# On the Nonsymmetric Longer Queue Model: Joint Distribution, Asymptotic Properties, and Heavy Traffic Limits

Authors: Charles Knessl; Haishen Yao
Journal: Advances in Operations Research
 (2013)

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## Abstract

We consider two parallel queues, each with independent
Poisson arrival rates, that are tended by a single server. The exponential server devotes all of its capacity to the longer of the queues. If both
queues are of equal length, the server devotesν of its capacity to the first queue and the remaining 1−ν to the second. We obtain exact integral representations for the joint probability distribution of the number of
customers in this two-node network. Then we evaluate this distribution
in various asymptotic limits, such as large numbers of customers
in either/both of the queues, light traffic where arrivals are infrequent,
and heavy traffic where the system is nearly unstable.


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## Body

## 1. Introduction

We consider a nonsymmetric version of the longer queue model. Here there are two parallel queues, each fed by a Poisson arrival stream. There is but a single server who tends to the longer of the two queues. If the number of customers in each queue is the same, then the server devotesν of its capacity to the first queue and 1-ν to the second queue, with 0<ν<1. We let N1 (N2) denote the number of customers in the first (second) queue, the two arrival rates are λ1 and λ2, and the server works at rate μ. Note that the total number of customers, N1+N2, in the two-node network behaves as the standard M/M/1 model, so in the steady state we have the geometric distribution Prob[N1+N2=J]=(1-ρ1-ρ2)(ρ1+ρ2)J, where ρ1=λ1/μ,ρ2=λ2/μ, assuming the stability condition ρ1+ρ2<1. The “symmetric case” corresponds to λ1=λ2 (thus ρ1=ρ2) and ν=1/2, and this was analyzed in detail by Flatto [1].Such models were proposed by Zheng and Zipkin [2] to study problems in inventory control. In [2] finite capacities were assumed in the two queues, and the authors studied numerically the steady state probabilities p(m,n)=Prob[N1=m,N2=n], in terms of the capacity size and also for different service disciplines, such as the longer queue (LQ) discipline here, and also the first-come-first-served discipline.In [1] the author used two-dimensional generating functions and analyticity arguments and obtained explicit expressions for p(m,n), in the symmetric case, as contour integrals. Then asymptotic results were derived for the joint distribution p(m,n), as m and/or n becomes large, and also the marginal tails and various conditional limit laws were obtained. In this paper we generalize some of the results of [1] to the nonsymmetric model, and we will show that now many of the asymptotic results become quite different. As in [1] we assume that the model is preemptive, so if N1=N2 (and thus the server works on both queues) and a new arrival occurs to the first queue, then the server switches immediately all its capacity to the first queue. The more difficult nonpreemptive version of the LQ model was studied by Cohen [3], in the case of Poisson arrivals and general service times. The problem is reduced in [3], using generating functions, to a functional equation which is furthermore converted to a Riemann-Hilbert boundary value problem.The present problem corresponds to a random walk in the quarter plane (asN1,N2⩾0), and other examples of such problems include shortest queue (SQ) problems [4–6], fork-join models [7, 8], two coupled processors with generalized processor sharing [9–11], and two coupled tandem queues [12]. General techniques for solving such problems are discussed in [13–15] and they involve functional equations, analyticity arguments, and singular integral equation methods [16].For the present model the analyticity arguments are fairly simple, and we focus mostly on the asymptotic properties of the solution. We will show that these asymptotics are quite different from those of the symmetric model in [1]. After obtaining exact integral representations for p(m,n), and also the marginal probabilities P(m)=Prob[N1=m] and 𝒫(n)=Prob[N2=n], we asymptotically evaluate these integrals for m and/or n large. We use standard techniques, such as the Laplace method, saddle point method, singular analysis, and the Euler-MacLaurin formula. Good general references on the asymptotic evaluation of integrals and sums are the books [17–21].In addition to this model being interesting on its own, many variants of shortest queue problems, such as ones with multiple servers and finite capacities, can be asymptotically reduced to LQ models of the type considered here (see [22, 23]). For example, in [23] we showed that the finite capacity version of the standard symmetric SQ model (analyzed in [4, 5]), where N1,N2⩽K and K is the capacity, asymptotically reduces to the symmetric LQ model in [1], if we consider the process (K-N1,K-N2), which measures the number of spots available in the two waiting rooms. Then having a thorough understanding of the nonsymmetric LQ model and its asymptotics will allow us to analyze, at least in some asymptotic limits, nonsymmetric variants of SQ models.The remainder of this paper is organized as follows. In Section2 we summarize all of the main results, both exact and asymptotic. They are listed in Theorems 1–5, and some discussion/interpretation appears following each theorem. In Section 3 we briefly derive the exact expressions for p(m,n); in Section 4 we derive asymptotic properties of p(m,n) for m,n large. In Section 5 we derive light traffic (where ρ1,ρ2→0) and heavy traffic (where ρ1+ρ2↑1) results.


