, which holds for m-k=ρ(X0-X)=O(1) and 0<Y<Y0. However, we note that since X0>1 we have Ψ+(X0,Y)<0 and thus π(k,r) in (73) is exponentially small in ρ. This is true for the entire domains D0 and D+, as there is very little probability mass in these ranges if X0>1.We next consider regionsR2 and R3 in parameter space, where it will become necessary to break up the state space into the three regions D+, D-, and D0. These regions are sketched in Figures 4 and 5. The curve that separates D+ from D0 is again given by (56), while the curve separating D+ from D- will be(75)Y=Y∗X=Y∗X;X0=122X0-X0-X-X0-X2-X02-Xand thus, since now X0<1,(76)Y∗0=0,Y∗X0=X0-X0∈0,Y0.Figure 4
RegionR2.Figure 5
RegionR3.We thus defineD- as(77)D-=X,Y:0<X<X0,0<Y<Y∗Xand, for regions R2 and R3, D+ now becomes(78)D+=X,Y:0<X<X0,Y∗X<Y<Y~X,with D0 still defined by (59).Proposition 6.
For(X0,Y0)∈R2∪R3 the asymptotic expansions of π(k,r) are as follows: (i)
(X,Y)∈D-(0<Y<Y∗(X))(79)πk,r~C1ρLX,YeρΦX,Yeρ1/3Φ1X,Y,where(80)ΦX,Y=W0log⁡W0+s1+Ylog⁡Y+W0-Ylog⁡Y-Xlog⁡1-W0W0+s1W0+Y-W0W0+s1W0+Y-12W0log⁡W0,(81)s1=s1X,Y=W0+Y+W02W0W0+1-X+W0+1-X2-4W0,(82)Φ1X,Y=Φ∗s1=-r01-X02/3X01/6log⁡s1+W01-X0,(83)LX,Y=W0+s13/2YW0+Y1s1-Ys1+W0X0-s12W0+s1+Y1-W0W0+s1W0+Y-1/212πX0-1/61-X0-5/6Ai′r0-2exp⁡-1+X02X0log⁡W0+s11-X0,withW0=1-X02 and r0 is the maximal root of the Airy function Ai(·). In (79), C1~ρ-2/3 for (X0,Y0)∈R2, C1 is given by (35) for (X0,Y0)∈R2∩R3 (then X0+Y0=1+γ/ρ), and C1 is given by (36) for (X0,Y0)∈R3.(ii)
(X,Y)∈D+(Y∗(X)<Y<Y~(X)).The expression in (60) applies with C~ρ-1/2 for (X0,Y0)∈R2, C is given by (32) for (X0,Y0)∈R2∩R3, and C is given by (33) for (X0,Y0)∈R3.(iii)
(X,Y)∈D0(Y~(X)<Y<Y0).The expression in (65) applies with C0 given by (45) for (X0,Y0)∈R2, C0 is given by (44) for (X0,Y0)∈R2∩R3, and C0~ρ-1/2(1-X0-Y0) for (X0,Y0)∈R3.In contrast to regionsD+ and D0, the expansion (79) in D- is a completely explicit function of X,Y, and X0. We also note that s1(X,Y) has a simple linear dependence upon Y, and s1(X0,Y)=Y. In Section 4 we give a more geometric interpretation of this expansion, and we also observe that the form of (79) is slightly different from the expansions in D+ and D0, as the former contains an additional factor that is of order exp[O(ρ1/3)], and thus gives an additional subexponential dependence on ρ. While the forms of C, C0, and C1 change according to whether (X0,Y0) lies in the regions R2, R3 or R2∩R3 of parameter space, the ratios C1 : C : C0 remain the same for these three cases.The expansion in (79) is valid only in the interior of D-. As Y→0 there is a singularity due to the factor 1/Y in L(X,Y) in (83). For X→X0 there is also a singularity due to the factor 1/s1-Y in (83), and as X→X0, we find that (s1-Y)-1/2=O(X0-X)-1/4. The curve Y=Y∗(X) that separates D+ from D- corresponds to s1(X,Y)=X0-X0, and along Y=Y∗(X) the factor (s1+W0)X0-s1=-1-X0[s1+X0-X0] vanishes. Thus (83) shows that L(X,Y∗(X))=0 which also indicates a nonuniformity in the asymptotics. Later we will give appropriate expansions near the three bounding curves (Y=0, X=X0, and Y=Y∗(X)) of region D-.In Figure4 we also indicate the curve(84)Y=Y1X=Y1X;X0=1-X021-X,which lies entirely within D+ (when (X0,Y0)∈R2) and corresponds to ∂Ψ+/∂Y=0, where Ψ+ is in (61). We can also show that Ψ+,YY(X,Y1(X))<0 so that, for a fixed X∈(0,X0), Ψ+ achieves a local maximum along Y=Y1(X). Note that Y=Y1(X) corresponds to s(X,Y)=1-X0(>X0-X0) in (62), for then X=1+(X0-1)et and Y=(1-X0)e-t.For(X0,Y0)∈R3 the curve in (84) plays no role, as it lies outside of D+, but now the curve(85)Y=Y2X=Y2X;X0,Y0=Y01-X01-Xlies within D0, connecting the points (0,Y0(1-X0)) and (X0,Y0). Along Y=Y2(X) we have ΨY=0 and then A=1-X0 and B=0, so that (68) becomes X=1-(1-X0)eτ and Y=Y0e-τ. Then for region R3, Ψ will have a local maximum along Y=Y2 in D0.For(X0,Y0)∈R3 most of the mass will lie near the corner point (X,Y)=(X0,Y0), where D0 and D+ meet, but neither (60) nor (65) (with the appropriate C0 and C) are valid there. For (X0,Y0)∈R2, π(k,r) will be maximal near the point (X,Y)=(X0,1-X0), and (73) applies for k=m-O(1) (or X=X0-O(ρ-1)) for both regions R1 and R2. By expanding (73) about Y=1-X0, which is where Ψ+(X0,Y) is maximal, we obtain precisely the expression in (49).Next we consider regionR4 of parameter space. Now the state space will be split into D0 and D-, and D+ will be absent. This is sketched in Figure 6, and we also observe that as X0+Y0↓X0 in region R3, the curves Y∗(X) and Y~(X) in (75) and (56) become identical, and thus D+ shrinks to this curve. For R4 there is a new curve that comes into play; namely,(86)Y=YcX=YcX;X0,Y0=-W0+Y0+W02W0W0+1-X-W0+1-X2-4W0.We have Yc(X0)=Y0 and the curve hits the x-axis (Y=0) when(87)YcX=0⟺X=X0-Y02Y0+W01W0. The curve Y=Yc(X) now separates D0 from D- and corresponds to A=Amin=1-X0 in (68), and also s1(X,Y)=Y0 in (81).Figure 6
RegionR4.Proposition 7.
For(X0,Y0)∈R4 (thus 0<Y0<X0-X0) the asymptotic expansions of π(k,r) are as follows: (i)
(X,Y)∈D-, 0<Y<Yc(X), X0-Y02/W0Y0+W0<X<X0.The expansion in (79) applies with now C1(ρ) given by (41).(ii)
(X,Y)∈D0(max{0,Yc(X)}<Y<Y0).The expansion in (65) applies for A∈(Amin,Amax) with C0(ρ)~ρ-1/2(1-X0-Y0).Again, different expansions must be given nearY=Yc(X) and near all boundaries of the state space. The curve Y=Y2(X) in (85) still lies within D0 and is sketched in Figure 6, with again ΨY=0 along this curve.We conclude by noting that, for all regions of parameter spaceRj, the expansion in D0 depends upon the secondary storage capacity Y0 (or R). For regions R1 and R2 the expansions in D+ and D- (R2 only) are independent of Y0, except through the curve Y~(X) that bounds D+. Thus if X0+Y0>1 the effects of the finite storage capacity appear in D0 only, and then letting Y0→∞ will recover the results for the storage model in [10, 11], which assumes that R=∞. For (X0,Y0)∈R3, hence X0<X0+Y0<1, the expansions in D- and D+ depend upon Y0 only through the multiplicative constants C1 and C, which now depend on Y0 in view of (33) and (36). For (X0,Y0)∈R4, hence 0<Y0<X0-X0, again the D- result depends on Y0 only through the constant C1 (cf. (41)). For (X0,Y0)∈R3∩R4, that is, along and near the curve X0+Y0=X0, we have Y3(X)=Yc(X)=Y~(X)=Y∗(X) so all four of these curves coalesce. A special analysis is required for X0+Y0-X0=O(ρ-1/3) for values of (X,Y) near this curve(s).This completes our summary of the asymptotic expansions in the three main regions,D0, D+, and D-, of the state space.


### 3.3. Joint Distribution: Boundary, Corner, and Transition Regions

We next analyze the four boundary segments of the state space rectangle, namely, the line segments{Y=0,0<X<X0}, {X=X0,0<Y<Y0}, {X=0,0<Y<Y0}, and {Y=Y0,0<X<X0}. As we previously discussed, the expansions in the three main regions become invalid near the boundary of the state space, with the exception of the D+ expansion in (60), which remains valid near X=X0, reducing to (73) when k=m-O(1)(X=X0-O(ρ-1)), at least for (X0,Y0)∈R1∪R2. For the four boundary segments we will typically consider the respective scales r=O(1), k=m-O(1), k=O(1), and r=R-O(1), though at times a different scaling must be considered in addition to these discrete scales. Note also that a particular point on a boundary segment may also require a separate expansion, and this occurs in R2∪R3, when Y∗(X) hits X=X0 at the point (X0,X0-X0). Also, in R1∪R2∪R3, the curve Y~(X) that separates D0 from D+ hits the y-axis when Y=Y01-X0/(X0+Y0)2(>0) and this point requires a separate analysis. Finally, in R4 the curve Yc(X) hits the x-axis when X=X0-Y02/[(Y0+W0)W0].After treating the four boundary segments we will give asymptotic results that are valid near the four corner points,(0,0), (X0,0), (X0,Y0), and (0,Y0), of the state space. Finally we give results for points (X,Y) that lie on or near the transition curves Y~(X) (for R1∪R2∪R3), Y∗(X) (for R2∪R3), and Yc(X) (for R4). Unlike the previous subsections where we listed the results by region Rj of parameter space, here we go by region of state space, and each proposition will correspond to one such region and the different results for the different Rj will be collected in that proposition.Proposition 8.
Forr=O(1) and k=ρX, 0<X<X0, one has the following expansions: (i)
R1:X0>1, 0<X<1(88)πk,r~ρ-rX0-12πXX01-X-rX0-1-2rr-1!eρ-2+X+X0-XlogX-X0logX0,r⩾1,(89)πk,0-ρke-ρk!~ρ-1-Xeρ-2+X+X0-XlogX-X0logX02πXX0X0-11-X.(ii)
R1:X0>1, X=1+α/ρ, α=O(1)(90)πk,r~ρ-r/2X0-12πX0X0-1-2r∫-α∞α+ur-1e-u2/2dueρ-1+X0-X0logX0,r⩾1,(91)πk,0-ρke-ρk!~ρ-1/2-eρ-1+X0-X0logX02πX0X0-1∫-α∞e-u2/2du.(iii)
R1:X0>1, 1<X<X0(92)πk,r~ρ-1/2X0-132πX0X-1r-1X-1+X0-12-r-1eρ-1+X0-X0logX0,r⩾1,(93)πk,0-ρke-ρk!~ρ-1/2-X0-1eρ-1+X0-X0logX02πX0X-1+X0-12.(iv)
R1∩R2:X0=1+β/ρ, β=O(1), 0<X<1(94)πk,r~ρ-1/21Xeρ-1+X-XlogX12πi∫Br+Γz+r1-Xzr!Dz-βDz-1-βdz,whereBr+ is the imaginary axis in the z-plane if r⩾1, and if r=0 the contour is to be indented to the right of the pole at z=0. Here Γ(·) is the gamma function. When β=0 the expression in (94) simplifies to(95)πk,r~ρ-1/22π1-XX2-X-r-1eρ-1+X-XlogX.(v)
R2∪R3:0<X0<1, X0+Y0>X0, 0<X<X0(96)πk,r~ρC1ρeρΦX,0eρ1/3Φ∗s10ρrW0rr!1-X01/6W0+s10X0-X0-s102πAi′r02X01/6s10s10+2W0W0s10+W01/2X0,whereW0=1-X02,(97)s10=-W0+W02W0+1-X+W0+1-X2-4W0,so thats10 corresponds to s1(X,0) in (81), Φ∗(·) is defined in (82), and Φ(X,0) can be computed from (80) by setting Y=0 and replacing s1 by s10. The value of C1 in (96) is given by (34) for R2, by (35) for R2∩R3(X0+Y0=1+O(ρ-1/2)), and by (36) for R3.(vi)
R4:0<Y0<X0-X0, X0-Y02/W0Y0+W0<X<X0.The expression in (96) applies with C1 now given by (41). For region R3∩R4 (hence X0+Y0=X0+O(ρ-1/3)), (96) holds for 0<X<X0, where C1 can be computed from (37).(vii)
R4:0<Y0<X0-X0, 0<X<X0-Y02/W0Y0+W0≡XL(98)πk,r~1-X0-Y0ρrr!Y0B01-B0rgXeρΨX,0,(99)ΨX,0=A01-1B0+X0log⁡1-A0+Y0log⁡1-B0-Xlog⁡1-A0B0,whereA0=A0(X)=A0(X;X0,Y0) is the solution of the algebraic equation(100)A0-1+X0+Y0Y01-A02+X0A01-X0-A0=-X1-X0-A01-A0A0+X0+Y0-X0that satisfiesA0(0)=1-X0-Y0 and A0(XL)=1-X0=W0,(101)B0=A01-X0-A01-A0A0+X0+Y0-X0,so thatB0(0)=1-X0-Y0 and B0(XL)=W0/(W0+Y0), and(102)gX=A01-A01-A02-X0Y01-B0B0-A0Y0-A0X0+Y0Y0-A02+1-X0-Y0A0A0+X0+Y0-X01-A0+A0B0X01-A02-1+B01-2X01-A0+X0X0+Y01-A02-1/2.ForX→0 and X→XL one can obtain g(X) more explicitly, with(103)gX~X0+Y0X,X⟶0,(104)gX~2Y0W0+Y03/21-X05/2XL-X2W0+Y03/2X0-X0-Y03/2,X⟶XL.(viii)
R4:0<Y0<X0-X0, X-XL=ρ-1/3Λ=O(ρ-1/3)(105)πk,r~1-X0-Y0ρrW0rr!ρ-1/3W0+Y0X01/61-X05/62W0+Y0X0-X0-Y03/2exp⁡ρΨXL,0+ρ2/3ΨXXL,0Λ+12ρ1/3ΨXXXL,0Λ2exp⁡Λ36W0Y0+W03Y03+3W0Y02+4W02Y0-2X0W05/2Y03Y0+2W03X0-X0-Y0212πi∫-i∞i∞1Aiθexp⁡Y0+W01-X05/3Y0Y0+2W0X01/6Λθdθ,where(106)ΨXL,0=Y0log⁡Y0W0+Y0-Y0W0+12X0log⁡X0+Y02Y0+W01W0-X0log⁡X0-X0-Y0W0,(107)ΨXXL,0=-log⁡X0-X0-Y01-X0,(108)ΨXXXL,0=-Y0+W02W0Y0Y0+2W0X0-X0-Y0.We note that the regions of validity of the expansions in Proposition8 are such that the four corner points of the state space are excluded. For regions R1 and R2 the expansions for r=O(1) are independent of the secondary storage capacity Y0. For region R3 the result in (96) does depend upon Y0, but only through the multiplicative constant C1, which now depends on Y0 in view of (36). This is also true for region R4 (and R3∩R4) when X>XL, as then C1 is given by (41) or (37). However, for 0<X<XL and X≈XL, the expressions in (98) and (105) show that now π(k,r) depends in an intricate way on Y0. Thus the finiteness of the secondary storage capacity affects the probabilities that there are even a few secondary spaces occupied.For regionR1, π(k,r) is exponentially small for r⩾1 and π(k,0) approximately follows a Poisson distribution, as indicated previously in Proposition 3. The results in (88)–(93) better quantify (47) and estimate the exponentially small error from the Poisson approximation. For regions R2, R3, and R4, π(k,r) is always exponentially small for r=O(1), and the approximations do not distinguish r=0 from r⩾1. By evaluating the contour integral in (105) for Λ→+∞ we can verify that (105) asymptotically matches to (96) (with C1 given by (41)), as X↓XL. Similarly, for Λ→-∞ (105) will match to (98) as X↑XL.Next we consider points(k,r) near X=X0, with X0-X=o(1). Note that only regions D+ and D- can be bounded by X=X0, for 0<Y<Y0.Proposition 9.
ForX~X0, one uses the scales k=m-n and X=X0-ρ-2/3ν1, so that n=ρ1/3ν1=ρ(X0-X) denotes the number of empty primary storage spaces. The expansions are now the following: (i)
R2∪R3:0<X0<1, X0+Y0>X0; n=O(1), 0<Y<X0-X0(109)πk,r~ρ-1/6C1ρeρΦX0,Yeρ1/3Φ∗YX0n12π1Ai′r01-X0-1/6X0-5/6n+X0Y+W0Y+W0X0-YY+W0X0-YYexp⁡-1+X02X0log⁡W0+Y1-X0,whereC1 is given by (34), (35), and (36) for parameter regions R2, R2∩R3, and R3, respectively, Φ∗(Y) is obtained by replacing s1 by s1(X0,Y)=Y in (82), and(110)ΦX0,Y=Y+W0log⁡Y+W0-Ylog⁡Y+X0-1-12X0log⁡X0-12W0log⁡W0.(ii)
R2∪R3:ν1=O(1) with ν1>0, 0<Y<X0-X0(111)πk,r~ρ-1/6C1ρeρΦX0,Yeρ1/3ν1logX0+Φ∗Y12π1Ai′r02Y+W0X0-YYX01-X01/2exp⁡-1+X02X0log⁡W0+Y1-X0Aiν+r0,whereν=1-X01/3X0-1/3ν1 and C1 is again given in Proposition 2 for the different ranges Rj.(iii)
R3∩R4 or R4: n=O(1) or ν1=O(1).Now the expressions in (109) and (111) apply over all Y(0<Y<Y0), with C1 given by (37) for R3∩R4 and (41) for R4.(iv)
R2∪R3: ν1=O(1), Y-(X0-X0)=ρ-1/3ω1=O(ρ-1/3)(112)πk,r~ρ-1/6C1ρeρ-1+X0-X0logX0exp⁡-ρ2/3ω1log⁡X0+ρ1/3ν1log⁡X0-ω122X01-X01/32πX07/12exp⁡1+X06X01-X0ω1312πi∫-i∞i∞Aiν+θAi2θe-ωθdθ,whereω=1-X0-1/3X0-1/6ω1, so that ν1ω1=X0νω, and C1 is given again in Proposition 2 for the three cases R2, R2∩R3, and R3.(v)
R1∪R2∪R3: 0<Y<Y0(R1) or X0-X0<Y<Y0(R2∪R3), n=O(1)(113)πk,r~CρX0+Y2-X02πX0+Y5/2X0X0+YneρΨ+X0,Y,whereΨ+(X0,Y) is given in (74) and C has the expansions in (31)–(33).Note that the asymptotics are most complicated in (112), which occurs when (X,Y)≈(X0,X0-X0) and this is where the curve Y∗(X) hits the line X=X0. This intersection occurs only for regions R2 and R3. The integrand in (112) is a meromorphic function of θ, having double poles at all roots of the Airy function Ai(·).Proposition 10.
Fork=O(1) and r=ρY, 0<Y<Y0, one has the following expansions: (i)
R1∪R2∪R3:0<Y<Y01-X0/(X0+Y0)2≡YU(114)πk,r~ρCρeρΨ+0,Yρkk!1+Y-D02k1-D0YD0X0Y+2D021-D0,whereD0=D0(Y) is the root of the cubic equation(115)D03-2D02+1-X0-YD0+Y=0that satisfiesD0(0)=1-X0=W0 and(116)Ψ+0,Y=D0+X01-D0-2-X0log⁡X0-Ylog⁡Y+2X0log⁡1-D0+2Ylog⁡D0.The expansion ofC is given by (31)–(33) according to subregions of R1∪R2∪R3.(ii)
R1∪R2∪R3∪R4: YU<Y<Y0(R1∪R2∪R3) or 0<Y<Y0(R4)(117)πk,r~Q0,0eρΨ0,Yρkk!A1A1+X0-2+11-A1kY0A1A1A1+X0-2+11-A12-X0YY0-YΔLYY0-A1X0+Y0,where(118)A1=A1Y=A1Y;X0,Y0=121-X0-Y0+X0+Y0-12+4Y,(119)Ψ0,Y=A1-1+X0log⁡1-A1+Y0-Ylog⁡1-A1+Ylog⁡A1-Ylog⁡Y-Y0-Ylog⁡Y0-Y+Y0log⁡Y0,(120)-ΔLY=1-A1A1+X0+Y0+X01-A12-X01-A1-1+A11-2X01-A1+X0X0+Y01-A12and the expansion ofQ(0,0) is given in Proposition 1 for the three cases R1∪R2, R2∩R3, and R3∪R4 (when Q(0,0)~1-X0-Y0).(iii)
R1∪R2∪R3: Y-YU=y∗/ρ=O(ρ-1/2)(121)πk,r~Q0,0eρΨ0,Y~0eρy∗ΨY0,Y~0ρρkk!Y0+X0X0+Y0kexp⁡12y∗2Ψ+,YY0,Y~0X0Y0X0+Y02-X0X0+Y03+X0Y0-X0∫acy∗∞e-u2/2du,where(122)Ψ0,Y~0=X0log⁡X0+Y0log⁡Y0-X0+Y0log⁡X0+Y0-X0X0+Y0+X0Y0X0+Y022log⁡X0+Y0-log⁡Y0+Y0X0X0+Y02-1log⁡1-X0X0+Y02,(123)ΨY0,Y~0=Ψ+,Y0,Y~0=log⁡Y0-log⁡X0+Y02-X0,(124)Ψ+,YY0,Y~0=X0Y0X0+Y02X0+Y03+X0Y0-X01-Y0X0X0+Y02+X0X0+Y02-X0,(125)ac=X0+Y05/2X0+Y02-X0X0Y0X0+Y03+X0Y0-X0X0+Y02+Y0-X0.We see that for each of the three cases the dependence ofπ(k,r) on k is of the form ρk[A∗(Y)]k/k! (times a function of Y or y∗), where the geometric ratio A∗ depends upon Y=r/ρ and undergoes a transition when Y increases past YU=Y~(0), which can occur only for regions R1, R2, and R3. We note that A1(YU)=Y0/(X0+Y0) and D0(YU)=Y0/(X0+Y0), so that A∗ is continuous along Y=YU, with A∗(YU)=1+Y0-X0/(X0+Y0), as indicated in (121). The expansion in (117) develops a singularity as Y↓YU, in view of the factor [Y0-A1(X0+Y0)]-1. The expansion is also singular as Y↑Y0 (for all regions Rj) and as Y↓0 (for region R4 only). Then we are approaching the corner points (0,Y0) or (0,0) of the state space. Using the expansions(126)∫acy∗∞e-u2/2du~2π,y∗⟶-∞acy∗-1exp⁡-ac2y∗22,y∗⟶+∞we can easily show that (121) matches to (117) (for region R1∪R2∪R3 in the intermediate limit where y∗→+∞ and Y↓YU) and to (114) (with now y∗→-∞ and Y↑YU). Note also that the ratio Q(0,0)/C(ρ) is asymptotically the same for each of the three regions R1, R2, and R3, in view of (28)–(33). For any region Rj, for k=O(1) and Y∈(0,Y0), π(k,r) is exponentially small in ρ.Proposition 11.
ForR-r=p=O(1) and 0<X<X0 one has (for all regions Rj in parameter space)(127)πk,r~Q0,0eρΨX,Y0ρpp!Y0eτU-1p1-Amaxe-τU-Amax1-X0-Y02+4Y01-Y0-X2-4Amax1-X0/1-Amax1/4,where τU=τU(X)=τU(X;X0,Y0) is given by(128)τU=-log⁡1-Y0-X-1-Y0-X2-4Amax1-X01-Amax+log⁡21-X01-Amax,where Amax is given in (69), Q(0,0) has the respective expansions in (28)–(30), and(129)ΨX,Y0=Amax1-eτU+X0log⁡1-Amax-Y0τU-Xlog⁡1-AmaxeτU.Thus (127) gives the expansion when there are only a few secondary spaces empty and a fraction X/X0=k/m∈(0,1) of primary spaces occupied. The expressions can be simplified in the limits X→0 (then τU(0)=-log(Amax)) and X→X0 (then τU(X0)=0), but then other “corner” expansions will apply. Note that (127) is a completely explicit expression in terms of p and X.Next we examine the four corners of the state space, where(X,Y)=(0,Y0), (0,0), (X0,0), and (X0,Y0). We recall that these ranges are important in that for region R1∩R2(X0≈1) most of the mass concentrates near the corner (X0,0), while for region R3∪R4(X0+Y0<1) most of the mass occurs near (X0,Y0). We will start with the corner (0,Y0) and proceed about the perimeter of the state space in a counterclockwise manner.Proposition 12.
Fork=O(1), p=R-r=O(1), and any parameter region Rj one has(130)πk,r~2πρQ0,0ρk+pk!p!1+Y0kY01-AmaxAmaxpeρΨ0,Y01-AmaxAmax1-X0-Y02+4Y01/4, where Q(0,0) is given in Proposition 1, Amax is in (69), and(131)Ψ0,Y0=Amax-1+X0log⁡1-Amax+Y0log⁡Amax.This gives the expansion when there are but a few primary spaces occupied and a few secondary spaces empty. Next we consider the corner point(0,0), and this will typically correspond to k,r=O(1). The results will be very different for region R1 compared to those for the remaining regions.Proposition 13.
Fork,r=O(1) and ρ→∞ one has (i)
(X0,Y0)∈R1(X0>1)(132)πk,r~ρ2πr-1!k!ρk-rX0-11-2rX0eρ-2+X0-X0logX0,r⩾1,(133)πk,0-ρke-ρk!~-ρ2πρkk-1!eρ-2+X0-X0logX0X0-1.(ii)
(X0,Y0)∈R1∩R2(X0-1=β/ρ=O(ρ-1/2)), r⩾0(134)πk,r~ρke-ρk!r!12πi∫Br+Γz+rDz-βDz-1-βdz,whereBr+ is as in (94), and when β=0(135)πk,r~ρke-ρk!2-rβ=0.(iii)
(X0,Y0)∈R2∪R3X0<1,Y0+X0>X0(136)πk,r~ρ2/3C1ρρk+re-ρk!r!W0r1-W0k,whereC1 is given by (34), (35), or (36) for R2, R2∩R3, or R3, respectively.(iv)
(X0,Y0)∈R4(0<Y0<X0-X0)(137)πk,r~2πρX0+Y01-X0-Y0eρ-X0-Y0+X0+Y0logX0+Y0ρk+rk!r!Y0+X0X0+Y0kY0X0+Y0-Y0r.(v)
(X0,Y0)∈R2∪R3; k=ρ2/3X′, r=ρ2/3Y′; X=ρ-1/3X′, Y=ρ-1/3Y′(138)πk,r~ρ2/3C1ρρk+rk!r!W0r1-W0k12πi∫-i∞i∞e-ξ′zAiz2dzexp⁡12ρ1/3Y′2W0-W01-W02X′2-16Y′3W02+W0W0+21-W04X′3,(139)ξ′=1-X02/3X01/6Y′W0-X′1-W0.We note that (136) and (137) are continuous along the curve X0+Y0=X0, corresponding to R3∩R4, in view of the expansion in (36) for C1 in R3. Thus the leading term for k,r=O(1) for (X0,Y0)∈R3∩R4 can be obtained by either using (36) to compute C1 in (136) or replacing Y0 by X0-X0 in (137). The scale k,r=O(ρ2/3) must be considered (cf. (138)), as the approximation in (136) cannot directly match to those for k=O(1), 0<Y<YU (cf. (114)) or r=O(1), 0<X<X0 (cf. (96)). We can view (136) as a special case of (138), letting X′,Y′→0 in the latter and noting that(140)2πi-1∫-i∞i∞Aiz-2dz=1. For region R4 we see from Figure 6 that only state space region D0 meets the corner (X,Y)=(0,0). Then (137) matches directly to (98), in the limits k→∞, X→0, and to (117), in the limits r→∞, Y→0. Note that, for region R4, A1(0)=1-X0-Y0, which follows from (118) (this is true also for R3, but then (117) applies only for Y>YU and thus not near the corner (0,0)). We can obtain a result analogous to (138) for the transitional range R3∩R4, but we do not give that for the sake of brevity.Proposition 14.
Fork=m-O(1) and r=O(1), or for k=m-O(ρ) and r=O(ρ), the expansions of π(k,r) are as follows: (i)
(X0,Y0)∈R1, m-k=n⩾0(141)πk,r~12πρX0eρ-1+X0-X0logX0X0-1X0-r-1,r⩾1πk,0~12πρX0e-1+X0-X0logX0X0n-1X0.(ii)
(X0,Y0)∈R1∩R2(X0-1=β/ρ), m-k=n⩾0(142)πk,r~ρ-1ρ-z0/2e-β2/4r!Δ0βΓz0+r+1nz0z0+r+1,wherez0 is the minimal root of the parabolic cylinder function Dz(·); that is, z0=z0(β)=min{z:Dz(-β)=0}, and Δ0(β)=-d/dzDz(-β)z=z0β.(iii)
(X0,Y0)∈R1∩R2(X0-1=β/ρ), k=ρ+ρα, and r=ρΩ, with α⩽β and Ω>0(143)πk,r~ρ-112πi∫-i∞i∞e-α2/4Dz-αΩz-1Dz-βDz-1-βdzwhich whenβ=0 simplifies to(144)πk,r~ρ-12πΩ-αe-Ω-α2/2;α<0,Ω>0.(iv)
(X0,Y0)∈R1∩R2(X0-1=β/ρ), k=ρ+ρα, α<β, r=O(1)(145)πk,r~ρ-1/2ρ-z0/2e-α2/4Γr+z0r!Dz0-αΔ0βDz0-1-β.(v)
(X0,Y0)∈R1∩R2 with now X0-1=β∗ρ-1/4=O(ρ-1/4) and β∗>0, r=O(1), k=ρ+ρα, α=O(1)(146)πk,r~β∗32πρ1/4⊗∗∫-α∞u+αr-1u+α+β∗2r+1e-u2/2du,r⩾1,(147)πk,0-ρke-ρk!~-β∗2πρ1/4⊗∗∫-α∞e-u2/2u+α+β∗2du,(148)⊗∗=exp⁡-ρ2β∗2+ρ1/46β∗3-112β∗4.(vi)
(X0,Y0)∈R2∪R3∪R4, m-k=n⩾0(149)πk,r~ρ1/3C1ρρrW0rr!X0nn+1eρΦX0,0eρ1/3Φ∗01-X04/3Ai′r0X01/3exp⁡-12X0log⁡1-X0,(150)Φ∗0=-r01-X02/3X01/6log⁡1-X0,(151)ΦX0,0=12W0log⁡X0-12X0log⁡X0+X0-1,andC1 is given in Proposition 2 for the different regions Rj.For regionR1, (141) show that the dependence of π(k,r) on m-k and r is quite simple, but we have to distinguish the cases r=0 and r⩾1. The same is true for regions R2, R3, and R4, where (149) applies for all r⩾0, except now the form of C1 in (149) is different for the five cases in Proposition 2. However, when X0≈1, the asymptotic structure of π(k,r) is quite complicated, and we must consider separately the scales X0-1=O(ρ-1/2)(m-ρ=O(ρ)) and X0-1=O(ρ-1/4)(m-ρ=O(ρ3/4)). For X0-1=O(ρ-1/2) we obtain the limiting density in (53) or (143), as the limit of ρπ(ρ+ρα,ρΩ)=ρπ(m+ρ(α-β),ρΩ) for ρ→∞. This expansion applies for Ω>0, for any α⩽β, but becomes invalid as Ω→0. For Ω→0 the asymptotic behavior of the contour integral in (143) is determined by the singularity at z=z0(β), and the density behaves as O(Ωz0-1) for Ω→0 and α<β. This corresponds to either an integrable singularity or a zero of the density (unless β=0 then z0=1) and in either case indicates a nonuniformity in the asymptotics. Thus we need the expansion in (145) for r=O(1). For r→∞ we have, by Stirling’s formula,(152)Γr+z0r!=Γr+z0Γr+1~rz0-1=ρΩz0-1 and then (145) matches to (143) in the intermediate limit where r→∞ but Ω=r/ρ→0. The expansion in (145) itself breaks down when α→β, since by the definition of z0(β) we have Dz0(β)(-α)→0 as α↑β. Then for α~β we have the expansion in (142), which holds for n=O(1) and we note that n=m-k=(β-α)/ρ so that α=β-nρ-1/2.We can show also that the expansions forn,r=O(1) (cf. (141), (142), and (149)) match in appropriate intermediate limits. Consider (142) for β→-∞ (then we are moving into the range X0<1). We now have z0→∞ and more precisely(153)z0β=β24-r0-β22/3-12+o1,β⟶-∞so the approximation to the minimal root of the parabolic cylinder function Dz(-β) involves the maximal root of the Airy function Ai(·). Thus, for β→-∞ we have(154)Δ0β~z0z0/2e-z0/22-β1/32πAi′r0which can be obtained by approximated Dz(-β) by Airy functions in the double limit where β→-∞ and z→∞, with z-β2/4=O(-β)2/3. Using (154) in (142) and expanding Γ(z0(β)+r+1) by Stirling’s formula (since now z0→∞) for a fixed r, (142) becomes(155)ρ-1ρ-z0/2n+1-β24/3e-β2/4Ai′r0z0z0/2e-z0/2z0rr!.Then (155) must agree with the expansion of (149) as X0↑1. Then we use the fact that ρ1/3C1~ρ-1/3 in region R2. Also, as X0→1, W0=1-X02~β2/(4ρ),(156)ρ-1+X0-12X0log⁡X0+W0log⁡1-X0=β24log⁡-β2ρ-38β2+o1,-ρ1/3r0X0-1/61-X02/3log⁡1-X0~-r0-β22/3log⁡-β2ρ.Using (156) we see that as X0↑1, (149) becomes(157)ρ-1/31r!β24rn+1Ai′r0-β2ρ4/3exp⁡β24-r0-β22/3-12log⁡-β2ρ-38β2.But by using (153) in (155) we obtain again the expression in (157), which verifies the matching.Now consider (141) for X0↓1 and (142) for β→+∞. We now have(158)z0β~β2πe-β2/2,Δ0β~2πβeβ2/4,β⟶+∞so that z0 is exponentially small and Δ0 is exponentially large. The last factor in (142), namely, nz0/(z0+r)+1, can be approximated by 1 if r⩾1 and is equal to n+1 if r=0. Now also Γ(z0+r+1)/r!→1 and (142) becomes, for β→+∞,(159)β2πe-β2/2ρ-11,r⩾1n+1,r=0, and the above clearly agrees with (141), when we expand these for X0→1.The results in (146)–(148) assume the scaling X0-1=β∗ρ-1/4, and these are needed to asymptotically connect the parameter ranges X0-1=O(ρ-1/2) and X0>1. It is only near the corner (X,Y)=(X0,0) that we must consider this scaling (and indeed also the β-scale). For other ranges of X and r=O(1), we can get the expansions of π(k,r) as limiting cases of other expansions, as they lie in the asymptotic matching range where X0→1 but (X0-1)ρ(=β)→∞. For a fixed β∗>0 and 0<X<1 we have(160)πk,r~r-1!β∗1-2rρ-r/2⊗∗2πXρ1/41-Xreρ-1+X-XlogX,r⩾1;for X-1=α∗ρ-1/4=O(ρ-1/4) with 0<α∗<β∗ we have(161)πk,r~⊗∗2πρ3/4β∗3α∗2,r⩾1;and for the scale m-k=n=O(1) we have α∗~β∗ so the last factor in (161) may be approximated by β∗3α∗-2~β∗. We can easily verify that as X↑1, (159) matches to (146) as α→-∞. Note that in this limit we have(162)∫-∞∞u+αr-1u+α+β∗2r+1e-u2/2du~β∗-2-2rr-1!e-α2/2-α-r. For α→+∞ we have(163)∫-α∞u+αr-1u+α+β∗2r+1e-u2/2du~2πα2=2πρα∗2 and then (146) clearly matches to (160) and in fact contains the latter as a limiting case. When r=0 we have, for 0<X<1,(164)πk,0-ρke-ρk!~-X1-X⊗∗2πρ3/4β∗eρ-1+X-XlogX;for X-1=α∗ρ-1/4, α∗>0, we have(165)πk,0-ρke-ρk!~-⊗∗2πρ1/2β∗α∗;and for n=m-k=ρ(β-α)=ρ3/4(β∗-α∗)=O(1)(166)πk,0~⊗∗2πρ3/4β∗n+1.We can again easily verify that (147) matches to (164) for X→1 and α→-∞ and to (165) for α→+∞ and α∗=αρ-1/4→0. Recalling that k=ρX=ρ+ρ3/4α∗=ρ+ρ3/4β∗-n, we have (167)ρke-ρk!-⊗∗2πρ1/2β∗k!=ρke-ρk!-⊗∗2πρβ∗β∗-ρ-3/4n=⊗∗2πρexp⁡nβ∗ρ1/4-1+Oρ-3/4~⊗∗2πρ3/4β∗n and this agrees with (166) for n→∞, which verifies the matching between (165) and (166). Thus we have given π(k,r) for all ranges of k for r=O(1) and the scaling m=ρ+ρ3/4β∗, β∗>0. However, only for the range k=ρ+O(ρ) do we get the new results in (146) and (147).Next we examine the corner(X,Y)=(X0,Y0), so both primary and secondary spaces will be nearly full. For regions R3∪R4 most of the mass is in the range. We will need to consider the scales m-k,R-r=O(1) and also m-k=O(ρ1/3). Consider R-r=O(ρ2/3).Proposition 15.
Fork=m-o(ρ) and r=R-o(ρ) one uses the variables m=m-k=ρ1/3ν1 and p=R-r=ρ2/3T1, and the expansions are as follows: (i)
(X0,Y0)∈R1∪R2∪R3∪R4, n=O(1), p=O(1)(168)πk,r~Q0,0X012πi∮1-wz--wz+-n-1w-p-1dw,wherez±(w) are defined in (50) and the expansions for Q(0,0) are in Proposition 1; in particular for region R3∪R4 one has Q(0,0)~1-X0-Y0>0 and then (168) is a proper discrete distribution over the range(s) n⩾0, p⩾0.(ii)
(X0,Y0)∈R4, m-k=ρ1/3ν1=O(ρ1/3), ν1>0, R-r=ρ2/3T1=O(ρ2/3)(169)πk,r~ρ-2/31-X0-Y01-X05/3X0-1/6X0-X0-Y0W0+Y0Fν,TX0nY0Y0+W0pexp⁡-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13,where(170)Fν,T=12πiddT∫-i∞i∞Aiν+θθAiθeθTdθ=∑j=0∞Aiν+rjAi′rjerjT,T>0,(171)ν1=νX01-X01/3,T1=W0+Y0X01/61-X02/3T,andrj are the roots of Ai(·), ordered as 0>r0>r1>⋯.(iii)
(X0,Y0)∈R4, m-k=n=O(1), R-r=p=ρ2/3T1=O(ρ2/3)(172)πk,r~ρ-11-X0-Y0W0X0-1/2X0-X0-Y0W0+Y0X0nY0Y0+W0pn+X0Y0+W0Y0+W0X0-Y0∑j=0∞erjTexp⁡-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13,T>0.(iv)
(X0,Y0)∈R3∩R4, m-k=n=ρ1/3ν1=O(ρ1/3), ν1⩾0, R-r=p=ρ2/3T1=O(ρ2/3)(173)πk,r~ρ-1/31-X04/3X0-1/3X0nY0Y0+W0pexp⁡-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13F¯ν,T;δ1,where(174)F¯ν,T;δ1=12πi∫-i∞i∞Aiν+θAiθ2eθT-δ1∫θ∞eδ1uAiududθ,(175)X0+Y0=X0+ρ-1/3δ∗,δ∗=X01/61-X01/3δ1and(ν,T) is given by (171) in terms of (ν1,T1).When(X0,Y0)∈R3∪R4 most mass concentrates on the (n,p) scale so there tend to be but a few available primary and secondary spaces. For (X0,Y0)∈R1∪R2(X0+Y0>1), the result in (168) still applies but now Q(0,0) is exponentially small (cf. (30)). For X0+Y0-1=O(ρ-1/2) we have Q(0,0)=O(ρ-1/2). Later we will study the behavior of the contour integral in (168) as n and/or p→∞, and we will see that for R1∪R2 we can get exponential growth in certain sectors, such as if p→∞ with n=O(1). From Proposition 15 we also see that the probabilities of finding O(ρ1/3) empty primary spaces and O(ρ2/3) secondary spaces are quite complicated, and their estimation involves contour integrals of Airy functions. These probabilities are however quite small, in view of the factors X0n=X0ρ1/3ν1 and [Y0/(Y0+W0)]p=[Y0/(Y0+W0)]ρ2/3T1 in (169) and (173). In (173) we can use (175) and rewrite the result in terms of X0 and δ∗, thus eliminating Y0.Finally we give results that apply near the transition curves that separateD0, D-, and D+. Note D0 meets D+ for region(s) R1∪R2∪R3, D+ meets D- for region R2∪R3, and D0 meets D- for region R4 (see also Figures 3–6).Proposition 16.
For(X0,Y0)∈R1∪R2∪R3, Y-Y~(X)≡η∗/ρ=O(ρ-1/2), and 0<X<X0 one has(176)πk,r~CρK+X,Y~X12π∫η∗/bt∗∞e-u2/2duexp⁡ρΨ+X,Y~X+ρΨ+,YX,Y~Xη∗+12Ψ+,YYX,Y~Xη∗2,where(177)Ψ+X,Y~X=X0log⁡X0+X0+Y0-1-2X0log⁡X0+Y0+Y0X0+Y01-et∗-Xlog⁡1-Y0et∗X0+Y0-Y~Xlog⁡1-1-X0-Y0et∗,(178)t∗=log⁡Y0-log⁡Y0-X0+Y0Y0-Y~X,(179)Ψ+,YX,Y~X=log⁡Y0-X0+Y0Y0-Y~XX0+Y0Y~X,(180)Ψ+,YYX,Y~X=-et∗sY0-1-X0-Y0tY01-1-X0-Y0et∗,(181)sY0=X0+Y0X0+Y0+Y0e-t∗tY0X0+Y02+e-t∗-1X0,(182)tY0=et∗1-et∗X0+Y0X0et∗+X0+Y02X0+Y0X02-Y02-X0+et∗X0-X0+Y022+e2t∗2-et∗X0Y0+e3t∗Y0X+Y02-1,(183)K+X,Y~X=Y0X0X0+Y02-X0322πΔ+012X0+Y03Y~XX0Y0-X0+Y02Y0-Y~X,(184)bt∗=X0Y0Δ0Δ+0X0+Y0X0+Y02-X01-X0-Y0e-t∗-Y0et∗X0+Y0-2,(185)Δ+0=Δ+0X=11-E∗D∗E∗-D∗et∗-2D∗1-D∗-D∗-E∗2e-t∗+D∗+E∗-2D∗E∗e-2t∗,(186)D∗=Y0X0+Y0,E∗=1-X0-Y0,(187)Δ0=Δ0X=e-t∗-1Y0X0+Y0+Y0X0+Y02-X0X0X0+Y0et∗+e-t∗1-2X0+Y0+X0+Y03X0-1.Y~(X) is given by (56), and C(ρ) has the expansions in (31)–(33) for the different regions Rj.Forη∗→-∞ we are moving into region D+ and the integral in (176) approaches 2π, and then (176) becomes simply the expansion of the D+ result in (60), about Y=Y~(X). Thus the matching of (176) to D+ is immediate, and we can also verify the matching to (65), by expanding the latter as Y↓Y~(X), and (176) for η∗→+∞. We recall that Proposition 2 shows that the expansion of C0(ρ)/C(ρ) is the same for regions R1, R2, and R3. The function t∗=t∗(X) in (178) is obtained by setting s=Y0 and then Y=Y~(X) in (62). The function Δ+0(X) is obtained by setting s=Y0 and t=t∗(X) in (64) and corresponds to the values of this Jacobian along the curve Y=Y~(X). Note also that t∗(X) can be obtained by setting A=Y0/(X0+Y0) and Y=Y~(X) in (68), and Δ0(X) corresponds to the Jacobian in (71) along the curve Y=Y~(X), with τ=t∗(X).Proposition 17.
For(X0,Y0)∈R2∪R3, Y-Y∗(X)=ρ-1/3ξ1=O(ρ-1/3), and 0<X<X0 one has(188)πk,r~ρ-1/2Cρexp⁡ρΦX,Y∗X+ρ-1/3ξ112πW0Y∗X1W0-W0+Y∗X212πi∫-i∞i∞1Aiθ2exp⁡-1-X02/3X01/6W0+Y∗Xξ1θdθ,where(189)ρΦX,Y∗+ρ-1/3ξ1=ρΦX,Y∗+ρ2/3ΦYX,Y∗ξ1+12ρ1/3ΦYYX,Y∗ξ12+16ΦYYYX,Y∗ξ13+o1,ΦX,Y∗=Y∗log⁡1+W0Y∗-W0W0+Y∗-Xlog⁡Y∗W0+Y∗,ΦYX,Y∗=log⁡1+W0Y∗,ΦYYX,Y∗=-W0Y∗W0+Y∗,ΦYYYX,Y∗=W0W0+2Y∗Y∗2W0+Y∗2,and C(ρ) has the expansions in (31)–(33) (one can replace ρ-1/2C(ρ) by ρ-1/3C1(ρ) in (188)), and Y∗=Y∗(X)=Y∗(X;X0) is given in (75).Forξ1→+∞ we can expand the integral in (188) by the saddle point method, after shifting the integration contour far to the right, with Re(θ)≫1. Then the Airy function in the integrand may be approximated using(190)Aiz~12πz-1/4e-2/3z3/2;z⟶∞,arg⁡z<πand we can verify that (188) for ξ1→+∞ matches to the D+ result in (60), as Y↓Y∗(X). For ξ1→-∞ the behavior of the integral in (188) is determined by the singularity with the largest real part, which is the double pole at θ=r0(<0). Then standard singularity analysis can be used to show that (188) for ξ1→-∞ agrees with the expansion of (79) as Y↑Y∗(X). Note that in this limit ρ1/3Φ∗(s1) becomes O(1) and proportional to ξ1.Proposition 18.
For(X0,Y0)∈R4, Y-Yc(X)=ρ-1/3η∗∗=O(ρ-1/3), and XL<X<X0 (with XL defined in item (vii) of Proposition 8) one has(191)πk,r~1-X0-Y0ρ5/62π1-X04/3X01/6X0-X0-Y0Y0YcXW0+Y0Y0-YcX2W0+Y0+YcXW0+YcX-W0W0+Y0-1/2expρΦX,YcX+ρ-1/3η∗∗-ΦX0,Y012πi∫-i∞i∞1Aiθexp-1-X02/3X01/6W0+Y0Xη∗∗θdθ,where Yc(X)=Yc(X;X0,Y0) is defined in (86).Thus the transition fromD0 to D- involves a slightly different integral (cf. (191)) compared to the transition from D+ to D- (cf. (188)), as the former has simple poles at the Airy roots. Using standard asymptotic analysis we can show that(192)12πi∫-i∞i∞e-⋆θAiθdθ~2⋆e-⋆3/3,⋆⟶+∞,(193)12πi∫-i∞i∞e-⋆θAiθdθ~e-⋆r0Ai′r0,⋆⟶-∞and (192) can be used to verify the matching between (191) and the D0 result in (65), where C0~ρ-1/2(1-X0-Y0) in R4. Similarly (193) can be used to verify matching to the D- result in (79), with C1 now given by (41). The factor involving ρΦ(X,Yc+ρ-1/3η∗∗) in (191) can be expanded in Taylor series, similarly to (189), by replacing Y∗(X) by Yc(X) and ξ1 by η∗∗.As we approachR3∩R4 (where X0+Y0=X0) from within R4 we see that (191) develops a singularity, in view of the factor X0-X0-Y0-1. As we approach R3∩R4 from within R3, the expression for K+(X,Y~(X)) in (183) vanishes and that for b(t∗) in (184) develops a singularity. Thus the expansions in both Propositions 16 and 18 become invalid, and below we give a new result that applies for R3∩R4, where D0 meets D-.Proposition 19.
For(X0,Y0)∈R3∩R4, with X0+Y0=X0+δ∗ρ-1/3 and δ∗=O(1),(194)πk,r~1-X03X01/4expρΦX,YcX+ρ-1/3η∗∗-ΦX0,Y02πρY∗XX0-X0-Y∗XY∗X+1-X02-X012πi∫-i∞i∞e-θδ1∫θ∞eδ1uAiuduAiθ2exp-1-X02/3X01/6W0+Y∗Xη∗∗θdθ,where δ1=X0-1/61-X0-1/3δ∗ and Yc(X) is in (86) and Y∗(X) in (75).With the scalingX0+Y0-X0=O(ρ-1/3), the curves Yc(X), Y∗(X), and Y~(X) nearly coincide, with the differences being O(ρ-1/3). In (194) we can again replace Φ(X,Yc+ρ-1/3η∗∗) by its Taylor expansion and replace Y0 everywhere by X0-X0+δ∗ρ-1/3, thus writing the result in terms of X0 and δ∗ (or δ1), along with X and η∗∗. Note that η∗∗ is still defined by η∗∗=ρ1/3[Y-Yc(X)], which differs from ρ1/3[Y-Y∗(X)] by an amount that is O(1). Thus, for region R3∩R4 the transition range in state space, between D- and D0, involves a somewhat more complicated integral than those in Propositions 18 and 19.This completes our summary and discussion of the various regions of state space, which carry zero volume in(X,Y) space but are necessary since the results in D0, D+, and D- do not always apply.