## 2. Problem Statement and Summary of Results

We let(N1,N2) be the numbers of customers in the two parallel queues, and let p(m,n)=Prob[N1=m,N2=n] be the joint queue length distribution in the steady state. The two arrival rates are λ1 and λ2, the exponential server works at rate μ, and ρ1=λ1/μ,ρ2=λ2/μ. If N1>N2(N1<N2) the server works on the first (second) queue, but if N1=N2 the server works at rate μν on the first queue and rate μ(1-ν) on the second, with 0<ν<1. The symmetric case corresponds to λ1=λ2 and ν=1/2. We henceforth assume the stability condition ρ1+ρ2<1.In Figure1 we sketch the transition rates for the random walk (N1,N2), which illustrates the discontinuity along the “interface” N1=N2. The main balance equations are
(1)(ρ1+ρ2+1)p(m,n)=ρ1p(m-1,n)+ρ2p(m,n-1)+p(m+1,n),m>n+1,n⩾1,(2)(ρ1+ρ2+1)p(m,n)=ρ1p(m-1,n)+ρ2p(m,n-1)+p(m,n+1),n>m+1,m⩾1.
We also have the following three interface equations:
(3)(ρ1+ρ2+1)p(n+1,n)=ρ1p(n,n)+ρ2p(n+1,n-1)+p(n+2,n)+(1-ν)p(n+1,n+1),n⩾1,(4)(ρ1+ρ2+1)p(n-1,n)=ρ1p(n-2,n)+ρ2p(n-1,n-1)+p(n-1,n+1)+νp(n,n),n⩾2,(5)(ρ1+ρ2+1)p(n,n)=ρ1p(n-1,n)+ρ2p(n,n-1)+p(n+1,n)+p(n,n+1),n⩾1,
the two boundary conditions
(6)(ρ1+ρ2+1)p(m,0)=ρ1p(m-1,0)+p(m+1,0),m⩾2,(7)(ρ1+ρ2+1)p(0,n)=ρ2p(0,n-1)+p(0,n+1),n⩾2,
and the corner conditions
(8)(ρ1+ρ2+1)p(1,0)=ρ1p(0,0)+(1-ν)p(1,1)+p(2,0),(ρ1+ρ2+1)p(0,1)=ρ2p(0,0)+νp(1,1)+p(0,2),(ρ1+ρ2)p(0,0)=p(1,0)+p(0,1).
The normalization is
(9)∑n=0∞‍∑m=0∞‍p(m,n)=∑n=0∞‍p(n,n)+∑m>n‍p(m,n)+∑m<n‍p(m,n)=1.
Note that the elementary difference equations (6) and (7) may be solved immediately to obtain p(m,0) and p(0,n), up to multiplicative constants. In Section 3 we analyze (1)–(9) to obtain p(m,n) in the forms of contour integrals, which we summarize below.Figure 1
A sketch of the transition rates for the random walk.Theorem 1.
Forρ1+ρ2<1, the steady state distribution is as follows:(i)
m>n:
(10)p(m,n)=1-ρ1-ρ22πi∮‍f(w)wn+1[Z+(w)]m-ndw,where the integral is over a small loop aboutw=0,
(11)Z+(w)=12ρ1[ρ1+ρ2+1+(ρ1+ρ2+1)2-4ρ1(ρ2w+1)],(12)f(w)=numf(w)den(w),(13)numf(w)=-2ρ1ρ2w+ρ22(1-ν)+ρ1ρ2-ρ2(1-ν)+(ρ12+ρ1)ν+[ρ2-(ρ1+ρ2)ν](ρ1+ρ2+1)2-4ρ2(ρ1w+1),(14)den(w)=[ρ1ν+ρ2(1-ν)](ρ1+ρ2+1)-ρ1(ρ2w+ν)(ρ1+ρ2+1)2-4ρ2(ρ1w+1)-ρ2(ρ1w+1-ν)(ρ1+ρ2+1)2-4ρ1(ρ2w+1).(ii)
m<n:
(15)p(m,n)=1-ρ1-ρ22πi∮‍g(w)wm+1[Z+*(w)]n-mdw,(16)g(w)=numg(w)den(w),(17)numg(w)=-2ρ1ρ2w+ρ22(1-ν)+ρ1ρ2+ρ2(1-ν)+(ρ12-ρ1)ν+[(ρ1+ρ2)ν-ρ2]×(ρ1+ρ2+1)2-4ρ1(ρ2w+1),(18)Z+*(w)=12ρ2[ρ1+ρ2+1+(ρ1+ρ2+1)2-4ρ2(ρ1w+1)].(iii)
m=n:
(19)p(n,n)=1-ρ1-ρ22πi∮‍h(w)wn+1dw,(20)h(w)=num(w)den(w),(21)num(w)=-2ρ1ρ2w+(ρ1+ρ2+1)[ρ1ν+ρ2(1-ν)]-ρ1ν(ρ1+ρ2+1)2-4ρ2(ρ1w+1)-ρ2(1-ν)(ρ1+ρ2+1)2-4ρ1(ρ2w+1).We next evaluatep(m,n) in various asymptotic limits, to gain more insight into the structure of the joint distribution. Writing p(m,n)=p(m,n;ρ1,ρ2,ν) to emphasize the dependence on the model parameters, we clearly have the symmetry relation
(22)p(m,n;ρ1,ρ2,ν)=p(n,m;ρ2,ρ1,1-ν).
Thus it is sufficient for the asymptotics to assume that
(23)ρ1⩾ρ2,
and we also note that the expressions in Theorem 1 are consistent with (22).We will show that the asymptotics are quite different whether(ρ1+ρ2)2<ρ1-ρ2, (ρ1+ρ2)2>ρ1-ρ2 or (ρ1+ρ2)2≈ρ1-ρ2. In Figure 2 we sketch the curve (ρ1+ρ2)2=ρ1-ρ2 in the (ρ1,ρ2) parameter plane, for ρ1+ρ2<1. Note that the curve may also be written as
(24)ρ2=12[-1-2ρ1+1+8ρ1]
and passes through the points (0,0) and (1,0). We first give results for p(m,n) when (ρ1+ρ2)2>ρ1-ρ2, and note that this includes the symmetric case of Flatto [1].Figure 2
A sketch of the parameter domain and the transition curve(ρ1+ρ2)2=ρ1-ρ2.Theorem 2.
Forρ1⩾ρ2 and (ρ1+ρ2)2>ρ1-ρ2 one lets R=m/n be the queue length ratio. The following asymptotic expressions then hold:(i)
n=O(1), m→∞(25)p(m,n)~(1-ρ1-ρ2)f(0)mnn!(ρ2(ρ1+ρ2+1)2-4ρ1)n×(2ρ1ρ1+ρ2+1+(ρ1+ρ2+1)2-4ρ1)m.(ii)
m,n→∞ with R=m/n∈(Rc,∞),
(26)Rc=1+[(ρ1+ρ2)2+ρ2][(ρ1+ρ2)2+ρ2-ρ1]ρ1ρ2>1,(27)p(m,n)~1-ρ1-ρ22πnf0(R)M(R)(1ws)n[1Z+(ws)]m-n,where(28)ws=ws(R)=({R2}(ρ1+ρ2+1)2R-4(R+1)ρ1+(ρ1+ρ2+1)R2(ρ1+ρ2+1)2-4ρ1(R2-1))×(2ρ1ρ2(R+1)2)-1,(29)Z+(ws)=12ρ1(R+1)×[{R2(ρ1+ρ2+1)2-4ρ1(R2-1)}R(ρ1+ρ2+1)+R2(ρ1+ρ2+1)2-4ρ1(R2-1)],(30)M(R)=R+1R-1×R2(ρ1+ρ2+1)2-4ρ1(R2-1)R2(ρ1+ρ2+1)2-4ρ1(R2-1)-(ρ1+ρ2+1)andf0(R)=f(ws)=f(ws(R)) can be computed from (12)–(14).(iii)
m,n→∞ with R=Rc+ξ/n (thus R-Rc=O(n-1/2))
(31)p(m,n)~C+(ρ1+ρ2)2n[ρ1(ρ1+ρ2)(ρ1+ρ2)2+ρ2]m-n×12π∫ξ1∞e-u2/2du,(32)ξ1=ξ1Rc-1[RcRc-1+2ρ1ρ2((ρ1+ρ2)2+ρ2-ρ1)2]-1/2,(33)C+=(1-ρ1-ρ2)×(ρ1+ρ2)4-(ρ1-ρ2)2ρ1(ρ1+ρ2)2×(ρ1+ρ2)2(1-ν)+ρ1(ρ1+ρ2)2+ρ2ν+ρ1(1-ν).(iv)
m,n→∞ with R=m/n∈(1,Rc)(34)p(m,n)~C+(ρ1+ρ2)2n[ρ1(ρ1+ρ2)(ρ1+ρ2)2+ρ2]m-n.(v)
m=n→∞(35)p(n,n)~C0(ρ1+ρ2)2n,(36)C0=1-ρ1-ρ2(ρ1+ρ2)2(ρ1+ρ2)4-(ρ1-ρ2)2(ρ1+ρ2)2+ρ2ν+ρ1(1-ν).(vi)
m,n→∞ with R=m/n∈(Rc*,1),
(37)Rc*≡ρ1ρ2ρ1ρ2+[(ρ1+ρ2)2+ρ1-ρ2][(ρ1+ρ2)2+ρ1]<1,(38)p(m,n)~C-(ρ1+ρ2)2m[ρ2(ρ1+ρ2)(ρ1+ρ2)2+ρ1]n-m,(39)C-=(1-ρ1-ρ2)(ρ1+ρ2)4-(ρ1-ρ2)2ρ2(ρ1+ρ2)2×ρ2+ν(ρ1+ρ2)2(ρ1+ρ2)2+ρ2ν+ρ1(1-ν).(vii)
m,n→∞ with R=Rc*+η/n (thus R-Rc*=O(n-1/2))
(40)p(m,n)~C-(ρ1+ρ2)2m×[ρ2(ρ1+ρ2)(ρ1+ρ2)2+ρ1]n-m12π∫-∞η1e-u2/2du,(41)η1=η11-Rc*[Rc*1-Rc*+2ρ1ρ2Rc*((ρ1+ρ2)2+ρ1-ρ2)2]-1/2.(viii)
m,n→∞ with R=m/n∈(0,Rc*)(42)p(m,n)~1-ρ1-ρ22πng0(R)M*(R)(1ws*)m[1Z+*(ws*)]n-m,(43)ws*=ws*(R)=(R[{(ρ1+ρ2+1)2-4ρ2(1-R2)}(ρ1+ρ2+1)2-4ρ2(R+1)+(ρ1+ρ2+1)(ρ1+ρ2+1)2-4ρ2(1-R2)])×(2ρ1ρ2(R+1)2)-1,(44)Z+*(ws*)=12ρ2(R+1)×[1+ρ1+ρ2+(1+ρ1+ρ2)2-4ρ2(1-R2)],(45)M*(R)=R1+R1-R×(ρ1+ρ2+1)2-4ρ2(1-R2)(ρ1+ρ2+1)2-4ρ2(1-R2)-R(ρ1+ρ2+1)andg0(R)=g(ws*)=g(ws*(R)) can be computed from (14), (16), and (17).(ix)
m=O(1), n→∞(46)p(m,n)~(1-ρ1-ρ2)g(0)nmm!(ρ1(ρ1+ρ2+1)2-4ρ2)m×(2ρ21+ρ1+ρ2+(1+ρ1+ρ2)2-4ρ2)n.Note that, in view of (12) and (16),
(47)f(0)=({(p+1)2}ρ22(1-ν)+ρ1ρ2-ρ2(1-ν)+(ρ12+ρ1)ν+[ρ2-(ρ1+ρ2)ν](ρ1+ρ2+1)2-4ρ2)×({(p+1)2}(ρ1+ρ2+1)[ρ1ν+ρ2(1-ν)]-ρ1ν(ρ1+ρ2+1)2-4ρ2-ρ2(1-ν)(ρ1+ρ2+1)2-4ρ1)-1,g(0)=({(p+1)2}ρ22(1-ν)+ρ1ρ2+ρ2(1-ν)+(ρ12-ρ1)ν+[(ρ1+ρ2)ν-ρ2](ρ1+ρ2+1)2-4ρ2)×({(p+1)2}(ρ1+ρ2+1)[ρ1ν+ρ2(1-ν)]-ρ1ν(ρ1+ρ2+1)2-4ρ2-ρ2(1-ν)(ρ1+ρ2+1)2-4ρ1)-1.In Theorem2 we listed the expansions of p(m,n) in order of decreasing R, from R=∞ in (25) to R=0 in (46). In the symmetric case ρ1=ρ2 and ν=1/2, and then p(m,n)=p(n,m) and Theorem 2 is consistent with the results of Flatto in [1] (there the cases m=O(1), n→∞ and n=O(1), m→∞ were not considered, and the limits where R≈Rc,Rc* were not treated in as much detail). Note that (34) and (38) correspond to “product form” approximations to