## 3.1. Joint Distribution and Its Limits

Now we consider the joint distribution;π(k,r)=π(k,r;ρ,m,R)=P(X,Y)=P(X,Y;ρ,X0,Y0) for ρ→∞. We recall that X0=m/ρ and Y0=R/ρ are the scaled numbers of primary and secondary spaces. The state space of the random walk is the lattice rectangle in Figure 1, and on the coarse (X,Y) spatial scale this can be viewed as the continuous rectangle {(X,Y):0⩽X⩽X0,0⩽Y⩽Y0}. Our goal is to give a complete asymptotic description of the joint distribution for ρ→∞, including ranges of the state space where there is appreciable mass and also ranges where π(k,r) is asymptotically small. This corresponds to the tails of the distribution and in such ranges π(k,r)=P(X,Y) is typically exponentially small for large ρ. We first discuss the ranges where there is significant mass, and this will lead to certain limiting distributions, which will be very different for regions R1–R4 of parameter space in (19)–(25).Proposition 3.
Forρ→∞ one has the following limiting distributions:(i)
X0=m/ρ>1 (thus (X0,Y0)∈R1)(47)πk,r≈δ0re-ρρkk!,kρ=X<X0,which can be recast as the limit(48)ρπρ+ρX′,r⟶δ0r12πe-X′2/2,ρ⟶∞.(ii)
0<X0<1,X0+Y0>1 (thus (X0,Y0)∈R2)(49)πk,r~1-X02πρX0nexp⁡-ρ2Y-1-X02and this applies forn=m-k=O(1) and Y-(1-X0)=O(ρ-1/2) (i.e., r=ρ-m+O(ρ)).(iii)
0<X0+Y0<1 (thus (X0,Y0)∈R3∪R4)(50)πk,r~1-X0-Y0X012πi∮1-wz-w-wz+w-n-1w-p-1dw,(51)z±w=12X01+X0+Y0-Y0w±1+X0+Y0-Y0w2-4X0,which holds forn=m-k=O(1) and p=R-r=O(1).WhenX0>1 we have m>ρ so the secondary storage spaces will be rarely needed, and then π(k,r) approximately follows the Poisson distribution in (47), which has also the Gaussian limit in (48). The results in (47) and (48) provide no information on π(k,r) for r⩾1, but later we will estimate precisely these probabilities. We also note that when r=0, (47) ceases to be valid for k=ρX0-O(1)=m=O(1), for then if almost all primary spaces are full there may well be some secondary spaces also occupied, and thus π(k,0) may become comparable to π(k,r) for r⩾1, for this range of k. If X0<1 and X0+Y0>1 the primary storage spaces are insufficient to meet the demand, but the total number of spaces does suffice. Then (49) shows that m-O(1) primary spaces and ρ(1-X0)+O(ρ)=ρ-m+O(ρ) secondary spaces will tend to be occupied, with the joint distribution being a product of a geometric and a Gaussian. This also shows that, to leading order for large ρ, the processes N1 and N2 decouple. When X0+Y0<1 we have m+R<ρ and the totality of storage spaces is not enough to meet the demand. Then typically all but a few spaces, both primary and secondary, will tend to be occupied, with the numbers n and p of available spaces following the discrete joint distribution in (50). From (50) we can easily show that(52)∑k+r=Lπk,r~1-X0-Y0X0+Y0L so that the total number of empty spaces is geometrically distributed; this result also follows easily from the exact expression in (12). We will later see that the tail behavior of (50), for n and/or p→∞, is quite different according as (X0,Y0)∈R3 or (X0,Y0)∈R4, which again will indicate that the triangle 0<X0+Y0<1 in parameter space needs to be split into the two regions R3 and R4.We next study the transitions between the three limiting results in Proposition4.Proposition 4.
Forρ→∞ one has the limiting distributions: (i)
X0-1=β/ρ=O(ρ-1/2), X-1=α/ρ, and Y=Ω/ρ (thus k=m+αρ and r=ρΩ)(53)πk,r~ρ-1e-α2/412πi∫-i∞i∞Ωz-1Dz-αDz-βDz-1-βdz,-∞<α<β,Ω>0,whereDz(-β) is the parabolic cylinder function of order Z and argument -β. When β=0(X0=1) the above simplifies to(54)πk,r~ρ-12πΩ-αe-Ω-α2/2,α<0,Ω>0.(ii)
X0+Y0-1=γ/ρ=O(ρ-1/2), n=m-k=O(1), and Y0-Y=ρ-1/2y~=O(ρ-1/2)(55)πk,r~ρ-1/21-X0X0n∫-∞γe-u2/2du-1e-y~-γ2/2,y~>0.Asγ→+∞ the truncated Gaussian distribution in (55) approaches the free space Gaussian in (49), which applies for 1-Y0<X0<1. For γ→-∞, (55) asymptotically matches to (50), when the latter is expanded for X0+Y0↑1 and simultaneously p→∞, with the product (1-X0-Y0)p=-γy~ held fixed.The complicated distribution in (53) is a necessary intermediate result since (47) and (49) do not asymptotically match. The right-hand side of (53) is of the form ρ-1× (density in (α,Ω)), with the density having support in the quarter plane α<β, Ω>0. Thus if m=ρ+O(ρ) there will tend to be O(ρ) empty primary spaces and O(ρ) full secondary spaces, with now an intricate coupling between the processes N1 and N2. Finally, we note that the results in items (i) and (ii) in Proposition 3 and in item (i) of Proposition 4 are independent of the secondary storage capacity Y0=R/ρ, while item (iii) in Proposition 3 and item (ii) in Proposition 4 do depend upon Y0.


## 3.2. Joint Distribution: Main Regions of State Space

The asymptotic expansion ofπ(k,r)=P(X,Y) will be different for the four parameter ranges indicated in Figure 2 and also for three main regions of the state space, which we call D0, D+, and D-, and we define/discuss these below.First consider regionR1 of parameter space, so that X0>1, and define the curve(56)Y=Y~X=Y~X;X0,Y0=Y0+Y02X0+Y02X0+Y0-1X0-X0+Y02+X0+Y0X-X0+X0+Y02+X0+Y0X0-X-X02-4X0+Y02X0+Y0-1X0-X.This curve depends on both X0 and Y0 and thus on both of the total numbers, m and R, of primary and secondary storage spaces. The curve is defined for 0⩽X⩽X0 and we have(57)Y~0=Y01-X0X0+Y02,Y~X0=Y0.For region R1 (and indeed also for R2 and R3) we have Y~(0)>0 so that (56) connects the point 0,Y01-X0/X0+Y02 to the corner point (X0,Y0) in the scaled state space. The curve Y~(X) divides the state space into the two regions D0 and D+, with(58)D+=X,Y:0<X⩽X0,0<Y<Y~X,(59)D0=X,Y:0<X<X0,Y~X<Y<Y0.Here we defined D0 as an open set, while D+ is bounded by the four curves X=0, Y=0, X=X0, and Y=Y~(X) and we include only the third of these (X=X0) as a part of D+. This is because the asymptotic expansion that will apply in the interior of D+ will remain valid near X=X0, but not near the other three bounding curves. The expansion valid in D0 will break down if either X≈0, Y≈Y0, or Y≈Y~(X). We sketch in Figure 3 the curve Y~(X) and we recall that if X0>1 most of the mass in π(k,r) is concentrated in the range r=0 and k=ρ+O(ρ)(X=1+O(ρ-1/2)) (see Proposition 3), and this corresponds to the lower bounding curve for D+.Figure 3
RegionR1.Proposition 5.
For(X0,Y0)∈R1 (X0>1) the asymptotic expansions of π(k,r)=π(ρX,ρY) are as follows: (i)
(X,Y)∈D+(0<Y<Y~(X))(60)πk,r~CρK+X,YeρΨ+X,Y,whereCρ~ρ-1/2,(61)Ψ+X,Y=X0log⁡X0+X0-1+s-2X0log⁡s+X0+ss+X01-et-ss+X0s+X0-1+e-tlog⁡1+s+X0-1et+e-t-ss+X01-X0-s-etlog⁡1-ss+X0et,where(s,t) are related to (X,Y) via the mapping, for 0<s<Y0,(62)X=1-ss+X0et1+s+X0-1e-t,Y=ss+X0e-t-1+X0+s,(63)K+X,Y=e-t+s+X0-1-1/2e-t-ss+X0-1/2Δ+s,t-1/2X02πs+X02-X03/2s+X03,whereΔ+ is the Jacobian associated with (62); that is,(64)Δ+s,t=XtYs-XsYt=11-EDE-Det-2D1-D-D-E2e-t+E+D-2EDe-2t,D=ss+X0=1-X0s+X0,E=1-X0-s.(ii)
(X,Y)∈D0(Y~(X)<Y<Y0)(65)πk,r~C0ρKX,YeρΨX,Y,whereC0(ρ) is given by (45) for region R1(66)ΨX,Y=A1-eτ+X0log⁡1-A+Y0log⁡1-B-Xlog⁡1-Aeτ-Ylog⁡1-Beτ=-Aeτ+Aeτ-1+e-τ-A1-X01-Alog⁡1-Aeτ+Y0e-τ-BB-1log⁡1-Beτ+A+X0log⁡1-A+Y0log⁡1-B,(67)B=A1-A-X01-AA+X0+Y0-X0,where(A,τ) are related to (X,Y) by(68)X=e-τ-AX01-A-1+eτ,Y=Y0-1-e-τA+X0+Y0-X01-A=Y0B-e-τB-1,whereA∈(Amin,Amax) with(69)Amin=1-X0,Amax=121-X0-Y0+X0+Y0+12-4X0,withA∈(Y0/(X0+Y0),Amax) corresponding to D0, and(70)KX,Y=e-τ-Ae-τ-B-1/2Δ-1/2A1-AY01-A2-X02πY0-AX0+Y0Y0+1-X0-Y0A-A2,whereΔ is the Jacobian associated with (68), so that(71)Δ=XτYA-XAYτ=e-τ-11-X01-A2-Aeτ+1-X01-Ae-τ+Y01-B1+X01-A2e-τ=e-τ-1A+X0+Y0-X01-A+AeτX01-A2-1+e-τ1-2X01-A+X0X0+Y01-A2.We can view (68) as representing a family of curves in the (X,Y) plane, with A indexing the family and τ increasing along a curve. When τ=0 the curves in (68) meet at the corner point (X,Y)=(X0,Y0) and we also note that the Jacobian in (71) vanishes when τ=0, indicating a singularity in the transformation in (68). When A=Amax the curve becomes the horizontal segment Y=Y0. But then (69) shows that (1-Amax)(Amax+X0+Y0)=X0 so that B=∞ in (67). Thus near Y=Y0 the expansion in (68) becomes invalid. When A=Y0/(X0+Y0) the curve in (68) becomes(72)Y=Y01-1-e-τX0+Y0,X=X0+Y0-1+eτe-τ-Y0X0+Y0and eliminating τ we see that (72) is precisely the curve Y=Y~(X) in (56) that separates D0 from D+. For A∈(Amin,Y0/(X0+Y0)) the curves in (68) fill a portion of D+, but then the leading term for π(k,r) is given by (60), and (65) corresponds to only an exponentially small correction to (60). When A=Amin=1-X0 the curve in (68) is tangent to the line X=X0 at the point Y=Y0, which will have significance for the parameter region R4. We also note that for A∈(Y0/(X0+Y0),Amax) when τ=-log⁡A the curves in (68) hit the y-axis (then X=0 by (68)) and then the first factor in (70) becomes singular, which indicates that the asymptotics become invalid. Along A=Y0/(X0+Y0) corresponding to Y=Y~(X), K(X,Y) in (70) is again singular. Thus (70) is singular when A=Y0/(X0+Y0), A=Amax(B=∞), and e-τ=A, corresponding to the three curves (Y=Y~(X), Y=Y0, and X=0) that bound the region D0. We will give the appropriate expansions near these bounding curves in Section 3.3.For0<s<Y0 the curves in (62) fill the entire region D+, with s=0 corresponding to the line segment Y=0, 0<X<X0, and s=Y0 corresponding to the curve Y=Y~(X) (then (62) coincides with (72)). When X=X0 we have t=0 and the curves in (62) hit the line X=X0 at finite and nonzero slopes, for all 0⩽s⩽Y0. As t increases each curve will hit first either the x-axis or the y-axis. When s=0 the x-axis is hit first. For 1<X<X0 this occurs at a finite value of t, when t=log[(X0-1)/(X-1)], but if 0<X<1 in order to approach the x-axis we must let s→0 and t→∞ in such a way that set is held fixed. In this limit (62) may be approximated by X~1-set/X0 and Y~(X0-1)s/X0. We discuss in more detail the behavior of (60) as Y→0 later, when we give the asymptotic expansion(s) for π(k,r) that apply for X0>1 and r=O(1). When s>0 the curves in (62) hit the y-axis when t=log(1+X0/s), for then X=0. In particular if s=Y0 the corresponding curve hits the y-axis at Y~(0) in (57). Near both the X- and Y-axes, (60) will have singular behaviors and other expansions must be constructed. Note, however, that (60) is not singular along the curve Y=Y~(X), whereas (65) is singular. We can simplify (60) near X=X0, and then we obtain the more explicit form(73)πk,r~ρ-1/2X0X0+Ym-kX0+Y2-X0X0+Y5/22πeρΨ+X0,Y,(74)Ψ+X0,Y=Y+X0-1-X0+Ylog⁡X0+Y, which holds for m-k=ρ(X0-X)=O(1) and 0<Y<Y0. However, we note that since X0>1 we have Ψ+(X0,Y)<0 and thus π(k,r) in (73) is exponentially small in ρ. This is true for the entire domains D0 and D+, as there is very little probability mass in these ranges if X0>1.We next consider regionsR2 and R3 in parameter space, where it will become necessary to break up the state space into the three regions D+, D-, and D0. These regions are sketched in Figures 4 and 5. The curve that separates D+ from D0 is again given by (56), while the curve separating D+ from D- will be(75)Y=Y∗X=Y∗X;X0=122X0-X0-X-X0-X2-X02-Xand thus, since now X0<1,(76)Y∗0=0,Y∗X0=X0-X0∈0,Y0.Figure 4
RegionR2.Figure 5
RegionR3.We thus defineD- as(77)D-=X,Y:0<X<X0,0<Y<Y∗Xand, for regions R2 and R3, D+ now becomes(78)D+=X,Y:0<X<X0,Y∗X<Y<Y~X,with D0 still defined by (59).Proposition 6.
For(X0,Y0)∈R2∪R3 the asymptotic expansions of π(k,r) are as follows: (i)
(X,Y)∈D-(0<Y<Y∗(X))(79)πk,r~C1ρLX,YeρΦX,Yeρ1/3Φ1X,Y,where(80)ΦX,Y=W0log⁡W0+s1+Ylog⁡Y+W0-Ylog⁡Y-Xlog⁡1-W0W0+s1W0+Y-W0W0+s1W0+Y-12W0log⁡W0,(81)s1=s1X,Y=W0+Y+W02W0W0+1-X+W0+1-X2-4W0,(82)Φ1X,Y=Φ∗s1=-r01-X02/3X01/6log⁡s1+W01-X0,(83)LX,Y=W0+s13/2YW0+Y1s1-Ys1+W0X0-s12W0+s1+Y1-W0W0+s1W0+Y-1/212πX0-1/61-X0-5/6Ai′r0-2exp⁡-1+X02X0log⁡W0+s11-X0,withW0=1-X02 and r0 is the maximal root of the Airy function Ai(·). In (79), C1~ρ-2/3 for (X0,Y0)∈R2, C1 is given by (35) for (X0,Y0)∈R2∩R3 (then X0+Y0=1+γ/ρ), and C1 is given by (36) for (X0,Y0)∈R3.(ii)
(X,Y)∈D+(Y∗(X)<Y<Y~(X)).The expression in (60) applies with C~ρ-1/2 for (X0,Y0)∈R2, C is given by (32) for (X0,Y0)∈R2∩R3, and C is given by (33) for (X0,Y0)∈R3.(iii)
(X,Y)∈D0(Y~(X)<Y<Y0).The expression in (65) applies with C0 given by (45) for (X0,Y0)∈R2, C0 is given by (44) for (X0,Y0)∈R2∩R3, and C0~ρ-1/2(1-X0-Y0) for (X0,Y0)∈R3.In contrast to regionsD+ and D0, the expansion (79) in D- is a completely explicit function of X,Y, and X0. We also note that s1(X,Y) has a simple linear dependence upon Y, and s1(X0,Y)=Y. In Section 4 we give a more geometric interpretation of this expansion, and we also observe that the form of (79) is slightly different from the expansions in D+ and D0, as the former contains an additional factor that is of order exp[O(ρ1/3)], and thus gives an additional subexponential dependence on ρ. While the forms of C, C0, and C1 change according to whether (X0,Y0) lies in the regions R2, R3 or R2∩R3 of parameter space, the ratios C1 : C : C0 remain the same for these three cases.The expansion in (79) is valid only in the interior of D-. As Y→0 there is a singularity due to the factor 1/Y in L(X,Y) in (83). For X→X0 there is also a singularity due to the factor 1/s1-Y in (83), and as X→X0, we find that (s1-Y)-1/2=O(X0-X)-1/4. The curve Y=Y∗(X) that separates D+ from D- corresponds to s1(X,Y)=X0-X0, and along Y=Y∗(X) the factor (s1+W0)X0-s1=-1-X0[s1+X0-X0] vanishes. Thus (83) shows that L(X,Y∗(X))=0 which also indicates a nonuniformity in the asymptotics. Later we will give appropriate expansions near the three bounding curves (Y=0, X=X0, and Y=Y∗(X)) of region D-.In Figure4 we also indicate the curve(84)Y=Y1X=Y1X;X0=1-X021-X,which lies entirely within D+ (when (X0,Y0)∈R2) and corresponds to ∂Ψ+/∂Y=0, where Ψ+ is in (61). We can also show that Ψ+,YY(X,Y1(X))<0 so that, for a fixed X∈(0,X0), Ψ+ achieves a local maximum along Y=Y1(X). Note that Y=Y1(X) corresponds to s(X,Y)=1-X0(>X0-X0) in (62), for then X=1+(X0-1)et and Y=(1-X0)e-t.For(X0,Y0)∈R3 the curve in (84) plays no role, as it lies outside of D+, but now the curve(85)Y=Y2X=Y2X;X0,Y0=Y01-X01-Xlies within D0, connecting the points (0,Y0(1-X0)) and (X0,Y0). Along Y=Y2(X) we have ΨY=0 and then A=1-X0 and B=0, so that (68) becomes X=1-(1-X0)eτ and Y=Y0e-τ. Then for region R3, Ψ will have a local maximum along Y=Y2 in D0.For(X0,Y0)∈R3 most of the mass will lie near the corner point (X,Y)=(X0,Y0), where D0 and D+ meet, but neither (60) nor (65) (with the appropriate C0 and C) are valid there. For (X0,Y0)∈R2, π(k,r) will be maximal near the point (X,Y)=(X0,1-X0), and (73) applies for k=m-O(1) (or X=X0-O(ρ-1)) for both regions R1 and R2. By expanding (73) about Y=1-X0, which is where Ψ+(X0,Y) is maximal, we obtain precisely the expression in (49).Next we consider regionR4 of parameter space. Now the state space will be split into D0 and D-, and D+ will be absent. This is sketched in Figure 6, and we also observe that as X0+Y0↓X0 in region R3, the curves Y∗(X) and Y~(X) in (75) and (56) become identical, and thus D+ shrinks to this curve. For R4 there is a new curve that comes into play; namely,(86)Y=YcX=YcX;X0,Y0=-W0+Y0+W02W0W0+1-X-W0+1-X2-4W0.We have Yc(X0)=Y0 and the curve hits the x-axis (Y=0) when(87)YcX=0⟺X=X0-Y02Y0+W01W0. The curve Y=Yc(X) now separates D0 from D- and corresponds to A=Amin=1-X0 in (68), and also s1(X,Y)=Y0 in (81).Figure 6
RegionR4.Proposition 7.
For(X0,Y0)∈R4 (thus 0<Y0<X0-X0) the asymptotic expansions of π(k,r) are as follows: (i)
(X,Y)∈D-, 0<Y<Yc(X), X0-Y02/W0Y0+W0<X<X0.The expansion in (79) applies with now C1(ρ) given by (41).(ii)
(X,Y)∈D0(max{0,Yc(X)}<Y<Y0).The expansion in (65) applies for A∈(Amin,Amax) with C0(ρ)~ρ-1/2(1-X0-Y0).Again, different expansions must be given nearY=Yc(X) and near all boundaries of the state space. The curve Y=Y2(X) in (85) still lies within D0 and is sketched in Figure 6, with again ΨY=0 along this curve.We conclude by noting that, for all regions of parameter spaceRj, the expansion in D0 depends upon the secondary storage capacity Y0 (or R). For regions R1 and R2 the expansions in D+ and D- (R2 only) are independent of Y0, except through the curve Y~(X) that bounds D+. Thus if X0+Y0>1 the effects of the finite storage capacity appear in D0 only, and then letting Y0→∞ will recover the results for the storage model in [10, 11], which assumes that R=∞. For (X0,Y0)∈R3, hence X0<X0+Y0<1, the expansions in D- and D+ depend upon Y0 only through the multiplicative constants C1 and C, which now depend on Y0 in view of (33) and (36). For (X0,Y0)∈R4, hence 0<Y0<X0-X0, again the D- result depends on Y0 only through the constant C1 (cf. (41)). For (X0,Y0)∈R3∩R4, that is, along and near the curve X0+Y0=X0, we have Y3(X)=Yc(X)=Y~(X)=Y∗(X) so all four of these curves coalesce. A special analysis is required for X0+Y0-X0=O(ρ-1/3) for values of (X,Y) near this curve(s).This completes our summary of the asymptotic expansions in the three main regions,D0, D+, and D-, of the state space.