p(m,n). The expressions in (25) and (46) are actually exact when n=0 and m=0, respectively, in view of our comments below (9). Our analysis in Section 4 will also indicate how to compute higher order correction terms in the various asymptotic series.Next we take(ρ1+ρ2)2<ρ1-ρ2. Now some of the asymptotic results for p(m,n) will be very different, while in other ranges of R=m/n they will be similar to those in Theorem 2. We emphasize below the formulas that are different.Theorem 3.
Forρ1>ρ2 and (ρ1+ρ2)2<ρ1-ρ2 one has the following asymptotic expansions:(i)
m,n→∞ with m>n and m-n=O(1)(48)p(m,n)~C¯+wbπn-3/2wb-n(2ρ1ρ1+ρ2+1)m-n×[2ρ1ρ2ρ1+ρ2+1(m-n)+Δ+],(49)wb=(ρ1+ρ2+1)2-4ρ14ρ1ρ2,(50)C¯+=1-ρ1-ρ221-ρ1-ρ2ρ1-ρ2-(ρ1+ρ2)×ρ1-ρ2+1/2-(ν-1/2)(ρ1+ρ2)(1/2)(ρ1+ρ2+1)-(ν-1)ρ1-ρ2,(51)Δ+=(2ρ1ρ2[{14(ρ1+ρ2+1)2}ρ2(1-ν)-ρ1+14(ρ1+ρ2+1)2])×([{12}ρ1-ρ2-(ρ1+ρ2)]×[1-ρ1-ρ2]×[12(ρ1+ρ2+1)-(ν-1)ρ1-ρ2])-1.(ii)
m=n→∞(52)p(n,n)~C¯wbπn-3/2wb-n,(53)C¯=({[12]}(1-ρ1-ρ2)wb(ρ1ρ2)3/2×[12+ρ1-ρ2-(ν-12)(ρ1+ρ2)])×({[12ρ1-ρ2]2}[ρ1-ρ2-(ρ1+ρ2)]2×[12(ρ1+ρ2+1)-(ν-1)ρ1-ρ2]2)-1.(iii)
m,n→∞ with R=m/n∈(Rd,1) (including the limit n-m=O(1)),
(54)Rd=(ρ1+ρ2+1)2-4ρ1(ρ1+ρ2+1+2ρ1-ρ2)2-4ρ2<1,(55)p(m,n)~C¯-(R-Rd)-3/2wbπn-3/2wb-m×(2ρ2ρ1+ρ2+1+2ρ1-ρ2)n-m,(56)C¯-=(ρ1-ρ2)3/4(1-ρ1-ρ2)ρ1ρ2×[ρ1+ρ2+1+2ρ1-ρ2(1/4)(ρ1+ρ2+1+2ρ1-ρ2)2-ρ2]3/2×([12+ρ1-ρ2+(ρ1+ρ2)(12-ν)]×[14(ρ1+ρ2+1)2-ρ1(1-ν)])×({[(12)]2}[ρ1-ρ2-(ρ1+ρ2)]2×[12(ρ1+ρ2+1)+ρ1-ρ2(1-ν)]2)-1.(iv)
m,n→∞ with R=Rd+S/n (thus R-Rd=O(n-1/2))
(57)p(m,n)=1-ρ1-ρ22πnwb-m[Z+*(wb)]m-n×[g(wb)Mde-S12/2-KwbMd3/4e-S12/4×D1/2(-S1)n1/4+O(n-1/2)],(58)Md=Rd1-Rd+Rdρ1ρ2wb2(ρ1-ρ2),(59)S1=11-Rd1MdS,(60)K=(2ρ1ρ2[12+ρ1-ρ2+(ρ1+ρ2)(12-ν)]×[14(ρ1+ρ2+1)2-ρ1(1-ν)])×({[12(ρ1+ρ2+1)+ρ1-ρ2(1-ν)]2}[ρ1-ρ2-(ρ1+ρ2)]2×[12(ρ1+ρ2+1)+ρ1-ρ2(1-ν)]2)-1,andD1/2(·) is the parabolic cylinder function of order 1/2.(v)
Forn=O(1) and m→∞, (25) applies. For m,n→∞ with R=m/n∈(1,∞), (27)–(30) hold. For m,n→∞ with R=m/n∈(0,Rd), (42)–(45) hold. For m=O(1) and n→∞, (46) applies.Thus when(ρ1+ρ2)2<ρ1-ρ2 the asymptotics of p(m,n) are different in eight ranges of R=m/n, but the final result in four of these is the same as the case (ρ1+ρ2)2>ρ1-ρ2. Note however that if (ρ1+ρ2)2<ρ1-ρ2 (27) applies for all R>1, so the transition for R≈Rc is now absent. Also, if (ρ1+ρ2)2<ρ1-ρ2, (42) applies for R∈(0,Rd) rather than R∈(0,Rc*). In (57), g(wb) is computed by setting w=wb, cf. (49), in (16), and we also note that 2ρ2Z+*(wb)=ρ1+ρ2+1+2ρ1-ρ2.The results in (49)–(60) are very different in form from the symmetric case asymptotics in [1]. For example, the diagonal probabilities in (52) contain the additional algebraic factor n-3/2. In (57) the parabolic cylinder function can be computed, for example, from
(61)Dp(Z)=eZ2/4i2π∫Bre-Zueu2/2updu,