## 3.3. Joint Distribution: Boundary, Corner, and Transition Regions

We next analyze the four boundary segments of the state space rectangle, namely, the line segments{Y=0,0<X<X0}, {X=X0,0<Y<Y0}, {X=0,0<Y<Y0}, and {Y=Y0,0<X<X0}. As we previously discussed, the expansions in the three main regions become invalid near the boundary of the state space, with the exception of the D+ expansion in (60), which remains valid near X=X0, reducing to (73) when k=m-O(1)(X=X0-O(ρ-1)), at least for (X0,Y0)∈R1∪R2. For the four boundary segments we will typically consider the respective scales r=O(1), k=m-O(1), k=O(1), and r=R-O(1), though at times a different scaling must be considered in addition to these discrete scales. Note also that a particular point on a boundary segment may also require a separate expansion, and this occurs in R2∪R3, when Y∗(X) hits X=X0 at the point (X0,X0-X0). Also, in R1∪R2∪R3, the curve Y~(X) that separates D0 from D+ hits the y-axis when Y=Y01-X0/(X0+Y0)2(>0) and this point requires a separate analysis. Finally, in R4 the curve Yc(X) hits the x-axis when X=X0-Y02/[(Y0+W0)W0].After treating the four boundary segments we will give asymptotic results that are valid near the four corner points,(0,0), (X0,0), (X0,Y0), and (0,Y0), of the state space. Finally we give results for points (X,Y) that lie on or near the transition curves Y~(X) (for R1∪R2∪R3), Y∗(X) (for R2∪R3), and Yc(X) (for R4). Unlike the previous subsections where we listed the results by region Rj of parameter space, here we go by region of state space, and each proposition will correspond to one such region and the different results for the different Rj will be collected in that proposition.Proposition 8.
Forr=O(1) and k=ρX, 0<X<X0, one has the following expansions: (i)
R1:X0>1, 0<X<1(88)πk,r~ρ-rX0-12πXX01-X-rX0-1-2rr-1!eρ-2+X+X0-XlogX-X0logX0,r⩾1,(89)πk,0-ρke-ρk!~ρ-1-Xeρ-2+X+X0-XlogX-X0logX02πXX0X0-11-X.(ii)
R1:X0>1, X=1+α/ρ, α=O(1)(90)πk,r~ρ-r/2X0-12πX0X0-1-2r∫-α∞α+ur-1e-u2/2dueρ-1+X0-X0logX0,r⩾1,(91)πk,0-ρke-ρk!~ρ-1/2-eρ-1+X0-X0logX02πX0X0-1∫-α∞e-u2/2du.(iii)
R1:X0>1, 1<X<X0(92)πk,r~ρ-1/2X0-132πX0X-1r-1X-1+X0-12-r-1eρ-1+X0-X0logX0,r⩾1,(93)πk,0-ρke-ρk!~ρ-1/2-X0-1eρ-1+X0-X0logX02πX0X-1+X0-12.(iv)
R1∩R2:X0=1+β/ρ, β=O(1), 0<X<1(94)πk,r~ρ-1/21Xeρ-1+X-XlogX12πi∫Br+Γz+r1-Xzr!Dz-βDz-1-βdz,whereBr+ is the imaginary axis in the z-plane if r⩾1, and if r=0 the contour is to be indented to the right of the pole at z=0. Here Γ(·) is the gamma function. When β=0 the expression in (94) simplifies to(95)πk,r~ρ-1/22π1-XX2-X-r-1eρ-1+X-XlogX.(v)
R2∪R3:0<X0<1, X0+Y0>X0, 0<X<X0(96)πk,r~ρC1ρeρΦX,0eρ1/3Φ∗s10ρrW0rr!1-X01/6W0+s10X0-X0-s102πAi′r02X01/6s10s10+2W0W0s10+W01/2X0,whereW0=1-X02,(97)s10=-W0+W02W0+1-X+W0+1-X2-4W0,so thats10 corresponds to s1(X,0) in (81), Φ∗(·) is defined in (82), and Φ(X,0) can be computed from (80) by setting Y=0 and replacing s1 by s10. The value of C1 in (96) is given by (34) for R2, by (35) for R2∩R3(X0+Y0=1+O(ρ-1/2)), and by (36) for R3.(vi)
R4:0<Y0<X0-X0, X0-Y02/W0Y0+W0<X<X0.The expression in (96) applies with C1 now given by (41). For region R3∩R4 (hence X0+Y0=X0+O(ρ-1/3)), (96) holds for 0<X<X0, where C1 can be computed from (37).(vii)
R4:0<Y0<X0-X0, 0<X<X0-Y02/W0Y0+W0≡XL(98)πk,r~1-X0-Y0ρrr!Y0B01-B0rgXeρΨX,0,(99)ΨX,0=A01-1B0+X0log⁡1-A0+Y0log⁡1-B0-Xlog⁡1-A0B0,whereA0=A0(X)=A0(X;X0,Y0) is the solution of the algebraic equation(100)A0-1+X0+Y0Y01-A02+X0A01-X0-A0=-X1-X0-A01-A0A0+X0+Y0-X0that satisfiesA0(0)=1-X0-Y0 and A0(XL)=1-X0=W0,(101)B0=A01-X0-A01-A0A0+X0+Y0-X0,so thatB0(0)=1-X0-Y0 and B0(XL)=W0/(W0+Y0), and(102)gX=A01-A01-A02-X0Y01-B0B0-A0Y0-A0X0+Y0Y0-A02+1-X0-Y0A0A0+X0+Y0-X01-A0+A0B0X01-A02-1+B01-2X01-A0+X0X0+Y01-A02-1/2.ForX→0 and X→XL one can obtain g(X) more explicitly, with(103)gX~X0+Y0X,X⟶0,(104)gX~2Y0W0+Y03/21-X05/2XL-X2W0+Y03/2X0-X0-Y03/2,X⟶XL.(viii)
R4:0<Y0<X0-X0, X-XL=ρ-1/3Λ=O(ρ-1/3)(105)πk,r~1-X0-Y0ρrW0rr!ρ-1/3W0+Y0X01/61-X05/62W0+Y0X0-X0-Y03/2exp⁡ρΨXL,0+ρ2/3ΨXXL,0Λ+12ρ1/3ΨXXXL,0Λ2exp⁡Λ36W0Y0+W03Y03+3W0Y02+4W02Y0-2X0W05/2Y03Y0+2W03X0-X0-Y0212πi∫-i∞i∞1Aiθexp⁡Y0+W01-X05/3Y0Y0+2W0X01/6Λθdθ,where(106)ΨXL,0=Y0log⁡Y0W0+Y0-Y0W0+12X0log⁡X0+Y02Y0+W01W0-X0log⁡X0-X0-Y0W0,(107)ΨXXL,0=-log⁡X0-X0-Y01-X0,(108)ΨXXXL,0=-Y0+W02W0Y0Y0+2W0X0-X0-Y0.We note that the regions of validity of the expansions in Proposition8 are such that the four corner points of the state space are excluded. For regions R1 and R2 the expansions for r=O(1) are independent of the secondary storage capacity Y0. For region R3 the result in (96) does depend upon Y0, but only through the multiplicative constant C1, which now depends on Y0 in view of (36). This is also true for region R4 (and R3∩R4) when X>XL, as then C1 is given by (41) or (37). However, for 0<X<XL and X≈XL, the expressions in (98) and (105) show that now π(k,r) depends in an intricate way on Y0. Thus the finiteness of the secondary storage capacity affects the probabilities that there are even a few secondary spaces occupied.For regionR1, π(k,r) is exponentially small for r⩾1 and π(k,0) approximately follows a Poisson distribution, as indicated previously in Proposition 3. The results in (88)–(93) better quantify (47) and estimate the exponentially small error from the Poisson approximation. For regions R2, R3, and R4, π(k,r) is always exponentially small for r=O(1), and the approximations do not distinguish r=0 from r⩾1. By evaluating the contour integral in (105) for Λ→+∞ we can verify that (105) asymptotically matches to (96) (with C1 given by (41)), as X↓XL. Similarly, for Λ→-∞ (105) will match to (98) as X↑XL.Next we consider points(k,r) near X=X0, with X0-X=o(1). Note that only regions D+ and D- can be bounded by X=X0, for 0<Y<Y0.Proposition 9.
ForX~X0, one uses the scales k=m-n and X=X0-ρ-2/3ν1, so that n=ρ1/3ν1=ρ(X0-X) denotes the number of empty primary storage spaces. The expansions are now the following: (i)
R2∪R3:0<X0<1, X0+Y0>X0; n=O(1), 0<Y<X0-X0(109)πk,r~ρ-1/6C1ρeρΦX0,Yeρ1/3Φ∗YX0n12π1Ai′r01-X0-1/6X0-5/6n+X0Y+W0Y+W0X0-YY+W0X0-YYexp⁡-1+X02X0log⁡W0+Y1-X0,whereC1 is given by (34), (35), and (36) for parameter regions R2, R2∩R3, and R3, respectively, Φ∗(Y) is obtained by replacing s1 by s1(X0,Y)=Y in (82), and(110)ΦX0,Y=Y+W0log⁡Y+W0-Ylog⁡Y+X0-1-12X0log⁡X0-12W0log⁡W0.(ii)
R2∪R3:ν1=O(1) with ν1>0, 0<Y<X0-X0(111)πk,r~ρ-1/6C1ρeρΦX0,Yeρ1/3ν1logX0+Φ∗Y12π1Ai′r02Y+W0X0-YYX01-X01/2exp⁡-1+X02X0log⁡W0+Y1-X0Aiν+r0,whereν=1-X01/3X0-1/3ν1 and C1 is again given in Proposition 2 for the different ranges Rj.(iii)
R3∩R4 or R4: n=O(1) or ν1=O(1).Now the expressions in (109) and (111) apply over all Y(0<Y<Y0), with C1 given by (37) for R3∩R4 and (41) for R4.(iv)
R2∪R3: ν1=O(1), Y-(X0-X0)=ρ-1/3ω1=O(ρ-1/3)(112)πk,r~ρ-1/6C1ρeρ-1+X0-X0logX0exp⁡-ρ2/3ω1log⁡X0+ρ1/3ν1log⁡X0-ω122X01-X01/32πX07/12exp⁡1+X06X01-X0ω1312πi∫-i∞i∞Aiν+θAi2θe-ωθdθ,whereω=1-X0-1/3X0-1/6ω1, so that ν1ω1=X0νω, and C1 is given again in Proposition 2 for the three cases R2, R2∩R3, and R3.(v)
R1∪R2∪R3: 0<Y<Y0(R1) or X0-X0<Y<Y0(R2∪R3), n=O(1)(113)πk,r~CρX0+Y2-X02πX0+Y5/2X0X0+YneρΨ+X0,Y,whereΨ+(X0,Y) is given in (74) and C has the expansions in (31)–(33).Note that the asymptotics are most complicated in (112), which occurs when (X,Y)≈(X0,X0-X0) and this is where the curve Y∗(X) hits the line X=X0. This intersection occurs only for regions R2 and R3. The integrand in (112) is a meromorphic function of θ, having double poles at all roots of the Airy function Ai(·).Proposition 10.
Fork=O(1) and r=ρY, 0<Y<Y0, one has the following expansions: (i)
R1∪R2∪R3:0<Y<Y01-X0/(X0+Y0)2≡YU(114)πk,r~ρCρeρΨ+0,Yρkk!1+Y-D02k1-D0YD0X0Y+2D021-D0,whereD0=D0(Y) is the root of the cubic equation(115)D03-2D02+1-X0-YD0+Y=0that satisfiesD0(0)=1-X0=W0 and(116)Ψ+0,Y=D0+X01-D0-2-X0log⁡X0-Ylog⁡Y+2X0log⁡1-D0+2Ylog⁡D0.The expansion ofC is given by (31)–(33) according to subregions of R1∪R2∪R3.(ii)
R1∪R2∪R3∪R4: YU<Y<Y0(R1∪R2∪R3) or 0<Y<Y0(R4)(117)πk,r~Q0,0eρΨ0,Yρkk!A1A1+X0-2+11-A1kY0A1A1A1+X0-2+11-A12-X0YY0-YΔLYY0-A1X0+Y0,where(118)A1=A1Y=A1Y;X0,Y0=121-X0-Y0+X0+Y0-12+4Y,(119)Ψ0,Y=A1-1+X0log⁡1-A1+Y0-Ylog⁡1-A1+Ylog⁡A1-Ylog⁡Y-Y0-Ylog⁡Y0-Y+Y0log⁡Y0,(120)-ΔLY=1-A1A1+X0+Y0+X01-A12-X01-A1-1+A11-2X01-A1+X0X0+Y01-A12and the expansion ofQ(0,0) is given in Proposition 1 for the three cases R1∪R2, R2∩R3, and R3∪R4 (when Q(0,0)~1-X0-Y0).(iii)
R1∪R2∪R3: Y-YU=y∗/ρ=O(ρ-1/2)(121)πk,r~Q0,0eρΨ0,Y~0eρy∗ΨY0,Y~0ρρkk!Y0+X0X0+Y0kexp⁡12y∗2Ψ+,YY0,Y~0X0Y0X0+Y02-X0X0+Y03+X0Y0-X0∫acy∗∞e-u2/2du,where(122)Ψ0,Y~0=X0log⁡X0+Y0log⁡Y0-X0+Y0log⁡X0+Y0-X0X0+Y0+X0Y0X0+Y022log⁡X0+Y0-log⁡Y0+Y0X0X0+Y02-1log⁡1-X0X0+Y02,(123)ΨY0,Y~0=Ψ+,Y0,Y~0=log⁡Y0-log⁡X0+Y02-X0,(124)Ψ+,YY0,Y~0=X0Y0X0+Y02X0+Y03+X0Y0-X01-Y0X0X0+Y02+X0X0+Y02-X0,(125)ac=X0+Y05/2X0+Y02-X0X0Y0X0+Y03+X0Y0-X0X0+Y02+Y0-X0.We see that for each of the three cases the dependence ofπ(k,r) on k is of the form ρk[A∗(Y)]k/k! (times a function of Y or y∗), where the geometric ratio A∗ depends upon Y=r/ρ and undergoes a transition when Y increases past YU=Y~(0), which can occur only for regions R1, R2, and R3. We note that A1(YU)=Y0/(X0+Y0) and D0(YU)=Y0/(X0+Y0), so that A∗ is continuous along Y=YU, with A∗(YU)=1+Y0-X0/(X0+Y0), as indicated in (121). The expansion in (117) develops a singularity as Y↓YU, in view of the factor [Y0-A1(X0+Y0)]-1. The expansion is also singular as Y↑Y0 (for all regions Rj) and as Y↓0 (for region R4 only). Then we are approaching the corner points (0,Y0) or (0,0) of the state space. Using the expansions(126)∫acy∗∞e-u2/2du~2π,y∗⟶-∞acy∗-1exp⁡-ac2y∗22,y∗⟶+∞we can easily show that (121) matches to (117) (for region R1∪R2∪R3 in the intermediate limit where y∗→+∞ and Y↓YU) and to (114) (with now y∗→-∞ and Y↑YU). Note also that the ratio Q(0,0)/C(ρ) is asymptotically the same for each of the three regions R1, R2, and R3, in view of (28)–(33). For any region Rj, for k=O(1) and Y∈(0,Y0), π(k,r) is exponentially small in ρ.Proposition 11.
ForR-r=p=O(1) and 0<X<X0 one has (for all regions Rj in parameter space)(127)πk,r~Q0,0eρΨX,Y0ρpp!Y0eτU-1p1-Amaxe-τU-Amax1-X0-Y02+4Y01-Y0-X2-4Amax1-X0/1-Amax1/4,where τU=τU(X)=τU(X;X0,Y0) is given by(128)τU=-log⁡1-Y0-X-1-Y0-X2-4Amax1-X01-Amax+log⁡21-X01-Amax,where Amax is given in (69), Q(0,0) has the respective expansions in (28)–(30), and(129)ΨX,Y0=Amax1-eτU+X0log⁡1-Amax-Y0τU-Xlog⁡1-AmaxeτU.Thus (127) gives the expansion when there are only a few secondary spaces empty and a fraction X/X0=k/m∈(0,1) of primary spaces occupied. The expressions can be simplified in the limits X→0 (then τU(0)=-log(Amax)) and X→X0 (then τU(X0)=0), but then other “corner” expansions will apply. Note that (127) is a completely explicit expression in terms of p and X.Next we examine the four corners of the state space, where(X,Y)=(0,Y0), (0,0), (X0,0), and (X0,Y0). We recall that these ranges are important in that for region R1∩R2(X0≈1) most of the mass concentrates near the corner (X0,0), while for region R3∪R4(X0+Y0<1) most of the mass occurs near (X0,Y0). We will start with the corner (0,Y0) and proceed about the perimeter of the state space in a counterclockwise manner.Proposition 12.
Fork=O(1), p=R-r=O(1), and any parameter region Rj one has(130)πk,r~2πρQ0,0ρk+pk!p!1+Y0kY01-AmaxAmaxpeρΨ0,Y01-AmaxAmax1-X0-Y02+4Y01/4, where Q(0,0) is given in Proposition 1, Amax is in (69), and(131)Ψ0,Y0=Amax-1+X0log⁡1-Amax+Y0log⁡Amax.This gives the expansion when there are but a few primary spaces occupied and a few secondary spaces empty. Next we consider the corner point(0,0), and this will typically correspond to k,r=O(1). The results will be very different for region R1 compared to those for the remaining regions.Proposition 13.
Fork,r=O(1) and ρ→∞ one has (i)
(X0,Y0)∈R1(X0>1)(132)πk,r~ρ2πr-1!k!ρk-rX0-11-2rX0eρ-2+X0-X0logX0,r⩾1,(133)πk,0-ρke-ρk!~-ρ2πρkk-1!eρ-2+X0-X0logX0X0-1.(ii)
(X0,Y0)∈R1∩R2(X0-1=β/ρ=O(ρ-1/2)), r⩾0(134)πk,r~ρke-ρk!r!12πi∫Br+Γz+rDz-βDz-1-βdz,whereBr+ is as in (94), and when β=0(135)πk,r~ρke-ρk!2-rβ=0.(iii)
(X0,Y0)∈R2∪R3X0<1,Y0+X0>X0(136)πk,r~ρ2/3C1ρρk+re-ρk!r!W0r1-W0k,whereC1 is given by (34), (35), or (36) for R2, R2∩R3, or R3, respectively.(iv)
(X0,Y0)∈R4(0<Y0<X0-X0)(137)πk,r~2πρX0+Y01-X0-Y0eρ-X0-Y0+X0+Y0logX0+Y0ρk+rk!r!Y0+X0X0+Y0kY0X0+Y0-Y0r.(v)
(X0,Y0)∈R2∪R3; k=ρ2/3X′, r=ρ2/3Y′; X=ρ-1/3X′, Y=ρ-1/3Y′(138)πk,r~ρ2/3C1ρρk+rk!r!W0r1-W0k12πi∫-i∞i∞e-ξ′zAiz2dzexp⁡12ρ1/3Y′2W0-W01-W02X′2-16Y′3W02+W0W0+21-W04X′3,(139)ξ′=1-X02/3X01/6Y′W0-X′1-W0.We note that (136) and (137) are continuous along the curve X0+Y0=X0, corresponding to R3∩R4, in view of the expansion in (36) for C1 in R3. Thus the leading term for k,r=O(1) for (X0,Y0)∈R3∩R4 can be obtained by either using (36) to compute C1 in (136) or replacing Y0 by X0-X0 in (137). The scale k,r=O(ρ2/3) must be considered (cf. (138)), as the approximation in (136) cannot directly match to those for k=O(1), 0<Y<YU (cf. (114)) or r=O(1), 0<X<X0 (cf. (96)). We can view (136) as a special case of (138), letting X′,Y′→0 in the latter and noting that(140)2πi-1∫-i∞i∞Aiz-2dz=1. For region R4 we see from Figure 6 that only state space region D0 meets the corner (X,Y)=(0,0). Then (137) matches directly to (98), in the limits k→∞, X→0, and to (117), in the limits r→∞, Y→0. Note that, for region R4, A1(0)=1-X0-Y0, which follows from (118) (this is true also for R3, but then (117) applies only for Y>YU and thus not near the corner (0,0)). We can obtain a result analogous to (138) for the transitional range R3∩R4, but we do not give that for the sake of brevity.Proposition 14.
Fork=m-O(1) and r=O(1), or for k=m-O(ρ) and r=O(ρ), the expansions of π(k,r) are as follows: (i)
(X0,Y0)∈R1, m-k=n⩾0(141)πk,r~12πρX0eρ-1+X0-X0logX0X0-1X0-r-1,r⩾1πk,0~12πρX0e-1+X0-X0logX0X0n-1X0.(ii)
(X0,Y0)∈R1∩R2(X0-1=β/ρ), m-k=n⩾0(142)πk,r~ρ-1ρ-z0/2e-β2/4r!Δ0βΓz0+r+1nz0z0+r+1,wherez0 is the minimal root of the parabolic cylinder function Dz(·); that is, z0=z0(β)=min{z:Dz(-β)=0}, and Δ0(β)=-d/dzDz(-β)z=z0β.(iii)
(X0,Y0)∈R1∩R2(X0-1=β/ρ), k=ρ+ρα, and r=ρΩ, with α⩽β and Ω>0(143)πk,r~ρ-112πi∫-i∞i∞e-α2/4Dz-αΩz-1Dz-βDz-1-βdzwhich whenβ=0 simplifies to(144)πk,r~ρ-12πΩ-αe-Ω-α2/2;α<0,Ω>0.(iv)
(X0,Y0)∈R1∩R2(X0-1=β/ρ), k=ρ+ρα, α<β, r=O(1)(145)πk,r~ρ-1/2ρ-z0/2e-α2/4Γr+z0r!Dz0-αΔ0βDz0-1-β.(v)
(X0,Y0)∈R1∩R2 with now X0-1=β∗ρ-1/4=O(ρ-1/4) and β∗>0, r=O(1), k=ρ+ρα, α=O(1)(146)πk,r~β∗32πρ1/4⊗∗∫-α∞u+αr-1u+α+β∗2r+1e-u2/2du,r⩾1,(147)πk,0-ρke-ρk!~-β∗2πρ1/4⊗∗∫-α∞e-u2/2u+α+β∗2du,(148)⊗∗=exp⁡-ρ2β∗2+ρ1/46β∗3-112β∗4.(vi)
(X0,Y0)∈R2∪R3∪R4, m-k=n⩾0(149)πk,r~ρ1/3C1ρρrW0rr!X0nn+1eρΦX0,0eρ1/3Φ∗01-X04/3Ai′r0X01/3exp⁡-12X0log⁡1-X0,(150)Φ∗0=-r01-X02/3X01/6log⁡1-X0,(151)ΦX0,0=12W0log⁡X0-12X0log⁡X0+X0-1,andC1 is given in Proposition 2 for the different regions Rj.For regionR1, (141) show that the dependence of π(k,r) on m-k and r is quite simple, but we have to distinguish the cases r=0 and r⩾1. The same is true for regions R2, R3, and R4, where (149) applies for all r⩾0, except now the form of C1 in (149) is different for the five cases in Proposition 2. However, when X0≈1, the asymptotic structure of π(k,r) is quite complicated, and we must consider separately the scales X0-1=O(ρ-1/2)(m-ρ=O(ρ)) and X0-1=O(ρ-1/4)(m-ρ=O(ρ3/4)). For X0-1=O(ρ-1/2) we obtain the limiting density in (53) or (143), as the limit of ρπ(ρ+ρα,ρΩ)=ρπ(m+ρ(α-β),ρΩ) for ρ→∞. This expansion applies for Ω>0, for any α⩽β, but becomes invalid as Ω→0. For Ω→0 the asymptotic behavior of the contour integral in (143) is determined by the singularity at z=z0(β), and the density behaves as O(Ωz0-1) for Ω→0 and α<β. This corresponds to either an integrable singularity or a zero of the density (unless β=0 then z0=1) and in either case indicates a nonuniformity in the asymptotics. Thus we need the expansion in (145) for r=O(1). For r→∞ we have, by Stirling’s formula,(152)Γr+z0r!=Γr+z0Γr+1~rz0-1=ρΩz0-1 and then (145) matches to (143) in the intermediate limit where r→∞ but Ω=r/ρ→0. The expansion in (145) itself breaks down when α→β, since by the definition of z0(β) we have Dz0(β)(-α)→0 as α↑β. Then for α~β we have the expansion in (142), which holds for n=O(1) and we note that n=m-k=(β-α)/ρ so that α=β-nρ-1/2.We can show also that the expansions forn,r=O(1) (cf. (141), (142), and (149)) match in appropriate intermediate limits. Consider (142) for β→-∞ (then we are moving into the range X0<1). We now have z0→∞ and more precisely(153)z0β=β24-r0-β22/3-12+o1,β⟶-∞so the approximation to the minimal root of the parabolic cylinder function Dz(-β) involves the maximal root of the Airy function Ai(·). Thus, for β→-∞ we have(154)Δ0β~z0z0/2e-z0/22-β1/32πAi′r0which can be obtained by approximated Dz(-β) by Airy functions in the double limit where β→-∞ and z→∞, with z-β2/4=O(-β)2/3. Using (154) in (142) and expanding Γ(z0(β)+r+1) by Stirling’s formula (since now z0→∞) for a fixed r, (142) becomes(155)ρ-1ρ-z0/2n+1-β24/3e-β2/4Ai′r0z0z0/2e-z0/2z0rr!.Then (155) must agree with the expansion of (149) as X0↑1. Then we use the fact that ρ1/3C1~ρ-1/3 in region R2. Also, as X0→1, W0=1-X02~β2/(4ρ),(156)ρ-1+X0-12X0log⁡X0+W0log⁡1-X0=β24log⁡-β2ρ-38β2+o1,-ρ1/3r0X0-1/61-X02/3log⁡1-X0~-r0-β22/3log⁡-β2ρ.Using (156) we see that as X0↑1, (149) becomes(157)ρ-1/31r!β24rn+1Ai′r0-β2ρ4/3exp⁡β24-r0-β22/3-12log⁡-β2ρ-38β2.But by using (153) in (155) we obtain again the expression in (157), which verifies the matching.Now consider (141) for X0↓1 and (142) for β→+∞. We now have(158)z0β~β2πe-β2/2,Δ0β~2πβeβ2/4,β⟶+∞so that z0 is exponentially small and Δ0 is exponentially large. The last factor in (142), namely, nz0/(z0+r)+1, can be approximated by 1 if r⩾1 and is equal to n+1 if r=0. Now also Γ(z0+r+1)/r!→1 and (142) becomes, for β→+∞,(159)β2πe-β2/2ρ-11,r⩾1n+1,r=0, and the above clearly agrees with (141), when we expand these for X0→1.The results in (146)–(148) assume the scaling X0-1=β∗ρ-1/4, and these are needed to asymptotically connect the parameter ranges X0-1=O(ρ-1/2) and X0>1. It is only near the corner (X,Y)=(X0,0) that we must consider this scaling (and indeed also the β-scale). For other ranges of X and r=O(1), we can get the expansions of π(k,r) as limiting cases of other expansions, as they lie in the asymptotic matching range where X0→1 but (X0-1)ρ(=β)→∞. For a fixed β∗>0 and 0<X<1 we have(160)πk,r~r-1!β∗1-2rρ-r/2⊗∗2πXρ1/41-Xreρ-1+X-XlogX,r⩾1;for X-1=α∗ρ-1/4=O(ρ-1/4) with 0<α∗<β∗ we have(161)πk,r~⊗∗2πρ3/4β∗3α∗2,r⩾1;and for the scale m-k=n=O(1) we have α∗~β∗ so the last factor in (161) may be approximated by β∗3α∗-2~β∗. We can easily verify that as X↑1, (159) matches to (146) as α→-∞. Note that in this limit we have(162)∫-∞∞u+αr-1u+α+β∗2r+1e-u2/2du~β∗-2-2rr-1!e-α2/2-α-r. For α→+∞ we have(163)∫-α∞u+αr-1u+α+β∗2r+1e-u2/2du~2πα2=2πρα∗2 and then (146) clearly matches to (160) and in fact contains the latter as a limiting case. When r=0 we have, for 0<X<1,(164)πk,0-ρke-ρk!~-X1-X⊗∗2πρ3/4β∗eρ-1+X-XlogX;for X-1=α∗ρ-1/4, α∗>0, we have(165)πk,0-ρke-ρk!~-⊗∗2πρ1/2β∗α∗;and for n=m-k=ρ(β-α)=ρ3/4(β∗-α∗)=O(1)(166)πk,0~⊗∗2πρ3/4β∗n+1.We can again easily verify that (147) matches to (164) for X→1 and α→-∞ and to (165) for α→+∞ and α∗=αρ-1/4→0. Recalling that k=ρX=ρ+ρ3/4α∗=ρ+ρ3/4β∗-n, we have (167)ρke-ρk!-⊗∗2πρ1/2β∗k!=ρke-ρk!-⊗∗2πρβ∗β∗-ρ-3/4n=⊗∗2πρexp⁡nβ∗ρ1/4-1+Oρ-3/4~⊗∗2πρ3/4β∗n and this agrees with (166) for n→∞, which verifies the matching between (165) and (166). Thus we have given π(k,r) for all ranges of k for r=O(1) and the scaling m=ρ+ρ3/4β∗, β∗>0. However, only for the range k=ρ+O(ρ) do we get the new results in (146) and (147).Next we examine the corner(X,Y)=(X0,Y0), so both primary and secondary spaces will be nearly full. For regions R3∪R4 most of the mass is in the range. We will need to consider the scales m-k,R-r=O(1) and also m-k=O(ρ1/3). Consider R-r=O(ρ2/3).Proposition 15.
Fork=m-o(ρ) and r=R-o(ρ) one uses the variables m=m-k=ρ1/3ν1 and p=R-r=ρ2/3T1, and the expansions are as follows: (i)
(X0,Y0)∈R1∪R2∪R3∪R4, n=O(1), p=O(1)(168)πk,r~Q0,0X012πi∮1-wz--wz+-n-1w-p-1dw,wherez±(w) are defined in (50) and the expansions for Q(0,0) are in Proposition 1; in particular for region R3∪R4 one has Q(0,0)~1-X0-Y0>0 and then (168) is a proper discrete distribution over the range(s) n⩾0, p⩾0.(ii)
(X0,Y0)∈R4, m-k=ρ1/3ν1=O(ρ1/3), ν1>0, R-r=ρ2/3T1=O(ρ2/3)(169)πk,r~ρ-2/31-X0-Y01-X05/3X0-1/6X0-X0-Y0W0+Y0Fν,TX0nY0Y0+W0pexp⁡-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13,where(170)Fν,T=12πiddT∫-i∞i∞Aiν+θθAiθeθTdθ=∑j=0∞Aiν+rjAi′rjerjT,T>0,(171)ν1=νX01-X01/3,T1=W0+Y0X01/61-X02/3T,andrj are the roots of Ai(·), ordered as 0>r0>r1>⋯.(iii)
(X0,Y0)∈R4, m-k=n=O(1), R-r=p=ρ2/3T1=O(ρ2/3)(172)πk,r~ρ-11-X0-Y0W0X0-1/2X0-X0-Y0W0+Y0X0nY0Y0+W0pn+X0Y0+W0Y0+W0X0-Y0∑j=0∞erjTexp⁡-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13,T>0.(iv)
(X0,Y0)∈R3∩R4, m-k=n=ρ1/3ν1=O(ρ1/3), ν1⩾0, R-r=p=ρ2/3T1=O(ρ2/3)(173)πk,r~ρ-1/31-X04/3X0-1/3X0nY0Y0+W0pexp⁡-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13F¯ν,T;δ1,where(174)F¯ν,T;δ1=12πi∫-i∞i∞Aiν+θAiθ2eθT-δ1∫θ∞eδ1uAiududθ,(175)X0+Y0=X0+ρ-1/3δ∗,δ∗=X01/61-X01/3δ1and(ν,T) is given by (171) in terms of (ν1,T1).When(X0,Y0)∈R3∪R4 most mass concentrates on the (n,p) scale so there tend to be but a few available primary and secondary spaces. For (X0,Y0)∈R1∪R2(X0+Y0>1), the result in (168) still applies but now Q(0,0) is exponentially small (cf. (30)). For X0+Y0-1=O(ρ-1/2) we have Q(0,0)=O(ρ-1/2). Later we will study the behavior of the contour integral in (168) as n and/or p→∞, and we will see that for R1∪R2 we can get exponential growth in certain sectors, such as if p→∞ with n=O(1). From Proposition 15 we also see that the probabilities of finding O(ρ1/3) empty primary spaces and O(ρ2/3) secondary spaces are quite complicated, and their estimation involves contour integrals of Airy functions. These probabilities are however quite small, in view of the factors X0n=X0ρ1/3ν1 and [Y0/(Y0+W0)]p=[Y0/(Y0+W0)]ρ2/3T1 in (169) and (173). In (173) we can use (175) and rewrite the result in terms of X0 and δ∗, thus eliminating Y0.Finally we give results that apply near the transition curves that separateD0, D-, and D+. Note D0 meets D+ for region(s) R1∪R2∪R3, D+ meets D- for region R2∪R3, and D0 meets D- for region R4 (see also Figures 3–6).Proposition 16.
For(X0,Y0)∈R1∪R2∪R3, Y-Y~(X)≡η∗/ρ=O(ρ-1/2), and 0<X<X0 one has(176)πk,r~CρK+X,Y~X12π∫η∗/bt∗∞e-u2/2duexp⁡ρΨ+X,Y~X+ρΨ+,YX,Y~Xη∗+12Ψ+,YYX,Y~Xη∗2,where(177)Ψ+X,Y~X=X0log⁡X0+X0+Y0-1-2X0log⁡X0+Y0+Y0X0+Y01-et∗-Xlog⁡1-Y0et∗X0+Y0-Y~Xlog⁡1-1-X0-Y0et∗,(178)t∗=log⁡Y0-log⁡Y0-X0+Y0Y0-Y~X,(179)Ψ+,YX,Y~X=log⁡Y0-X0+Y0Y0-Y~XX0+Y0Y~X,(180)Ψ+,YYX,Y~X=-et∗sY0-1-X0-Y0tY01-1-X0-Y0et∗,(181)sY0=X0+Y0X0+Y0+Y0e-t∗tY0X0+Y02+e-t∗-1X0,(182)tY0=et∗1-et∗X0+Y0X0et∗+X0+Y02X0+Y0X02-Y02-X0+et∗X0-X0+Y022+e2t∗2-et∗X0Y0+e3t∗Y0X+Y02-1,(183)K+X,Y~X=Y0X0X0+Y02-X0322πΔ+012X0+Y03Y~XX0Y0-X0+Y02Y0-Y~X,(184)bt∗=X0Y0Δ0Δ+0X0+Y0X0+Y02-X01-X0-Y0e-t∗-Y0et∗X0+Y0-2,(185)Δ+0=Δ+0X=11-E∗D∗E∗-D∗et∗-2D∗1-D∗-D∗-E∗2e-t∗+D∗+E∗-2D∗E∗e-2t∗,(186)D∗=Y0X0+Y0,E∗=1-X0-Y0,(187)Δ0=Δ0X=e-t∗-1Y0X0+Y0+Y0X0+Y02-X0X0X0+Y0et∗+e-t∗1-2X0+Y0+X0+Y03X0-1.Y~(X) is given by (56), and C(ρ) has the expansions in (31)–(33) for the different regions Rj.Forη∗→-∞ we are moving into region D+ and the integral in (176) approaches 2π, and then (176) becomes simply the expansion of the D+ result in (60), about Y=Y~(X). Thus the matching of (176) to D+ is immediate, and we can also verify the matching to (65), by expanding the latter as Y↓Y~(X), and (176) for η∗→+∞. We recall that Proposition 2 shows that the expansion of C0(ρ)/C(ρ) is the same for regions R1, R2, and R3. The function t∗=t∗(X) in (178) is obtained by setting s=Y0 and then Y=Y~(X) in (62). The function Δ+0(X) is obtained by setting s=Y0 and t=t∗(X) in (64) and corresponds to the values of this Jacobian along the curve Y=Y~(X). Note also that t∗(X) can be obtained by setting A=Y0/(X0+Y0) and Y=Y~(X) in (68), and Δ0(X) corresponds to the Jacobian in (71) along the curve Y=Y~(X), with τ=t∗(X).Proposition 17.
For(X0,Y0)∈R2∪R3, Y-Y∗(X)=ρ-1/3ξ1=O(ρ-1/3), and 0<X<X0 one has(188)πk,r~ρ-1/2Cρexp⁡ρΦX,Y∗X+ρ-1/3ξ112πW0Y∗X1W0-W0+Y∗X212πi∫-i∞i∞1Aiθ2exp⁡-1-X02/3X01/6W0+Y∗Xξ1θdθ,where(189)ρΦX,Y∗+ρ-1/3ξ1=ρΦX,Y∗+ρ2/3ΦYX,Y∗ξ1+12ρ1/3ΦYYX,Y∗ξ12+16ΦYYYX,Y∗ξ13+o1,ΦX,Y∗=Y∗log⁡1+W0Y∗-W0W0+Y∗-Xlog⁡Y∗W0+Y∗,ΦYX,Y∗=log⁡1+W0Y∗,ΦYYX,Y∗=-W0Y∗W0+Y∗,ΦYYYX,Y∗=W0W0+2Y∗Y∗2W0+Y∗2,and C(ρ) has the expansions in (31)–(33) (one can replace ρ-1/2C(ρ) by ρ-1/3C1(ρ) in (188)), and Y∗=Y∗(X)=Y∗(X;X0) is given in (75).Forξ1→+∞ we can expand the integral in (188) by the saddle point method, after shifting the integration contour far to the right, with Re(θ)≫1. Then the Airy function in the integrand may be approximated using(190)Aiz~12πz-1/4e-2/3z3/2;z⟶∞,arg⁡z<πand we can verify that (188) for ξ1→+∞ matches to the D+ result in (60), as Y↓Y∗(X). For ξ1→-∞ the behavior of the integral in (188) is determined by the singularity with the largest real part, which is the double pole at θ=r0(<0). Then standard singularity analysis can be used to show that (188) for ξ1→-∞ agrees with the expansion of (79) as Y↑Y∗(X). Note that in this limit ρ1/3Φ∗(s1) becomes O(1) and proportional to ξ1.Proposition 18.
For(X0,Y0)∈R4, Y-Yc(X)=ρ-1/3η∗∗=O(ρ-1/3), and XL<X<X0 (with XL defined in item (vii) of Proposition 8) one has(191)πk,r~1-X0-Y0ρ5/62π1-X04/3X01/6X0-X0-Y0Y0YcXW0+Y0Y0-YcX2W0+Y0+YcXW0+YcX-W0W0+Y0-1/2expρΦX,YcX+ρ-1/3η∗∗-ΦX0,Y012πi∫-i∞i∞1Aiθexp-1-X02/3X01/6W0+Y0Xη∗∗θdθ,where Yc(X)=Yc(X;X0,Y0) is defined in (86).Thus the transition fromD0 to D- involves a slightly different integral (cf. (191)) compared to the transition from D+ to D- (cf. (188)), as the former has simple poles at the Airy roots. Using standard asymptotic analysis we can show that(192)12πi∫-i∞i∞e-⋆θAiθdθ~2⋆e-⋆3/3,⋆⟶+∞,(193)12πi∫-i∞i∞e-⋆θAiθdθ~e-⋆r0Ai′r0,⋆⟶-∞and (192) can be used to verify the matching between (191) and the D0 result in (65), where C0~ρ-1/2(1-X0-Y0) in R4. Similarly (193) can be used to verify matching to the D- result in (79), with C1 now given by (41). The factor involving ρΦ(X,Yc+ρ-1/3η∗∗) in (191) can be expanded in Taylor series, similarly to (189), by replacing Y∗(X) by Yc(X) and ξ1 by η∗∗.As we approachR3∩R4 (where X0+Y0=X0) from within R4 we see that (191) develops a singularity, in view of the factor X0-X0-Y0-1. As we approach R3∩R4 from within R3, the expression for K+(X,Y~(X)) in (183) vanishes and that for b(t∗) in (184) develops a singularity. Thus the expansions in both Propositions 16 and 18 become invalid, and below we give a new result that applies for R3∩R4, where D0 meets D-.Proposition 19.
For(X0,Y0)∈R3∩R4, with X0+Y0=X0+δ∗ρ-1/3 and δ∗=O(1),(194)πk,r~1-X03X01/4expρΦX,YcX+ρ-1/3η∗∗-ΦX0,Y02πρY∗XX0-X0-Y∗XY∗X+1-X02-X012πi∫-i∞i∞e-θδ1∫θ∞eδ1uAiuduAiθ2exp-1-X02/3X01/6W0+Y∗Xη∗∗θdθ,where δ1=X0-1/61-X0-1/3δ∗ and Yc(X) is in (86) and Y∗(X) in (75).With the scalingX0+Y0-X0=O(ρ-1/3), the curves Yc(X), Y∗(X), and Y~(X) nearly coincide, with the differences being O(ρ-1/3). In (194) we can again replace Φ(X,Yc+ρ-1/3η∗∗) by its Taylor expansion and replace Y0 everywhere by X0-X0+δ∗ρ-1/3, thus writing the result in terms of X0 and δ∗ (or δ1), along with X and η∗∗. Note that η∗∗ is still defined by η∗∗=ρ1/3[Y-Yc(X)], which differs from ρ1/3[Y-Y∗(X)] by an amount that is O(1). Thus, for region R3∩R4 the transition range in state space, between D- and D0, involves a somewhat more complicated integral than those in Propositions 18 and 19.This completes our summary and discussion of the various regions of state space, which carry zero volume in(X,Y) space but are necessary since the results in D0, D+, and D- do not always apply.


## 4. Asymptotic Expansion in RegionD0

In this section we will construct the expansion for(X,Y)∈D0, that is, (65). The analysis for the complementary regions D+ and D- is virtually identical to that for the model with an infinite secondary capacity (R=∞), and the detailed calculations can be found in [10]. Here we will only discuss, for regions D+ and D-, those aspects that change when R<∞. In order to uniquely determine the expansion in D0, we will need to use asymptotic matching to the corner expansion that applies on the scale k=m-O(1), r=R-O(1), and this is discussed in Section 4.2. For parameter region R4 (and also for R3∩R4) we will also need to carefully analyze the scale k=m-O(ρ1/3) and r=R-O(ρ2/3), which will be necessary to determine the multiplicative constant C1=C1(ρ;X0,Y0) that arises in the asymptotic expansion in D-; this analysis is done in Section 4.3.


### 4.1. Ray Expansion in the Interior ofD0

We analyze the scaled equation in (17) using the ray method of geometrical optics, where we assume an expansion of the form(195)PX,Y=C0ρeρΨX,YKX,Y+ρ-1K1X,Y+ρ-2K2X,Y+⋯.Then we have(196)PX±ρ-1,YPX,Y=e±ΨXX,Y1+ρ-1±KXK+12ΨXXX,Y+Oρ-2and using (195) in (17) we obtain in the limit as ρ→∞ the “eikonal” equation(197)1+X+Y=YeΨY+XeΨX+e-ΨXand at the next order in ρ-1 the “transport” equation(198)eΨYKY+12YΨYYK+K+eΨXXKX+12XΨXXK+K+e-ΨX-KX+12ΨXXK=0.The first-order PDE in (197) can be solved by the method of characteristics (see [14]), where one must solve the five ODEs(199)dXdτ=e-ΨX-XeΨX,dYdτ=-YeΨY,(200)dΨdt=ΨXdXdτ+ΨYdYdτ=ΨXe-ΨX-XeΨX-YΨYeΨY,(201)dΨXdτ=eΨX-1,dΨYdτ=eΨY-1.Here τ is a parameter along a given characteristic curve, which is also called a “ray,” due to applications in optics.To uniquely specify the solutionΨ(X,Y) to (197), we must either specify Ψ along some curve, called the “initial manifold,” in the (X,Y) plane, or use a singular solution, where all the rays emanate from a single point. The appropriate solution to the eikonal equation must be determined for each individual problem, and we will see that for the present model we will need to use three different solutions Ψ to (197), with two having the boundary X=X0 as the initial manifold and the third corresponding to all the rays emanating from the corner point (X0,Y0). The first two solutions will correspond to regions D+ and D- and the third to D0. Since D+ and D- arose also in the infinite capacity model, where R=∞, we discuss these only briefly, and the details of the corresponding solutions to (197)–(201) can be found in [10].Let us denote the solution in regionD+ as C(ρ)K+(X,Y)eρΨ+(X,Y) to distinguish it from (195), which will apply in D0. If this expansion will satisfy the boundary equation along X=X0 (or k=m) in (4), or, equivalently, (18), we must have(202)X0eΨ+,XX0,Y=e-Ψ+,YX0,Y.Requiring Ψ+(X,Y) to satisfy (202) is equivalent, up to an additive constant in Ψ+ which can be incorporated into C(ρ), to specifying Ψ+ along the initial manifold X=X0. Solving (199)–(201) subject to (202) leads to(203)Ψ+,X=-log⁡1-ss+X0et,Ψ+,Y=-log⁡1+s+X0-1etand then the rays are given in parametric form by (62), which relates (s,t) to (X,Y). When t=0 we have (X,Y)=(X0,s) so that s is the point where a ray hits the line X=X0. We use now t instead of τ as the parameter along a given ray, and s is used to index the family. Finally, solving (200) with (Ψ,τ) replaced by (Ψ+,t) leads to the expression in (61). We also note that when t=0(X=X0) we have(204)Ψ+X0,Y=X0-1+Y-X0+Ylog⁡X0+Ywhich is an explicit function of Y. From the first expression in (62) we find that for t→0(205)X0-X=Y+X02-X0Y+X0t+Ot2.It follows that for t>0 the rays that start from X=X0 enter the state space, where X⩽X0, for t>0 only if (Y+X0)2-X0>0, or Y+X0>X0. If X0>1 this condition holds for all Y⩾0 and hence the rays fill the region D+ indicated in Figure 3. If X0<1 and Y0+X0>X0 then the condition holds only for Y in the interval X0-X0<Y<Y0. Then these rays fill the domain D+ indicated in Figures 4 and 5. But if X0<1 and X0+Y0<X0, which is true for parameter region R4, the condition never holds and then this ray expansion plays no role in the analysis (see also Figure 6). Once we compute Ψ+ in D+, we can integrate (198) to obtain K+(X,Y) in (63). Thus we have shown that (63), up to the constant C(ρ) which we have yet to determine, holds in the portion D+ of the state space, for parameter regions R1∪R2∪R3.Now we observe that ifX0>1, Ψ+(X0,Y) is maximal at Y=0 and by evaluating also K+ at X=X0 and Y=0 we find that near the corner (X,Y)=(X0,0) with X=X0(k=m) and Y=r/ρ, and we have(206)CK+eρΨ+~C12πX0-1X03/2X0-reρX0-1-X0logX0.The scale r=O(1) for X0>1 must be analyzed separately and the details are carried out in [10]. For r=O(1) and m-k=n=O(1) the result in (141) applies, and by asymptotically matching this to (206) (for r⩾1) we conclude that(207)Cρ~ρ-1/2,X0,Y0∈R1.If X0<1 and X0+Y0>1 then Ψ+(X0,Y) in (204) is maximal at Y=1-X0, which lies in the range (0,Y0) precisely for parameter region R2. Expanding Ψ+(X,Y) in (61) about X=X0 and Y=1-X0 leads to(208)CK+eρΨ+~C1-X0X0n12πexp⁡-12ρr-ρ-m2.For parameter region R2 the expansions in D- and D0 will be uniformly exponentially small in ρ, as there is little mass in these state space ranges. Thus the main contribution to the double sum in the normalization condition (10) will come from D+ and in particular from the scale m-k=n=O(1) and r=ρ-m+O(ρ) (corresponding to Y=1-X0+O(ρ-1/2)). Then normalizing the approximation in (208), after approximating the sum over r by an integral over Y, we conclude that C(ρ)~ρ-1/2, and hence (207) applies also for parameter region R2 (and, by continuity, the relation holds in R1∩R2 (X0-1=O(ρ-1/2)) also).We have thus shown that the ray expansion inD+ has the state space regions D- and D0 as “shadows,” and thus the other solutions to (197) must apply. To construct the solution in D- we must slightly modify the ansatz in (195) and now expand the joint distribution as(209)PX,Y=C1ρeρΦX,Yeρ1/3Φ1X,YLX,Y+ρ-1/3L1X,Y+ρ-2/3L2X,Y+⋯.Now Φ will satisfy (197) and L will satisfy (198), and the subexponential term Φ1 will satisfy the PDE(210)YeΦYΦ1,Y+XeΦX-e-ΦXΦ1,X=0.Again the detailed analysis can be found in [10]. We now find that(211)ΦX=-log⁡1-1-X0et1,ΦY=-log⁡1-W0W0+s1et1and the rays are given in parametric form by(212)X=1-1-X0et11-1-X0e-t1,Y=-W0+W0+s1e-t1,where we recall that W0=1-X02. When t1=0 we have X=X0 and Y=s1, but unlike the rays in D+, those in D- are all tangent to the boundary X=X0. Thus the boundary is a “caustic boundary,” for 0<Y<X0-X0 for region R2∪R3 and for all Y with 0<Y<Y0 for region R4. The solution Φ to (197) is now given by(213)ΦX,Y=W0-W0+s1e-t1log⁡1-W0et1W0+s1-1-X0-e-t11-X0-et1log⁡1-W0et1-W0et1+W0log⁡W0+s1-12W0log⁡W0.For region D- we can invert transformation (212) and write s1 and t1 explicitly in terms of X and Y. This leads to the expression in (81) for s1=s1(X,Y), and then (80) gives Φ in terms of X and Y.Equation (210) implies that Φ1(X,Y) is constant along a caustic ray, so we write Φ1(X,Y)=Φ∗(s1). To determine Φ∗(·) and also completely determine L(X,Y) in (209), we need to construct two “nested” boundary layer corrections to the expansion in (209), near X=X0, corresponding to the scales X0-X=O(ρ-2/3) (m-k=O(ρ1/3)) and X0-X=O(ρ-1) (m-k=O(1)). The boundary condition in (18) can only be imposed on the expansion that applies for n=m-k=O(1). Using asymptotic matching between the expansions for X0-X=O(ρ-1), X0-X=O(ρ-2/3), and the ray expansion in (209) allows us to determine (209) up to the multiplicative constant C1.To determineC1 we need to, for regions R2 and R3, first relate C1(ρ) to the constant C(ρ) in the D+ expansion. This can be done by analyzing the transition curve Y=Y∗(X), with the scaling Y-Y∗(X)=O(ρ-1/3). The details are again presented in [10], and this leads to the conclusion that(214)C1ρCρ~ρ-1/6,X0,Y0∈R2∪R3.In R2 we have C(ρ)~ρ-1/2 so that C1(ρ)~ρ-2/3, but we have yet to determine either C or C1 for region R3. The conclusion in (214) can also be reached by constructing an expansion near the point (X,Y)=(X0,X0-X0), with the scaling X0-X=O(ρ-2/3) and Y-(X0-X0)=O(ρ-1/3) (see subsection  5.3 in [10] for that analysis). Note also that the curve Y=Y∗(X) corresponds to the ray with s1=X0-X0 in (212) and also the ray with s=X0-X0 in (62). Then we have Ψ+(X,Y∗(X))=Φ(X,Y∗(X)) so that the exponential factors in the expansions in (60) and (209) are continuous along Y=Y∗(X), which separates D+ from D-.The regionD0 is a shadow of both the D+ and D- rays for parameter regions R2∪R3 and a shadow of the D- rays for parameter region R4. For R4, the caustic rays that fill D- correspond to 0<s1<Y0 in (212), and s1=Y0 corresponds to the curve Y=Yc(X) in (86), which separates D- from D0. To fill the shadow D0 and thus obtain an approximation to P(X,Y) for (X,Y)∈D0, we must use a singular solution to (197), one that has all rays start from the corner point (X,Y)=(X0,Y0). To construct this solution we first integrate the two ODEs in (201), which yields(215)ΨX=-log⁡1-Aeτ,ΨY=-log⁡1-Beτ,where A and B are constant along a given ray. Evaluating the eikonal equation in (197) along the corner X=X0,Y=Y0 and using (215) with τ=0 gives(216)1+X0+Y0=Y01-B+X01-A+1-Aand this leads to the relation between A and B in (67). Here we take τ=0 to correspond to when a ray starts from the corner point X=X0,Y=Y0. Using (215) we solve the two ODEs in (199), subject to Xτ=0=X0 and Yτ=0=Y0, and we thus obtain the expressions in (68), which give the corner rays in parametric form. Using (215) and (68) we then integrate (200), and choosing Ψ(X0,Y0)=0 for convenience, we obtain (66). Note that Ψ can only be determined from (197) up to an additive constant, but such a constant can be incorporated into C0=C0(ρ;X0,Y0) in (195).We can viewA as indexing this family of rays. If A is such that (A+X0+Y0)(1-A)-X0=0 then B in (216) and (67) becomes infinite, and this corresponds to A=Amax in (69). This critical ray corresponds to Y=Y0, and for τ∈(0,-log(Amax)) we have X∈(0,X0), so the ray is the upper boundary of the state space rectangle, where all secondary storage spaces are full. The ray A=Amin=1-X0(for X0<1) corresponds to the curve Y=Yc(X)=Yc(X;X0,Y0) in (86). For regions R1∪R2∪R3 we have Y0/(X0+Y0)>1-X0 and then the ray with A=Y0/(X0+Y0) is the curve Y~(X) in (56). Thus for regions R1∪R2∪R3 it suffices to consider A in the range (Y0/(X0+Y0),Amax) to fill region D0 in state space. We also note that in order for the rays, which all start from (X,Y)=(X0,Y0), to enter the state space we need Xττ=0<0 and Yττ=0<0, which implies that 1-A-X0/(1-A)<0 and X0/(1-A)-(A+X0+Y0)<0. Adding the last two inequalities implies that A>1/2(1-X0-Y0) and this is certainly true if A>Amin.We next solve the transport equation (198) for K(X,Y). This equation can be written as an ODE along a ray, with(217)KτK=12YΨYY+1eΨY+12XΨXX+1eΨX+12ΨXXe-ΨX=eΨX+eΨY-12Xττ+XτeΨX-12Yττ+YτeΨY,where we used (199), and the last equality in (217) follows by differentiating (199) with respect to τ. Introducing the Jacobian Δ≡XτYA-XAYτ associated with the mapping from (τ,A) to (X,Y) variables, after some calculation we find that (217) becomes(218)KτK=1211-Aeτ+1211-Beτ-12ΔτΔand the most general solution to (218) is given by(219)KX,Y=e-τ-Ae-τ-BΔτ,A-1/2K0A.Here K0(A) is an arbitrary function of the parameter that indexes the rays and is thus constant along any particular ray. We have thus determined the expansion in (195) up to the constant C0 and the function K0(·). To complete the D0 ray expansion we will need to use asymptotic matching to a local expansion valid near the corner (X,Y)=(X0,Y0). This is constructed in Section 4.2. In order to accomplish the matching we will need the behavior of (195) as X↑X0, Y↑Y0, and this is examined next.We note that(X,Y)→(X0,Y0) corresponds to τ→0 and in this limit the Jacobian in (71) vanishes, and(220)Δ~-τ1+X0+Y01+X01-A2-4X01-A,τ⟶0+.For τ→0 we also have(221)X0-X~X01-A+A-1τ,Y0-Y~Y01-Bτso that(222)Y0-YX0-X=pn~A+X0+Y01-A-X0X0-1-A2,τ⟶0.Here we recall that Y=Y0-p/ρ and X=X0-n/ρ, and (222) gives the slope at which the ray indexed by A hits the corner point (X0,Y0).We can thus invert the transformation in (68) locally, with (222) corresponding to a quadratic equation for A=A(n/p), and hence(223)A~As≡1+121p-n1+X0+Y0n-1+X0+Y02n2+4p2-n2X0.Then we define Bs by replacing A by As in (67). Note that the right-hand side of (223) approaches Amax if n→∞, approaches Amin=1-X0 as p→∞, and is equal to Y0/(X0+Y0) if n/p=(X0+Y0)2-X0/Y0. For τ→0 we also have Ψ(X,Y)→0 with(224)ΨX,Y=A+X01-A-1log⁡1-A+Y01-Blog⁡1-Bτ+Oτ2=X0-Xlog⁡1-As+Y0-Ylog⁡1-Bs+Oτ2so that eρΨ(X,Y)~(1-As)n(1-Bs)p. We have thus shown that as τ→0 we have(225)C0KX,YeρΨX,Y~C0K0As1-As1-Bs1-Asn1-Bsp1τ1+X0+Y01+X01-As2-4X01-As-1/2,where A~As was approximated by (223), and by (221) we have(226)τ~1-BsY0-YY0~1-BsY0pρ,so that (225) becomes an explicit function of n and p.


### 4.2. Analysis of the Scalek=m-O(1), r=R-O(1)

We use the variablesn=m-k and p=R-r and obtain an approximation to π(k,r) that is valid when all but a few of the primary and secondary spaces are occupied. This is likely for parameter regions R3 and R4 but unlikely for R1 and R2. We define Q by(227)πk,r=πk,r;ρ,m,R=Qn,p;ρ,X0,Y0.We thus denote by Q(n,p) the exact probability that there are n (resp., p) empty primary (resp., secondary) spaces, and we will later denote by QL(n,p) the leading term in an asymptotic expansion of this probability, which is valid for the scale n,p=O(1). Then clearly Q(0,0)~QL(0,0) so in asymptotic relations involving Q(0,0) we can drop the subscript L.Writing the balance equations in (3), (4), and (5) in terms of (n,p) leads to(228)1+X0+Y0-ρ-1n+pQn,p=Y0+ρ-11-pQn,p-1+X0+ρ-11-nQn-1,p+Qn+1,p;n⩾1,p⩾1,(229)1+X0+Y0-ρ-1pQ0,p=Y0+ρ-11-pQ0,p-1+Q1,p+Q0,p+1,p⩾1,(230)1+X0+Y0-ρ-1nQn,0=X0+ρ-11-nQn-1,0+Qn+1,0,n⩾1,and the corner condition in (9) becomes(231)X0+Y0Q0,0=Q1,0+Q0,1.Note that near this corner only the boundaries X=X0 and Y=Y0 of the state space rectangle are relevant, and the problem in (228)–(231) corresponds to a random walk in a quarter plane, in (n,p) space. For ρ→∞ the problem in (228)–(231) may be further approximated by(232)1+X0+Y0QLn,p=Y0QLn,p-1+X0QLn-1,p+QLn+1,p;n⩾1,p⩾1,(233)1+X0+Y0QL0,p=Y0QL0,p-1+QL1,p+QL0,p+1,p⩾1,(234)1+X0+Y0QLn,0=X0QLn-1,0+QLn+1,0,n⩾1.Here QL(n,p) in (232)–(234) is understood to be the leading term in an asymptotic expansion of Q(n,p;ρ), for ρ→∞, and the corner equation in (231) must also be satisfied by this leading term.We introduce the double generating function(235)Q~z,w=∑n=0∞∑p=0∞QLn,pznwpand from (231)–(234) we obtain(236)1+X01-z+Y01-w-1zQ~z,w=1w-1zQ~0,w+1-1wQL0,0.From (236), by setting w=1 we obtain(237)Q~z,1=Q~0,11-zX0and if z=w we find that Q~(z,z)=QL(0,0)/[1-(X0+Y0)z]. Setting z=w then in (235) leads to(238)∑n+p=lQLn,p=QL0,0X0+Y0l,l⩾0.The expression in (238) shows that the total number of empty spaces (=m+R-k-r=n+p) follows asymptotically a geometric distribution if X0+Y0<1, and this could be also deduced from (12), by expanding the exact (truncated Poisson) distribution for ρ→∞. Note, however, that (238) holds also for X0+Y0⩾1. The factor in brackets in the left-hand side of (236) vanishes when z=z±(w), where z± are given in (51). Requiring Q~(z,w) to be analytic at the smaller root z-(w) determines Q~(0,w) as(239)Q~0,w=z+w1-wz-w-wX0QL0,0.Then using (239) in (236) and partially inverting the transform in (236) lead to(240)∑p=0∞wpQLn,p=1-wz-w-wX0z+w-n-1QL0,0,n⩾0,and hence by the Cauchy integral formula(241)QLn,p=QL0,0X02πi∮1-wz-w-wz+w-n-1w-p-1dw,where the contour is over a small loop about w=0.WhenX0+Y0<1 (m+R<ρ, region R3∪R4), most primary and secondary spaces will be full, and then (241) becomes a normalized discrete distribution with(242)Q0,0~QL0,0=1-X0-Y0,R3∪R4,leading to the limit law in (50).For regionsR1∪R2, (241) still represents a local approximation to π(k,r), but now different arguments must be used to determine QL(0,0), and indeed now QL(0,0) will turn out to be exponentially small for large ρ. We proceed to relate (241) to the ray expansions in D0, D+, and D-, noting that, for parameter region R1∪R2∪R3, D0 and D+ both border the corner point, while, for R4, D0 and D- border this point. We will need to expand (241) asymptotically, for n and/or p→∞. The integrand in (241) has branch points at w=(1+X0+Y0±2X0)/Y0 and possible poles at solutions of z-(w)=w. Now, z-(1)=1 for X0>1 but w=1 is not a pole in view of the factor 1-w in the numerator. The only other possible solution to z-(w)=w occurs at w=1/(X0+Y0) and this is a pole if(243)2X0X0+Y0=1+X0+Y0-Y0X0+Y0-1+X0+Y0-Y0X0+Y02-4X0=X0+Y0+X0X0+Y0-X0+Y02-X0X0+Y0.But (243) holds precisely when (X0+Y0)2>X0, which is true for regions R1∪R2∪R3. Thus for R4 the pole is absent and along the transition curve R3∩R4 (cf. (25)) the pole coalesces with the lower branch point, both being at w=1/X0 (>1). We can recast the integral in (241) by using the conformal map u=z+(w), so that the inverse is(244)w=1+1Y01+X0-X0u-1u and (241) becomes(245)QLn,p=QL0,02πi∫Γ1-uX0u2-1Y0pup-n-2X0+Y0-X0u1+X0+Y0u-X0u2-1p+1du,where Γ is a small loop about u=z+(0), with(246)z+0=12X01+X0+Y0+1+X0+Y02-4X0,and the integrand in (245) has a pole of order p+1 at u=z+(0). Below we collect some asymptotic results for QL(n,p) that will be used in the matching calculations.Proposition 20.
Forn and/or p→∞ the function QL(n,p) has the following asymptotic expansions:
(i)
p = O ( 1 ), n→∞(247)QLn,p~Q0,0z+0-np!npY01+X0+Y02-4X0p. (ii)
p , n → ∞ with 0⩽n/p<θ1, θ1=(X0+Y0)2-X0/Y0, for (X0,Y0)∈R1∪R2∪R3(248)QLn,p~Q0,0X0+Y02-X0X0nX0+Y0p-n-2. (iii)
p , n → ∞ with θ1<n/p<∞, for R1∪R2∪R3(249)QLn,p~Q0,02πn21+X0+Y02+4p2-n2X0-1/41-usX0us2-1us3/2X0+Y0-X0us1+X0+Y0us-1-X0us2usp-nY0p1+X0+Y0us-1-X0us2-p, where(250)us=usnp=12X0n+p1+X0+Y0n+1+X0+Y02n2+4X0p2-n2. (iv)
p , n → ∞, n/p≈θ1 (with n-pθ1=O(p)), for R1∪R2∪R3(251)QLn,p~Q0,0X0nX0+Y0p-n-2X0+Y02-X012π∫-∞ηe-v2/2dv,(252)η=Y0pX0+Y0X0+Y03+X0+X0+Y0Y0-2X0θ1-np=O1. (v)
p , n → ∞, 0<n/p⩽∞, for region R4. The expression in (249) holds for all 0<n/p<∞ and (247) holds for n→∞ with p=O(1). (vi)
n = O ( 1 ), p→∞, for region R4(253)QLn,p~Q0,02πX0n/2-1/4Y0Y0+W0pW0+Y01-X0X0-X0-Y0p-3/2n+X0X0-1+X0X0-X0-Y0. (vii)
n , p → ∞ with n=O(p), for region R3∩R4, with Y=X0-X0+ρ-1/2δ, δ=O(1)(254)QLn,p~Q0,0πX0n/2Y0Y0+W0p1-X0X01/4pexp⁡n24p1-1X0+δρ2X0exp⁡pδ2ρX01-X0-nδρX0∫η0∞e-v2/2dv,(255)η0=1-X02X01/4np-21-X0X01/4pδρ.The results in (247)–(255) may be obtained by expanding the integrals in (241) or (245) by a combination of singularity analysis and the saddle point method. Good references on asymptotic expansion of integrals are the books [15–19], but since these methods are now well established, we merely sketch the proof of Proposition 20.To obtain (247) we approximate the integrand in (241) for w=O(n-1), scaling w=w1/n and using (256)z+w-n-1~z+0-n-11+wz+′0z+0+Ow2-n-1~z+0-n-1exp⁡-z+′0z+0w1. Note that (247) is not only asymptotically true but also an exact expression when p=0, since Q(n,0) satisfies the boundary equation in (234). Turning to (245) we see that the integrand has a simple pole at u=u∗≡(X0+Y0)/X0 and saddle point(s) where(257)ddup-nlog⁡u-plog⁡1+X0+Y0u-X0u2-1=0, and (257) is equivalent to the quadratic equation(258)np+1X0u2-np1+X0+Y0u+np-1=0. One root of (258) is given by us=us(n/p) in (250), and the complementary root, with a minus sign in front of the square root, will correspond to a second saddle which will not play any role in the analysis. The pole u∗ and saddle us, whose location depends on the ratio n/p, coalesce when n/p=θ1, for regions R1, R2, and R3. Note also that as n/p→∞ we have us→z+(0) and that the integrand in (245) has a zero at u=1/X0. For region R1∪R2∪R3 and n/p∈(0,θ1) we have the ordering(259)1X0<us<u∗<z+0 and we note that u∗<z+(0) is always true. Then we dilate the small loop Γ about u=z+(0) to the saddle point contour u-z+(0)=us-z+(0), which is a circular contour that traverses the saddle us in the steepest descent directions, which are arg(u-us)=±π/2. But in doing the dilation we must take into account the contribution from the residue at the pole u=u∗, in view of (259). It turns out that the residue dominates the saddle point contribution, and we thus obtain the expression in (248). For n/p∈(θ1,∞) we have the ordering(260)1X0<u∗<us<z+0 and then the pole does not contribute as we dilate Γ to the saddle point contour. Setting(261)Gu=1-uX0u2-1X0+Y0u-11u21+X0+Y0u-X0u2-1,Fu=Fu;np=log⁡Y0+1-nplog⁡u-log⁡1+X0+Y0u-X0u2-1 we use for (245) the standard saddle point estimate (262)12πi∫ΓGuepFudu=12πi∫ΓsGuepFudu~12πpGusF′′usepFus, and this leads to the expression in (249). The transitional result in (251) corresponds to n/p≈θ1 and then we have u∗≈us, so the saddle is close to a simple pole. Such situations are discussed in detail in [15], and by expanding the integrand in (245) about u∗ we ultimately obtain a simpler integrand that is related to D-1(·), the parabolic cylinder function of order -1, which can in turn be expressed in terms of the standard error function, leading to (251). Note that as η→+∞ we approach the region n/p<θ1 and then (251) reduces to (248).The pole atu∗ and zero at 1/X0 coalesce when X0+Y0=X0, which is precisely the curve R3∩R4 which separates R3 from R4 in parameter space. For (X0,Y0)∈R4 we have the ordering(263)u∗<1X0<us<z+0.Then for any n/p∈ (0,∞) we can deform Γ into the saddle point contour and obtain (249) as the approximation to QL(n,p). The limits n=O(1), p→∞, and n→∞, p=O(1) require a separate analysis. For the latter us becomes close to z+(0) and we again obtain (247), while for the former the saddle us gets close to zero at 1/X0. By expanding the integrand near u=1/X0, setting u=1/X0+v/p, we obtain(264)QLn,p=QL0,0X0nY0Y0+W0pX0Y0+W01X0+Y0-X012πi∫-i∞i∞vp2X0X0-1+Bvp+Op-11-X0nvp1-X02Y0+W0v3pexp⁡X03/2v2Y0+W0dv,where(265)B=-5X03/2+3X0+2X03/2X0-1X0+Y0-X0. Evaluating the integral(s) in (264) we obtain (253). Note that the coefficient of the O(p-1) term in the integral is zero, so the result is O(p-3/2).WhenY0+X0≈X0 the pole at u∗ and zero at 1/X0 are close together. If n,p→∞ so that n/p remains fixed and positive, the saddle lies well to the right of these, and then (249) holds. But if n,p→∞ with n=O(p) then the saddle, pole, and zero are all close. Then we must reexamine the integrand in (245) and expand it about u=1/X0. Again setting v=ρu-1/X0=O(1) and replacing Y0 by X0-X0+δ/ρ (with δ=O(1)), (245) becomes asymptotically(266)QLn,p~Q0,0X0n/2Y0Y0+W0p2X0ρ12πi∫Br+X0vX0v-δexp⁡-nρX0v+pρX03/2Y0+W0v2dv, where Br+ is a vertical contour in the complex v-plane, which is to the right of the pole at v=δ/X0. The integral in (266) may be evaluated as a combination of a Gaussian and an error function, and this ultimately leads to (254) with (255). Since the large parameter ρ appears explicitly in (254) and (266) we can define the asymptotic limit more precisely, as ρ→∞ with Y0+X0-X0=O(ρ-1/2), p=O(ρ), and n=O(ρ).With Proposition20 we are now ready to relate the corner approximation QL(n,p) to the various ray expansions, via asymptotic matching. First consider parameter region R1∪R2∪R3, so that D0 and D+ come together at the corner point (X0,Y0). In D+ the function Ψ+(X,Y) can be expanded in Taylor series about (X,Y)=(X0,Y0) and as (X,Y)→(X0,Y0) we obtain(267)CρK+X,YeρΨ+X,Y~CρK+X0,Y0eρΨ+X0,Y0exp⁡ρX-X0Ψ+,XX0,Y0+ρY-Y0Ψ+,YX0,Y0=CρX0+Y02-X0X0+Y05/22πeρX0+Y0-1-X0+Y0logX0+Y0X0X0+Y0nX0+Y0p,where we used ρ(X-X0)=-n and ρ(Y-Y0)=-p. Near X=X0, the curve Y=Y~(X) that separates D0 from D+ has the slope Y~′(X0)=Y0/(X0+Y0)2-X0 so that Y~(X) can be approximated by the straight line Y-Y0=Y~′(X0)(X-X0), which is the same as p/n=1/θ1. Then D+ meets the corner in the sector n/p∈[0,θ1), and this corresponds precisely to where the asymptotic result (248) applies. Comparing (267) to (248) we see that the matching is possible, if Q(0,0) and C(ρ) are related by(268)Cρ~2πX0+Y0eρ1-X0-Y0+X0+Y0logX0+Y0Q0,0and this holds throughout R1∪R2∪R3. For R1∪R2 we have previously determined that C(ρ)~ρ-1/2 and thus (268) determines Q(0,0), as in (30). For region R3 we have Q(0,0)~1-X0-Y0 and then (268) leads to (33). For R1∩R2, corresponding to X0+Y0-1=γ/ρ=O(ρ-1/2), the approximation in (208) holds for n=O(1) and Y-(1-X0)=O(ρ-1/2), but now the upper boundary Y=Y0 lies within this range. Then applying the normalization condition in (10) to (208) leads to(269)1~∑p=0∞∑n=0∞C2π1-X0X0nexp⁡-ρ2Y+X0-12=C2π∑p=0∞exp⁡-ρ2γρ-pρ2~ρ2πC∫0∞e-γ-v2/2dv=ρ2πC∫-∞γe-v2/2dvand then C is given by (33). Then QL(0,0)~Q(0,0) can be computed from (268), and since ρ[1-X0-Y0+(X0+Y0)log(X0+Y0)]~γ2/2, we obtain the expression in (29). We have thus determined both Q(0,0) and C(ρ) for all cases of the parameters X0,Y0.To matchQ(n,p) to the ray expansion in D0 we note that, for regions R1∪R2∪R3, D0 meets the corner over the sector n/p∈ (θ1,∞), while for region R4 this holds for all n/p∈ (0,∞). We recall that for R4 the curve Y=Yc(X) separates D0 from D-, but this curve has infinite slope at X=X0 and is thus tangent to the line X=X0 at Y=Y0. So if we approach the corner along any straight line (excluding slopes 0 and ∞) we are approaching from within D0. By the asymptotic matching principle, the expansion of QL(n,p) for n,p→∞ should agree with the expansion of C0KeρΨ as X→X0, Y→Y0. Thus in D0 we must compare (249) to (225). But from (250) and (223) we see that (1-As)us=1 and from (67) we obtain(270)1-Bs=Y0usX0+Y0-1us-1-X0us2.It follows that(271)1-Asn1-Bsp=us-nusY01+X0+Y0us-1-X0us2p and the matching is possible if(272)C0K0AsY01-As1-Bsρp1+X0+Y01+X01-As2-4X01-As-1/2~Q0,02π1-usX0us2-1n21+X0+Y02+4p2-n2X0-1/4us3/2X0+Y0-X0us1+X0+Y0us-1-X0us2.Using (258) to express n/p in terms of us, after some calculation we find that(273)1+X0+Y02n2p2+4X01-n2p2=X0us2+11+X0+Y0-4X0usX0us2+1-1+X0+Y0us2.Then choosing C0(ρ)=QL(0,0)/ρ we have(274)C0ρ~1ρQ0,0R1∪R2∪R3∪R4.Then we use (273) to simplify (272) to(275)K0AsY01-As1-Bs=12π1-usus3/2X0us2-1X0+Y0-X0us=12πAs1-As2-X01-AsY0-X0+Y0Asso we have determined the functional form of K0(·), and (275) along with (219) leads to (70). We have thus completely determined the ray expansion in D0, as C0 is known for all parameter regions Rj via (274).It remains only to determine the constantC1 in the caustic ray expansion, for regions R4 and R3∩R4(X0+Y0≈X0) (for R3, C1 can be inferred from (214)). For this we must relate Q(0,0) to C1(ρ) by asymptotic matching, but this matching will require the analysis of another scale, which is intermediate to the (X,Y) and (n,p) scales. This analysis is carried out in the next subsection.


### 4.3. Analysis of the Scalek=m-O(ρ1/3), r=R-O(ρ2/3)

We will considerm-k=ν1ρ1/3=O(ρ1/3) and R-r=T1ρ2/3=O(ρ2/3). Then X0-X=O(ρ-2/3) and Y0-Y=O(ρ-1/3) so we are examining the vicinity of the corner point (X0,Y0) along parabolas, where (Y0-Y)2/(X0-X) is constant. We note that the scale k=m-O(ρ1/3) (with r=ρY and Y>0) was also important in the analysis of the model with R=∞ (see [10]). There, a more geometric interpretation is given, in terms of caustic rays and caustic boundaries and also in terms of sample paths of large deviations. Since the caustic rays in region D- cannot fill the entire domain (for parameter regions R4 and R3∩R4) we would expect a boundary effect near Y=Y0 or r=R. The corner scale k=m-O(1),r=R-O(1) that we analyzed in Section 4.2 is insufficient for fully understanding this boundary effect, and hence we analyze the scaling r=R-O(ρ2/3), which will connect the cases r=R-O(1) and r=ρY, Y<Y0. We cannot give an a priori probabilistic argument of why this scale is needed but mention that it leads to an interesting PDE, namely, (285), and such problems arise in many other areas, including queues with time-dependent arrival rates (see [20]) and steady two-dimensional convection-diffusion problems past curved obstacles (see [21]).We first set(276)Qn,p=X0nY0Y0+W0pQn,pwith which the main balance equation (228) becomes(277)1+X0+Y0-ρ-1n+pQn,p=W0+Y0+ρ-11-pW0+Y0Y0Qn,p-1+X0+ρ-11-n1X0Qn-1,p+X0Qn+1,p.To obtain a limiting PDE on the (ν1,T1) scale, we first formally expand (277) using Q(n±1,p)=Q(n,p)±Qn(n,p)+1/2Qnn(n+p)+⋯, where we anticipate that n will be scaled to be large, with n=O(ρ1/3) and also p=O(ρ2/3). We thus rewrite (277) as(278)0=ρ-1n+p+W0+Y0Y01-p+1X01-nQn,p-W0+Y0+ρ-1W0+Y0Y01-pQpn,p+12W0+Y0Qppn,p+ρ-11X0n-1Qnn,p+X0Qnnn,p+⋯,where the error term(s) involve derivatives of Q of order ⩾3 and terms of order O(ρ-1pQpp). We furthermore set(279)Qn,p=exp⁡-W02Y0W0+Y0p2ρQ¯n,p=exp⁡-W02Y0W0+Y0ρ1/3T12Q¯n,pso that(280)Qp=2apρQ¯+Q¯peap2/ρ,a=-W02Y0W0+Y0,(281)Qpp=4a2p2ρ2Q¯+4apρQ¯p+2aρQ¯+Q¯ppeap2/ρ~W02T12Y02W0+Y02ρ-2/3Q¯eaρ1/3T12,where we isolated dominant term in (281) on the scale p=O(ρ2/3). Setting p=ρ2/3T1 and n=ρ1/3ν1, multiplying (278) by ρ2/3e-aρ1/3T12, and using (279)–(281), we obtain the limiting PDE(282)X0∂2Q¯∂ν12+ν11-1X0Q¯-W0+Y0∂Q¯∂T1-W0W0+2Y02Y02W0+Y0T12Q¯=0,which applies over the quarter plane ν1>0, T1>0, and we now view Q¯ as a function of (ν1,T1) instead of (n,p). By rescaling ν1 and T1 using(283)ν1=X01-X01/3ν,T1=Y0+W0X01/61-X02/3Tand letting(284)Q¯ν1,T1=ρ-2/31-X05/3Q0,0X0-X0-Y0X01/6W0+Y0exp⁡-T136W0W0+2Y0Y02W0+Y02Fν,Twe obtain from (282) the separable, parabolic PDE(285)Fνν-νF=FT;ν>0,T>0.Including the exponential factor in (284) allows us to eliminate the last term in the left-hand side of (282), while the other factors (such as ρ-2/3 and Q(0,0)) are purely for convenience. We will show, by asymptotic matching between the (n,p) and (ν,T) scales, that Q¯ must be proportional to Q(0,0), and including the factor ρ-2/3 in (284) will lead to F being asymptotically O(1).Next we examine the boundary condition in (18), which can be also written as (X0+ρ-1)P(X0+ρ-1,Y)=P(X0,Y-ρ-1), or, using the (n,p) variables, as(286)X0+ρ-1Q-1,p=Q0,p+1.In view of (276) and (279) we can also write (286) as(287)X0+1X0ρ-1Q-1,p=Y0Y0+W0Q0,p+1or(288)X0+1X0ρ-1Q¯-1,p=Y0Y0+W0exp⁡aρ2p+1Q¯0,p+1=Y0Y0+W01-W0Y0W0+Y0ρ-1/3T1+Oρ-2/3Q¯0,p+1,where we used also (280) and (284).On the(ν1,T1) scale we have(289)Q¯-1,p=Q¯-ρ-1/3,T1=Q¯0,T1+Oρ-1/3,Q¯0,p+1=Q¯0,T1+ρ-2/3=Q¯0,T1+Oρ-2/3 so as long as X0≠Y0/(Y0+W0) we conclude from (288) that Q¯ must vanish along ν1=0 for all T1>0, and thus, in view of (284) so must F; hence(290)F0,T=0,T>0.The case where X0=Y0/(Y0+W0) corresponds to R3∩R4, and then the boundary condition will become more complicated. First we analyze the interior of R4 where (290) holds.To analyze (285) with (290), we first derive the behavior of F as ν,T→0, by matching the expansions on the (n,p) and (ν,T) scales. To this end we need, for region R4, the behavior of QL(n,p) for n,p→∞ with n=O(p). This asymptotic limit lies in the asymptotic matching region between (249) (which applies for 0<n/p<∞ for region R4) and (253), which applies for p→∞ but with n=O(1). Setting u=1/X0+v/p in the integral in (245) and shifting the contour Γ toward Re(u)=1/X0 we obtain(291)QLn,p~Q0,0X0nY0W0+Y0p2X03/2X0-1pW0+Y0X0+Y0-X012πi∫-i∞i∞vexp⁡-npX0v+X03/2W0+Y0v2dv.Here we used(292)u-n=1X0+vp-n~X0n/2exp-npX0vand approximated Γ by a vertical contour. Evaluating the integral in (291) explicitly we conclude that(293)QLn,p~Q0,02πX0nY0Y0+W0p1-X0W0+Y0X01/4X0-X0-Y0np3/2exp⁡-W0+Y04X0n2p;n,p⟶∞withn=Op.By asymptotic matching the expansion on the (ν1,T1) scale must behave as the right side of (293), when ν1,T1→0 with ν1/T1 fixed. But n=ρ1/3ν1, p=ρ2/3T1, and (283) shows that(294)n2p=ν12T1=X0W0+Y0ν2T,np3/2=1-X02/3X01/12ρ2/3W0+Y03/2νT3/2.Then comparing (293) with (276) (using (279) and (284)) we conclude that(295)Fν,T~ν2πT3/2exp⁡-ν24T;ν,T⟶0withν=OT.Note that (295) is consistent with the boundary condition in (290) along ν=0. The exponential factors in (279) and (284) do not enter the matching condition, since T1=O(1) and ρ1/3T12=ρ-1p2, and we can choose p so that p=o(ρ1/2) in the matching region.To solve (285) subject to (295) it is useful to view the function (πT)-1/2exp-ν2/(4T) as being an approximation to the delta function δ(ν) for T→0, with the mass concentrated in the range ν>0. Then the right side of (295) corresponds to the dipole -δ′(ν). Consider the problem(296)Fνν∗-νF∗=FT∗;ν>0,T>0,(297)F∗ν,0=δν-ν0,ν0>0,(298)F∗0,T=0,T>0.Introducing the Laplace transform(299)F^=F^ν,θ;ν0=∫0∞e-θTF∗ν,T;ν0dT,(296) becomes(300)F^νν-ν+θF^=-δν-ν0,ν>0and (298) implies that F^(0,θ)=0. The problem in (300) is a standard Green’s function problem with solution(301)F^=-πAiν0+θAiθAiν+θBiθ-Biν+θAiθ,0<ν<ν0,(302)F^=-πAiν+θAiθAiν0+θBiθ-Biν0+θAiθ,ν>ν0,where Ai(·) and Bi(·) are the Airy functions. Then the solution F to (285) will be(303)Fν,T=∂∂ν0Fν,T;ν0ν0=0.By differentiating (302) with respect to ν0, setting ν0=0, and noting the Wronskian identity AiθBi′(θ)-Ai′(θ)Bi(θ)=π-1 we find that(304)∂F^∂ν0ν0=0=Aiν+θAiθ,ν>0.Inverting the Laplace transform in (299) leads to F(ν,T) in (170), so we have established the expansion on the (ν1,T1) scale. We can also easily verify that (295) is indeed satisfied, since(305)Fν,T=12πiddT∫-i∞i∞Aiν+θAiθeθTdθ~12πiddT∫-i∞i∞eθT-νθθ1+Oν2θdθ~1πi∫-i∞i∞eω2T-ωνωdω=ν2πT3/2exp⁡-ν24T.Here we approximated the first integrand for θ→∞ using asymptotic properties of Ai(·).We next relate the constantC1 in D- to Q(0,0) (~1-X0-Y0 in R4), having expressed the expansions near the corner point (X0,Y0) in terms of Q(0,0) on both the (n,p) and (ν1,T1) scales. As we discussed previously, the caustic ray expansion in D- does not satisfy the boundary condition along X=X0, and different expansions must be constructed on the ν1 and n scales, corresponding to X0-X=O(ρ-2/3) and X0-X=O(ρ-1). For region R4, these expansions are given, respectively, in (111) and (109) (with (110)), up to the constant C1. The details of their construction are given in [10]. We will need to carefully estimate π(k,r) for n=m-k=O(1). For region R4, (109) applies for n=O(1) and all Y∈(0,Y0) (but breaks down as either Y↓0 or Y↑Y0). For n=O(1) and p=O(1), (241) holds as then π(k,r)~QL(n,p), and in (253) we gave the asymptotics for Q(n,p) for p→∞ and n=O(1). However, we must derive yet another expansion on the scale T1=O(1)(Y0-Y=O(ρ-1/3)) with T1>0 and n=O(1). The expansion in (169) develops a nonuniformity as ν→0, since F(0,T)=0. Since by definition of the Airy roots rj we have Ai(rj)=0, from the infinite series form in (170) we conclude that(306)Fν,T~ν∑j=0∞erjT=1-X0X01/3nρ1/3∑j=0∞e-rjT,ν⟶0,and this holds for any fixed T>0. Consider the scale n=O(1) and T1>0. Viewing Q in (277) now as a function of n and T1 and noting that p→p±1 corresponds to T1→T1∓ρ-2/3 the limiting form of (277) becomes(307)0=W0-1-X0Qn,T1+X0Qn-1,T1+Qn+1,T1.But W0=(1-X0)2=1+X0-2X0 so that the general solution of (307) is the linear function a(T1)+nb(T1), where a(·) and b(·) are arbitrary functions of T1. Note also that (307) applies asymptotically whether the exponential factors in (279) and (284) are included or excluded. The boundary condition in (287) can be applied on the n=O(1) scale and leads asymptotically to (W0+Y0)X0Q(-1,T1)=Y0Q(0,T1). This restricts the functions a(·) and b(·) in the linear solution to (307), by(308)Y0aT1=W0+Y0X0aT1-bT1.We have thus shown that the expansion on the (n,T1) scale is given by(309)πk,r~Q0,0ρ-11-X02W0+Y0X0X0-X0-Y0X0nY0Y0+W0pn+X0Y0+W0Y0+W0X0-Y0f∗Texp⁡-W02Y0W0+Y0ρ1/3T12-W0W0+2Y0Y0W0+Y02T13.Here we made a(·) proportional to f∗(·), which we will determine shortly, and computed b(·) from (308). Now we require (309), for n→∞, to match to the (ν1,T1) scale result, for ν1→0. But in view of (306) we see that the matching is possible provided that(310)f∗T=∑j=0∞e-rjT.Thus we now have three expansions valid for n=O(1), on the scales p=O(1), p=O(ρ2/3), and p=O(ρ) with Y<Y0. The matching between the p=O(1) result and (309) follows by letting p→∞ in (241) (corresponding to (253) for region R4) and letting T→0. The matching condition is satisfied if(311)12πp-3/2W0+Y01-X0X01/4X0-X0-Y0~1-X02ρ-1W0+Y0X0X0-X0-Y0f∗T,in some intermediate limit where p→∞ and T1=ρ-2/3p→0. Simplifying (311) by using (283) we must show that(312)f∗T~12πT-3/2,T⟶0.The Airy roots rj are well known to satisfy -rj~3jπ/22/3, j→∞, so for T→0 we approximate the sum in (310) by an integral to get(313)∑j=0∞e-rjT~∫0∞exp⁡-32ξπ2/3Tdξ=23πT-3/2∫0∞e-u32udu=1πT-3/2Γ32which verifies (312).Now we match (309) for T→∞ with (109) as Y→Y0. For T→∞ we have f∗(T)~e-|r0|T as the zeroth term in the sum in (310) dominates. By expanding (109) for Y→Y0, after canceling some common factors the matching implies that(314)ρ-1Q0,0W0W0+Y0X0X0-X0-Y0Y0Y0+W0perTexp⁡-W02Y0W0+Y0ρ1/3T12-W0W0+2Y0Y02W0+Y02T13~ρ-1/6C1ρeρΦX0,Y0eρΦYX0,Y0Y-Y0eρ1/3Φ∗Y0exp⁡12ρΦYYX0,Y0Y-Y02+16ρΦYYYX0,Y0Y-Y03+ρ1/3Φ∗′Y0Y-Y0exp⁡-1+X02X0log⁡Y0+W01-X012π1Ai′r0Y0+W0X0-Y0Y0X05/61-X01/6.Now,(315)eρΦYX0,Y0Y-Y0=e-pΦYX0,Y0=Y0Y0+W0p,expρ1/3Φ∗′Y0Y-Y0=exp⁡-T1Φ∗′Y0=exp⁡r0T11-X02/3W0+Y0X01/6=er0T. By using (110) to compute ΦYY and ΦYYY and noting that(316)ρY-Y02=ρ1/3T12,ρY-Y03=-T13 we see that the dependence on T1 is exactly the same in the left- and right-hand sides of (314). Thus we have a relation between the constants Q(0,0) and C1 for region R4, and since Q(0,0)~1-X0-Y0 we have derived the expression for C1 in (41).The analysis for regionR4 breaks down as Y0→X0-X0, as then the expressions in (284), (293), (309), and (314) all become singular. We thus examine the transitional case R3∩R4, where X0+Y0=X0+O(ρ-1/3). Then we certainly have Q(0,0)~1-X0-Y0 and (241) applies for any Rj. But now on parabolic scales, where n,p→∞ with n2/p fixed, Q(n,p) has the expansion in (254). We now define F¯(ν,T) by setting(317)πk,r~Q0,0ρ-1/31-X0X01/3X0nY0Y0+W0pexp⁡-W02Y0W0+Y0ρ1/3T12-W02Y0+W06Y02W0+Y02T13F¯ν,Tsince the factors in (317) correspond to the same sequence of transformations we made for the analysis of the interior of R4, and F¯ will again satisfy the PDE(318)F¯νν-ρF¯=F¯T;ν>0,T>0.Note that in (317) we included the factor ρ-1/3 whereas in (284) we have the two factors ρ-2/3 and X0-X0-Y0-1, as the product of these becomes, in region R3∩R4, O(ρ-2/3)×O(ρ1/3)=O(ρ-1/3). Including the factor ρ-1/3 will insure that F¯ is O(1) for ρ→∞, and the factor 1-X01/3X0-1/3 is purely for convenience. But for region R3∩R4 we must reexamine the boundary condition in (288), which in terms of F¯ becomes(319)X0+Oρ-1F¯-ρ1/3,T1=Y0Y0+W01-W0Y0W0+Y0ρ-1/3T1+Oρ-2/3F¯0,T1+ρ-2/3.But now (320)Y0+W0X0-Y0=1-X0X0-X0-Y0=-1-X0δ∗ρ-1/3 so that the limiting boundary condition from (319) becomes(321)X0∂F¯∂ν1+δ∗-T1F¯=0,ν1=0.Then scaling δ∗=X01/6(1-X0)1/3δ1, in terms of (ν,T) (321) becomes(322)F¯ν0,T+δ1-TF¯0,T=0,T>0.A matching condition for F¯, as ν,T→0 with ν=O(T), is obtained by comparing (317) to (254). From the definitions of δ and δ∗ we have δρ-1/2=δ∗ρ-1/3 so that δ=δ∗ρ1/6. Then also(323)pδ2ρ=T1δ∗2,nδρ=ν1δ∗,n2p=ν12T1are all O(1), while p-1/2=ρ-1/3T1-1/2 and δρ-1/2=δ∗ρ-1/3, so in the matching region, where T1 is small but δ∗=O(1), the Gaussian first term in the braces in (254) dominates the error function second term, and the matching will imply that(324)1-X0X01/3F¯ν,T~1π1-X01/2X01/4T1exp⁡-1-X04X0ν12T1and in terms of (ν,T) this simplifies to(325)F¯ν,T~1πTexp⁡-ν24T;ν,T⟶0withν=OT.Here we also used Y0+W0~1-X0 in region R3∩R4, so that T1~(1-X0)1/3X01/6T.The matching condition in (325) may be replaced by the condition F¯T=0=δ(ν) and we analyzed such problems in [20] where it was found that the solution to (318) with (322) is given by(326)F¯ν,T=12πi∫-i∞i∞Aiν+θAi2θeθT-δ1∫θ∞eδ1uAiududθ.The right-hand side of (326) clearly satisfies (318), since this PDE is separable and any function of the form g(θ)Ai(ν+θ)eθT is a solution, for any function g(·) and any complex θ. We can also verify that (322) holds along ν=0, by using(327)Ai′θAiθ2eθT-δ1+δ1-TAiθeθT-δ1=-ddθeθT-δ1Aiθ and some integration by parts in (326). To verify the matching condition in (325) we expand the integrand in (326) for ν→0, T→0, and θ→∞, using (190) to approximate the Airy function. We have Ai(ν+θ)/Ai(θ)~e-νθ and(328)e-θδ1∫θ∞eδ1uAiudu=∫0∞eδ1ξAiθ+ξdξ~e-2/3θ3/214π∫0∞exp⁡-θξ+Oξ2θ1+δ1ξ+Oξ2θ-1/41-ξ4θ+Oξ2θ2dξ=e-2/3θ3/214πθ-1/41θ+δ1θ+oθ-1=Aiθ1θ+δ1θ+oθ-1.Here we also expanded the integrand for small ξ, since by Watson’s lemma the main contribution comes from the range ξ=O(θ-1/2). Using (328) in (326) and noting that(329)12πi∫-i∞i∞eθT-θνdθθ=1πTexp⁡-ν24T,12πi∫-i∞i∞eθT-θνdθθ=12π∫ν/2T∞e-u2/2du we obtain, for ν,T→0 with ν=O(T),(330)F¯ν,T=1πTexp⁡-ν24T+2πδ1∫ν/2T∞e-u2/2du+o1.The expression in (330) is a two-term asymptotic approximation in this limit, and the leading term verifies the matching condition in (325). The correction term in (330), which is proportional to δ1, will asymptotically match to the term proportional to the error function in (254), as in the matching region pδ2/ρ=T1δ∗2 and nδ/ρ=ν1δ∗ are o(1), and(331)η0~1-X01/22X01/4ν1T1~ν2T,2X0δρ=ρ-1/31-X0X01/32δ1.We proceed to determineC1. Now the approximation in (209) (with (326)) remains valid on the scale n=O(1), since F¯(0,T)≠0. Thus for n=O(1)π(k,r) can be approximated by (317) with F¯(ν,T) replaced by F¯(0,T) and for T→∞ the expansion of (326) is determined by the pole at θ=r0<0, which is a simple pole if ν=0. Hence,(332)F¯0,T~er0T-δ1Ai′r0∫r0∞eδ1uAiudu,T⟶∞.For n=O(1) and 0<Y<Y0, the result in (109) holds, and this remains finite as Y↑Y0 even if X0+Y0=X0. In parameter region R3∩R4 we have Y0+W0~1-X0 and then as Y↑Y0, (109) becomes(333)ρ-1/6C1ρX0neρΦX0,Y0e-pΦYX0,Y0eρ1/3Φ∗Y0er0Texp⁡12ρ1/3ΦYYX0,Y0T12-16ΦYYYX0,Y0T1312π1Ai′r01-X01/3X0-7/12.Comparing (333) to (317) with (332), the exponential parts agree as for region R4, and we now have(334)ρ1/3Φ∗Y0=-ρ1/3r01-X02/3X01/6log⁡Y0+W01-X0=-ρ1/3r01-X02/3X01/6log⁡1+ρ-1/3δ∗1-X0~-r01-X0-1/3X0-1/6δ∗=-r0δ1.Thus (333) agrees with (317), with ν=0 and T→∞, if(335)Q0,0ρ-1/3X0-1/3∫r0∞eδ1uAiudu~C1ρ2πρ-1/6X0-7/12eρΦX0,Y0.But now Q(0,0)~1-X0-Y0~1-X0 and thus (335) leads to the result in (37).We have now determined the values ofQ(0,0), C0(ρ), C(ρ), and C1(ρ) for all possible regions Rj of parameter space, thus establishing Propositions 1 and 2. We also have analyzed all of the relevant scales near the corner point (X0,Y0) of the state space and thus established all parts of Proposition 15. As a final step we briefly discuss the asymptotic matching between the ray expansion in D0 and the expansion on the (ν,T) scale in (169). We only consider region R4 and note that this matching will involve letting (X,Y)→(X0,Y0) along parabolic scales with (Y0-Y)2/(X0-X)(=T12/ν1) held fixed and letting ν1,T1→∞ in (169). We have already verified that along linear scales, where (X,Y)→(X0,Y0) with (Y0-Y)/(X0-X) (=p/n) fixed, the D0 ray expansion matches directly to (168). The matching of the D0 and (ν,T) scale expansions will show that there are no “gaps” in the asymptotics, which would require the analysis of yet other scales.We first expandF(ν,T) in (170) for ν,T→∞, and we note that, along parabolic scales with ν=O(T2), ν/T2>1 corresponds to moving from the corner (X0,Y0) in region D0, while ν/T2<1 corresponds to moving into D-. By using (190) to approximate the integrand in (170) and shifting the contour to the right, to the range Re(θ)≫1, we are led to(336)Fν,T~12πi∫Brθν+θ1/4eθTexp⁡23θ3/2-θ+ν3/2dθ.Here we can view θ as being scaled to be O(ν) and T to be O(ν). The integrand in (336) has a saddle point where(337)ddθθT+23θ3/2-23θ+ν3/2=T+θ-θ+ν=0and this occurs when(338)θ=θs≡ν-T224T2=ν24T2-ν2+T24.Taking ν/T2>1 then a standard saddle point estimate leads to(339)F~12π12θs-12θs+ν-1/2θsθs+ν1/4exp⁡θsT+23θs3/2-θs+ν3/2=ν-T22πT3/2exp⁡112T3-12νT-14ν2T;ν,T⟶∞.By using (339) in (169) we match the result to the D0 ray expansion, which is ρ-1/2Q(0,0)K(X,Y)eρΨ(X,Y), as (X,Y)→(X0,Y0). By separating the exponential factors from the algebraic ones, the matching will hold if both(340)ρ-1/2KX,Y~ρ-2/31-X05/3X0-1/62πW0+Y0X0-X0-Y0ν-T2T3/2,(341)eρΨX,Y~X0nY0Y0+W0pexp⁡-ρ1/3W02Y0W0+Y0T12exp⁡-W0W0+2Y06Y02W0+Y02T13+112T3-12νT-14ν2T.We thus proceed to evaluate K and Ψ as (X,Y)→(X0,Y0) along parabolic scales, where (Y0-Y)2/(X0-X) is fixed. In this limit we have A→Amin=1-X0 and B→Bmin=W0/(Y0+W0), and it is useful to rewrite (68) as(342)X0-X=1-e-τX01-A+Aeτ-1,Y0-Y=Y01-B1-e-τ.The function K0(A) in (275) vanishes in this limit, in view of the factor (1-A)2-X0, and we have(343)K0A~2Y01-X0A-AminπX0-Y0-X0W0+Y0,A⟶1-X0.Expanding the Jacobian Δ in (71) for τ→0 and A→Amin leads to(344)Δ~-2τW0+Y0;τ⟶0,A⟶1-X0.But from (342) we have Y0-Y=ρ-1/3T1~(W0+Y0)τ so that(345)Δ-1/2~ρ1/612T1-X01/3X01/12Y0+W0,where we wrote the result in terms of T rather than T1. From the first expression in (342) we have(346)X0-X~τ2A-Amin+Aminτ so that(347)A-Amin~12τX0-X-1-X0τ2~12W0+Y0Y0-YX0-X-1-X0Y0-Y2W0+Y2=12ρ-1/3W0+Y0T1ν1-1-X0W0+Y02T12=12ρ-1/3X01/61-X01/3ν-T2T.Now (348)KX,Y=e-τ-A-1/2e-τ-B-1/2Δ-1/2K0A~X0-1/4Y0+W01/2Y0-1/2Δ-1/2K0A, so using (347) and (345) we can easily verify that (340) is satisfied.To verify (341) we first note that Ψ(X0,Y0)=0,(349)exp⁡ρX-X0ΨXX0,Y0=exp⁡ρX0-Xlog⁡1-Amin=X0n,exp⁡ρY-Y0ΨYX0,Y0=Y0Y0+W0p. Then peeling off the linear part of Ψ near X=X0,Y=Y0, (341) is equivalent to showing that(350)ρΨX,Y+X0-XΨXX0,Y0+Y0-YΨYX0,Y0~-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13+112T3-12νT-14ν2T.We can rewrite Ψ in (66) as(351)ΨX,Y+X0-XΨXX0,Y0+Y0-YΨYX0,Y0=X0-Xlog⁡1-Aeτ1-Amin+Y0-Ylog⁡1-Beτ1-Bmin-X0log⁡1-Aeτ1-A-Y0log⁡1-Beτ1-B+A1-eτ,where we have partially expressed the right-hand side of (351) in terms of X0-X=ρ-2/3ν1 and Y0-Y=ρ-1/3T1. Next we expand (351) as a triple Taylor series about τ=0, A=Amin, and B=Bmin. We have(352)A1-eτ-X0log⁡1-Aeτ-11-A-Y0log⁡1-Beτ-11-B=X0A1-A+Y0B1-B-Aeτ-1+X0A221-A2+Y0B221-B2eτ-12+X0A331-A3+Y0B331-B3eτ-13+Oτ4.But the relation in (216) implies that the coefficient of (eτ-1) in (352) vanishes and also that(353)B-Bmin~-1X0Y0Y0+W02A-Amin2.By (347), on the (ν,T) scale, A-Amin is O(ρ-1/3) so that B-Bmin=O(ρ-2/3), and thus (352) may be further approximated by (using (eτ-1)2=τ2+τ3+O(τ4))(354)X0Amin221-Amin2+Y0Bmin221-Bmin2+X0Amin331-Amin3+Y0Bmin331-Bmin3τ3+X0Amin221-Amin2+Y0Bmin221-Bmin2τ2+X0Amin1-Amin2+X0Amin21-Amin3A-Aminτ2with an O(τ4) error, which is not needed for the matching verification since exp[O(ρτ4)]=exp[O(ρ-1/3)]~1. We also have(355)X0-Xlog⁡1-Aeτ1-Amin=ρ-2/3ν1log⁡1-A-Amin1-Amin-Aminτ1-Amin+Oτ2,τA-Amin~-ρ-2/3ν1X0A-Amin+Aminτ~-ρ-1νX01/61-X01/312X01/61-X01/3ν-T2T+1-X0W0+Y0T1=-ν2ρν+T2T,Y0-Ylog⁡1-Beτ1-Bmin=Y0-Ylog⁡1-B-Bmin1-Bmin-Bmineτ-11-Bmin+OB-Bmin2=ρ-1/3T1-B-Bmin1-Bmin-Bminτ1-Bmin-Bminτ221-Bmin2+Oτ3.From (342) we have (356)Y0-Y1-B=ρ-1/3T11-Bmin+Oρ-2/3=Y01-e-τ=Y0τ-12τ2+Oτ3 so we can refine the estimate above (345) to(357)τ=ρ-1/3T1W0+Y0+12ρ-2/3T12W0+Y02+Oρ-1.Adding the expressions in (354) and (355) and using the estimates in (347), (353), and (357), (351) becomes, when multiplied by ρ,(358)1-X022+W022Y0+1-X033X0+W033Y02T13W0+Y03+1-X0+1-X02X0X01/61-X01/32W0+Y02T12ν-T2T+1-X022+W022Y0ρ1/3T12W0+Y02+T13W0+Y03-ν22T-νT2+W0+Y0Y0T1Y0W0+Y021X0X01/31-X02/34νT-T2-W0W0+Y0wY02T1T1W0+Y02-W0Y0ρ1/3T12W0+Y0+T132W0+Y02.Comparing (358) to (350) we find, after some simplification, that they are identical. The comparison is facilitated by separately comparing terms proportional to ρ1/3T12, ν2/T, νT, and T3. This verifies the matching between the D0 ray expansion and that on the (ν,T) scale. The Airy functions that arose on the (ν,T) scale disappear in the limit where ν,T→∞. If we let ν,T→∞ with ν/T2<1, then the expansion of F(ν,T) is much different, and now we have(359)Fν,T~12πAi′r01ν1/4exp⁡-23ν3/2+r0T-ν,as the pole at θ=r0 determines the asymptotic behavior. We can show that then (169) with (359) agrees with the ray expansion in D-, that is, C1L(X,Y)eρΦ(X,Y)eρ1/3Φ∗(s1), as the latter is expanded for (X,Y)→(X0,Y0), for parameter region R4. The case where ν/T2≈1 will be important in determining the transition layer expansion that applies where D0 meets D-, which is along the curve Y=Yc(X). Then neither (339) nor (359) apply.