where Br is a vertical contour in the complex u-plane with Re(u)>0, and we have p=1/2 for the correction term in (57). Also, D0(Z)=e-Z2/4 and for p=N=0,1,2,3,… we can express DN(Z) in terms of a Hermite polynomial. In (57) we give a two-term asymptotic approximation in the transition range R≈Rd. Also, it may be shown that the leading term in (57) is just the limiting form of (42), as R→Rd. The expression in (55) applies both to m,n→∞ with Rd<R<1, and to m,n→∞ with n-m=O(1). In the latter case we can approximate the factor (R-Rd)-3/2 by (1-Rd)-3/2. The parabolic cylinder function D1/2(-S1) has the asymptotic behaviors
(62)D1/2(-S1)~e-S12/4-S1,S1⟶-∞,D1/2(-S1)~eS12/4S1-3/2-12,S1⟶+∞,
so as S1→+∞ in (57) (corresponding to R>Rd) the O(n-1/4) correction term begins to dominate the leading term (which occurs for e-S12/2≪n-1/4 or |S1|≫(1/2)logn). Then we can show that the expression in (57) for S1→+∞ agrees with (55) for R↓Rd, and hence the two expansions asymptotically match. Note also that the scaling in (57) has m=nRd+O(n).Next we study the transition range(ρ1+ρ2)2≈ρ1-ρ2 in parameter space. This will lead to a new set of asymptotic results which will show, for example, how the formula in (52) for p(n,n) changes to the purely geometric approximation in (35). To quantify the closeness to the curve (ρ1+ρ2)2=ρ1-ρ2 in Figure 2 we write
(63)(ρ1+ρ2)2-(ρ1-ρ2)=δβ,
where δ→0+ and β can have either sign. If β=0 we are exactly on the transition curve. For small δ, in certain ranges of R=m/n, the results in Theorems 2 and 3 still apply. For example, if m=O(1) and n→∞ or n=O(1) and m→∞ Theorems 2 and 3 agree, and then no transition range result is needed. As (ρ1+ρ2)2↓ρ1-ρ2, Rc↓1 in (26), and then (27) will hold for all R∈(1,∞) in the transition case. Thus for R=m/n>1 the transition case will require a new asymptotic result only if R≈1. For R<1 the asymptotic result in (42) will apply for R<Rc*, where now, since (ρ1+ρ2)2≈ρ1-ρ2, Rc*≈Rd (cf. (37) and (54)). Then if R<1, we will need different asymptotic results only when R∈[Rd0,1], including R≈Rd0 and R≈1, where
(64)Rd0=1-a21+4a+11a2,a=ρ1+ρ2.
Note that Rd0 is the limit of both Rd and Rc*, if we replace ρ1-ρ2 by (ρ1+ρ2)2=a2, as (37) leads to (setting a=ρ1+ρ2, b=ρ1-ρ2)
(65)Rc*=ρ1ρ2ρ1ρ2+[(ρ1+ρ2)2+ρ1-ρ2][(ρ1+ρ2)2+ρ1]=(1/4)(a2-b4)(1/4)(a2-b4)+(a2+b2)[a2+(1/2)(a+b2)]⟶(1/4)a2(1-a2)(1/4)a2(1-a2)+a2(3a2+a)=1-a21+4a+11a2,
since δ→0+ corresponds to b→a. Similarly, (54) leads to Rd→Rd0 in this limit.Since we will now have the state variablesm and n large, and also δ small, it is necessary to relate these. In Theorem 4 we summarize the transition case results, scaling m and n in terms of δ.Theorem 4.