## 4.1. Ray Expansion in the Interior ofD0

We analyze the scaled equation in (17) using the ray method of geometrical optics, where we assume an expansion of the form(195)PX,Y=C0ρeρΨX,YKX,Y+ρ-1K1X,Y+ρ-2K2X,Y+⋯.Then we have(196)PX±ρ-1,YPX,Y=e±ΨXX,Y1+ρ-1±KXK+12ΨXXX,Y+Oρ-2and using (195) in (17) we obtain in the limit as ρ→∞ the “eikonal” equation(197)1+X+Y=YeΨY+XeΨX+e-ΨXand at the next order in ρ-1 the “transport” equation(198)eΨYKY+12YΨYYK+K+eΨXXKX+12XΨXXK+K+e-ΨX-KX+12ΨXXK=0.The first-order PDE in (197) can be solved by the method of characteristics (see [14]), where one must solve the five ODEs(199)dXdτ=e-ΨX-XeΨX,dYdτ=-YeΨY,(200)dΨdt=ΨXdXdτ+ΨYdYdτ=ΨXe-ΨX-XeΨX-YΨYeΨY,(201)dΨXdτ=eΨX-1,dΨYdτ=eΨY-1.Here τ is a parameter along a given characteristic curve, which is also called a “ray,” due to applications in optics.To uniquely specify the solutionΨ(X,Y) to (197), we must either specify Ψ along some curve, called the “initial manifold,” in the (X,Y) plane, or use a singular solution, where all the rays emanate from a single point. The appropriate solution to the eikonal equation must be determined for each individual problem, and we will see that for the present model we will need to use three different solutions Ψ to (197), with two having the boundary X=X0 as the initial manifold and the third corresponding to all the rays emanating from the corner point (X0,Y0). The first two solutions will correspond to regions D+ and D- and the third to D0. Since D+ and D- arose also in the infinite capacity model, where R=∞, we discuss these only briefly, and the details of the corresponding solutions to (197)–(201) can be found in [10].Let us denote the solution in regionD+ as C(ρ)K+(X,Y)eρΨ+(X,Y) to distinguish it from (195), which will apply in D0. If this expansion will satisfy the boundary equation along X=X0 (or k=m) in (4), or, equivalently, (18), we must have(202)X0eΨ+,XX0,Y=e-Ψ+,YX0,Y.Requiring Ψ+(X,Y) to satisfy (202) is equivalent, up to an additive constant in Ψ+ which can be incorporated into C(ρ), to specifying Ψ+ along the initial manifold X=X0. Solving (199)–(201) subject to (202) leads to(203)Ψ+,X=-log⁡1-ss+X0et,Ψ+,Y=-log⁡1+s+X0-1etand then the rays are given in parametric form by (62), which relates (s,t) to (X,Y). When t=0 we have (X,Y)=(X0,s) so that s is the point where a ray hits the line X=X0. We use now t instead of τ as the parameter along a given ray, and s is used to index the family. Finally, solving (200) with (Ψ,τ) replaced by (Ψ+,t) leads to the expression in (61). We also note that when t=0(X=X0) we have(204)Ψ+X0,Y=X0-1+Y-X0+Ylog⁡X0+Ywhich is an explicit function of Y. From the first expression in (62) we find that for t→0(205)X0-X=Y+X02-X0Y+X0t+Ot2.It follows that for t>0 the rays that start from X=X0 enter the state space, where X⩽X0, for t>0 only if (Y+X0)2-X0>0, or Y+X0>X0. If X0>1 this condition holds for all Y⩾0 and hence the rays fill the region D+ indicated in Figure 3. If X0<1 and Y0+X0>X0 then the condition holds only for Y in the interval X0-X0<Y<Y0. Then these rays fill the domain D+ indicated in Figures 4 and 5. But if X0<1 and X0+Y0<X0, which is true for parameter region R4, the condition never holds and then this ray expansion plays no role in the analysis (see also Figure 6). Once we compute Ψ+ in D+, we can integrate (198) to obtain K+(X,Y) in (63). Thus we have shown that (63), up to the constant C(ρ) which we have yet to determine, holds in the portion D+ of the state space, for parameter regions R1∪R2∪R3.Now we observe that ifX0>1, Ψ+(X0,Y) is maximal at Y=0 and by evaluating also K+ at X=X0 and Y=0 we find that near the corner (X,Y)=(X0,0) with X=X0(k=m) and Y=r/ρ, and we have(206)CK+eρΨ+~C12πX0-1X03/2X0-reρX0-1-X0logX0.The scale r=O(1) for X0>1 must be analyzed separately and the details are carried out in [10]. For r=O(1) and m-k=n=O(1) the result in (141) applies, and by asymptotically matching this to (206) (for r⩾1) we conclude that(207)Cρ~ρ-1/2,X0,Y0∈R1.If X0<1 and X0+Y0>1 then Ψ+(X0,Y) in (204) is maximal at Y=1-X0, which lies in the range (0,Y0) precisely for parameter region R2. Expanding Ψ+(X,Y) in (61) about X=X0 and Y=1-X0 leads to(208)CK+eρΨ+~C1-X0X0n12πexp⁡-12ρr-ρ-m2.For parameter region R2 the expansions in D- and D0 will be uniformly exponentially small in ρ, as there is little mass in these state space ranges. Thus the main contribution to the double sum in the normalization condition (10) will come from D+ and in particular from the scale m-k=n=O(1) and r=ρ-m+O(ρ) (corresponding to Y=1-X0+O(ρ-1/2)). Then normalizing the approximation in (208), after approximating the sum over r by an integral over Y, we conclude that C(ρ)~ρ-1/2, and hence (207) applies also for parameter region R2 (and, by continuity, the relation holds in R1∩R2 (X0-1=O(ρ-1/2)) also).We have thus shown that the ray expansion inD+ has the state space regions D- and D0 as “shadows,” and thus the other solutions to (197) must apply. To construct the solution in D- we must slightly modify the ansatz in (195) and now expand the joint distribution as(209)PX,Y=C1ρeρΦX,Yeρ1/3Φ1X,YLX,Y+ρ-1/3L1X,Y+ρ-2/3L2X,Y+⋯.Now Φ will satisfy (197) and L will satisfy (198), and the subexponential term Φ1 will satisfy the PDE(210)YeΦYΦ1,Y+XeΦX-e-ΦXΦ1,X=0.Again the detailed analysis can be found in [10]. We now find that(211)ΦX=-log⁡1-1-X0et1,ΦY=-log⁡1-W0W0+s1et1and the rays are given in parametric form by(212)X=1-1-X0et11-1-X0e-t1,Y=-W0+W0+s1e-t1,where we recall that W0=1-X02. When t1=0 we have X=X0 and Y=s1, but unlike the rays in D+, those in D- are all tangent to the boundary X=X0. Thus the boundary is a “caustic boundary,” for 0<Y<X0-X0 for region R2∪R3 and for all Y with 0<Y<Y0 for region R4. The solution Φ to (197) is now given by(213)ΦX,Y=W0-W0+s1e-t1log⁡1-W0et1W0+s1-1-X0-e-t11-X0-et1log⁡1-W0et1-W0et1+W0log⁡W0+s1-12W0log⁡W0.For region D- we can invert transformation (212) and write s1 and t1 explicitly in terms of X and Y. This leads to the expression in (81) for s1=s1(X,Y), and then (80) gives Φ in terms of X and Y.Equation (210) implies that Φ1(X,Y) is constant along a caustic ray, so we write Φ1(X,Y)=Φ∗(s1). To determine Φ∗(·) and also completely determine L(X,Y) in (209), we need to construct two “nested” boundary layer corrections to the expansion in (209), near X=X0, corresponding to the scales X0-X=O(ρ-2/3) (m-k=O(ρ1/3)) and X0-X=O(ρ-1) (m-k=O(1)). The boundary condition in (18) can only be imposed on the expansion that applies for n=m-k=O(1). Using asymptotic matching between the expansions for X0-X=O(ρ-1), X0-X=O(ρ-2/3), and the ray expansion in (209) allows us to determine (209) up to the multiplicative constant C1.To determineC1 we need to, for regions R2 and R3, first relate C1(ρ) to the constant C(ρ) in the D+ expansion. This can be done by analyzing the transition curve Y=Y∗(X), with the scaling Y-Y∗(X)=O(ρ-1/3). The details are again presented in [10], and this leads to the conclusion that(214)C1ρCρ~ρ-1/6,X0,Y0∈R2∪R3.In R2 we have C(ρ)~ρ-1/2 so that C1(ρ)~ρ-2/3, but we have yet to determine either C or C1 for region R3. The conclusion in (214) can also be reached by constructing an expansion near the point (X,Y)=(X0,X0-X0), with the scaling X0-X=O(ρ-2/3) and Y-(X0-X0)=O(ρ-1/3) (see subsection  5.3 in [10] for that analysis). Note also that the curve Y=Y∗(X) corresponds to the ray with s1=X0-X0 in (212) and also the ray with s=X0-X0 in (62). Then we have Ψ+(X,Y∗(X))=Φ(X,Y∗(X)) so that the exponential factors in the expansions in (60) and (209) are continuous along Y=Y∗(X), which separates D+ from D-.The regionD0 is a shadow of both the D+ and D- rays for parameter regions R2∪R3 and a shadow of the D- rays for parameter region R4. For R4, the caustic rays that fill D- correspond to 0<s1<Y0 in (212), and s1=Y0 corresponds to the curve Y=Yc(X) in (86), which separates D- from D0. To fill the shadow D0 and thus obtain an approximation to P(X,Y) for (X,Y)∈D0, we must use a singular solution to (197), one that has all rays start from the corner point (X,Y)=(X0,Y0). To construct this solution we first integrate the two ODEs in (201), which yields(215)ΨX=-log⁡1-Aeτ,ΨY=-log⁡1-Beτ,where A and B are constant along a given ray. Evaluating the eikonal equation in (197) along the corner X=X0,Y=Y0 and using (215) with τ=0 gives(216)1+X0+Y0=Y01-B+X01-A+1-Aand this leads to the relation between A and B in (67). Here we take τ=0 to correspond to when a ray starts from the corner point X=X0,Y=Y0. Using (215) we solve the two ODEs in (199), subject to Xτ=0=X0 and Yτ=0=Y0, and we thus obtain the expressions in (68), which give the corner rays in parametric form. Using (215) and (68) we then integrate (200), and choosing Ψ(X0,Y0)=0 for convenience, we obtain (66). Note that Ψ can only be determined from (197) up to an additive constant, but such a constant can be incorporated into C0=C0(ρ;X0,Y0) in (195).We can viewA as indexing this family of rays. If A is such that (A+X0+Y0)(1-A)-X0=0 then B in (216) and (67) becomes infinite, and this corresponds to A=Amax in (69). This critical ray corresponds to Y=Y0, and for τ∈(0,-log(Amax)) we have X∈(0,X0), so the ray is the upper boundary of the state space rectangle, where all secondary storage spaces are full. The ray A=Amin=1-X0(for X0<1) corresponds to the curve Y=Yc(X)=Yc(X;X0,Y0) in (86). For regions R1∪R2∪R3 we have Y0/(X0+Y0)>1-X0 and then the ray with A=Y0/(X0+Y0) is the curve Y~(X) in (56). Thus for regions R1∪R2∪R3 it suffices to consider A in the range (Y0/(X0+Y0),Amax) to fill region D0 in state space. We also note that in order for the rays, which all start from (X,Y)=(X0,Y0), to enter the state space we need Xττ=0<0 and Yττ=0<0, which implies that 1-A-X0/(1-A)<0 and X0/(1-A)-(A+X0+Y0)<0. Adding the last two inequalities implies that A>1/2(1-X0-Y0) and this is certainly true if A>Amin.We next solve the transport equation (198) for K(X,Y). This equation can be written as an ODE along a ray, with(217)KτK=12YΨYY+1eΨY+12XΨXX+1eΨX+12ΨXXe-ΨX=eΨX+eΨY-12Xττ+XτeΨX-12Yττ+YτeΨY,where we used (199), and the last equality in (217) follows by differentiating (199) with respect to τ. Introducing the Jacobian Δ≡XτYA-XAYτ associated with the mapping from (τ,A) to (X,Y) variables, after some calculation we find that (217) becomes(218)KτK=1211-Aeτ+1211-Beτ-12ΔτΔand the most general solution to (218) is given by(219)KX,Y=e-τ-Ae-τ-BΔτ,A-1/2K0A.Here K0(A) is an arbitrary function of the parameter that indexes the rays and is thus constant along any particular ray. We have thus determined the expansion in (195) up to the constant C0 and the function K0(·). To complete the D0 ray expansion we will need to use asymptotic matching to a local expansion valid near the corner (X,Y)=(X0,Y0). This is constructed in Section 4.2. In order to accomplish the matching we will need the behavior of (195) as X↑X0, Y↑Y0, and this is examined next.We note that(X,Y)→(X0,Y0) corresponds to τ→0 and in this limit the Jacobian in (71) vanishes, and(220)Δ~-τ1+X0+Y01+X01-A2-4X01-A,τ⟶0+.For τ→0 we also have(221)X0-X~X01-A+A-1τ,Y0-Y~Y01-Bτso that(222)Y0-YX0-X=pn~A+X0+Y01-A-X0X0-1-A2,τ⟶0.Here we recall that Y=Y0-p/ρ and X=X0-n/ρ, and (222) gives the slope at which the ray indexed by A hits the corner point (X0,Y0).We can thus invert the transformation in (68) locally, with (222) corresponding to a quadratic equation for A=A(n/p), and hence(223)A~As≡1+121p-n1+X0+Y0n-1+X0+Y02n2+4p2-n2X0.Then we define Bs by replacing A by As in (67). Note that the right-hand side of (223) approaches Amax if n→∞, approaches Amin=1-X0 as p→∞, and is equal to Y0/(X0+Y0) if n/p=(X0+Y0)2-X0/Y0. For τ→0 we also have Ψ(X,Y)→0 with(224)ΨX,Y=A+X01-A-1log⁡1-A+Y01-Blog⁡1-Bτ+Oτ2=X0-Xlog⁡1-As+Y0-Ylog⁡1-Bs+Oτ2so that eρΨ(X,Y)~(1-As)n(1-Bs)p. We have thus shown that as τ→0 we have(225)C0KX,YeρΨX,Y~C0K0As1-As1-Bs1-Asn1-Bsp1τ1+X0+Y01+X01-As2-4X01-As-1/2,where A~As was approximated by (223), and by (221) we have(226)τ~1-BsY0-YY0~1-BsY0pρ,so that (225) becomes an explicit function of n and p.


## 4.2. Analysis of the Scalek=m-O(1), r=R-O(1)

We use the variablesn=m-k and p=R-r and obtain an approximation to π(k,r) that is valid when all but a few of the primary and secondary spaces are occupied. This is likely for parameter regions R3 and R4 but unlikely for R1 and R2. We define Q by(227)πk,r=πk,r;ρ,m,R=Qn,p;ρ,X0,Y0.We thus denote by Q(n,p) the exact probability that there are n (resp., p) empty primary (resp., secondary) spaces, and we will later denote by QL(n,p) the leading term in an asymptotic expansion of this probability, which is valid for the scale n,p=O(1). Then clearly Q(0,0)~QL(0,0) so in asymptotic relations involving Q(0,0) we can drop the subscript L.Writing the balance equations in (3), (4), and (5) in terms of (n,p) leads to(228)1+X0+Y0-ρ-1n+pQn,p=Y0+ρ-11-pQn,p-1+X0+ρ-11-nQn-1,p+Qn+1,p;n⩾1,p⩾1,(229)1+X0+Y0-ρ-1pQ0,p=Y0+ρ-11-pQ0,p-1+Q1,p+Q0,p+1,p⩾1,(230)1+X0+Y0-ρ-1nQn,0=X0+ρ-11-nQn-1,0+Qn+1,0,n⩾1,and the corner condition in (9) becomes(231)X0+Y0Q0,0=Q1,0+Q0,1.Note that near this corner only the boundaries X=X0 and Y=Y0 of the state space rectangle are relevant, and the problem in (228)–(231) corresponds to a random walk in a quarter plane, in (n,p) space. For ρ→∞ the problem in (228)–(231) may be further approximated by(232)1+X0+Y0QLn,p=Y0QLn,p-1+X0QLn-1,p+QLn+1,p;n⩾1,p⩾1,(233)1+X0+Y0QL0,p=Y0QL0,p-1+QL1,p+QL0,p+1,p⩾1,(234)1+X0+Y0QLn,0=X0QLn-1,0+QLn+1,0,n⩾1.Here QL(n,p) in (232)–(234) is understood to be the leading term in an asymptotic expansion of Q(n,p;ρ), for ρ→∞, and the corner equation in (231) must also be satisfied by this leading term.We introduce the double generating function(235)Q~z,w=∑n=0∞∑p=0∞QLn,pznwpand from (231)–(234) we obtain(236)1+X01-z+Y01-w-1zQ~z,w=1w-1zQ~0,w+1-1wQL0,0.From (236), by setting w=1 we obtain(237)Q~z,1=Q~0,11-zX0and if z=w we find that Q~(z,z)=QL(0,0)/[1-(X0+Y0)z]. Setting z=w then in (235) leads to(238)∑n+p=lQLn,p=QL0,0X0+Y0l,l⩾0.The expression in (238) shows that the total number of empty spaces (=m+R-k-r=n+p) follows asymptotically a geometric distribution if X0+Y0<1, and this could be also deduced from (12), by expanding the exact (truncated Poisson) distribution for ρ→∞. Note, however, that (238) holds also for X0+Y0⩾1. The factor in brackets in the left-hand side of (236) vanishes when z=z±(w), where z± are given in (51). Requiring Q~(z,w) to be analytic at the smaller root z-(w) determines Q~(0,w) as(239)Q~0,w=z+w1-wz-w-wX0QL0,0.Then using (239) in (236) and partially inverting the transform in (236) lead to(240)∑p=0∞wpQLn,p=1-wz-w-wX0z+w-n-1QL0,0,n⩾0,and hence by the Cauchy integral formula(241)QLn,p=QL0,0X02πi∮1-wz-w-wz+w-n-1w-p-1dw,where the contour is over a small loop about w=0.WhenX0+Y0<1 (m+R<ρ, region R3∪R4), most primary and secondary spaces will be full, and then (241) becomes a normalized discrete distribution with(242)Q0,0~QL0,0=1-X0-Y0,R3∪R4,leading to the limit law in (50).For regionsR1∪R2, (241) still represents a local approximation to π(k,r), but now different arguments must be used to determine QL(0,0), and indeed now QL(0,0) will turn out to be exponentially small for large ρ. We proceed to relate (241) to the ray expansions in D0, D+, and D-, noting that, for parameter region R1∪R2∪R3, D0 and D+ both border the corner point, while, for R4, D0 and D- border this point. We will need to expand (241) asymptotically, for n and/or p→∞. The integrand in (241) has branch points at w=(1+X0+Y0±2X0)/Y0 and possible poles at solutions of z-(w)=w. Now, z-(1)=1 for X0>1 but w=1 is not a pole in view of the factor 1-w in the numerator. The only other possible solution to z-(w)=w occurs at w=1/(X0+Y0) and this is a pole if(243)2X0X0+Y0=1+X0+Y0-Y0X0+Y0-1+X0+Y0-Y0X0+Y02-4X0=X0+Y0+X0X0+Y0-X0+Y02-X0X0+Y0.But (243) holds precisely when (X0+Y0)2>X0, which is true for regions R1∪R2∪R3. Thus for R4 the pole is absent and along the transition curve R3∩R4 (cf. (25)) the pole coalesces with the lower branch point, both being at w=1/X0 (>1). We can recast the integral in (241) by using the conformal map u=z+(w), so that the inverse is(244)w=1+1Y01+X0-X0u-1u and (241) becomes(245)QLn,p=QL0,02πi∫Γ1-uX0u2-1Y0pup-n-2X0+Y0-X0u1+X0+Y0u-X0u2-1p+1du,where Γ is a small loop about u=z+(0), with(246)z+0=12X01+X0+Y0+1+X0+Y02-4X0,and the integrand in (245) has a pole of order p+1 at u=z+(0). Below we collect some asymptotic results for QL(n,p) that will be used in the matching calculations.Proposition 20.
Forn and/or p→∞ the function QL(n,p) has the following asymptotic expansions:
(i)
p = O ( 1 ), n→∞(247)QLn,p~Q0,0z+0-np!npY01+X0+Y02-4X0p. (ii)
p , n → ∞ with 0⩽n/p<θ1, θ1=(X0+Y0)2-X0/Y0, for (X0,Y0)∈R1∪R2∪R3(248)QLn,p~Q0,0X0+Y02-X0X0nX0+Y0p-n-2. (iii)
p , n → ∞ with θ1<n/p<∞, for R1∪R2∪R3(249)QLn,p~Q0,02πn21+X0+Y02+4p2-n2X0-1/41-usX0us2-1us3/2X0+Y0-X0us1+X0+Y0us-1-X0us2usp-nY0p1+X0+Y0us-1-X0us2-p, where(250)us=usnp=12X0n+p1+X0+Y0n+1+X0+Y02n2+4X0p2-n2. (iv)
p , n → ∞, n/p≈θ1 (with n-pθ1=O(p)), for R1∪R2∪R3(251)QLn,p~Q0,0X0nX0+Y0p-n-2X0+Y02-X012π∫-∞ηe-v2/2dv,(252)η=Y0pX0+Y0X0+Y03+X0+X0+Y0Y0-2X0θ1-np=O1. (v)
p , n → ∞, 0<n/p⩽∞, for region R4. The expression in (249) holds for all 0<n/p<∞ and (247) holds for n→∞ with p=O(1). (vi)
n = O ( 1 ), p→∞, for region R4(253)QLn,p~Q0,02πX0n/2-1/4Y0Y0+W0pW0+Y01-X0X0-X0-Y0p-3/2n+X0X0-1+X0X0-X0-Y0. (vii)
n , p → ∞ with n=O(p), for region R3∩R4, with Y=X0-X0+ρ-1/2δ, δ=O(1)(254)QLn,p~Q0,0πX0n/2Y0Y0+W0p1-X0X01/4pexp⁡n24p1-1X0+δρ2X0exp⁡pδ2ρX01-X0-nδρX0∫η0∞e-v2/2dv,(255)η0=1-X02X01/4np-21-X0X01/4pδρ.The results in (247)–(255) may be obtained by expanding the integrals in (241) or (245) by a combination of singularity analysis and the saddle point method. Good references on asymptotic expansion of integrals are the books [15–19], but since these methods are now well established, we merely sketch the proof of Proposition 20.To obtain (247) we approximate the integrand in (241) for w=O(n-1), scaling w=w1/n and using (256)z+w-n-1~z+0-n-11+wz+′0z+0+Ow2-n-1~z+0-n-1exp⁡-z+′0z+0w1. Note that (247) is not only asymptotically true but also an exact expression when p=0, since Q(n,0) satisfies the boundary equation in (234). Turning to (245) we see that the integrand has a simple pole at u=u∗≡(X0+Y0)/X0 and saddle point(s) where(257)ddup-nlog⁡u-plog⁡1+X0+Y0u-X0u2-1=0, and (257) is equivalent to the quadratic equation(258)np+1X0u2-np1+X0+Y0u+np-1=0. One root of (258) is given by us=us(n/p) in (250), and the complementary root, with a minus sign in front of the square root, will correspond to a second saddle which will not play any role in the analysis. The pole u∗ and saddle us, whose location depends on the ratio n/p, coalesce when n/p=θ1, for regions R1, R2, and R3. Note also that as n/p→∞ we have us→z+(0) and that the integrand in (245) has a zero at u=1/X0. For region R1∪R2∪R3 and n/p∈(0,θ1) we have the ordering(259)1X0<us<u∗<z+0 and we note that u∗<z+(0) is always true. Then we dilate the small loop Γ about u=z+(0) to the saddle point contour u-z+(0)=us-z+(0), which is a circular contour that traverses the saddle us in the steepest descent directions, which are arg(u-us)=±π/2. But in doing the dilation we must take into account the contribution from the residue at the pole u=u∗, in view of (259). It turns out that the residue dominates the saddle point contribution, and we thus obtain the expression in (248). For n/p∈(θ1,∞) we have the ordering(260)1X0<u∗<us<z+0 and then the pole does not contribute as we dilate Γ to the saddle point contour. Setting(261)Gu=1-uX0u2-1X0+Y0u-11u21+X0+Y0u-X0u2-1,Fu=Fu;np=log⁡Y0+1-nplog⁡u-log⁡1+X0+Y0u-X0u2-1 we use for (245) the standard saddle point estimate (262)12πi∫ΓGuepFudu=12πi∫ΓsGuepFudu~12πpGusF′′usepFus, and this leads to the expression in (249). The transitional result in (251) corresponds to n/p≈θ1 and then we have u∗≈us, so the saddle is close to a simple pole. Such situations are discussed in detail in [15], and by expanding the integrand in (245) about u∗ we ultimately obtain a simpler integrand that is related to D-1(·), the parabolic cylinder function of order -1, which can in turn be expressed in terms of the standard error function, leading to (251). Note that as η→+∞ we approach the region n/p<θ1 and then (251) reduces to (248).The pole atu∗ and zero at 1/X0 coalesce when X0+Y0=X0, which is precisely the curve R3∩R4 which separates R3 from R4 in parameter space. For (X0,Y0)∈R4 we have the ordering(263)u∗<1X0<us<z+0.Then for any n/p∈ (0,∞) we can deform Γ into the saddle point contour and obtain (249) as the approximation to QL(n,p). The limits n=O(1), p→∞, and n→∞, p=O(1) require a separate analysis. For the latter us becomes close to z+(0) and we again obtain (247), while for the former the saddle us gets close to zero at 1/X0. By expanding the integrand near u=1/X0, setting u=1/X0+v/p, we obtain(264)QLn,p=QL0,0X0nY0Y0+W0pX0Y0+W01X0+Y0-X012πi∫-i∞i∞vp2X0X0-1+Bvp+Op-11-X0nvp1-X02Y0+W0v3pexp⁡X03/2v2Y0+W0dv,where(265)B=-5X03/2+3X0+2X03/2X0-1X0+Y0-X0. Evaluating the integral(s) in (264) we obtain (253). Note that the coefficient of the O(p-1) term in the integral is zero, so the result is O(p-3/2).WhenY0+X0≈X0 the pole at u∗ and zero at 1/X0 are close together. If n,p→∞ so that n/p remains fixed and positive, the saddle lies well to the right of these, and then (249) holds. But if n,p→∞ with n=O(p) then the saddle, pole, and zero are all close. Then we must reexamine the integrand in (245) and expand it about u=1/X0. Again setting v=ρu-1/X0=O(1) and replacing Y0 by X0-X0+δ/ρ (with δ=O(1)), (245) becomes asymptotically(266)QLn,p~Q0,0X0n/2Y0Y0+W0p2X0ρ12πi∫Br+X0vX0v-δexp⁡-nρX0v+pρX03/2Y0+W0v2dv, where Br+ is a vertical contour in the complex v-plane, which is to the right of the pole at v=δ/X0. The integral in (266) may be evaluated as a combination of a Gaussian and an error function, and this ultimately leads to (254) with (255). Since the large parameter ρ appears explicitly in (254) and (266) we can define the asymptotic limit more precisely, as ρ→∞ with Y0+X0-X0=O(ρ-1/2), p=O(ρ), and n=O(ρ).With Proposition20 we are now ready to relate the corner approximation QL(n,p) to the various ray expansions, via asymptotic matching. First consider parameter region R1∪R2∪R3, so that D0 and D+ come together at the corner point (X0,Y0). In D+ the function Ψ+(X,Y) can be expanded in Taylor series about (X,Y)=(X0,Y0) and as (X,Y)→(X0,Y0) we obtain(267)CρK+X,YeρΨ+X,Y~CρK+X0,Y0eρΨ+X0,Y0exp⁡ρX-X0Ψ+,XX0,Y0+ρY-Y0Ψ+,YX0,Y0=CρX0+Y02-X0X0+Y05/22πeρX0+Y0-1-X0+Y0logX0+Y0X0X0+Y0nX0+Y0p,where we used ρ(X-X0)=-n and ρ(Y-Y0)=-p. Near X=X0, the curve Y=Y~(X) that separates D0 from D+ has the slope Y~′(X0)=Y0/(X0+Y0)2-X0 so that Y~(X) can be approximated by the straight line Y-Y0=Y~′(X0)(X-X0), which is the same as p/n=1/θ1. Then D+ meets the corner in the sector n/p∈[0,θ1), and this corresponds precisely to where the asymptotic result (248) applies. Comparing (267) to (248) we see that the matching is possible, if Q(0,0) and C(ρ) are related by(268)Cρ~2πX0+Y0eρ1-X0-Y0+X0+Y0logX0+Y0Q0,0and this holds throughout R1∪R2∪R3. For R1∪R2 we have previously determined that C(ρ)~ρ-1/2 and thus (268) determines Q(0,0), as in (30). For region R3 we have Q(0,0)~1-X0-Y0 and then (268) leads to (33). For R1∩R2, corresponding to X0+Y0-1=γ/ρ=O(ρ-1/2), the approximation in (208) holds for n=O(1) and Y-(1-X0)=O(ρ-1/2), but now the upper boundary Y=Y0 lies within this range. Then applying the normalization condition in (10) to (208) leads to(269)1~∑p=0∞∑n=0∞C2π1-X0X0nexp⁡-ρ2Y+X0-12=C2π∑p=0∞exp⁡-ρ2γρ-pρ2~ρ2πC∫0∞e-γ-v2/2dv=ρ2πC∫-∞γe-v2/2dvand then C is given by (33). Then QL(0,0)~Q(0,0) can be computed from (268), and since ρ[1-X0-Y0+(X0+Y0)log(X0+Y0)]~γ2/2, we obtain the expression in (29). We have thus determined both Q(0,0) and C(ρ) for all cases of the parameters X0,Y0.To matchQ(n,p) to the ray expansion in D0 we note that, for regions R1∪R2∪R3, D0 meets the corner over the sector n/p∈ (θ1,∞), while for region R4 this holds for all n/p∈ (0,∞). We recall that for R4 the curve Y=Yc(X) separates D0 from D-, but this curve has infinite slope at X=X0 and is thus tangent to the line X=X0 at Y=Y0. So if we approach the corner along any straight line (excluding slopes 0 and ∞) we are approaching from within D0. By the asymptotic matching principle, the expansion of QL(n,p) for n,p→∞ should agree with the expansion of C0KeρΨ as X→X0, Y→Y0. Thus in D0 we must compare (249) to (225). But from (250) and (223) we see that (1-As)us=1 and from (67) we obtain(270)1-Bs=Y0usX0+Y0-1us-1-X0us2.It follows that(271)1-Asn1-Bsp=us-nusY01+X0+Y0us-1-X0us2p and the matching is possible if(272)C0K0AsY01-As1-Bsρp1+X0+Y01+X01-As2-4X01-As-1/2~Q0,02π1-usX0us2-1n21+X0+Y02+4p2-n2X0-1/4us3/2X0+Y0-X0us1+X0+Y0us-1-X0us2.Using (258) to express n/p in terms of us, after some calculation we find that(273)1+X0+Y02n2p2+4X01-n2p2=X0us2+11+X0+Y0-4X0usX0us2+1-1+X0+Y0us2.Then choosing C0(ρ)=QL(0,0)/ρ we have(274)C0ρ~1ρQ0,0R1∪R2∪R3∪R4.Then we use (273) to simplify (272) to(275)K0AsY01-As1-Bs=12π1-usus3/2X0us2-1X0+Y0-X0us=12πAs1-As2-X01-AsY0-X0+Y0Asso we have determined the functional form of K0(·), and (275) along with (219) leads to (70). We have thus completely determined the ray expansion in D0, as C0 is known for all parameter regions Rj via (274).It remains only to determine the constantC1 in the caustic ray expansion, for regions R4 and R3∩R4(X0+Y0≈X0) (for R3, C1 can be inferred from (214)). For this we must relate Q(0,0) to C1(ρ) by asymptotic matching, but this matching will require the analysis of another scale, which is intermediate to the (X,Y) and (n,p) scales. This analysis is carried out in the next subsection.


## 4.3. Analysis of the Scalek=m-O(ρ1/3), r=R-O(ρ2/3)

We will considerm-k=ν1ρ1/3=O(ρ1/3) and R-r=T1ρ2/3=O(ρ2/3). Then X0-X=O(ρ-2/3) and Y0-Y=O(ρ-1/3) so we are examining the vicinity of the corner point (X0,Y0) along parabolas, where (Y0-Y)2/(X0-X) is constant. We note that the scale k=m-O(ρ1/3) (with r=ρY and Y>0) was also important in the analysis of the model with R=∞ (see [10]). There, a more geometric interpretation is given, in terms of caustic rays and caustic boundaries and also in terms of sample paths of large deviations. Since the caustic rays in region D- cannot fill the entire domain (for parameter regions R4 and R3∩R4) we would expect a boundary effect near Y=Y0 or r=R. The corner scale k=m-O(1),r=R-O(1) that we analyzed in Section 4.2 is insufficient for fully understanding this boundary effect, and hence we analyze the scaling r=R-O(ρ2/3), which will connect the cases r=R-O(1) and r=ρY, Y<Y0. We cannot give an a priori probabilistic argument of why this scale is needed but mention that it leads to an interesting PDE, namely, (285), and such problems arise in many other areas, including queues with time-dependent arrival rates (see [20]) and steady two-dimensional convection-diffusion problems past curved obstacles (see [21]).We first set(276)Qn,p=X0nY0Y0+W0pQn,pwith which the main balance equation (228) becomes(277)1+X0+Y0-ρ-1n+pQn,p=W0+Y0+ρ-11-pW0+Y0Y0Qn,p-1+X0+ρ-11-n1X0Qn-1,p+X0Qn+1,p.To obtain a limiting PDE on the (ν1,T1) scale, we first formally expand (277) using Q(n±1,p)=Q(n,p)±Qn(n,p)+1/2Qnn(n+p)+⋯, where we anticipate that n will be scaled to be large, with n=O(ρ1/3) and also p=O(ρ2/3). We thus rewrite (277) as(278)0=ρ-1n+p+W0+Y0Y01-p+1X01-nQn,p-W0+Y0+ρ-1W0+Y0Y01-pQpn,p+12W0+Y0Qppn,p+ρ-11X0n-1Qnn,p+X0Qnnn,p+⋯,where the error term(s) involve derivatives of Q of order ⩾3 and terms of order O(ρ-1pQpp). We furthermore set(279)Qn,p=exp⁡-W02Y0W0+Y0p2ρQ¯n,p=exp⁡-W02Y0W0+Y0ρ1/3T12Q¯n,pso that(280)Qp=2apρQ¯+Q¯peap2/ρ,a=-W02Y0W0+Y0,(281)Qpp=4a2p2ρ2Q¯+4apρQ¯p+2aρQ¯+Q¯ppeap2/ρ~W02T12Y02W0+Y02ρ-2/3Q¯eaρ1/3T12,where we isolated dominant term in (281) on the scale p=O(ρ2/3). Setting p=ρ2/3T1 and n=ρ1/3ν1, multiplying (278) by ρ2/3e-aρ1/3T12, and using (279)–(281), we obtain the limiting PDE(282)X0∂2Q¯∂ν12+ν11-1X0Q¯-W0+Y0∂Q¯∂T1-W0W0+2Y02Y02W0+Y0T12Q¯=0,which applies over the quarter plane ν1>0, T1>0, and we now view Q¯ as a function of (ν1,T1) instead of (n,p). By rescaling ν1 and T1 using(283)ν1=X01-X01/3ν,T1=Y0+W0X01/61-X02/3Tand letting(284)Q¯ν1,T1=ρ-2/31-X05/3Q0,0X0-X0-Y0X01/6W0+Y0exp⁡-T136W0W0+2Y0Y02W0+Y02Fν,Twe obtain from (282) the separable, parabolic PDE(285)Fνν-νF=FT;ν>0,T>0.Including the exponential factor in (284) allows us to eliminate the last term in the left-hand side of (282), while the other factors (such as ρ-2/3 and Q(0,0)) are purely for convenience. We will show, by asymptotic matching between the (n,p) and (ν,T) scales, that Q¯ must be proportional to Q(0,0), and including the factor ρ-2/3 in (284) will lead to F being asymptotically O(1).Next we examine the boundary condition in (18), which can be also written as (X0+ρ-1)P(X0+ρ-1,Y)=P(X0,Y-ρ-1), or, using the (n,p) variables, as(286)X0+ρ-1Q-1,p=Q0,p+1.In view of (276) and (279) we can also write (286) as(287)X0+1X0ρ-1Q-1,p=Y0Y0+W0Q0,p+1or(288)X0+1X0ρ-1Q¯-1,p=Y0Y0+W0exp⁡aρ2p+1Q¯0,p+1=Y0Y0+W01-W0Y0W0+Y0ρ-1/3T1+Oρ-2/3Q¯0,p+1,where we used also (280) and (284).On the(ν1,T1) scale we have(289)Q¯-1,p=Q¯-ρ-1/3,T1=Q¯0,T1+Oρ-1/3,Q¯0,p+1=Q¯0,T1+ρ-2/3=Q¯0,T1+Oρ-2/3 so as long as X0≠Y0/(Y0+W0) we conclude from (288) that Q¯ must vanish along ν1=0 for all T1>0, and thus, in view of (284) so must F; hence(290)F0,T=0,T>0.The case where X0=Y0/(Y0+W0) corresponds to R3∩R4, and then the boundary condition will become more complicated. First we analyze the interior of R4 where (290) holds.To analyze (285) with (290), we first derive the behavior of F as ν,T→0, by matching the expansions on the (n,p) and (ν,T) scales. To this end we need, for region R4, the behavior of QL(n,p) for n,p→∞ with n=O(p). This asymptotic limit lies in the asymptotic matching region between (249) (which applies for 0<n/p<∞ for region R4) and (253), which applies for p→∞ but with n=O(1). Setting u=1/X0+v/p in the integral in (245) and shifting the contour Γ toward Re(u)=1/X0 we obtain(291)QLn,p~Q0,0X0nY0W0+Y0p2X03/2X0-1pW0+Y0X0+Y0-X012πi∫-i∞i∞vexp⁡-npX0v+X03/2W0+Y0v2dv.Here we used(292)u-n=1X0+vp-n~X0n/2exp-npX0vand approximated Γ by a vertical contour. Evaluating the integral in (291) explicitly we conclude that(293)QLn,p~Q0,02πX0nY0Y0+W0p1-X0W0+Y0X01/4X0-X0-Y0np3/2exp⁡-W0+Y04X0n2p;n,p⟶∞withn=Op.By asymptotic matching the expansion on the (ν1,T1) scale must behave as the right side of (293), when ν1,T1→0 with ν1/T1 fixed. But n=ρ1/3ν1, p=ρ2/3T1, and (283) shows that(294)n2p=ν12T1=X0W0+Y0ν2T,np3/2=1-X02/3X01/12ρ2/3W0+Y03/2νT3/2.Then comparing (293) with (276) (using (279) and (284)) we conclude that(295)Fν,T~ν2πT3/2exp⁡-ν24T;ν,T⟶0withν=OT.Note that (295) is consistent with the boundary condition in (290) along ν=0. The exponential factors in (279) and (284) do not enter the matching condition, since T1=O(1) and ρ1/3T12=ρ-1p2, and we can choose p so that p=o(ρ1/2) in the matching region.To solve (285) subject to (295) it is useful to view the function (πT)-1/2exp-ν2/(4T) as being an approximation to the delta function δ(ν) for T→0, with the mass concentrated in the range ν>0. Then the right side of (295) corresponds to the dipole -δ′(ν). Consider the problem(296)Fνν∗-νF∗=FT∗;ν>0,T>0,(297)F∗ν,0=δν-ν0,ν0>0,(298)F∗0,T=0,T>0.Introducing the Laplace transform(299)F^=F^ν,θ;ν0=∫0∞e-θTF∗ν,T;ν0dT,(296) becomes(300)F^νν-ν+θF^=-δν-ν0,ν>0and (298) implies that F^(0,θ)=0. The problem in (300) is a standard Green’s function problem with solution(301)F^=-πAiν0+θAiθAiν+θBiθ-Biν+θAiθ,0<ν<ν0,(302)F^=-πAiν+θAiθAiν0+θBiθ-Biν0+θAiθ,ν>ν0,where Ai(·) and Bi(·) are the Airy functions. Then the solution F to (285) will be(303)Fν,T=∂∂ν0Fν,T;ν0ν0=0.By differentiating (302) with respect to ν0, setting ν0=0, and noting the Wronskian identity AiθBi′(θ)-Ai′(θ)Bi(θ)=π-1 we find that(304)∂F^∂ν0ν0=0=Aiν+θAiθ,ν>0.Inverting the Laplace transform in (299) leads to F(ν,T) in (170), so we have established the expansion on the (ν1,T1) scale. We can also easily verify that (295) is indeed satisfied, since(305)Fν,T=12πiddT∫-i∞i∞Aiν+θAiθeθTdθ~12πiddT∫-i∞i∞eθT-νθθ1+Oν2θdθ~1πi∫-i∞i∞eω2T-ωνωdω=ν2πT3/2exp⁡-ν24T.Here we approximated the first integrand for θ→∞ using asymptotic properties of Ai(·).We next relate the constantC1 in D- to Q(0,0) (~1-X0-Y0 in R4), having expressed the expansions near the corner point (X0,Y0) in terms of Q(0,0) on both the (n,p) and (ν1,T1) scales. As we discussed previously, the caustic ray expansion in D- does not satisfy the boundary condition along X=X0, and different expansions must be constructed on the ν1 and n scales, corresponding to X0-X=O(ρ-2/3) and X0-X=O(ρ-1). For region R4, these expansions are given, respectively, in (111) and (109) (with (110)), up to the constant C1. The details of their construction are given in [10]. We will need to carefully estimate π(k,r) for n=m-k=O(1). For region R4, (109) applies for n=O(1) and all Y∈(0,Y0) (but breaks down as either Y↓0 or Y↑Y0). For n=O(1) and p=O(1), (241) holds as then π(k,r)~QL(n,p), and in (253) we gave the asymptotics for Q(n,p) for p→∞ and n=O(1). However, we must derive yet another expansion on the scale T1=O(1)(Y0-Y=O(ρ-1/3)) with T1>0 and n=O(1). The expansion in (169) develops a nonuniformity as ν→0, since F(0,T)=0. Since by definition of the Airy roots rj we have Ai(rj)=0, from the infinite series form in (170) we conclude that(306)Fν,T~ν∑j=0∞erjT=1-X0X01/3nρ1/3∑j=0∞e-rjT,ν⟶0,and this holds for any fixed T>0. Consider the scale n=O(1) and T1>0. Viewing Q in (277) now as a function of n and T1 and noting that p→p±1 corresponds to T1→T1∓ρ-2/3 the limiting form of (277) becomes(307)0=W0-1-X0Qn,T1+X0Qn-1,T1+Qn+1,T1.But W0=(1-X0)2=1+X0-2X0 so that the general solution of (307) is the linear function a(T1)+nb(T1), where a(·) and b(·) are arbitrary functions of T1. Note also that (307) applies asymptotically whether the exponential factors in (279) and (284) are included or excluded. The boundary condition in (287) can be applied on the n=O(1) scale and leads asymptotically to (W0+Y0)X0Q(-1,T1)=Y0Q(0,T1). This restricts the functions a(·) and b(·) in the linear solution to (307), by(308)Y0aT1=W0+Y0X0aT1-bT1.We have thus shown that the expansion on the (n,T1) scale is given by(309)πk,r~Q0,0ρ-11-X02W0+Y0X0X0-X0-Y0X0nY0Y0+W0pn+X0Y0+W0Y0+W0X0-Y0f∗Texp⁡-W02Y0W0+Y0ρ1/3T12-W0W0+2Y0Y0W0+Y02T13.Here we made a(·) proportional to f∗(·), which we will determine shortly, and computed b(·) from (308). Now we require (309), for n→∞, to match to the (ν1,T1) scale result, for ν1→0. But in view of (306) we see that the matching is possible provided that(310)f∗T=∑j=0∞e-rjT.Thus we now have three expansions valid for n=O(1), on the scales p=O(1), p=O(ρ2/3), and p=O(ρ) with Y<Y0. The matching between the p=O(1) result and (309) follows by letting p→∞ in (241) (corresponding to (253) for region R4) and letting T→0. The matching condition is satisfied if(311)12πp-3/2W0+Y01-X0X01/4X0-X0-Y0~1-X02ρ-1W0+Y0X0X0-X0-Y0f∗T,in some intermediate limit where p→∞ and T1=ρ-2/3p→0. Simplifying (311) by using (283) we must show that(312)f∗T~12πT-3/2,T⟶0.The Airy roots rj are well known to satisfy -rj~3jπ/22/3, j→∞, so for T→0 we approximate the sum in (310) by an integral to get(313)∑j=0∞e-rjT~∫0∞exp⁡-32ξπ2/3Tdξ=23πT-3/2∫0∞e-u32udu=1πT-3/2Γ32which verifies (312).Now we match (309) for T→∞ with (109) as Y→Y0. For T→∞ we have f∗(T)~e-|r0|T as the zeroth term in the sum in (310) dominates. By expanding (109) for Y→Y0, after canceling some common factors the matching implies that(314)ρ-1Q0,0W0W0+Y0X0X0-X0-Y0Y0Y0+W0perTexp⁡-W02Y0W0+Y0ρ1/3T12-W0W0+2Y0Y02W0+Y02T13~ρ-1/6C1ρeρΦX0,Y0eρΦYX0,Y0Y-Y0eρ1/3Φ∗Y0exp⁡12ρΦYYX0,Y0Y-Y02+16ρΦYYYX0,Y0Y-Y03+ρ1/3Φ∗′Y0Y-Y0exp⁡-1+X02X0log⁡Y0+W01-X012π1Ai′r0Y0+W0X0-Y0Y0X05/61-X01/6.Now,(315)eρΦYX0,Y0Y-Y0=e-pΦYX0,Y0=Y0Y0+W0p,expρ1/3Φ∗′Y0Y-Y0=exp⁡-T1Φ∗′Y0=exp⁡r0T11-X02/3W0+Y0X01/6=er0T. By using (110) to compute ΦYY and ΦYYY and noting that(316)ρY-Y02=ρ1/3T12,ρY-Y03=-T13 we see that the dependence on T1 is exactly the same in the left- and right-hand sides of (314). Thus we have a relation between the constants Q(0,0) and C1 for region R4, and since Q(0,0)~1-X0-Y0 we have derived the expression for C1 in (41).The analysis for regionR4 breaks down as Y0→X0-X0, as then the expressions in (284), (293), (309), and (314) all become singular. We thus examine the transitional case R3∩R4, where X0+Y0=X0+O(ρ-1/3). Then we certainly have Q(0,0)~1-X0-Y0 and (241) applies for any Rj. But now on parabolic scales, where n,p→∞ with n2/p fixed, Q(n,p) has the expansion in (254). We now define F¯(ν,T) by setting(317)πk,r~Q0,0ρ-1/31-X0X01/3X0nY0Y0+W0pexp⁡-W02Y0W0+Y0ρ1/3T12-W02Y0+W06Y02W0+Y02T13F¯ν,Tsince the factors in (317) correspond to the same sequence of transformations we made for the analysis of the interior of R4, and F¯ will again satisfy the PDE(318)F¯νν-ρF¯=F¯T;ν>0,T>0.Note that in (317) we included the factor ρ-1/3 whereas in (284) we have the two factors ρ-2/3 and X0-X0-Y0-1, as the product of these becomes, in region R3∩R4, O(ρ-2/3)×O(ρ1/3)=O(ρ-1/3). Including the factor ρ-1/3 will insure that F¯ is O(1) for ρ→∞, and the factor 1-X01/3X0-1/3 is purely for convenience. But for region R3∩R4 we must reexamine the boundary condition in (288), which in terms of F¯ becomes(319)X0+Oρ-1F¯-ρ1/3,T1=Y0Y0+W01-W0Y0W0+Y0ρ-1/3T1+Oρ-2/3F¯0,T1+ρ-2/3.But now (320)Y0+W0X0-Y0=1-X0X0-X0-Y0=-1-X0δ∗ρ-1/3 so that the limiting boundary condition from (319) becomes(321)X0∂F¯∂ν1+δ∗-T1F¯=0,ν1=0.Then scaling δ∗=X01/6(1-X0)1/3δ1, in terms of (ν,T) (321) becomes(322)F¯ν0,T+δ1-TF¯0,T=0,T>0.A matching condition for F¯, as ν,T→0 with ν=O(T), is obtained by comparing (317) to (254). From the definitions of δ and δ∗ we have δρ-1/2=δ∗ρ-1/3 so that δ=δ∗ρ1/6. Then also(323)pδ2ρ=T1δ∗2,nδρ=ν1δ∗,n2p=ν12T1are all O(1), while p-1/2=ρ-1/3T1-1/2 and δρ-1/2=δ∗ρ-1/3, so in the matching region, where T1 is small but δ∗=O(1), the Gaussian first term in the braces in (254) dominates the error function second term, and the matching will imply that(324)1-X0X01/3F¯ν,T~1π1-X01/2X01/4T1exp⁡-1-X04X0ν12T1and in terms of (ν,T) this simplifies to(325)F¯ν,T~1πTexp⁡-ν24T;ν,T⟶0withν=OT.Here we also used Y0+W0~1-X0 in region R3∩R4, so that T1~(1-X0)1/3X01/6T.The matching condition in (325) may be replaced by the condition F¯T=0=δ(ν) and we analyzed such problems in [20] where it was found that the solution to (318) with (322) is given by(326)F¯ν,T=12πi∫-i∞i∞Aiν+θAi2θeθT-δ1∫θ∞eδ1uAiududθ.The right-hand side of (326) clearly satisfies (318), since this PDE is separable and any function of the form g(θ)Ai(ν+θ)eθT is a solution, for any function g(·) and any complex θ. We can also verify that (322) holds along ν=0, by using(327)Ai′θAiθ2eθT-δ1+δ1-TAiθeθT-δ1=-ddθeθT-δ1Aiθ and some integration by parts in (326). To verify the matching condition in (325) we expand the integrand in (326) for ν→0, T→0, and θ→∞, using (190) to approximate the Airy function. We have Ai(ν+θ)/Ai(θ)~e-νθ and(328)e-θδ1∫θ∞eδ1uAiudu=∫0∞eδ1ξAiθ+ξdξ~e-2/3θ3/214π∫0∞exp⁡-θξ+Oξ2θ1+δ1ξ+Oξ2θ-1/41-ξ4θ+Oξ2θ2dξ=e-2/3θ3/214πθ-1/41θ+δ1θ+oθ-1=Aiθ1θ+δ1θ+oθ-1.Here we also expanded the integrand for small ξ, since by Watson’s lemma the main contribution comes from the range ξ=O(θ-1/2). Using (328) in (326) and noting that(329)12πi∫-i∞i∞eθT-θνdθθ=1πTexp⁡-ν24T,12πi∫-i∞i∞eθT-θνdθθ=12π∫ν/2T∞e-u2/2du we obtain, for ν,T→0 with ν=O(T),(330)F¯ν,T=1πTexp⁡-ν24T+2πδ1∫ν/2T∞e-u2/2du+o1.The expression in (330) is a two-term asymptotic approximation in this limit, and the leading term verifies the matching condition in (325). The correction term in (330), which is proportional to δ1, will asymptotically match to the term proportional to the error function in (254), as in the matching region pδ2/ρ=T1δ∗2 and nδ/ρ=ν1δ∗ are o(1), and(331)η0~1-X01/22X01/4ν1T1~ν2T,2X0δρ=ρ-1/31-X0X01/32δ1.We proceed to determineC1. Now the approximation in (209) (with (326)) remains valid on the scale n=O(1), since F¯(0,T)≠0. Thus for n=O(1)π(k,r) can be approximated by (317) with F¯(ν,T) replaced by F¯(0,T) and for T→∞ the expansion of (326) is determined by the pole at θ=r0<0, which is a simple pole if ν=0. Hence,(332)F¯0,T~er0T-δ1Ai′r0∫r0∞eδ1uAiudu,T⟶∞.For n=O(1) and 0<Y<Y0, the result in (109) holds, and this remains finite as Y↑Y0 even if X0+Y0=X0. In parameter region R3∩R4 we have Y0+W0~1-X0 and then as Y↑Y0, (109) becomes(333)ρ-1/6C1ρX0neρΦX0,Y0e-pΦYX0,Y0eρ1/3Φ∗Y0er0Texp⁡12ρ1/3ΦYYX0,Y0T12-16ΦYYYX0,Y0T1312π1Ai′r01-X01/3X0-7/12.Comparing (333) to (317) with (332), the exponential parts agree as for region R4, and we now have(334)ρ1/3Φ∗Y0=-ρ1/3r01-X02/3X01/6log⁡Y0+W01-X0=-ρ1/3r01-X02/3X01/6log⁡1+ρ-1/3δ∗1-X0~-r01-X0-1/3X0-1/6δ∗=-r0δ1.Thus (333) agrees with (317), with ν=0 and T→∞, if(335)Q0,0ρ-1/3X0-1/3∫r0∞eδ1uAiudu~C1ρ2πρ-1/6X0-7/12eρΦX0,Y0.But now Q(0,0)~1-X0-Y0~1-X0 and thus (335) leads to the result in (37).We have now determined the values ofQ(0,0), C0(ρ), C(ρ), and C1(ρ) for all possible regions Rj of parameter space, thus establishing Propositions 1 and 2. We also have analyzed all of the relevant scales near the corner point (X0,Y0) of the state space and thus established all parts of Proposition 15. As a final step we briefly discuss the asymptotic matching between the ray expansion in D0 and the expansion on the (ν,T) scale in (169). We only consider region R4 and note that this matching will involve letting (X,Y)→(X0,Y0) along parabolic scales with (Y0-Y)2/(X0-X)(=T12/ν1) held fixed and letting ν1,T1→∞ in (169). We have already verified that along linear scales, where (X,Y)→(X0,Y0) with (Y0-Y)/(X0-X) (=p/n) fixed, the D0 ray expansion matches directly to (168). The matching of the D0 and (ν,T) scale expansions will show that there are no “gaps” in the asymptotics, which would require the analysis of yet other scales.We first expandF(ν,T) in (170) for ν,T→∞, and we note that, along parabolic scales with ν=O(T2), ν/T2>1 corresponds to moving from the corner (X0,Y0) in region D0, while ν/T2<1 corresponds to moving into D-. By using (190) to approximate the integrand in (170) and shifting the contour to the right, to the range Re(θ)≫1, we are led to(336)Fν,T~12πi∫Brθν+θ1/4eθTexp⁡23θ3/2-θ+ν3/2dθ.Here we can view θ as being scaled to be O(ν) and T to be O(ν). The integrand in (336) has a saddle point where(337)ddθθT+23θ3/2-23θ+ν3/2=T+θ-θ+ν=0and this occurs when(338)θ=θs≡ν-T224T2=ν24T2-ν2+T24.Taking ν/T2>1 then a standard saddle point estimate leads to(339)F~12π12θs-12θs+ν-1/2θsθs+ν1/4exp⁡θsT+23θs3/2-θs+ν3/2=ν-T22πT3/2exp⁡112T3-12νT-14ν2T;ν,T⟶∞.By using (339) in (169) we match the result to the D0 ray expansion, which is ρ-1/2Q(0,0)K(X,Y)eρΨ(X,Y), as (X,Y)→(X0,Y0). By separating the exponential factors from the algebraic ones, the matching will hold if both(340)ρ-1/2KX,Y~ρ-2/31-X05/3X0-1/62πW0+Y0X0-X0-Y0ν-T2T3/2,(341)eρΨX,Y~X0nY0Y0+W0pexp⁡-ρ1/3W02Y0W0+Y0T12exp⁡-W0W0+2Y06Y02W0+Y02T13+112T3-12νT-14ν2T.We thus proceed to evaluate K and Ψ as (X,Y)→(X0,Y0) along parabolic scales, where (Y0-Y)2/(X0-X) is fixed. In this limit we have A→Amin=1-X0 and B→Bmin=W0/(Y0+W0), and it is useful to rewrite (68) as(342)X0-X=1-e-τX01-A+Aeτ-1,Y0-Y=Y01-B1-e-τ.The function K0(A) in (275) vanishes in this limit, in view of the factor (1-A)2-X0, and we have(343)K0A~2Y01-X0A-AminπX0-Y0-X0W0+Y0,A⟶1-X0.Expanding the Jacobian Δ in (71) for τ→0 and A→Amin leads to(344)Δ~-2τW0+Y0;τ⟶0,A⟶1-X0.But from (342) we have Y0-Y=ρ-1/3T1~(W0+Y0)τ so that(345)Δ-1/2~ρ1/612T1-X01/3X01/12Y0+W0,where we wrote the result in terms of T rather than T1. From the first expression in (342) we have(346)X0-X~τ2A-Amin+Aminτ so that(347)A-Amin~12τX0-X-1-X0τ2~12W0+Y0Y0-YX0-X-1-X0Y0-Y2W0+Y2=12ρ-1/3W0+Y0T1ν1-1-X0W0+Y02T12=12ρ-1/3X01/61-X01/3ν-T2T.Now (348)KX,Y=e-τ-A-1/2e-τ-B-1/2Δ-1/2K0A~X0-1/4Y0+W01/2Y0-1/2Δ-1/2K0A, so using (347) and (345) we can easily verify that (340) is satisfied.To verify (341) we first note that Ψ(X0,Y0)=0,(349)exp⁡ρX-X0ΨXX0,Y0=exp⁡ρX0-Xlog⁡1-Amin=X0n,exp⁡ρY-Y0ΨYX0,Y0=Y0Y0+W0p. Then peeling off the linear part of Ψ near X=X0,Y=Y0, (341) is equivalent to showing that(350)ρΨX,Y+X0-XΨXX0,Y0+Y0-YΨYX0,Y0~-ρ1/3W02Y0W0+Y0T12-W0W0+2Y06Y02W0+Y02T13+112T3-12νT-14ν2T.We can rewrite Ψ in (66) as(351)ΨX,Y+X0-XΨXX0,Y0+Y0-YΨYX0,Y0=X0-Xlog⁡1-Aeτ1-Amin+Y0-Ylog⁡1-Beτ1-Bmin-X0log⁡1-Aeτ1-A-Y0log⁡1-Beτ1-B+A1-eτ,where we have partially expressed the right-hand side of (351) in terms of X0-X=ρ-2/3ν1 and Y0-Y=ρ-1/3T1. Next we expand (351) as a triple Taylor series about τ=0, A=Amin, and B=Bmin. We have(352)A1-eτ-X0log⁡1-Aeτ-11-A-Y0log⁡1-Beτ-11-B=X0A1-A+Y0B1-B-Aeτ-1+X0A221-A2+Y0B221-B2eτ-12+X0A331-A3+Y0B331-B3eτ-13+Oτ4.But the relation in (216) implies that the coefficient of (eτ-1) in (352) vanishes and also that(353)B-Bmin~-1X0Y0Y0+W02A-Amin2.By (347), on the (ν,T) scale, A-Amin is O(ρ-1/3) so that B-Bmin=O(ρ-2/3), and thus (352) may be further approximated by (using (eτ-1)2=τ2+τ3+O(τ4))(354)X0Amin221-Amin2+Y0Bmin221-Bmin2+X0Amin331-Amin3+Y0Bmin331-Bmin3τ3+X0Amin221-Amin2+Y0Bmin221-Bmin2τ2+X0Amin1-Amin2+X0Amin21-Amin3A-Aminτ2with an O(τ4) error, which is not needed for the matching verification since exp[O(ρτ4)]=exp[O(ρ-1/3)]~1. We also have(355)X0-Xlog⁡1-Aeτ1-Amin=ρ-2/3ν1log⁡1-A-Amin1-Amin-Aminτ1-Amin+Oτ2,τA-Amin~-ρ-2/3ν1X0A-Amin+Aminτ~-ρ-1νX01/61-X01/312X01/61-X01/3ν-T2T+1-X0W0+Y0T1=-ν2ρν+T2T,Y0-Ylog⁡1-Beτ1-Bmin=Y0-Ylog⁡1-B-Bmin1-Bmin-Bmineτ-11-Bmin+OB-Bmin2=ρ-1/3T1-B-Bmin1-Bmin-Bminτ1-Bmin-Bminτ221-Bmin2+Oτ3.From (342) we have (356)Y0-Y1-B=ρ-1/3T11-Bmin+Oρ-2/3=Y01-e-τ=Y0τ-12τ2+Oτ3 so we can refine the estimate above (345) to(357)τ=ρ-1/3T1W0+Y0+12ρ-2/3T12W0+Y02+Oρ-1.Adding the expressions in (354) and (355) and using the estimates in (347), (353), and (357), (351) becomes, when multiplied by ρ,(358)1-X022+W022Y0+1-X033X0+W033Y02T13W0+Y03+1-X0+1-X02X0X01/61-X01/32W0+Y02T12ν-T2T+1-X022+W022Y0ρ1/3T12W0+Y02+T13W0+Y03-ν22T-νT2+W0+Y0Y0T1Y0W0+Y021X0X01/31-X02/34νT-T2-W0W0+Y0wY02T1T1W0+Y02-W0Y0ρ1/3T12W0+Y0+T132W0+Y02.Comparing (358) to (350) we find, after some simplification, that they are identical. The comparison is facilitated by separately comparing terms proportional to ρ1/3T12, ν2/T, νT, and T3. This verifies the matching between the D0 ray expansion and that on the (ν,T) scale. The Airy functions that arose on the (ν,T) scale disappear in the limit where ν,T→∞. If we let ν,T→∞ with ν/T2<1, then the expansion of F(ν,T) is much different, and now we have(359)Fν,T~12πAi′r01ν1/4exp⁡-23ν3/2+r0T-ν,as the pole at θ=r0 determines the asymptotic behavior. We can show that then (169) with (359) agrees with the ray expansion in D-, that is, C1L(X,Y)eρΦ(X,Y)eρ1/3Φ∗(s1), as the latter is expanded for (X,Y)→(X0,Y0), for parameter region R4. The case where ν/T2≈1 will be important in determining the transition layer expansion that applies where D0 meets D-, which is along the curve Y=Yc(X). Then neither (339) nor (359) apply.


## 5. Asymptotic Expansions near State Space Boundaries

We discuss the four boundary segments,X=0, X=X0, Y=0, and Y=Y0, of the state space rectangle, avoiding for now the corner points. From Figures 3–6 we see that D+ will border X=0 and X=X0 for parameter regions R2 and R3 and also border Y=0 for region R1. As we previously discussed the ray expansion in D+ remains valid near X=X0 but breaks down near Y=0(R1) and X=0(R2∪R3). The analysis of Y=0 (corresponding to the scales r=O(1) and r=O(ρ)) is identical to that of the infinite capacity model, and we show in [10] how to construct appropriate boundary layer corrections; this leads to (88)–(95) in Proposition 8. The analysis near X=0 (with 0<Y<(X0+Y0)2-X0/(X0+Y0)2) is also very similar to that in [10], except that if X0+Y0<1(R2∪R3) we must use the values of C(ρ) in (32) and (33). Thus both the D+ ray expansion and the boundary layer expansion, which applies for k=O(1), are rescaled by a constant. We thus obtain (114).For parameter region(s)R2∪R3∪R4, D- meets the line X=X0, and there the expansions in (109)–(112) apply. Their derivation for R2 is identical to that in [10], while for the other subcases we must simply multiply both the D- ray expansion and the boundary layer corrections by the appropriate C1(ρ), from among (35), (36), (37), and (41). Similarly, the boundary layer correction along Y=0 is given by (96), with the appropriate value of C1(ρ). For R2∪R3 this covers the entire range 0<X<X0, but for R4 only the range XL<X<X0 (with XL defined in item (vii) in Proposition 8), as in the range 0<X<XL, D0 meets Y=0.Here we will only examine whereD0 meets the state space boundaries, and this occurs along Y=Y0 for all regions Rj, along X=0 with (X0+Y0)2-X0/(X0+Y0)2<Y<Y0 for R1∪R2∪R3, and all 0<Y<Y0 for R4, and along Y=0 with 0<X<XL for R4 only. We thus proceed to construct boundary layer corrections to the D0 ray expansion for these three boundary segments.


### 5.1. The Boundary SegmentX=0

We consider the scalek=O(1) and let(360)πk,r=C2ρeρΨ0,YρkΠkY;ρ.Then the main balance equation (3), after dividing by exp[ρΨ(0,Y)], becomes(361)ρ1+Y+kΠkY;ρ=ρY+1ΠkY+ρ-1;ρexpρΨ0,Y+ρ-1-Ψ0,Y+ρk+1Πk+1Y;ρ+Πk-1Y;ρ.Taking Πk(Y) to be the leading order approximation to Πk(Y;ρ), dividing (361) by ρ, and letting ρ→∞ lead to the limiting equation(362)1+Y-YeΨY0,YΠkY=k+1Πk+1Y,k⩾1.By examining the boundary condition in (6) we conclude that (362) holds also if k=0, and thus(363)ΠkY=Π0Yk!1+Y-YeΨY0,Yk.To determineC2(ρ) and Π0(Y) we asymptotically match (360) to the D0 ray expansion. We thus expand ρ-1/2Q(0,0)K(X,Y)eρΨ(X,Y) as X→0 and compare the result to the large k expansion of (360). By using Stirling’s formula we see that the matching is possible if, as X→0,(364)ΨX,Y=Ψ0,Y-Xlog⁡X+X1+log⁡1+Y-YeΨY0,Y+oX,(365)Q0,0ρKX,Y~C2ρ2πρXΠ0Y,X⟶0.Note that the exponential factor in (360) must be included for the matching to be possible. Thus we could set C2(ρ)=Q(0,0) and then Π0(Y) will be the limit of 2πXK(X,Y) as X→0, which we show below to be finite and nonzero.From (68) we see that X=0 corresponds to A=e-τ and we set A1(Y)=A(0,Y), where we view A as a function of (X,Y) via the mapping in (68). Then we also set B1(Y)=B(0,Y) and from (68) find that(366)A1-B1=Y1-A1Y0-Y.Using (366) in (67) leads to a quadratic equation for A1(Y), whose solution is given by (118). Also, replacing (eτ,A,B) by (1/A1,A1,B1) in (66) leads to the expression in (119) for Ψ(0,Y), which is an explicit function of Y. As X→0, X/(e-τ-A) is finite and from (68) we obtain(367)e-τ-A~A11-A1A1A1+X0-2+1X,X⟶0.Evaluating the Jacobian Δ in (71) along X=0 leads to ΔL(Y) in (120). Thus as X→0 we have(368)e-τ-A-1/2e-τ-B-1/2Δ-1/2K0A~A1-B1-1/2ΔL-1/2K0A1A1A1+X0-2+1A11-A11X.We have thus identifiedΠ0(Y) as(369)Π0Y=A1-B1-1/2ΔL-1/22πK0A1A1A1+X0-2+1A11-A1.Using (366) and K0(·) in (275) we see that (360) agrees with (117) as X→0, if(370)1+Y-YeΨY0,Y=A1A1+X0-2+11-A1and Ψ has the small X behavior indicated in (364). Now, ΨY=-log(1-Beτ) so that ΨY(0,Y)=-log(1-B1/A1) and using (366) leads to(371)1+Y-YeΨY0,Y=1+Y-YA1A1-B1=1+Y-Y0-Y1-A1A1.But from (118) we see that A12+(X0+Y0-1)A1-Y=0 so that the right-hand sides of (370) and (371) agree.Using (68) we have, as X→0,(372)ΨX+log⁡X=-log⁡1-Aeτ+log⁡e-τ-AX01-A-1+eτ~log⁡X01-A-1e-τ+1~log⁡X0A11-A1+1-A1.Integrating the above, noting thatΨX+log⁡X is an analytic function of X, we conclude that(373)ΨX,Y+Xlog⁡X-X=Xlog⁡A12+X0-2A1+11-A1OX2 and this verifies (364). We have thus shown that the asymptotic matching holds and also determined Π0(Y). This completes the derivation of (117)–(120).For regionR4, (117) holds for all 0<Y<Y0, and then the expansion matches to the corner expansions that apply for Y=O(ρ-1) (r=O(1)) and Y=Y0-O(ρ-1) (p=R-r=O(1)). But for region R1∪R2∪R3 and k=O(1), (117) holds only for YU<Y<Y0, while (114) applies for 0<Y<YU. To connect these it is necessary to construct another expansion for Y-YU=O(ρ-1/2). Using the variables k and y∗≡ρ(Y-YU)=O(1) we conclude from (3) and (6) that(374)πk,r~C3ρeρΨ0,YU+ρ-1/2y∗ρkk!1+Y0-Y0X0+Y0kΠ∗y∗,where we note that 1+Y-YeΨY(0,Y)=1+Y0-Y0/(X0+Y0) when Y=YU, since A1(YU)=Y0/(X0+Y0). To determine Π∗(·) we first infer its behaviors as y∗→±∞ by asymptotically matching (374) to the expansions that apply for Y<YU and Y>YU.We expand the result in (114) as Y↑YU. We have D0(YU)=D0(Y~(0))=Y0/(X0+Y0) so that(375)1+YU-D02YU=A1YUA1YU+X0-2+11-A1YU=Y0+X0X0+Y0,Ψ+,Y0,YU=ΨY0,YU=-log⁡X0+Y02-X0Y0.After some calculation we find that(376)1-D0YUD0YUYUX0YU+2D02YU1-D0YU-1/2=X0Y01-X0X0+Y02X0+Y03/2X0+Y03+X0Y0-X0.Furthermore,(377)Ψ0,YU=Ψ+0,YU+1-X0-X0+X0+Y0log⁡X0+Y0and we note that, in view of Propositions 1 and 2,(378)Cρ~2πX0+Y0Q0,0eρΨ0,YU-Ψ+0,YUwhich holds in regions R1∪R2∪R3. Using (375)–(378) to infer the behavior of (114) as Y↑YU and comparing the result to (374) we can take C3(ρ)=2πρQ(0,0) and then, as y∗→-∞,(379)Π∗y∗exp⁡12ΨYY0,YUy∗2~X0+Y02-X0X0+Y03+X0Y0-X0X0Y0exp⁡12Ψ+,YY0,YUy∗2.Next we expand (117) as Y↓YU. Since (X0+Y0)A1(YU)=Y0, (117) is singular in this limit, and we use(380)Y0-A1YX0+Y0~-A1′YUX0+Y0Y-YU=-A1′YUX0+Y0y∗ρ-1/2 to obtain(381)Y0YY0-YA1A1+X0-2+11-A12-X0A1ΔLY1/2Y0-A1X0+Y0~ρy∗A1YUA1YU+X0-2+11-A1YU2-X0A1YUΔLYU1/2-X0+Y0A1′YU=ρy∗X0X0+Y05/2X0+Y02-X0X0+Y02+Y0-X0=ρy∗X0Y0X0+Y02-X0X0+Y03+X0Y0-X0-1/21ac,where ac was defined in (125). To obtain (381) we also used(382)A1′YU=X0+Y0X0+Y02+Y0-X0,ΔLYU=X0+Y02+Y0-X0X0+Y02X0+Y0Y0+X0,which follow from (118) and (120). Denoting the ratio of the left- and right-hand sides of (379) by f(y∗), the matching conditions as y∗→±∞ may be written as(383)fy∗⟶1,y∗⟶-∞fy∗~12πacy∗exp⁡y∗22Ψ+,YY0,YU-ΨYY0,YU,y∗⟶∞.After some calculation we can show that(384)Ψ+,YY0,YU-ac2=ΨYY0,YUso that the error function(385)fy∗=12π∫acy∗∞e-u2/2dusatisfies the conditions in (383). To determine f(y∗) completely we need to match (374), with C3=2πρQ(0,0), to the expansion in the transition layer where D+ and D0 meet, which is given by (176). This matching has X→0 in (176) and k→∞ in (374), with η∗ fixed. Noting that as X→0(386)η∗=ρY-Y~X=ρY-YU+OX=y∗+Oρ-1/2 we can show, using (178) and (184)–(187), that b(t∗)→ac-2 as X→0 and thus (385) determines f(y∗) for all y∗=O(1).


### 5.2. The Boundary SegmentY=0

We considerr=O(1), in ranges where D0 meets the x-axis, which can occur only for R4. The analysis for where D+ (region R1) or D- (regions R2 and R3) meet the x-axis is very similar to that in [10] so we omit it. By examining (3), noting that this equation holds even if r=0, on the scale r=O(1) with k=ρX, X>0, we find that(387)πk,r~C4ρρrΠ~rXeρΨX,0,where Π~r(X) will satisfy(388)1+X-e-ΨXX,0-XeΨXX,0Π~rX=r+1Π~r+1X.Noting that ΨX(X,0)=-log(1-Aeτ), we define A0(·) by A0(X)=A(X,0). A ray in D0 reaches the x-axis when e-τ=B so we also let B0(X)=B(X,0). Then evaluating (197) along Y=0 leads to(389)1+X-XeΨXX,0-e-ΨXX,0=limY→0⁡YeΨY=lime-τ→B⁡Y0B-e-τB-1exp⁡-log⁡1-Beτ=Y0B01-B0so that the solution to (388) is(390)Π~rX=Π~0X1r!Y0B0Y1-B0Yr.We proceed to determine the constant C4(ρ) and function Π~0(·) by asymptotically matching (387) to the D0 ray expansion.First we note that settinge-τ=B0 in (66) and replacing (A,B) by (A0,B0) lead to the expression in (99) for Ψ(X,0). As Y→0 we also have, in view of (389),(391)ΨYX,Y=-log⁡Y+log⁡Y0B01-B0+o1.By integrating (391) we see that the D0 ray expansion behaves for Y→0 as(392)Q0,0ρKX,Ye-ρYlogYeρYexp⁡ρYlog⁡Y0B01-B0=ρrrrerY0B01-B0r1-X0-Y0ρKX,Y.Here we used Q(0,0)~1-X0-Y0 (in R4) and ρY=r. Expanding r! in (390) by Stirling’s formula we see the matching is possible if C4=1-X0-Y0 and(393)KX,Y~12πYΠ~0X,Y⟶0.We denote by g(X) the limit of [2πYK(X,Y)] as Y→0. Writing(394)KX,Y=e-τ-B-1/2e-τ-A-1/2Δ-1/2K0A we simply replace A by A0(X) in the factors that are not singular and use(395)e-τ-B-1/2=YY01-B-1/2~1YY01-B0. Then g(X) becomes precisely the expression in (102). We have thus determined Π~0(X) as g(X) and thus derived (98)–(102).The approximation in (387) applies only for 0<X<XL, and (96) applies for XL<X<X0. But for X≈XL a new approximation is needed, as g(X) vanishes as X↑XL, due to the factor (1-A0)2-X0, since 1-A0(XL)=X0. Since B0(XL)=W0/(W0+Y0) we find from (102) that(396)gX~W0+Y02W0+Y01-X0X0X0-1-A02X0-X0-Y03/2,X⟶XL.By implicit differentiation of (100) we find that(397)A0′XL=-W0W0+Y0Y02W0+Y0.Using(398)X0-1-A02~21-A0A0′XLX-XL=2X0A0′XLX-XLand (397) we obtain from (396) the expression in (104), which will be used for asymptotic matching verifications.We consider the scaleX-XL=ρ-1/3Λ=O(ρ-1/3), retaining r=O(1). From the balance equations we can conclude that the expansion has the form(399)C5ρeρΨX,0Y0B0XL1-B0XLrρrr!gXL~C5ρeρΨXL,0ρrW0rr!exp⁡ρ2/3ΨXXL,0Λ+12ρ1/3ΨXXXL,0Λ2Π^Λ,where the last exponential factor follows from expanding Ψ(X,0) about X=XL, and these are necessary to have a chance of matching to the expansion for X<XL. By comparing (399) to the behavior of (98) as X↑XL we immediately conclude that C5(ρ)=O(ρ-1/3) so we set C5(ρ)=ρ-1/3(1-X0-Y0), and then(400)Π^Λ~exp⁡16ΨXXXXL,0Λ32W0+Y03/2W05/4Y02W0+Y03/2X0-X0-Y03/2-Λ,Λ⟶-∞.From (101) we find that(401)Y01-B02B0′X=1-X01-A02A0′Xso that B0′(XL)=0. Then from (99) we find that(402)ΨXXX,0=1B0-1B0-A0B0′+A0′B0-A0and thus ΨXX(XL,0)=A0′(XL)/[B0(XL)-1+X0] is as in (107). From (402) we also conclude that(403)ΨXXXXL,0=1B0XL-1B0XL-A0XLB0′′XL+A0′′XLB0XL-A0XL+A0′XLB0XL-A0XL2.By further differentiation of (100) and (401) we find that(404)B0′′XL=-2Y0X0W022W0+Y02,A0′′XL=-2W03/2Y0+W0Y0X02W0+Y02-2W03W0+Y02X0Y022W0+Y03,and thus(405)ΨXXXXL,0=-2X0W05/2W0+Y03Y022W0+Y031X0-X0-Y0+W0W0+Y04Y022W0+Y021X0-X0-Y02.We next derive a matching condition forΠ^(Λ) as Λ→∞. We first note from (97) that if we write s10+W0=W0f(X) then(406)fX=12W0+1-X+W0+1-X2-4W0,(407)fXL=W0+Y01-X0=W0+Y0W0⟹s10XL=Y0.From (406) we then obtain(408)f′XL=-Y0+W02Y0Y0+2W0,f′′XL=-2W05/2Y0+W03Y03Y0+2W03.From (80) with Y=0 we have ΦX(X,0)=-log[1-f(X)] from which we can show that(409)ΦXXL,0=ΨXXL,0,ΦXXXL,0=ΨXXXL,0,(410)ΦXXXXL,0=W0W0+Y03Y03+3W0Y02+4W02Y0-2X0W05/2Y032W0+Y03X0-X0-Y02.By using the expression for C1 in (41) we have(411)ΦXL,0-ΦX0,Y0=ΨXL,0,ρ1/3Φ∗s10-Φ∗Y0=-ρ1/3r01-X02/3X01/6log⁡s10+W0Y0+W0=-ρ1/3r01-X02/3X01/6log⁡fXfXL~-r0Λ1-X02/3X01/6f′XLfXL=r0ΛY0+W0Y0Y0+2W01-X05/3X01/6,ρC11-X01/6X01/6Ai′r02W0+s10s10s10+2W0W0s10+W01/2X0X0-X0-s10~ρ-1/31-X0-Y0Ai′r02πW0+Y0Y0+2W01-X05/6X01/6X0-X0-Y03/2exp⁡-ρΦX0,Y0-ρ1/3Φ∗Y0.With (409)–(411) we have obtained the behavior of (96) as X↓XL and thus derived the matching condition(412)Π^Λ~exp⁡16ΦXXXXL,0Λ3+r0ΛY0+W0Y0Y0+2W01-X05/3X01/61Ai′r0W0+Y0Y0+2W01-X05/6X01/6X0-X0-Y03/2,Λ⟶+∞.Thus (400) and (412) yield the behavior of Π^(Λ) as Λ→±∞, but to determine this function for all Λ we must use a third matching condition, to the transition layer in Proposition 18, which applies for Y-Yc(X)=ρ-1/3η∗∗=O(ρ-1/3). Thus we let X→XL in (191) and let r→∞ in (399), in such a way that Λ is fixed. Since Λ=ρ1/3(X-XL) this means that(413)η∗∗=ρ1/3Y-YcX~ρ1/3rρ-Yc′XLX-XL~-Yc′XLΛremains also O(1). From (86) we have(414)Yc′XL=W03/2Y0+W0Y0Y0+2W0.We can expand ρΦX,Yc+ρ-1/3η∗∗-Φ(X0,Y0) in (191) as X→XL which is equivalent to expanding ρΦ(X,Y)-Φ(X0,Y0) for (X,Y)→(XL,0) and show that the exponential factors in (191) agree with those in (399), after r! is approximated by Stirling’s formula, so that(415)ρrW0rr!~2π-1/2r-1/2exp⁡-ρYlog⁡Y+ρY1+log⁡W0. Then using C5=ρ-1/3(1-X0-Y0) we compare algebraic factors in (399) and (191), which yields(416)12π1-X0-Y0ρ-1/3ρY-1/2Π^Λ~1-X0-Y0ρ-5/612π1-X05/6X01/6X0-X0-Y03/2W0+Y02W0+Y01YC′XLX-XLexp⁡16Λ3ΦXXXXL,012πi∫-i∞i∞e⋆θAiθdθ,where(417)⋆=ΛY0+W0Y0Y0+2W01-X05/3X01/6.In this limit Y~Yc′(XL)(X-XL) and Λ is fixed, so that (416) determines Π^(·) as(418)Π^Λ=1-X05/6X01/6X0-X0-Y03/2W0+Y02W0+Y0exp16Λ3ΦXXXXL,012πi∫-i∞i∞e⋆θAiθdθ.We have thus derived the result in (105). By using the asymptotic results in (192) and (193) (with ⋆ replaced by -⋆) we see immediately that (412) holds. The matching condition in (400) will also hold since(419)16ΦXXXXL,0+13⋆Λ3=16ΨXXXXL,0, by (417), (405), and (410). This completes our analysis of the scale r=O(1).