Forρ1>ρ2 and (ρ1+ρ2)2-(ρ1-ρ2)=βδ with δ→0+, one has the following, where a=ρ1+ρ2 and b=ρ1-ρ2.(i)
m=n→∞, n=y/δ2=O(δ-2)(66)p(n,n)~1πn2(1-a)1-a23a+1-2aνwb-n×[1+2β1yeβ12y∫-∞β1ye-u2du],(67)β=a1-a2β1.(ii)
n=y/δ2=O(δ-2),m-n=z/δ=O(δ-1),m>n(68)p(m,n)~2(1-a)1-a1+a1πnwb-n(2ρ1ρ1+ρ2+1)m-n×[e-z~2/4+2β1yeβ12ye-z~β1y∫-∞β1y-z~/2e-u2du],(69)z~=1-a1+azy=1-a1+am-nn=O(1).(iii)
n=y/δ2=O(δ-2),R=m/n∈(Rd0,1],m<n(70)p(m,n)~2(1-a)1-a1+a1πn1-a+2νa3a+1-2νa1-Rd0R-Rd0×wb-m[Z+*(wb)]m-n×[1+2β2yeβ22∫-∞β2ye-u2du],(71)β=a1-a21-Rd0R-Rd0β2.(iv)
m,n→∞, R=m/n=Rd0+S0/n (thus R-Rd0=O(n-1/2))
(72)p(m,n)~2(1-a2)π1-a+2νa1+3a-2νawb-m[Z+*(wb)]m-n×[Rd0(11-Rd0+1-a28a2)]-1/4n-1/4e-(S10)2/4×D-1/2(-S10),(73)S10=1Rd0(1-Rd0)[11-Rd0+1-a28a2]-1/2S0,andD-1/2(·) is the parabolic cylinder function of order -1/2  (see (61)).(v)
m,n→∞, n=y1/δ4=O(δ-4), R=Rd0+S0/n=Rd0+O(δ2)(74)p(m,n)~21-a2n1/41-a+2νa1+3a-2νa[Rd0(11-Rd0+1-a28a2)]-1/4×wb-m[Z+*(wb)]m-nℒ(S10;y11/4β3),(75)β3=βa1-a2(Rd0)-1/4[11-Rd0+1-a28a2]-1/4andℒ is defined by the contour integral
(76)ℒ(Z;γ)=12πi∫Br+eZueu2/2-γ+udu,whereBr+ is to the right of all singularities of the integrand, including the pole at u=γ2 if γ>0.If(ρ1+ρ2)2=ρ1-ρ2 then β=β1=0 and the term with the integral in (66) is absent. In Theorem 4 we wrote most results in terms of a=ρ1+ρ2 and β. As β(orβ1)→+∞ it can be shown that (66) asymptotically matches to (35), and as β→-∞, (66) matches to (52). Similarly as β→∞ (68) matches to (27) while as β→-∞ (68) matches to (48). Some of these asymptotic matching relations are further discussed in Section 4. In (70), β2 depends on β and also on the ratio R=m/n, in view of (71). The results in Theorem 4 show that for δ→0+ it is natural to scale n=O(δ-2), and then either m=O(δ-2) independently, or m-n=O(δ-1). The expression in (74) is similar to, but somewhat more complicated than, that in (72). The function ℒ in (76) reduces to (2π)-1/2D-1/2(Z)e-Z2/4 in the limit γ→0, and thus (72) may be viewed as a special case of (74)–(76). Note also that the results in Theorem 4 can be rewritten without introducing β and δ in (63), as for example
(77)β1y=1a1-a2×[(ρ1+ρ2)2-(ρ1-ρ2)]n,a=ρ1+ρ2,
and β3y11/4 is proportional to βδn1/4, and βδ may be written in terms of the original parameters ρ1,ρ2.Next we consider some different asymptotic limits, those of “light” and “heavy” traffic. Light traffic corresponds to infrequent arrivals, whereρ1,ρ2→0. Heavy traffic corresponds to nearly unstable systems, where ρ1+ρ2↑1. It turns out that the present model has two possible heavy traffic limits. In the first, which we call HTL1, we have ρ1+ρ2↑1 with a fixed 0<ρ2<1. Then most of the probability mass will occur in the range where m and n are large, but with m-n=O(1). More precisely, if ε=1-ρ1-ρ2→0+ then m and n must be scaled to be O(ε-1), but with the difference m-n fixed. In the second heavy traffic limit (HTL2) we again set ε=1-ρ1-ρ2→0+ but now let ρ2→0+, with ρ2=O(ε). Now the probability mass will become more spread out, with appreciable mass anywhere in the range X>Y>0, where m=X/ε and n=Y/ε. The light and heavy traffic results are summarized below as Theorem 5.Theorem 5.
(i)
Light traffic: For ρ1,ρ2→0 one has
(78)p(m,n)~ρ1mρ2n(m+nn).
(ii)
Heavy traffic limit 1:ε=1-ρ1-ρ2→0+ with 0<ρ2<1, n=Y/ε=O(ε-1),m-n=ℓ=O(1)(79)p(m,n)~ε4ρ2(2-ν-ρ2)2-ρ2+(2ρ2-1)ν(1-ρ21+ρ2)ℓe-2Y,ℓ⩾1,(80)p(n,n)~ε4ρ2(1-ρ2)2-ρ2+(2ρ2-1)νe-2Y,ℓ=0,(81)p(m,n)~ε4(1-ρ2)(ρ2+ν)2-ρ2+(2ρ2-1)ν(ρ22-ρ2)-ℓe-2Y,ℓ⩽-1.