### 5.3. The Boundary SegmentY=Y0

We consider the scalep=ρ(Y0-Y)=O(1) and 0<X<X0. Now the analysis will be the same for any region Rj, and the expansion we construct will hold everywhere except near the corner points (X,Y)=(0,Y0) and (X0,Y0). By expanding (3) and the boundary condition in (5) along r=R, we find that an asymptotic solution in this range is given by(420)πk,r~C6ρρpp!Y0eτU-1peρΨX,Y0g¯X,where τU=τU(X) is given in (128). Again, (420) must contain the factor exp[ρΨ(X,Y0)] in order to have a chance of matching to the D0 ray expansion, and this factor determines the geometric factor in p. We must only determine C6(ρ) and g¯(X) by asymptotic matching.AsY→Y0 we have A→Amax and B→-∞ in (68). Also,(421)ΨY=-log⁡-B-τ+o1,Y⟶Y0and from (68) we obtain(422)-B~Y01-e-τUY0-Y,where τU=τU(X) is obtained by solving the first equation in (68) with A replaced by Amax. Using (421) and (422) we conclude that(423)ΨX,Y=ΨX,Y0+Y-Y0log⁡Y0-Y+Y0-Y+Y0-Ylog⁡Y0eτU-1+oY0-Y,Y⟶Y0.Expanding p! by Stirling’s formula and comparing the result to(424)Q0,0ρ-1/2KX,YeρΨX,Y as Y→YU we see that we can take C6=Q(0,0)~1-X0-Y0 and the matching will hold if(425)12πpg¯X=ρ-1/212πY0-Yg¯X~ρ-1/2KX,Y,Y⟶Y0.To expand (70) as Y→Y0 we note that in this limit e-τ-A-1/2 and Δ-1/2 are finite, while e-τ-B-1/2~(-B)-1/2 vanishes and K0(A) is singular, due to the factor Y0+(1-X0-Y0)A-A2 in the denominator. We thus have(426)KX,Y~1-Be-τU-Amax-1/2ΔUX-1/2Y02π1-Amax2-X01-X0-Y0-2Amax-1A-Amax,Y⟶Y0.Here ΔU(X) is the Jacobian in (71) evaluated along Y=Y0, and we also used the identity Amax(Amax-1)=Y0-Amax(X0+Y0). From (67) we see that(427)B~Amax1-Amax-X01-X0-Y0-2Amax1A-Amaxso that with (422) we have, as Y↑Y0,(428)KX,Y~12π1Y0-Y1-e-τUe-τU-AmaxX0-1-Amax21-AmaxΔU-1/2.Using (428) in (425) we can identify g¯(X) and then (420) becomes the same as (127).Finally we note that asX↑X0, (127) will asymptotically match to (168), the approximation valid for n,p=O(1). For n→∞ with p=O(1), (168) can be approximated by (247). As X→X0 we have τU(X)→0, Ψ(X0,Y0)=0, ΨX(X0,Y0)=-log(1-Amax), and z+(0)(1-Amax)=1. Then(429)Y0eτUX-1~Y0τU′X0X-X0=-nρY0τU′X0=nρY01+X0+Y02-4X0 and thus, for X→X0, (127) reduces to (247). This completes the analysis of the scale p=R-r=O(1).


## 5.1. The Boundary SegmentX=0

We consider the scalek=O(1) and let(360)πk,r=C2ρeρΨ0,YρkΠkY;ρ.Then the main balance equation (3), after dividing by exp[ρΨ(0,Y)], becomes(361)ρ1+Y+kΠkY;ρ=ρY+1ΠkY+ρ-1;ρexpρΨ0,Y+ρ-1-Ψ0,Y+ρk+1Πk+1Y;ρ+Πk-1Y;ρ.Taking Πk(Y) to be the leading order approximation to Πk(Y;ρ), dividing (361) by ρ, and letting ρ→∞ lead to the limiting equation(362)1+Y-YeΨY0,YΠkY=k+1Πk+1Y,k⩾1.By examining the boundary condition in (6) we conclude that (362) holds also if k=0, and thus(363)ΠkY=Π0Yk!1+Y-YeΨY0,Yk.To determineC2(ρ) and Π0(Y) we asymptotically match (360) to the D0 ray expansion. We thus expand ρ-1/2Q(0,0)K(X,Y)eρΨ(X,Y) as X→0 and compare the result to the large k expansion of (360). By using Stirling’s formula we see that the matching is possible if, as X→0,(364)ΨX,Y=Ψ0,Y-Xlog⁡X+X1+log⁡1+Y-YeΨY0,Y+oX,(365)Q0,0ρKX,Y~C2ρ2πρXΠ0Y,X⟶0.Note that the exponential factor in (360) must be included for the matching to be possible. Thus we could set C2(ρ)=Q(0,0) and then Π0(Y) will be the limit of 2πXK(X,Y) as X→0, which we show below to be finite and nonzero.From (68) we see that X=0 corresponds to A=e-τ and we set A1(Y)=A(0,Y), where we view A as a function of (X,Y) via the mapping in (68). Then we also set B1(Y)=B(0,Y) and from (68) find that(366)A1-B1=Y1-A1Y0-Y.Using (366) in (67) leads to a quadratic equation for A1(Y), whose solution is given by (118). Also, replacing (eτ,A,B) by (1/A1,A1,B1) in (66) leads to the expression in (119) for Ψ(0,Y), which is an explicit function of Y. As X→0, X/(e-τ-A) is finite and from (68) we obtain(367)e-τ-A~A11-A1A1A1+X0-2+1X,X⟶0.Evaluating the Jacobian Δ in (71) along X=0 leads to ΔL(Y) in (120). Thus as X→0 we have(368)e-τ-A-1/2e-τ-B-1/2Δ-1/2K0A~A1-B1-1/2ΔL-1/2K0A1A1A1+X0-2+1A11-A11X.We have thus identifiedΠ0(Y) as(369)Π0Y=A1-B1-1/2ΔL-1/22πK0A1A1A1+X0-2+1A11-A1.Using (366) and K0(·) in (275) we see that (360) agrees with (117) as X→0, if(370)1+Y-YeΨY0,Y=A1A1+X0-2+11-A1and Ψ has the small X behavior indicated in (364). Now, ΨY=-log(1-Beτ) so that ΨY(0,Y)=-log(1-B1/A1) and using (366) leads to(371)1+Y-YeΨY0,Y=1+Y-YA1A1-B1=1+Y-Y0-Y1-A1A1.But from (118) we see that A12+(X0+Y0-1)A1-Y=0 so that the right-hand sides of (370) and (371) agree.Using (68) we have, as X→0,(372)ΨX+log⁡X=-log⁡1-Aeτ+log⁡e-τ-AX01-A-1+eτ~log⁡X01-A-1e-τ+1~log⁡X0A11-A1+1-A1.Integrating the above, noting thatΨX+log⁡X is an analytic function of X, we conclude that(373)ΨX,Y+Xlog⁡X-X=Xlog⁡A12+X0-2A1+11-A1OX2 and this verifies (364). We have thus shown that the asymptotic matching holds and also determined Π0(Y). This completes the derivation of (117)–(120).For regionR4, (117) holds for all 0<Y<Y0, and then the expansion matches to the corner expansions that apply for Y=O(ρ-1) (r=O(1)) and Y=Y0-O(ρ-1) (p=R-r=O(1)). But for region R1∪R2∪R3 and k=O(1), (117) holds only for YU<Y<Y0, while (114) applies for 0<Y<YU. To connect these it is necessary to construct another expansion for Y-YU=O(ρ-1/2). Using the variables k and y∗≡ρ(Y-YU)=O(1) we conclude from (3) and (6) that(374)πk,r~C3ρeρΨ0,YU+ρ-1/2y∗ρkk!1+Y0-Y0X0+Y0kΠ∗y∗,where we note that 1+Y-YeΨY(0,Y)=1+Y0-Y0/(X0+Y0) when Y=YU, since A1(YU)=Y0/(X0+Y0). To determine Π∗(·) we first infer its behaviors as y∗→±∞ by asymptotically matching (374) to the expansions that apply for Y<YU and Y>YU.We expand the result in (114) as Y↑YU. We have D0(YU)=D0(Y~(0))=Y0/(X0+Y0) so that(375)1+YU-D02YU=A1YUA1YU+X0-2+11-A1YU=Y0+X0X0+Y0,Ψ+,Y0,YU=ΨY0,YU=-log⁡X0+Y02-X0Y0.After some calculation we find that(376)1-D0YUD0YUYUX0YU+2D02YU1-D0YU-1/2=X0Y01-X0X0+Y02X0+Y03/2X0+Y03+X0Y0-X0.Furthermore,(377)Ψ0,YU=Ψ+0,YU+1-X0-X0+X0+Y0log⁡X0+Y0and we note that, in view of Propositions 1 and 2,(378)Cρ~2πX0+Y0Q0,0eρΨ0,YU-Ψ+0,YUwhich holds in regions R1∪R2∪R3. Using (375)–(378) to infer the behavior of (114) as Y↑YU and comparing the result to (374) we can take C3(ρ)=2πρQ(0,0) and then, as y∗→-∞,(379)Π∗y∗exp⁡12ΨYY0,YUy∗2~X0+Y02-X0X0+Y03+X0Y0-X0X0Y0exp⁡12Ψ+,YY0,YUy∗2.Next we expand (117) as Y↓YU. Since (X0+Y0)A1(YU)=Y0, (117) is singular in this limit, and we use(380)Y0-A1YX0+Y0~-A1′YUX0+Y0Y-YU=-A1′YUX0+Y0y∗ρ-1/2 to obtain(381)Y0YY0-YA1A1+X0-2+11-A12-X0A1ΔLY1/2Y0-A1X0+Y0~ρy∗A1YUA1YU+X0-2+11-A1YU2-X0A1YUΔLYU1/2-X0+Y0A1′YU=ρy∗X0X0+Y05/2X0+Y02-X0X0+Y02+Y0-X0=ρy∗X0Y0X0+Y02-X0X0+Y03+X0Y0-X0-1/21ac,where ac was defined in (125). To obtain (381) we also used(382)A1′YU=X0+Y0X0+Y02+Y0-X0,ΔLYU=X0+Y02+Y0-X0X0+Y02X0+Y0Y0+X0,which follow from (118) and (120). Denoting the ratio of the left- and right-hand sides of (379) by f(y∗), the matching conditions as y∗→±∞ may be written as(383)fy∗⟶1,y∗⟶-∞fy∗~12πacy∗exp⁡y∗22Ψ+,YY0,YU-ΨYY0,YU,y∗⟶∞.After some calculation we can show that(384)Ψ+,YY0,YU-ac2=ΨYY0,YUso that the error function(385)fy∗=12π∫acy∗∞e-u2/2dusatisfies the conditions in (383). To determine f(y∗) completely we need to match (374), with C3=2πρQ(0,0), to the expansion in the transition layer where D+ and D0 meet, which is given by (176). This matching has X→0 in (176) and k→∞ in (374), with η∗ fixed. Noting that as X→0(386)η∗=ρY-Y~X=ρY-YU+OX=y∗+Oρ-1/2 we can show, using (178) and (184)–(187), that b(t∗)→ac-2 as X→0 and thus (385) determines f(y∗) for all y∗=O(1).


## 5.2. The Boundary SegmentY=0

We considerr=O(1), in ranges where D0 meets the x-axis, which can occur only for R4. The analysis for where D+ (region R1) or D- (regions R2 and R3) meet the x-axis is very similar to that in [10] so we omit it. By examining (3), noting that this equation holds even if r=0, on the scale r=O(1) with k=ρX, X>0, we find that(387)πk,r~C4ρρrΠ~rXeρΨX,0,where Π~r(X) will satisfy(388)1+X-e-ΨXX,0-XeΨXX,0Π~rX=r+1Π~r+1X.Noting that ΨX(X,0)=-log(1-Aeτ), we define A0(·) by A0(X)=A(X,0). A ray in D0 reaches the x-axis when e-τ=B so we also let B0(X)=B(X,0). Then evaluating (197) along Y=0 leads to(389)1+X-XeΨXX,0-e-ΨXX,0=limY→0⁡YeΨY=lime-τ→B⁡Y0B-e-τB-1exp⁡-log⁡1-Beτ=Y0B01-B0so that the solution to (388) is(390)Π~rX=Π~0X1r!Y0B0Y1-B0Yr.We proceed to determine the constant C4(ρ) and function Π~0(·) by asymptotically matching (387) to the D0 ray expansion.First we note that settinge-τ=B0 in (66) and replacing (A,B) by (A0,B0) lead to the expression in (99) for Ψ(X,0). As Y→0 we also have, in view of (389),(391)ΨYX,Y=-log⁡Y+log⁡Y0B01-B0+o1.By integrating (391) we see that the D0 ray expansion behaves for Y→0 as(392)Q0,0ρKX,Ye-ρYlogYeρYexp⁡ρYlog⁡Y0B01-B0=ρrrrerY0B01-B0r1-X0-Y0ρKX,Y.Here we used Q(0,0)~1-X0-Y0 (in R4) and ρY=r. Expanding r! in (390) by Stirling’s formula we see the matching is possible if C4=1-X0-Y0 and(393)KX,Y~12πYΠ~0X,Y⟶0.We denote by g(X) the limit of [2πYK(X,Y)] as Y→0. Writing(394)KX,Y=e-τ-B-1/2e-τ-A-1/2Δ-1/2K0A we simply replace A by A0(X) in the factors that are not singular and use(395)e-τ-B-1/2=YY01-B-1/2~1YY01-B0. Then g(X) becomes precisely the expression in (102). We have thus determined Π~0(X) as g(X) and thus derived (98)–(102).The approximation in (387) applies only for 0<X<XL, and (96) applies for XL<X<X0. But for X≈XL a new approximation is needed, as g(X) vanishes as X↑XL, due to the factor (1-A0)2-X0, since 1-A0(XL)=X0. Since B0(XL)=W0/(W0+Y0) we find from (102) that(396)gX~W0+Y02W0+Y01-X0X0X0-1-A02X0-X0-Y03/2,X⟶XL.By implicit differentiation of (100) we find that(397)A0′XL=-W0W0+Y0Y02W0+Y0.Using(398)X0-1-A02~21-A0A0′XLX-XL=2X0A0′XLX-XLand (397) we obtain from (396) the expression in (104), which will be used for asymptotic matching verifications.We consider the scaleX-XL=ρ-1/3Λ=O(ρ-1/3), retaining r=O(1). From the balance equations we can conclude that the expansion has the form(399)C5ρeρΨX,0Y0B0XL1-B0XLrρrr!gXL~C5ρeρΨXL,0ρrW0rr!exp⁡ρ2/3ΨXXL,0Λ+12ρ1/3ΨXXXL,0Λ2Π^Λ,where the last exponential factor follows from expanding Ψ(X,0) about X=XL, and these are necessary to have a chance of matching to the expansion for X<XL. By comparing (399) to the behavior of (98) as X↑XL we immediately conclude that C5(ρ)=O(ρ-1/3) so we set C5(ρ)=ρ-1/3(1-X0-Y0), and then(400)Π^Λ~exp⁡16ΨXXXXL,0Λ32W0+Y03/2W05/4Y02W0+Y03/2X0-X0-Y03/2-Λ,Λ⟶-∞.From (101) we find that(401)Y01-B02B0′X=1-X01-A02A0′Xso that B0′(XL)=0. Then from (99) we find that(402)ΨXXX,0=1B0-1B0-A0B0′+A0′B0-A0and thus ΨXX(XL,0)=A0′(XL)/[B0(XL)-1+X0] is as in (107). From (402) we also conclude that(403)ΨXXXXL,0=1B0XL-1B0XL-A0XLB0′′XL+A0′′XLB0XL-A0XL+A0′XLB0XL-A0XL2.By further differentiation of (100) and (401) we find that(404)B0′′XL=-2Y0X0W022W0+Y02,A0′′XL=-2W03/2Y0+W0Y0X02W0+Y02-2W03W0+Y02X0Y022W0+Y03,and thus(405)ΨXXXXL,0=-2X0W05/2W0+Y03Y022W0+Y031X0-X0-Y0+W0W0+Y04Y022W0+Y021X0-X0-Y02.We next derive a matching condition forΠ^(Λ) as Λ→∞. We first note from (97) that if we write s10+W0=W0f(X) then(406)fX=12W0+1-X+W0+1-X2-4W0,(407)fXL=W0+Y01-X0=W0+Y0W0⟹s10XL=Y0.From (406) we then obtain(408)f′XL=-Y0+W02Y0Y0+2W0,f′′XL=-2W05/2Y0+W03Y03Y0+2W03.From (80) with Y=0 we have ΦX(X,0)=-log[1-f(X)] from which we can show that(409)ΦXXL,0=ΨXXL,0,ΦXXXL,0=ΨXXXL,0,(410)ΦXXXXL,0=W0W0+Y03Y03+3W0Y02+4W02Y0-2X0W05/2Y032W0+Y03X0-X0-Y02.By using the expression for C1 in (41) we have(411)ΦXL,0-ΦX0,Y0=ΨXL,0,ρ1/3Φ∗s10-Φ∗Y0=-ρ1/3r01-X02/3X01/6log⁡s10+W0Y0+W0=-ρ1/3r01-X02/3X01/6log⁡fXfXL~-r0Λ1-X02/3X01/6f′XLfXL=r0ΛY0+W0Y0Y0+2W01-X05/3X01/6,ρC11-X01/6X01/6Ai′r02W0+s10s10s10+2W0W0s10+W01/2X0X0-X0-s10~ρ-1/31-X0-Y0Ai′r02πW0+Y0Y0+2W01-X05/6X01/6X0-X0-Y03/2exp⁡-ρΦX0,Y0-ρ1/3Φ∗Y0.With (409)–(411) we have obtained the behavior of (96) as X↓XL and thus derived the matching condition(412)Π^Λ~exp⁡16ΦXXXXL,0Λ3+r0ΛY0+W0Y0Y0+2W01-X05/3X01/61Ai′r0W0+Y0Y0+2W01-X05/6X01/6X0-X0-Y03/2,Λ⟶+∞.Thus (400) and (412) yield the behavior of Π^(Λ) as Λ→±∞, but to determine this function for all Λ we must use a third matching condition, to the transition layer in Proposition 18, which applies for Y-Yc(X)=ρ-1/3η∗∗=O(ρ-1/3). Thus we let X→XL in (191) and let r→∞ in (399), in such a way that Λ is fixed. Since Λ=ρ1/3(X-XL) this means that(413)η∗∗=ρ1/3Y-YcX~ρ1/3rρ-Yc′XLX-XL~-Yc′XLΛremains also O(1). From (86) we have(414)Yc′XL=W03/2Y0+W0Y0Y0+2W0.We can expand ρΦX,Yc+ρ-1/3η∗∗-Φ(X0,Y0) in (191) as X→XL which is equivalent to expanding ρΦ(X,Y)-Φ(X0,Y0) for (X,Y)→(XL,0) and show that the exponential factors in (191) agree with those in (399), after r! is approximated by Stirling’s formula, so that(415)ρrW0rr!~2π-1/2r-1/2exp⁡-ρYlog⁡Y+ρY1+log⁡W0. Then using C5=ρ-1/3(1-X0-Y0) we compare algebraic factors in (399) and (191), which yields(416)12π1-X0-Y0ρ-1/3ρY-1/2Π^Λ~1-X0-Y0ρ-5/612π1-X05/6X01/6X0-X0-Y03/2W0+Y02W0+Y01YC′XLX-XLexp⁡16Λ3ΦXXXXL,012πi∫-i∞i∞e⋆θAiθdθ,where(417)⋆=ΛY0+W0Y0Y0+2W01-X05/3X01/6.In this limit Y~Yc′(XL)(X-XL) and Λ is fixed, so that (416) determines Π^(·) as(418)Π^Λ=1-X05/6X01/6X0-X0-Y03/2W0+Y02W0+Y0exp16Λ3ΦXXXXL,012πi∫-i∞i∞e⋆θAiθdθ.We have thus derived the result in (105). By using the asymptotic results in (192) and (193) (with ⋆ replaced by -⋆) we see immediately that (412) holds. The matching condition in (400) will also hold since(419)16ΦXXXXL,0+13⋆Λ3=16ΨXXXXL,0, by (417), (405), and (410). This completes our analysis of the scale r=O(1).


## 5.3. The Boundary SegmentY=Y0

We consider the scalep=ρ(Y0-Y)=O(1) and 0<X<X0. Now the analysis will be the same for any region Rj, and the expansion we construct will hold everywhere except near the corner points (X,Y)=(0,Y0) and (X0,Y0). By expanding (3) and the boundary condition in (5) along r=R, we find that an asymptotic solution in this range is given by(420)πk,r~C6ρρpp!Y0eτU-1peρΨX,Y0g¯X,where τU=τU(X) is given in (128). Again, (420) must contain the factor exp[ρΨ(X,Y0)] in order to have a chance of matching to the D0 ray expansion, and this factor determines the geometric factor in p. We must only determine C6(ρ) and g¯(X) by asymptotic matching.AsY→Y0 we have A→Amax and B→-∞ in (68). Also,(421)ΨY=-log⁡-B-τ+o1,Y⟶Y0and from (68) we obtain(422)-B~Y01-e-τUY0-Y,where τU=τU(X) is obtained by solving the first equation in (68) with A replaced by Amax. Using (421) and (422) we conclude that(423)ΨX,Y=ΨX,Y0+Y-Y0log⁡Y0-Y+Y0-Y+Y0-Ylog⁡Y0eτU-1+oY0-Y,Y⟶Y0.Expanding p! by Stirling’s formula and comparing the result to(424)Q0,0ρ-1/2KX,YeρΨX,Y as Y→YU we see that we can take C6=Q(0,0)~1-X0-Y0 and the matching will hold if(425)12πpg¯X=ρ-1/212πY0-Yg¯X~ρ-1/2KX,Y,Y⟶Y0.To expand (70) as Y→Y0 we note that in this limit e-τ-A-1/2 and Δ-1/2 are finite, while e-τ-B-1/2~(-B)-1/2 vanishes and K0(A) is singular, due to the factor Y0+(1-X0-Y0)A-A2 in the denominator. We thus have(426)KX,Y~1-Be-τU-Amax-1/2ΔUX-1/2Y02π1-Amax2-X01-X0-Y0-2Amax-1A-Amax,Y⟶Y0.Here ΔU(X) is the Jacobian in (71) evaluated along Y=Y0, and we also used the identity Amax(Amax-1)=Y0-Amax(X0+Y0). From (67) we see that(427)B~Amax1-Amax-X01-X0-Y0-2Amax1A-Amaxso that with (422) we have, as Y↑Y0,(428)KX,Y~12π1Y0-Y1-e-τUe-τU-AmaxX0-1-Amax21-AmaxΔU-1/2.Using (428) in (425) we can identify g¯(X) and then (420) becomes the same as (127).Finally we note that asX↑X0, (127) will asymptotically match to (168), the approximation valid for n,p=O(1). For n→∞ with p=O(1), (168) can be approximated by (247). As X→X0 we have τU(X)→0, Ψ(X0,Y0)=0, ΨX(X0,Y0)=-log(1-Amax), and z+(0)(1-Amax)=1. Then(429)Y0eτUX-1~Y0τU′X0X-X0=-nρY0τU′X0=nρY01+X0+Y02-4X0 and thus, for X→X0, (127) reduces to (247). This completes the analysis of the scale p=R-r=O(1).


## 6. Approximations near State Space Corners

We consider the four state space corners. The upper right corner(X,Y)=(X0,Y0) was analyzed already in Section 4, since this was necessary to completely determine the ray expansion in D0. The analysis of the lower right corner (X,Y)=(X0,0) is essentially the same as that for the infinite capacity model in [10]. For regions R1 and R2 the leading terms for π(k,r) near this corner are the same as for the infinite capacity model, while for regions R3 and R4 the analysis differs only through the multiplicative constant C1. Thus we omit the analysis of these two corners, focusing on the lower and upper left corners, (0,0) and (0,Y0).


### 6.1. The Corner(X,Y)=(0,0)

For regionsR1∪R2∪R3 the analysis is very similar to that in [10], leading to (132)–(136) in Proposition 13. For region R4 we set(430)πk,r=C7ρeρΨ0,0ρk+rQ∗k,r;ρ,recalling that Ψ(0,0)=-X0-Y0+(X0+Y0)log(X0+Y0). Dividing the main balance equation in (3) by ρ, using (430), and letting Q∗(k,r) denote the leading term approximation for Q∗(k,r;ρ), we obtain in the limit ρ→∞(431)Q∗k,r=r+1Q∗k,r+1+k+1Q∗k+1,r;k⩾1,r⩾0.Note that (6) implies that (431) holds along k=0, r⩾0. Equation (431) has many different solutions, and anything of the form α0kβ0r/(k!r!) will be a solution provided that α0+β0=1. Let us write(432)Q∗k,r=1k!r!α0k1-α0rand we will show by asymptotic matching that only one value of α0 is needed, and the matching will also determine the appropriate value. For regions R1∪R2∪R3 the same argument was used in [10] to determine α0 as 1-W0=2X0-X0, leading to (136). For region R4 we can asymptotically match (430) to (117) (then r→∞ with k=O(1)), to (98) (then k→∞ with r=O(1)), or to the D0 ray expansion (then k,r→∞ with k/r fixed). We discuss only the first two matchings, as the third will lead to the same conclusion.For fixedk and r→∞, (430) and (432) become(433)πk,r~C7ρρk+rk!r!eρΨ0,0α0k1-α0r~C7ρ2πρYρkk!α0kexp⁡ρΨ0,0-Ylog⁡Y+Y+Ylog⁡1-α0,where the last formula holds in the matching region where r→∞ but Y=r/ρ→0, and we wrote the expression in terms of Y. To expand (117) as Y→0, we observe from (118) that A1(0)=1-X0-Y0. Then comparing the geometric factors in k, in (117) and (433), we must have(434)α0=A10A10+X0-2+11-A10=Y0+X0X0+Y0.From (119) we find that as Y→0(435)Ψ0,Y=Ψ0,0+Y-Ylog⁡Y+Ylog⁡Y01-X0-Y0X0+Y0+oYwhich implies that 1-α0=Y0(1-X0-Y0)/(X0+Y0), consistent with (434). The matching also implies that(436)C7ρ2πρY~Q0,0A10A10A10+X0-2+11-A102-X0YΔL0Y0-A10X0+Y0.After some calculation we find from (120) that(437)-ΔL0=1-X0-Y0X0+Y0X0+Y0X0+Y0and thus(438)C7ρ~ρ2πX0+Y0Q0,0.We have thus derived (137), since Q(0,0)~1-X0-Y0 for region R4.The same conclusions follow by matching to (98), as then, for fixed r and k→∞, (430) with (432) becomes(439)πk,r~C7ρ2πρXρrr!1-α0rexp⁡ρΨ0,0-Xlog⁡X+X+Xlog⁡α0.We expand (98) as X→0. We have A0(0)=1-X0-Y0 and from (100) find that(440)A0′0=-Y0X0+Y0X0+Y0.Then also B0(0)=1-X0-Y0 and, using (101),(441)B0′0=-X0+Y02-X0X0+Y0X0+Y0.We have, as X→0,(442)ΨXX,0+log⁡X=log⁡X-log1-A0XB0X=log⁡B0X-log⁡B0X-A0XX⟶log⁡B00-log⁡B0′0-A0′0=log⁡Y0+X0X0+Y0.By using (442) to infer the small X behavior of Ψ(X,0) and comparing the result to the exponential part of (439), we again conclude that α0 is as in (434). To expand g(X) in (102) as X→0 we note that it is singular due to the factor(443)B0X-A0X-1/2~X-1/2B0′0-A0′0-1/2. Then after some calculation, using (440) and (441) we find that g(X) has the expansion in (103) as X→0. Expanding (98) as X→0 and comparing the result to (439), noting that Y0B0(0)/1-B0(0)=1-α0, regain the expression in (438) for C7(ρ). We can also easily verify that (137) matches to the D0 ray expansion, by expanding the latter for X,Y→0 along lines of constant slope Y/X.


### 6.2. The Corner(X,Y)=(0,Y0)

Now we use the variablesk and p=R-r, with(444)πk,r=C8ρeρΨ0,Y0ρk+pQ^k,p;ρ.Near this corner only the balance equations in (3), (5), (6), and (7) apply. Using (444) in (3) we obtain(445)ρ+ρY0+k-pQ^k,p;ρ=Y0+1-pρ-1Q^k,p-1;ρ+ρk+1Q^k+1,p;ρ+Q^k-1,p;ρ.Letting Q^(k,p) be the limiting form of Q^(k,p;ρ) as ρ→∞, (445) leads to(446)1+Y0Q^k,p=k+1Q^k+1,pwhose most general solution is(447)Q^k,p=1+Y0kk!Q¯p.Equations (5)–(7) provide, in the limit ρ→∞, no additional information. To determine Q¯(p) in (447) we use asymptotic matching to the boundary layer expansion in (127), which applies for p=O(1) and 0<X<X0.AsX→0 we have τU(X)→τU(0)=-log(Amax), and then Ψ(0,Y0) is as in (131). Also, from (68) with (A,τ) replaced by (Amax,τU) we find that(448)1-AmaxeτUX~X1+Y0,X⟶0 and then ΨX(X,Y0)+logX→log(1+Y0) as X→0. Using the above in (127) we find that for X→0 we have(449)πk,r~Q0,0eρΨ0,Y0eρXlogX-XeρXlog1+Y01p!Y01Amax-1p1-AmaxAmax1-X0-Y02+4Y01/4.By comparing (449) to the large k expansion of (444), with (447), we can take(450)C8ρ=2πρQ0,0and then Q¯(·) is determined as(451)Q¯p=1p!Y01-AmaxAmaxp1-AmaxAmax1-X0-Y02+4Y01/4.We have thus established the result in (130).We conclude by showing that (130), for p→∞ and fixed k, matches asymptotically to (117), for Y↑Y0 and fixed k. In this limit, (117) is singular due to the factor 1/Y0-Y=ρ/p. First we note from (118) that A1(Y0)=Amax and from (119) we get(452)ΨY0,Y=log⁡Y0-YA1YY1-A1Y and thus(453)Ψ0,Y-Ψ0,Y0=Y-Y0log⁡Y0-Y+Y0-Y+Y-Y0log⁡Amax1-Amax1Y0+oY-Y0.Thus,(454)eρΨ0,Y~eρΨ0,Y01-AmaxAmaxY0pρppep,Y⟶Y0.After some calculation we find from (120) that(455)-ΔLY0=1+Y0X01-Amax2-11-Amax=1+Y01-X0-Y02+4Y01/2and also(456)AmaxAmaxAmax+X0-2+11-Amax2-X0Y0-X0+Y0Amax=AmaxAmax+X0-2+1Amax1-Amax1-Amax2-X0=1+Y01-AmaxAmax1-X0-Y02+4Y0.With (454)–(456) the expansion of (117) agrees precisely with the large p behavior of (444), with (447), (450), and (451). This completes the matching verification.


## 6.1. The Corner(X,Y)=(0,0)

For regionsR1∪R2∪R3 the analysis is very similar to that in [10], leading to (132)–(136) in Proposition 13. For region R4 we set(430)πk,r=C7ρeρΨ0,0ρk+rQ∗k,r;ρ,recalling that Ψ(0,0)=-X0-Y0+(X0+Y0)log(X0+Y0). Dividing the main balance equation in (3) by ρ, using (430), and letting Q∗(k,r) denote the leading term approximation for Q∗(k,r;ρ), we obtain in the limit ρ→∞(431)Q∗k,r=r+1Q∗k,r+1+k+1Q∗k+1,r;k⩾1,r⩾0.Note that (6) implies that (431) holds along k=0, r⩾0. Equation (431) has many different solutions, and anything of the form α0kβ0r/(k!r!) will be a solution provided that α0+β0=1. Let us write(432)Q∗k,r=1k!r!α0k1-α0rand we will show by asymptotic matching that only one value of α0 is needed, and the matching will also determine the appropriate value. For regions R1∪R2∪R3 the same argument was used in [10] to determine α0 as 1-W0=2X0-X0, leading to (136). For region R4 we can asymptotically match (430) to (117) (then r→∞ with k=O(1)), to (98) (then k→∞ with r=O(1)), or to the D0 ray expansion (then k,r→∞ with k/r fixed). We discuss only the first two matchings, as the third will lead to the same conclusion.For fixedk and r→∞, (430) and (432) become(433)πk,r~C7ρρk+rk!r!eρΨ0,0α0k1-α0r~C7ρ2πρYρkk!α0kexp⁡ρΨ0,0-Ylog⁡Y+Y+Ylog⁡1-α0,where the last formula holds in the matching region where r→∞ but Y=r/ρ→0, and we wrote the expression in terms of Y. To expand (117) as Y→0, we observe from (118) that A1(0)=1-X0-Y0. Then comparing the geometric factors in k, in (117) and (433), we must have(434)α0=A10A10+X0-2+11-A10=Y0+X0X0+Y0.From (119) we find that as Y→0(435)Ψ0,Y=Ψ0,0+Y-Ylog⁡Y+Ylog⁡Y01-X0-Y0X0+Y0+oYwhich implies that 1-α0=Y0(1-X0-Y0)/(X0+Y0), consistent with (434). The matching also implies that(436)C7ρ2πρY~Q0,0A10A10A10+X0-2+11-A102-X0YΔL0Y0-A10X0+Y0.After some calculation we find from (120) that(437)-ΔL0=1-X0-Y0X0+Y0X0+Y0X0+Y0and thus(438)C7ρ~ρ2πX0+Y0Q0,0.We have thus derived (137), since Q(0,0)~1-X0-Y0 for region R4.The same conclusions follow by matching to (98), as then, for fixed r and k→∞, (430) with (432) becomes(439)πk,r~C7ρ2πρXρrr!1-α0rexp⁡ρΨ0,0-Xlog⁡X+X+Xlog⁡α0.We expand (98) as X→0. We have A0(0)=1-X0-Y0 and from (100) find that(440)A0′0=-Y0X0+Y0X0+Y0.Then also B0(0)=1-X0-Y0 and, using (101),(441)B0′0=-X0+Y02-X0X0+Y0X0+Y0.We have, as X→0,(442)ΨXX,0+log⁡X=log⁡X-log1-A0XB0X=log⁡B0X-log⁡B0X-A0XX⟶log⁡B00-log⁡B0′0-A0′0=log⁡Y0+X0X0+Y0.By using (442) to infer the small X behavior of Ψ(X,0) and comparing the result to the exponential part of (439), we again conclude that α0 is as in (434). To expand g(X) in (102) as X→0 we note that it is singular due to the factor(443)B0X-A0X-1/2~X-1/2B0′0-A0′0-1/2. Then after some calculation, using (440) and (441) we find that g(X) has the expansion in (103) as X→0. Expanding (98) as X→0 and comparing the result to (439), noting that Y0B0(0)/1-B0(0)=1-α0, regain the expression in (438) for C7(ρ). We can also easily verify that (137) matches to the D0 ray expansion, by expanding the latter for X,Y→0 along lines of constant slope Y/X.


## 6.2. The Corner(X,Y)=(0,Y0)

Now we use the variablesk and p=R-r, with(444)πk,r=C8ρeρΨ0,Y0ρk+pQ^k,p;ρ.Near this corner only the balance equations in (3), (5), (6), and (7) apply. Using (444) in (3) we obtain(445)ρ+ρY0+k-pQ^k,p;ρ=Y0+1-pρ-1Q^k,p-1;ρ+ρk+1Q^k+1,p;ρ+Q^k-1,p;ρ.Letting Q^(k,p) be the limiting form of Q^(k,p;ρ) as ρ→∞, (445) leads to(446)1+Y0Q^k,p=k+1Q^k+1,pwhose most general solution is(447)Q^k,p=1+Y0kk!Q¯p.Equations (5)–(7) provide, in the limit ρ→∞, no additional information. To determine Q¯(p) in (447) we use asymptotic matching to the boundary layer expansion in (127), which applies for p=O(1) and 0<X<X0.AsX→0 we have τU(X)→τU(0)=-log(Amax), and then Ψ(0,Y0) is as in (131). Also, from (68) with (A,τ) replaced by (Amax,τU) we find that(448)1-AmaxeτUX~X1+Y0,X⟶0 and then ΨX(X,Y0)+logX→log(1+Y0) as X→0. Using the above in (127) we find that for X→0 we have(449)πk,r~Q0,0eρΨ0,Y0eρXlogX-XeρXlog1+Y01p!Y01Amax-1p1-AmaxAmax1-X0-Y02+4Y01/4.By comparing (449) to the large k expansion of (444), with (447), we can take(450)C8ρ=2πρQ0,0and then Q¯(·) is determined as(451)Q¯p=1p!Y01-AmaxAmaxp1-AmaxAmax1-X0-Y02+4Y01/4.We have thus established the result in (130).We conclude by showing that (130), for p→∞ and fixed k, matches asymptotically to (117), for Y↑Y0 and fixed k. In this limit, (117) is singular due to the factor 1/Y0-Y=ρ/p. First we note from (118) that A1(Y0)=Amax and from (119) we get(452)ΨY0,Y=log⁡Y0-YA1YY1-A1Y and thus(453)Ψ0,Y-Ψ0,Y0=Y-Y0log⁡Y0-Y+Y0-Y+Y-Y0log⁡Amax1-Amax1Y0+oY-Y0.Thus,(454)eρΨ0,Y~eρΨ0,Y01-AmaxAmaxY0pρppep,Y⟶Y0.After some calculation we find from (120) that(455)-ΔLY0=1+Y0X01-Amax2-11-Amax=1+Y01-X0-Y02+4Y01/2and also(456)AmaxAmaxAmax+X0-2+11-Amax2-X0Y0-X0+Y0Amax=AmaxAmax+X0-2+1Amax1-Amax1-Amax2-X0=1+Y01-AmaxAmax1-X0-Y02+4Y0.With (454)–(456) the expansion of (117) agrees precisely with the large p behavior of (444), with (447), (450), and (451). This completes the matching verification.


## 7. Approximations near Transition Layers

We analyze the vicinities of the curves where the regionsD0, D+, and D- meet. From Figures 3–6 we see that, for R1, D0 meets D+ along the curve Y=Y~(X). For regions R2∪R3, D0 also meets D+, while D+ and D- meet along Y=Y∗(X). For region R4, D0 and D- meet along Y=Yc(X). The analysis for Y=Y∗(X), with the scaling Y-Y∗(X)=O(ρ-1/3), is carried out in [10] and we omit it here. For R2 the analysis is exactly as in [10], while for R2∩R3 and R3 the D+ and D- ray expansions must be multiplied by the appropriate constants C and C1, but the analysis is otherwise unchanged. We thus obtain the result in (188).


### 7.1. Transition Layer nearY=Yc(X), Region R4

This layer arises only for parameter regionR4. We set ρ1/3Y-Yc(X)=η∗∗=O(1) and use (X,η∗∗) as the variables. Let us write L(X,Y) in the ray expansion in (79) as(457)LX,Y=W0+s12YW0+Y1s1-Y12W0+s1+Y1-W0W0+s1W0+Y-1/2L0s1≡L1X,YL0s1,where(458)L0s1=12πAi′r0-2X0-1/61-X0-5/6s1+W0X0-s1W0+s1exp⁡-12+12X0log⁡W0+s11-X0.The scaling Y-Yc(X)=O(ρ-1/3) corresponds to rays that have s1≈Y0 and more precisely s1-Y0=O(ρ-1/3). We thus set(459)s1=Y1+ρ-1/3s~1.In view of (75) and (86) we have(460)YcX+W0Y0+W0=Y∗X+W0W0which relates the two curves Yc and Y∗. From the definition of s1 in (81) we then have(461)s1+W0=Y0+W0+ρ-1/3s~1=W0Y+W0Y∗X+W0so that(462)s~1=ρ-1/3W0Y+W0Y∗X+W0-YcX+W0Y∗X+W0=ρ-1/3W0Y∗X+W0Y-YcX=Y0+W0YcX+W0η∗∗.Consider the balance equation (17) on the scale η∗∗=O(1), with 0<X<X0. An asymptotic solution is given by(463)C1eρΦX,Yeρ1/3Φ1X,YLX,YLX,η∗∗, where we can replace Y by Yc(X)+ρ-1/3η∗∗, so the expansion is in terms of X and η∗∗. We can also view L as being a function of the ray variables t and s~1. But then the product LL will satisfy the transport equation in (198). Since L is a particular solution, L must be constant along a ray and thus a function of s~1 but not t, so in view of (462) we write(464)LX,η∗∗=L0Y0+W0YcX+W0η∗∗.We thus write the expansion in the transition layer as(465)πk,r~C1LX,YcXL0Y0+W0YcX+W0η∗∗exp⁡ρΦX,YcX+ρ-1/3η∗∗+ρ1/3ΦX,YcX+ρ-1/3η∗∗,where we used L(X,Y)~L(X,Yc(X)). If (465) is to asymptotically match to the D- ray expansion we must have(466)L0-∞=1.Using the fact that Φ1(X,Y)=Φ∗(s1) we have(467)ρ1/3Φ∗s1-ΦY0=ρ1/3r01-X2/3X0-1/6log⁡W0+Y0W0+s1~-r0η∗∗1-X02/3X01/6W0+YcX.Next we examine the ray expansion inD0 near the curve Y=Yc(X). This will yield a matching condition for L0(·) as η∗∗→+∞. We can easily establish the following continuity conditions between Ψ(X,Y) and Φ(X,Y) across Y=Yc(X)(468)ΨX,YcX=Φ∗X,YcX-ΦX0,Y0,ΨYX,YcX=ΦYX,YcX,ΨYYX,YcX=ΦYYX,YcX,where we recall that Y=Yc(X) corresponds to the D0 ray with A=1-X0, B=W0/(W0+Y0). To this end we note that(469)ΨYX,Y=log⁡B+1-BYY0-log⁡1-BYY0,where we expressed τ in terms of B and Y using (68). Along Y=Yc we have ∂B/∂Y=0 so that from (469) we obtain(470)ΨYY=Y01-B1Y0B+1-BY∂B∂Y+1-BY0B+1-BY-1Yand hence(471)ΨYYX,YcX=-W0YcYc+W0.From the relation between B and A in (67) we have(472)Y01-B2∂B∂Y=1-X01-A2∂A∂Yand thus(473)∂2B∂Y2=-2Y0X0Y0+W02∂A∂Y2,atY=YcX.It also follows from (470) that(474)ΨYYYX,YcX=1Yc2-1W0+Yc2+Y0+W02Y0Yc+W0∂2B∂Y2X,YcX.Eliminating τ in (68) leads to(475)X+Y-1B+1-BYY0+A=A+X0+Y0-1B+1-BYY02.By implicit differentiation of (475) we obtain(476)∂A∂YX,YcX=W0W0+YcXand then using (473) in (474) leads to(477)ΨYYYX,YcX=1Yc2-1W0+Yc2-2W0X01W0+Yc3=ΦYYYX,YcX-2W0X01W0+Yc3.Combining (468) with (477) we have(478)ρΨX,Yc+ρ-1/3η∗∗-ΦX,Yc+ρ-1/3η∗∗-ΦX0,Y0~-131-X2X0η∗∗W0+YcX3.AsY→Yc(X), e-τ→(W0+Yc)/(W0+Y0),(479)e-τ-Ae-τ-B~YcY0+W02Yc+X0W0-W0Y0,-Δ~Y0-YcW0+Y0Y0+Yc+2W0.The function K0(A) vanishes as A→1-X0, in view of the factor (1-A)2-X0, and we have(480)K0A~2Y01-X022πY0+W0X0-X0-Y0W0+Ycρ-1/3η∗∗,Y⟶Yc.Combining (479) and (480) leads to, for Y→Yc(s),(481)C0KX,Y~21-X0-Y0ρ-5/6η∗∗1-X02Y0W0+Y0Y0-YcYc+Y0+2W012π1W0+Yc1X0-X0-Y01Yc+W0-W0W0+Y0.Using the expression in (41) for C1 and the continuity conditions in (468) and (477), we compare (465) to the D0 ray expansion as Y→Yc(X) to conclude that(482)L0⋆~2Ai′r01-X02/3X0-1/6⋆Y0+W0exp⁡r01-X02/3X01/6⋆Y0+W0-131-X02X0⋆Y0+W03,⋆⟶+∞.Thus (466) and (482) give the behaviors of L0(⋆) as ⋆→±∞. But to determine L0(·) completely we must use asymptotic matching to the corner approximation in (169), which applies on the (ν,T) scale. We thus expand (465) for X→X0 and asymptotically match this to (169), expanding the latter for ν→∞, T→∞ but with T-ν=O(1). In this limit we have(483)12πi∫-i∞i∞Aiν+θAiθeθTdθ~e-2/3ν3/22πν1/412πi∫-i∞i∞eT-νθAiθdθ.Recalling that X0-X=ρ-2/3ν1 we have(484)Y0-YcX~Y0+W0W01/4ρ-1/3ν1and then(485)η∗∗=ρ1/3Y-YcX=-T1+ρ1/3Y0-YcX~Y0+W0X01/61-X02/3ν-T,X,Y⟶X0,Y0.Apart from the exponential factor(486)exp⁡ρΦX,YcX+ρ-1/3η∗∗-ΦX0,Y0,as Y→Yc the expression in (465) becomes(487)1-X0-Y0ρ-5/612πAi′r0-11-X04/3X0-1/122W0+Y0X0-X0-Y0Y0+W0W01/4ρ-1/3ν1-1/2exp⁡-r0ν-TL0Y0+W0X01/61-X02/3ν-T.Here we note that L1(X,Yc(X)) is singular as X→X0, in view of the factor (s1-Yc)-1/2, and in this limit(488)s1-Yc=s1-Yc+Y0-YcX~Y0+W0W01/4X0-X. The limit of (169) as T,ν→∞, using (483), apart from some exponential factors, which will automatically match those from (486), is given by(489)ρ-2/31-X05/31-X0-Y0Y0+W0X0-X0-Y0ν-1/42π12πi∫-i∞i∞eT-νθAiθdθ.Comparing (487) with (489) determines L0(·) as(490)L0Y0+W0X01/61-X02/3U=Ai′r0er0U2πi∫-i∞i∞e-θUAiθdθ.With (490), (465) becomes the same as (191), so we have established Proposition 8. Note also that (490) is consistent with (466), since for U→-∞ the asymptotics of the contour integral are determined by the pole at θ=r0<0. For U→+∞ a standard saddle point calculation shows that (490) is asymptotic to(491)Ai′r0er0U2Ue-U3/3, which is consistent with (482).


### 7.2. Transition Layer nearY=Yc(X), Region R3∩R4

We takeX0+Y0=X0+ρ-1/3δ∗. Now the curves Y~(X), Y∗(X), and Yc(X) are all close to each other, coinciding if δ∗=0. Most of the analysis closely parallels that in R4, so we include here fewer of the details. We again use the variables X and η∗∗ to find that in the transition layer(492)πk,r~exp⁡ρΦX,Yc+ρ-1/3η∗∗-ΦX0,Y0exp⁡-r01-X02/3X01/6W0+Ycη∗∗ρ-1/2L1Y0+W0Yc+W0η∗∗1Yc0X1X0-X0-Yc0X1Yc0X+1-X02-X01-X05/3X01/1212πAi′r0-2∫r0∞eδ1uAiudu.The factors in (492) that precede L1 come from eρΦ(X,Y), C1 (now given by (37)), eρ1/3Φ1(X,Y), L1(X,Y), and L0(s1), except we exclude from L0(·) the factor (s1+W0)X0-s1, which vanishes if s1→Y0 and Y0→X0-X0. Instead we include an extra factor of ρ-1/3, and then ρ-1/2=ρ-1/3ρ-1/6, where ρ-1/6 comes from C1 in (37). Also, in (492)(493)Yc0X≡YcXY0=X0-X0. In region R3∩R4 we have W0+Y0-W0(W0+Y0)~Yc and Y0+2W0~2+X0-3X0, and also(494)L0s1s1+W0X0-s1⟶12πAi′r0-2X0-1/61-X0-4/3,s1⟶Y0. Thus apart from the factors ρ-1/3L1(·), (492) is just the D- ray expansion expanded near Y→Yc(X) (or s1→Y0), divided by (s1+W0)X0-s1. Thus (492) matches to the D- ray expansion if(495)ρ-1/3L1Y0+W0Yc+W0η∗∗η∗∗→-∞~s1+W0X0-s1s1→Y0.But the left side of (495), in view of (461) and (462), becomes(496)s1+W0X0-s1~1-X02ρ-1/3-η∗∗Yc+W0.Now Y0+W0~1-X0 so that (495) and (496) lead to(497)L1U~1-X0-U,U⟶-∞.Next we match (492) to the D0 ray expansion and thus infer the behavior of L1(U) as U→+∞. Now C0~(1-X0-Y0)ρ-1/2~1-X0ρ-1/2. First evaluating K(X,Y) at Y0=X0-X0 and then letting Y→Yc0 we get(498)KX,YY0=X0-X0Y→Yc0~2πX01/41-X02Yc0X1X0-X0-Yc0X1W0+W0+Yc0X.The estimate in (478) still holds so if C0K(X,Y)eρΨ(X,Y) is to agree, for Y→Yc, with the large η∗∗ asymptotics of (492), we must have(499)Ai′r0-2∫r0∞eδ1uAiudu1-X05/3X01/12L1Y0+W0Yc+W0η∗∗~2X01/41-X03exp⁡r01-X02/3X01/6W0+Y0η∗∗-131-X02X0η∗∗W0+Yc3.We thus have the matching condition(500)L1U~2X1/61-X04/3Ai′r02∫r0∞eδ1uAiudu-1exp⁡r0UX01/61-X01/3-U33X01-X0,U⟶+∞.To determine L1(·) completely we need to match (492) to the corner layer expansion in (173). Using(501)Aiν+θ~2-1π-1/2ν-1/4exp⁡-23ν3/2-νθ in the integral we expand (173) for ν,T→∞ with T-ν=O(1). The result contains some exponential factors and some algebraic ones, with the latter being(502)ρ-1/31-X04/3X0-1/3ν-1/412π12πi∫-i∞i∞eθT-ν-δ1Aiθ2∫θ∞eδ1uAiududθ.Now we evaluate the algebraic part of (492), for X→X0. From (460), since Y0+W0→1-X0=W0, we see that Yc0(X) is the same as Y∗(X), which is independent of Y0. We also have Y∗(X0)=X0-X0 so that (498) becomes singular as X→X0. We have(503)X0-X0-Y∗X=12X-X0+X0-X2-X02-X~1-X01/2ρ-1/3ν1.The result in (485) still applies, now simplifying to(504)η∗∗~X01/61-X01/3ν-T.With (503) and (504), the algebraic part of (492) becomes(505)ρ-1/2e-r0T-ν12πρ-1/6ν-1/4X0-1/4Ai′r0-2∫r0∞eδ1uAiuduL1X01/61-X01/3ν-T.Comparing (502) to (505) determines L1(·) as(506)L1U=X01/61-X04/3Ai′r02∫r0∞eδ1uAiudu-112πi∫-i∞i∞e-θδ1Ai2θ∫θ∞eδ1uAiuduexp⁡-UθX01/61-X01/3dθexp⁡r0UX01/61-X01/3.We have thus established the result in (194). By asymptotically expanding the contour integral in (506) in the limits U→±∞ we can easily verify that (497) and (500) are satisfied.


### 7.3. Transition Layer nearY=Y~(X), Region R1∪R2∪R3

We consider the curve that separatesD0 from D+. We use the scaling(507)Y=Y~X+ρ-1/2η∗,η∗=O1and expand the joint distribution as(508)πk,r~CρFX,η∗K+X,Y~exp⁡ρΨ+X,Y~+ρΨ+,YX,Y~η∗+12Ψ+,YYX,Y~η∗2.The exponential factor in (508) corresponds to the expansions of Ψ+(X,Y) about Y=Y~(X). If (508) as η∗→-∞ is to match to the D+ ray expansion we must have(509)FX,-∞=1,∀0<X<X0.We use (508) in the main balance equation (3) and after a lengthy calculation we find that F(X,η∗) satisfies the parabolic PDE(510)12Y~′2XeΨ+,X+e-Ψ+,X+Y~eΨ+,YFη∗η∗+η∗eΨ+,YFη∗+XeΨ+,X-e-Ψ+,XFX=0,for X∈(0,X0) and η∗∈(-∞,∞). In (510), Ψ+ is understood to be evaluated at (X,Y)=(X,Y~(X)). We can write the curve Y~(X) in parametric form, as(511)X=1-Y0X0+Y0et∗1+X0+Y0-1e-t∗,Y~X=Y0X0+Y0X0+Y0-1+e-t∗,with the latter equation corresponding to (178). Since t∗=t∗(X) we note that(512)XeΨ+,XX,Y~X-e-Ψ+,XX,Y~X=-dXdt∗and define M∗ from(513)M∗=Y~′X2XeΨ+,XX,Y~X+e-Ψ+,XX,Y~X+Y~XeΨ+,YX,Y~X.We can viewM∗ as being a function of either X or t∗. In terms of t∗ we have(514)M∗t∗=Y0e-t∗X0+Y0+2+X0+Y0-1e-t∗-Y0X0+Y0et∗Y0e-t∗Y0et∗+X0+Y0X0+Y0-1e-t∗2.Changing variables from (X,η∗) to (t∗,η∗) the PDE in (510) becomes(515)12M∗Fη∗η∗+η∗1+X0+Y0-1et∗Fη∗=Ft∗,t∗>0and note that t∗=0 corresponds to X=X0, as Y~(X0)=Y0.Next we assume thatF(t∗,η∗) will be a function of a single “similarity” variable, which we call ⋆=η∗a(t∗) and with which (515) becomes the ordinary differential equation(516)12M∗a2t∗F0′′⋆+Bt∗-a′t∗at∗⋆F0′⋆=0,where(517)F0η∗at∗=Ft∗,η∗,Bt∗=1+X0+Y0-1et∗-1. Such a similarity solution is possible if a(t∗) satisfies the nonlinear ODE(518)Bt∗-a′t∗at∗=12M∗t∗a2t∗.Setting a(t∗)=[b(t∗)]-1/2, the Bernoulli equation in (518) becomes the linear equation(519)b′t∗+2Bt∗bt∗=M∗t∗.Solving (519) subject to b(0)=0 leads to the expression in (184). With (518), (516) becomes(520)F0′′⋆+⋆F0′⋆=0and (509) implies that F(-∞)=1, and thus(521)F0η∗bt∗=12π∫η∗/bt∗∞e-u2/2du.We have thus derived (176).With (521) substituted for F(X,η∗) in (508), we can show that as η∗→+∞, (508) asymptotically matches to the D0 ray expansion. As η∗→∞ from (521) we have(522)F0η∗bt∗~bt∗2π1η∗exp⁡-η∗22bt∗,η∗⟶∞.We can easily establish the continuity conditions(523)ΨX,Y~X=Ψ+X,Y~X-Ψ+X0,Y0,ΨYX,Y~X=Ψ+,YX,Y~Xand, after a lengthy calculation, show that(524)Ψ+,YYX,Y~X=1bt∗+ΨYYX,Y~X.Thus expanding the D0 ray expansion, C0K(X,Y)eρΨ(X,Y), as Y↓Y~(X) and comparing this to the large η∗ expansion of (508), using (523) and (524), lead to(525)C0KX,Y~CK+X,Y~Xbt∗2πeρΨ+X0,Y0ρY-Y~X,Y⟶Y~X.But from Proposition 2 we see that, for R1∪R2∪R3,(526)CC0~2πρX0+Y0exp⁡ρ1-X0-Y0+X0+Y0log⁡X0+Y0,Ψ+X0,Y0=-1+X0+Y0-X0+Y0log⁡X0+Y0. The curve Y=Y~(X) corresponds to the ray A=Y0/(X0+Y0), and then B=1-X0-Y0. From (70) we see that K becomes singular as Y→Y~(X), due to the factor Y0-A(X0+Y0)-1. This singularity precisely matches that in the right-hand side of (525), and yet another lengthy calculation shows that(527)limY→Y~X⁡Y-Y~XKX,Y=X0+Y0bt∗K+X,Y~X, which establishes the matching.We note that the PDE (515) contains many solutions other than the similarity solution. An initial condition as t∗→0 can be obtained by asymptotically matching (508) to the corner layer valid on the (n,p) scale. This will determine the solution to (515) uniquely, but the matching will again lead to the conclusion that F is given by (521). Below we only briefly verify that the matching holds. Near the corner the curve Y=Y~(X) can be approximated by the straight line Y-Y0=Y~′(X0)(X-X0), which corresponds to(528)Y0-YX0-X=pn=Y0X0+Y02-X0=1θ1,where θ1 was defined in Proposition 20. Thus we must show that (508) for X→X0 is the same as (251). The exponential parts agree automatically and C(ρ)~ρ-1/2Q(0,0), so we must only show that, for X→X0,(529)K+X,Y~X0+Y0∫η∗/bt∗∞e-u2/2du~X0+Y02-X0X0+Y0212π∫-∞ηe-u2/2du.But(530)K+X0,Y0=2π-1/2X0+Y02-X0X0+Y0-5/2,η∗=ρY-Y~X=ρY-Y0+Y0-Y~X=-pρ+ρY0-Y~X~1ρ-p+Y0X0+Y02-X0n. From (184)–(187) we obtain, as X→X0 or t∗→0,(531)bt∗~Y0X0+Y0X0+Y03+X0+Y0Y0-2X0+X0X0+Y02-X03nρsince n/p~θ1, as t∗→0η∗/b(t∗) approaches -η in (252), which verifies (529). This completes the analysis of the transition from D+ to D0.


## 7.1. Transition Layer nearY=Yc(X), Region R4

This layer arises only for parameter regionR4. We set ρ1/3Y-Yc(X)=η∗∗=O(1) and use (X,η∗∗) as the variables. Let us write L(X,Y) in the ray expansion in (79) as(457)LX,Y=W0+s12YW0+Y1s1-Y12W0+s1+Y1-W0W0+s1W0+Y-1/2L0s1≡L1X,YL0s1,where(458)L0s1=12πAi′r0-2X0-1/61-X0-5/6s1+W0X0-s1W0+s1exp⁡-12+12X0log⁡W0+s11-X0.The scaling Y-Yc(X)=O(ρ-1/3) corresponds to rays that have s1≈Y0 and more precisely s1-Y0=O(ρ-1/3). We thus set(459)s1=Y1+ρ-1/3s~1.In view of (75) and (86) we have(460)YcX+W0Y0+W0=Y∗X+W0W0which relates the two curves Yc and Y∗. From the definition of s1 in (81) we then have(461)s1+W0=Y0+W0+ρ-1/3s~1=W0Y+W0Y∗X+W0so that(462)s~1=ρ-1/3W0Y+W0Y∗X+W0-YcX+W0Y∗X+W0=ρ-1/3W0Y∗X+W0Y-YcX=Y0+W0YcX+W0η∗∗.Consider the balance equation (17) on the scale η∗∗=O(1), with 0<X<X0. An asymptotic solution is given by(463)C1eρΦX,Yeρ1/3Φ1X,YLX,YLX,η∗∗, where we can replace Y by Yc(X)+ρ-1/3η∗∗, so the expansion is in terms of X and η∗∗. We can also view L as being a function of the ray variables t and s~1. But then the product LL will satisfy the transport equation in (198). Since L is a particular solution, L must be constant along a ray and thus a function of s~1 but not t, so in view of (462) we write(464)LX,η∗∗=L0Y0+W0YcX+W0η∗∗.We thus write the expansion in the transition layer as(465)πk,r~C1LX,YcXL0Y0+W0YcX+W0η∗∗exp⁡ρΦX,YcX+ρ-1/3η∗∗+ρ1/3ΦX,YcX+ρ-1/3η∗∗,where we used L(X,Y)~L(X,Yc(X)). If (465) is to asymptotically match to the D- ray expansion we must have(466)L0-∞=1.Using the fact that Φ1(X,Y)=Φ∗(s1) we have(467)ρ1/3Φ∗s1-ΦY0=ρ1/3r01-X2/3X0-1/6log⁡W0+Y0W0+s1~-r0η∗∗1-X02/3X01/6W0+YcX.Next we examine the ray expansion inD0 near the curve Y=Yc(X). This will yield a matching condition for L0(·) as η∗∗→+∞. We can easily establish the following continuity conditions between Ψ(X,Y) and Φ(X,Y) across Y=Yc(X)(468)ΨX,YcX=Φ∗X,YcX-ΦX0,Y0,ΨYX,YcX=ΦYX,YcX,ΨYYX,YcX=ΦYYX,YcX,where we recall that Y=Yc(X) corresponds to the D0 ray with A=1-X0, B=W0/(W0+Y0). To this end we note that(469)ΨYX,Y=log⁡B+1-BYY0-log⁡1-BYY0,where we expressed τ in terms of B and Y using (68). Along Y=Yc we have ∂B/∂Y=0 so that from (469) we obtain(470)ΨYY=Y01-B1Y0B+1-BY∂B∂Y+1-BY0B+1-BY-1Yand hence(471)ΨYYX,YcX=-W0YcYc+W0.From the relation between B and A in (67) we have(472)Y01-B2∂B∂Y=1-X01-A2∂A∂Yand thus(473)∂2B∂Y2=-2Y0X0Y0+W02∂A∂Y2,atY=YcX.It also follows from (470) that(474)ΨYYYX,YcX=1Yc2-1W0+Yc2+Y0+W02Y0Yc+W0∂2B∂Y2X,YcX.Eliminating τ in (68) leads to(475)X+Y-1B+1-BYY0+A=A+X0+Y0-1B+1-BYY02.By implicit differentiation of (475) we obtain(476)∂A∂YX,YcX=W0W0+YcXand then using (473) in (474) leads to(477)ΨYYYX,YcX=1Yc2-1W0+Yc2-2W0X01W0+Yc3=ΦYYYX,YcX-2W0X01W0+Yc3.Combining (468) with (477) we have(478)ρΨX,Yc+ρ-1/3η∗∗-ΦX,Yc+ρ-1/3η∗∗-ΦX0,Y0~-131-X2X0η∗∗W0+YcX3.AsY→Yc(X), e-τ→(W0+Yc)/(W0+Y0),(479)e-τ-Ae-τ-B~YcY0+W02Yc+X0W0-W0Y0,-Δ~Y0-YcW0+Y0Y0+Yc+2W0.The function K0(A) vanishes as A→1-X0, in view of the factor (1-A)2-X0, and we have(480)K0A~2Y01-X022πY0+W0X0-X0-Y0W0+Ycρ-1/3η∗∗,Y⟶Yc.Combining (479) and (480) leads to, for Y→Yc(s),(481)C0KX,Y~21-X0-Y0ρ-5/6η∗∗1-X02Y0W0+Y0Y0-YcYc+Y0+2W012π1W0+Yc1X0-X0-Y01Yc+W0-W0W0+Y0.Using the expression in (41) for C1 and the continuity conditions in (468) and (477), we compare (465) to the D0 ray expansion as Y→Yc(X) to conclude that(482)L0⋆~2Ai′r01-X02/3X0-1/6⋆Y0+W0exp⁡r01-X02/3X01/6⋆Y0+W0-131-X02X0⋆Y0+W03,⋆⟶+∞.Thus (466) and (482) give the behaviors of L0(⋆) as ⋆→±∞. But to determine L0(·) completely we must use asymptotic matching to the corner approximation in (169), which applies on the (ν,T) scale. We thus expand (465) for X→X0 and asymptotically match this to (169), expanding the latter for ν→∞, T→∞ but with T-ν=O(1). In this limit we have(483)12πi∫-i∞i∞Aiν+θAiθeθTdθ~e-2/3ν3/22πν1/412πi∫-i∞i∞eT-νθAiθdθ.Recalling that X0-X=ρ-2/3ν1 we have(484)Y0-YcX~Y0+W0W01/4ρ-1/3ν1and then(485)η∗∗=ρ1/3Y-YcX=-T1+ρ1/3Y0-YcX~Y0+W0X01/61-X02/3ν-T,X,Y⟶X0,Y0.Apart from the exponential factor(486)exp⁡ρΦX,YcX+ρ-1/3η∗∗-ΦX0,Y0,as Y→Yc the expression in (465) becomes(487)1-X0-Y0ρ-5/612πAi′r0-11-X04/3X0-1/122W0+Y0X0-X0-Y0Y0+W0W01/4ρ-1/3ν1-1/2exp⁡-r0ν-TL0Y0+W0X01/61-X02/3ν-T.Here we note that L1(X,Yc(X)) is singular as X→X0, in view of the factor (s1-Yc)-1/2, and in this limit(488)s1-Yc=s1-Yc+Y0-YcX~Y0+W0W01/4X0-X. The limit of (169) as T,ν→∞, using (483), apart from some exponential factors, which will automatically match those from (486), is given by(489)ρ-2/31-X05/31-X0-Y0Y0+W0X0-X0-Y0ν-1/42π12πi∫-i∞i∞eT-νθAiθdθ.Comparing (487) with (489) determines L0(·) as(490)L0Y0+W0X01/61-X02/3U=Ai′r0er0U2πi∫-i∞i∞e-θUAiθdθ.With (490), (465) becomes the same as (191), so we have established Proposition 8. Note also that (490) is consistent with (466), since for U→-∞ the asymptotics of the contour integral are determined by the pole at θ=r0<0. For U→+∞ a standard saddle point calculation shows that (490) is asymptotic to(491)Ai′r0er0U2Ue-U3/3, which is consistent with (482).


## 7.2. Transition Layer nearY=Yc(X), Region R3∩R4

We takeX0+Y0=X0+ρ-1/3δ∗. Now the curves Y~(X), Y∗(X), and Yc(X) are all close to each other, coinciding if δ∗=0. Most of the analysis closely parallels that in R4, so we include here fewer of the details. We again use the variables X and η∗∗ to find that in the transition layer(492)πk,r~exp⁡ρΦX,Yc+ρ-1/3η∗∗-ΦX0,Y0exp⁡-r01-X02/3X01/6W0+Ycη∗∗ρ-1/2L1Y0+W0Yc+W0η∗∗1Yc0X1X0-X0-Yc0X1Yc0X+1-X02-X01-X05/3X01/1212πAi′r0-2∫r0∞eδ1uAiudu.The factors in (492) that precede L1 come from eρΦ(X,Y), C1 (now given by (37)), eρ1/3Φ1(X,Y), L1(X,Y), and L0(s1), except we exclude from L0(·) the factor (s1+W0)X0-s1, which vanishes if s1→Y0 and Y0→X0-X0. Instead we include an extra factor of ρ-1/3, and then ρ-1/2=ρ-1/3ρ-1/6, where ρ-1/6 comes from C1 in (37). Also, in (492)(493)Yc0X≡YcXY0=X0-X0. In region R3∩R4 we have W0+Y0-W0(W0+Y0)~Yc and Y0+2W0~2+X0-3X0, and also(494)L0s1s1+W0X0-s1⟶12πAi′r0-2X0-1/61-X0-4/3,s1⟶Y0. Thus apart from the factors ρ-1/3L1(·), (492) is just the D- ray expansion expanded near Y→Yc(X) (or s1→Y0), divided by (s1+W0)X0-s1. Thus (492) matches to the D- ray expansion if(495)ρ-1/3L1Y0+W0Yc+W0η∗∗η∗∗→-∞~s1+W0X0-s1s1→Y0.But the left side of (495), in view of (461) and (462), becomes(496)s1+W0X0-s1~1-X02ρ-1/3-η∗∗Yc+W0.Now Y0+W0~1-X0 so that (495) and (496) lead to(497)L1U~1-X0-U,U⟶-∞.Next we match (492) to the D0 ray expansion and thus infer the behavior of L1(U) as U→+∞. Now C0~(1-X0-Y0)ρ-1/2~1-X0ρ-1/2. First evaluating K(X,Y) at Y0=X0-X0 and then letting Y→Yc0 we get(498)KX,YY0=X0-X0Y→Yc0~2πX01/41-X02Yc0X1X0-X0-Yc0X1W0+W0+Yc0X.The estimate in (478) still holds so if C0K(X,Y)eρΨ(X,Y) is to agree, for Y→Yc, with the large η∗∗ asymptotics of (492), we must have(499)Ai′r0-2∫r0∞eδ1uAiudu1-X05/3X01/12L1Y0+W0Yc+W0η∗∗~2X01/41-X03exp⁡r01-X02/3X01/6W0+Y0η∗∗-131-X02X0η∗∗W0+Yc3.We thus have the matching condition(500)L1U~2X1/61-X04/3Ai′r02∫r0∞eδ1uAiudu-1exp⁡r0UX01/61-X01/3-U33X01-X0,U⟶+∞.To determine L1(·) completely we need to match (492) to the corner layer expansion in (173). Using(501)Aiν+θ~2-1π-1/2ν-1/4exp⁡-23ν3/2-νθ in the integral we expand (173) for ν,T→∞ with T-ν=O(1). The result contains some exponential factors and some algebraic ones, with the latter being(502)ρ-1/31-X04/3X0-1/3ν-1/412π12πi∫-i∞i∞eθT-ν-δ1Aiθ2∫θ∞eδ1uAiududθ.Now we evaluate the algebraic part of (492), for X→X0. From (460), since Y0+W0→1-X0=W0, we see that Yc0(X) is the same as Y∗(X), which is independent of Y0. We also have Y∗(X0)=X0-X0 so that (498) becomes singular as X→X0. We have(503)X0-X0-Y∗X=12X-X0+X0-X2-X02-X~1-X01/2ρ-1/3ν1.The result in (485) still applies, now simplifying to(504)η∗∗~X01/61-X01/3ν-T.With (503) and (504), the algebraic part of (492) becomes(505)ρ-1/2e-r0T-ν12πρ-1/6ν-1/4X0-1/4Ai′r0-2∫r0∞eδ1uAiuduL1X01/61-X01/3ν-T.Comparing (502) to (505) determines L1(·) as(506)L1U=X01/61-X04/3Ai′r02∫r0∞eδ1uAiudu-112πi∫-i∞i∞e-θδ1Ai2θ∫θ∞eδ1uAiuduexp⁡-UθX01/61-X01/3dθexp⁡r0UX01/61-X01/3.We have thus established the result in (194). By asymptotically expanding the contour integral in (506) in the limits U→±∞ we can easily verify that (497) and (500) are satisfied.


## 7.3. Transition Layer nearY=Y~(X), Region R1∪R2∪R3

We consider the curve that separatesD0 from D+. We use the scaling(507)Y=Y~X+ρ-1/2η∗,η∗=O1and expand the joint distribution as(508)πk,r~CρFX,η∗K+X,Y~exp⁡ρΨ+X,Y~+ρΨ+,YX,Y~η∗+12Ψ+,YYX,Y~η∗2.The exponential factor in (508) corresponds to the expansions of Ψ+(X,Y) about Y=Y~(X). If (508) as η∗→-∞ is to match to the D+ ray expansion we must have(509)FX,-∞=1,∀0<X<X0.We use (508) in the main balance equation (3) and after a lengthy calculation we find that F(X,η∗) satisfies the parabolic PDE(510)12Y~′2XeΨ+,X+e-Ψ+,X+Y~eΨ+,YFη∗η∗+η∗eΨ+,YFη∗+XeΨ+,X-e-Ψ+,XFX=0,for X∈(0,X0) and η∗∈(-∞,∞). In (510), Ψ+ is understood to be evaluated at (X,Y)=(X,Y~(X)). We can write the curve Y~(X) in parametric form, as(511)X=1-Y0X0+Y0et∗1+X0+Y0-1e-t∗,Y~X=Y0X0+Y0X0+Y0-1+e-t∗,with the latter equation corresponding to (178). Since t∗=t∗(X) we note that(512)XeΨ+,XX,Y~X-e-Ψ+,XX,Y~X=-dXdt∗and define M∗ from(513)M∗=Y~′X2XeΨ+,XX,Y~X+e-Ψ+,XX,Y~X+Y~XeΨ+,YX,Y~X.We can viewM∗ as being a function of either X or t∗. In terms of t∗ we have(514)M∗t∗=Y0e-t∗X0+Y0+2+X0+Y0-1e-t∗-Y0X0+Y0et∗Y0e-t∗Y0et∗+X0+Y0X0+Y0-1e-t∗2.Changing variables from (X,η∗) to (t∗,η∗) the PDE in (510) becomes(515)12M∗Fη∗η∗+η∗1+X0+Y0-1et∗Fη∗=Ft∗,t∗>0and note that t∗=0 corresponds to X=X0, as Y~(X0)=Y0.Next we assume thatF(t∗,η∗) will be a function of a single “similarity” variable, which we call ⋆=η∗a(t∗) and with which (515) becomes the ordinary differential equation(516)12M∗a2t∗F0′′⋆+Bt∗-a′t∗at∗⋆F0′⋆=0,where(517)F0η∗at∗=Ft∗,η∗,Bt∗=1+X0+Y0-1et∗-1. Such a similarity solution is possible if a(t∗) satisfies the nonlinear ODE(518)Bt∗-a′t∗at∗=12M∗t∗a2t∗.Setting a(t∗)=[b(t∗)]-1/2, the Bernoulli equation in (518) becomes the linear equation(519)b′t∗+2Bt∗bt∗=M∗t∗.Solving (519) subject to b(0)=0 leads to the expression in (184). With (518), (516) becomes(520)F0′′⋆+⋆F0′⋆=0and (509) implies that F(-∞)=1, and thus(521)F0η∗bt∗=12π∫η∗/bt∗∞e-u2/2du.We have thus derived (176).With (521) substituted for F(X,η∗) in (508), we can show that as η∗→+∞, (508) asymptotically matches to the D0 ray expansion. As η∗→∞ from (521) we have(522)F0η∗bt∗~bt∗2π1η∗exp⁡-η∗22bt∗,η∗⟶∞.We can easily establish the continuity conditions(523)ΨX,Y~X=Ψ+X,Y~X-Ψ+X0,Y0,ΨYX,Y~X=Ψ+,YX,Y~Xand, after a lengthy calculation, show that(524)Ψ+,YYX,Y~X=1bt∗+ΨYYX,Y~X.Thus expanding the D0 ray expansion, C0K(X,Y)eρΨ(X,Y), as Y↓Y~(X) and comparing this to the large η∗ expansion of (508), using (523) and (524), lead to(525)C0KX,Y~CK+X,Y~Xbt∗2πeρΨ+X0,Y0ρY-Y~X,Y⟶Y~X.But from Proposition 2 we see that, for R1∪R2∪R3,(526)CC0~2πρX0+Y0exp⁡ρ1-X0-Y0+X0+Y0log⁡X0+Y0,Ψ+X0,Y0=-1+X0+Y0-X0+Y0log⁡X0+Y0. The curve Y=Y~(X) corresponds to the ray A=Y0/(X0+Y0), and then B=1-X0-Y0. From (70) we see that K becomes singular as Y→Y~(X), due to the factor Y0-A(X0+Y0)-1. This singularity precisely matches that in the right-hand side of (525), and yet another lengthy calculation shows that(527)limY→Y~X⁡Y-Y~XKX,Y=X0+Y0bt∗K+X,Y~X, which establishes the matching.We note that the PDE (515) contains many solutions other than the similarity solution. An initial condition as t∗→0 can be obtained by asymptotically matching (508) to the corner layer valid on the (n,p) scale. This will determine the solution to (515) uniquely, but the matching will again lead to the conclusion that F is given by (521). Below we only briefly verify that the matching holds. Near the corner the curve Y=Y~(X) can be approximated by the straight line Y-Y0=Y~′(X0)(X-X0), which corresponds to(528)Y0-YX0-X=pn=Y0X0+Y02-X0=1θ1,where θ1 was defined in Proposition 20. Thus we must show that (508) for X→X0 is the same as (251). The exponential parts agree automatically and C(ρ)~ρ-1/2Q(0,0), so we must only show that, for X→X0,(529)K+X,Y~X0+Y0∫η∗/bt∗∞e-u2/2du~X0+Y02-X0X0+Y0212π∫-∞ηe-u2/2du.But(530)K+X0,Y0=2π-1/2X0+Y02-X0X0+Y0-5/2,η∗=ρY-Y~X=ρY-Y0+Y0-Y~X=-pρ+ρY0-Y~X~1ρ-p+Y0X0+Y02-X0n. From (184)–(187) we obtain, as X→X0 or t∗→0,(531)bt∗~Y0X0+Y0X0+Y03+X0+Y0Y0-2X0+X0X0+Y02-X03nρsince n/p~θ1, as t∗→0η∗/b(t∗) approaches -η in (252), which verifies (529). This completes the analysis of the transition from D+ to D0.


## 8. Numerical Studies and Discussion

Next we show that our asymptotic results can also be used to accurately estimate the probabilitiesπ(k,r) in ranges where there is little mass. We will consider π(0,R) which is the (very unlikely) situation where no primary spaces are occupied but all of the secondary spaces are full. To estimate π(0,R) we must use the expansion that is valid for k=O(1) and p=R-r=O(1), and this corresponds to (130). Thus we set k=p=0 in (130) and note that Q(0,0) has the different expansions in Proposition 1, according to regions in parameter space. In Table 1 we take X0=2, Y0=1 and use the expression for Q(0,0) that holds in parameter region R1. We see that the exact and asymptotic results agree to three decimal places even if ρ is as small as 2. In Table 2 we have X0=0.8 and Y0=0.4, and Q(0,0) is again computed from (30). Now the agreement is not quite as good as in Table 1, and we typically have errors of about 10%. This is probably due to the fact that numerically X0+Y0=1.2, which exceeds one only slightly, so it may be preferable to use (29) to approximate Q(0,0). In Tables 3 and 4 we take, respectively, X0=0.4, Y0=0.4 and X0=0.4, Y0=0.2. Table 3 corresponds to region R3 and Table 4 to region R4. The agreement is certainly better for region R4, as again the sum X0+Y0 is further away from the critical value of one. Here we used Q(0,0)~1-X0-Y0 for region(s) R3∪R4.Table 1
π ( 0 , R ) for region R1, X0=2, Y0=1.
ρ Exact Asymptotic 2 5.41 (10-4) 5.41 (10-4) 4 2.54 (10-7) 2.54 (10-7) 6 1.19 (10-10) 1.19 (10-10) 8 5.58 (10-14) 5.58 (10-14) 10 2.62 (10-17) 2.62 (10-17)Table 2
π ( 0 , R ) for region R2, X0=0.8, Y0=0.4.
ρ Exact Asymptotic 5 1.47 (10-3) 1.13 (10-3) 10 1.70 (10-6) 1.35 (10-6) 15 1.96 (10-9) 1.61 (10-9) 20 2.27 (10-12) 1.92 (10-12) 25 2.65 (10-15) 2.29 (10-15) 30 3.09 (10-18) 2.73 (10-18) 35 3.63 (10-21) 3.25 (10-21)Table 3
π ( 0 , R ) for region R3, X0=0.4, Y0=0.4.
ρ Exact Asymptotic 10 1.86 (10-4) 1.08 (10-4) 20 2.29 (10-8) 1.56 (10-8) 30 2.64 (10-12) 1.94 (10-12) 40 2.96 (10-16) 2.28 (10-16) 50 3.25 (10-20) 2.59 (10-20) 60 3.53 (10-24) 2.89 (10-24) 70 3.81 (10-28) 3.18 (10-28)Table 4
π ( 0 , R ) for region R4, X0=0.4, Y0=0.2.
ρ Exact Asymptotic 10 5.13 (10-4) 4.17 (10-4) 20 1.33 (10-7) 1.17 (10-7) 30 3.12 (10-11) 2.83 (10-11) 40 6.99 (10-15) 6.48 (10-15) 50 1.53 (10-18) 1.43 (10-18) 60 3.29 (10-22) 3.11 (10-22) 70 6.98 (10-26) 6.66 (10-26)Tables1–4 show that the very small values of π(0,R) are well predicted by the asymptotic formula(s).To summarize, we have done a rather thorough asymptotic analysis for this storage allocation model. We have shown that as long asX0+Y0>1 (m+R>ρ) the effects of the finiteness of the secondary storage capacity occur only for (X,Y)∈D0 and for those state space regions that border D0. However, for X0+Y0<1 (m+R<ρ) the entire state space is affected by the finite capacity, as then there are not enough storage spaces to satisfy the demand for them. For X0+Y0<1 it proves useful to consider the numbers, m-N1 and R-N2, of empty primary and secondary spaces, and then the steady state distribution π(k,r)=π(m-n,R-p) has for ρ→∞ the limiting form in (50). Even though the marginal distributions of N1 and the sum N1+N2 are particularly simple, the joint distribution of (N1,N2) is quite complicated, as the analysis involves many different ranges of the parameter space (cf. Figure 2) and the state space (cf. Figures 3–6). In some ranges special functions such as Airy and parabolic cylinder functions play a key role.



---

*Source: 1925827-2016-08-31.xml*