(iii)
Heavy traffic limit 2:ε=1-ρ1-ρ2→0+ with ρ2=γε=O(ε), n=Y/ε, m=X/ε(82)p(m,n)~ε22πi∫Br+f0(ξ)eYξ×exp{Y-X2[2γ+1+(2γ+1)2+4γξ]2}dξ,X>Y>0,(83)f0(ξ)=ξξ+2γ+1-(2γ+1)2+4γξ,(84)p(n,n)~ε22πi∫Br+h0(ξ)eYξdξ,(85)h0(ξ)=(2-ν)ξ+(2γ+1)(1-ν)-(1-ν)(2γ+1)2+4γξ(2-ν)[ξ+2γ+1-(2γ+1)2+4γξ],(86)p(m,n)~ε(εγ2)n-m12πi∫Br+g¯(ξ)eYξdξ,n>m,(87)g¯(ε)=ν2-ν1γ(2γ+1)2+4γξ-(2γ+1)2γ+1+ξ-(2γ+1)2+4γξdξ.
Here Br+ is a vertical contour in the ξ-plane, which lies to the right of all singularities.The expression in (78) applies for m>n, m<n, and m=n, and in the light traffic limit the discontinuity of p(m,n) along the diagonal will appear only in the higher order terms. In HTL1, (79)–(81) show a piecewise geometric distribution in the ℓ variable, and an exponential density in Y. For HTL2, writing (82) as p(m,n)~ε2ℱ(X,Y) we can easily show that ∫0∞‍∫Y∞‍ℱ(X,Y)dXdY=1 so that to leading order the probability mass concentrates where m,n=O(ε-1) with X>Y. From (84) we have p(n,n)=O(ε2) but the total mass along the main diagonal is O(ε), which is smaller than the mass in (82). Then also p(n-1,n)=O(ε2) with total mass ∑n‍p(n-1,n)=O(ε), which is comparable to that along the main diagonal. The diagonals with n-m⩾2 have mass O(εn-m), which is smaller still. The integrands in (82), (84), and (86) have branch points at ξ=-(2γ+1)2/(4γ), are analytic at ξ=0, and may have poles at ξ=-2, if γ>1/2.This completes our summarization of the exact and asymptotic results. Despite the seeming complexity and the many separate cases, all the results follow from fairly standard asymptotic evaluations of the integrals in Theorems1, as we will show in Sections 4 and 5.


## 3. The Exact Solution

We solve the difference equation(s) in (1)–(9) and thus obtain Theorem 1. We begin by introducing the three generating functions:
(88)F(z,w)=∑n=0∞‍∑m=n+1∞‍zm-n-1wnp(m,n),(89)G(z,w)=∑n=1∞‍∑m=0n-1‍zn-m-1wnp(m,n),(90)H(w)=∑n=0∞‍p(n,n)wn,
and we note that H(0)=p(0,0), and
(91)F(0,w)=∑n=0∞‍p(n+1,n)wn,G(0,w)=∑n=1∞‍p(n-1,n)wn.
Then (9) shows that
(92)F(1,1)+G(1,1)+H(1)=1.From (1) and (3) we obtain, after some calculation,
(93)[ρ1+ρ2+1-ρ1z-1z-ρ2wz]F(z,w)=ρ1H(w)+(1-ν)H(w)-p(0,0)w-1+ρ2wzF(0,w),
and (2) and (4) lead to
(94)[ρ1+ρ2+1-ρ1z-ρ2wz-1wz]G(z,w)=ρ2wH(w)+ν[H(w)-p(0,0)]-1z(ρ1+1w)G(0,w).
Equation (5) along m=n then leads to the following relation between F(0,w), G(0,w), and H(w):
(95)(ρ1-ρ2+1)H(w)-p(0,0)=(ρ1+1w)G(0,w)+(ρ2w+1)F(0,w).If we define(96)𝒮(ξ)=∑N=0∞‍[∑m+n=N‍p(m,n)]ξN,
then from (96) and (88)–(90) we find that
(97)𝒮(ξ)=H(ξ2)+ξF(ξ,ξ2)+ξ-1G(ξ-1,ξ2).
Using (93) with (z,w)=(ξ,ξ2), (94) with (z,w)=(ξ-1,ξ2) and (95) with w=ξ2, we obtain for 𝒮(ξ) the simpler equation
(98)[ρ1+ρ2+1-(ρ1+ρ2)ξ-1ξ]𝒮(ξ)=(1-1ξ)p(0,0)
so that 𝒮(ξ)=p(0,0)/[1-(ρ1+ρ2)ξ] and then 𝒮(1)=1 by (92), and thus p(0,0)=1-ρ1-ρ2. Then from (96) we have
(99)∑m+n=N‍p(m,n)=(1-ρ1-ρ2)(ρ1+ρ2)N,
so that N1+N2 follows a geometric distribution with parameter ρ1+ρ2, and hence N1+N2 behaves as the standard M/M/1 model.The coefficient ofF(z,w) in (93) has roots at
(100)z=12ρ1[1+ρ1+ρ2±(ρ1+ρ2+1)2-4ρ1(ρ2w+1)]≡Z±(w),
while that of G(z,w) in (94) has roots at
(101)z=12ρ2w[1+ρ1+ρ2±(ρ1+ρ2+1)2-4ρ2(ρ1w+1)]≡Z±*(w)w.
By using (95) to express H(w) in terms of F(0,w) and G(0,w), eliminating H(w) in the right sides of (93) and (94), and requiring that F(z,w) be analytic at z=Z-(w) and that G(z,w) be analytic at z=Z-*(w)/w, we obtain two equations for F(0,w) and G(0,w):
(102)(ρ2w+1)(ρ2w+ν)ρ1+ρ2+1F(0,w)+[(ρ1w+1)(ρ2w+ν)w(ρ1+ρ2+1)-ρ1w+1Z-*(w)]G(0,w)=p(0,0)[ν-ρ2w+νρ1+ρ2+1],[(ρ2w+1)(ρ1w+1-ν)(ρ1+ρ2+1)w-1+ρ2wZ-(w)]F(0,w)+(ρ1w+1)(ρ1w+1-ν)w2(ρ1+ρ2+1)G(0,w)=p(0,0)[1-νw-ρ1w+1-ν(ρ1+ρ2+1)w].
Solving the algebraic system in (98) and (99), and then using the result in (95) to compute H(w), we hence obtain explicitly the right sides of (93) and (94). In particular,
(103)F(z,w)=(1-ρ1-ρ2)f(w)Z+(w)-z=(1-ρ1-ρ2){∑J=0∞‍zJ[Z+(w)]J+1}f(w),
where f(w) is as in (12)–(14). Comparing (100) to (88) we conclude that
(104)∑n=0∞‍p(n+1+J,n)wn=(1-ρ1-ρ2)f(w)[Z+(w)]J+1
for J⩾0. Setting J=m-n-1 and inverting the generating function in (101) leads to (10) for m>n. Similarly, inverting the double generating function for G(z,w) in (89) leads to (15), and we note that (10) and (15) are consistent with the symmetry (n,ρ1,ν,Z+,f)↔(m,ρ2,1-ν,Z+*,g). Finally, p(n,n) in (19) is obtained by inverting the generating function H(w)=(1-ρ1-ρ2)h(w). We have thus established Theorem 1.


## 4. Asymptotics of the Joint Distribution

We derive Theorems2–4 by expanding asymptotically the integrals in Theorem 1. We will use a combination of the saddle point method and singularity analysis. Good general references on techniques for asymptotically evaluating integrals can be found in [17–21].We need to understand the singularities of the integrands in (10), (15), and (19). There are clearly branch points where w=wb and w=wb*, with
(105)wb=(ρ1+ρ2+1)2-4ρ14ρ1ρ2,wb*=(ρ1+ρ2+1)2-4ρ24ρ1ρ2.
Since ρ1⩾ρ2 we have wb⩽wb* and for ρ1>ρ2 the branch point at wb* is farther from the origin than the one at wb. In fact, wb* will never play a role in the asymptotics. The integrands are also singular at w=0, where (10) has a pole of order n+1, and (15) has a pole of order m+1. The only possible other singular points are at the zeros of den(w). We can easily verify that w=1 is a simple zero of all four functions numf(w), numg(w), num(w), and den(w), so all the integrands are analytic at w=1. In the appendix we study in detail the algebraic equation den(w)=0, and show that the only possible zero is at w=wp(106)wp=(ρ1+ρ2)-2,(ρ1+ρ2)2>ρ1-ρ2,
but only if (ρ1+ρ2)2>ρ1-ρ2 (if (ρ1+ρ2)2=ρ1-ρ2 then den(w) has a branch point at w=wp(=wb)). If (ρ1+ρ2)2<ρ1-ρ2 then the two branch points are the only singularities of the functions f(w), g(w), and h(w) in (12), (16), and (19). If (ρ1+ρ2)2>ρ1-ρ2, which is clearly true in the symmetric case, then w=wp is a simple pole of these functions (since den′(wp)≠0). In view of (105) and (106), we have 1<wp<wb<wb*, if the pole is present and the stability condition ρ1+ρ2<1 holds.Consider first (19). If (ρ1+ρ2)2>ρ1-ρ2 the pole at wp determines the asymptotic behavior, as it is closer to w=0 than the branch point at wb. Hence as n→∞(107)p(n,n)~-(1-ρ1-ρ2)limw→wp[(w-wp)h(w)wn+1]=-(1-ρ1-ρ2)wp-n-1num(wp)den′(wp).
Straightforward computations, using (14) and (21), show that
(108)den′(wp)=2ρ1ρ2(ρ1+ρ2)2[(ρ1+ρ2)2+ρ2ν+ρ1(1-ν)](1-ρ1-ρ2)(ρ1+ρ2)4-(ρ1-ρ2)2,(109)num(wp)=-2ρ1ρ2(1-ρ1-ρ2)(ρ1+ρ2)2.
Using (107)–(109) leads to (35), with (36).Next we letm,n→∞ simultaneously, setting R=m/n, still taking (ρ1+ρ2)2>ρ1-ρ2, and first consider R>1, where (10) applies. The integrand in (10) again has singularities at wp, wb, and wb*. But now we write
(110)w-n[Z+(w)]n-m=exp{n[(1-R)log(Z+(w))-logw]}
and let F(w)=F(w;R)=(1-R)log(Z+(w))-logw. Then with (110), (10) has saddle points where F′(w)=0, or
(111)(1-R)Z+′(w)Z+(w)-1w=0.
Solving (111) using (11), after some algebra we are led to the saddle ws=ws(R) in (28). There is also a second saddle at w¯s, where w¯s corresponds to replacing …  in (28) by -…. By evaluating F′′(ws) and F′′(w¯s) we can easily show that the directions of steepest descent at the saddle(s) are arg(w-ws)=±π/2 and arg(w-w¯s)=0,π. From (28) we have ws(1)=wb, so as R=m/n↓1 the saddle approaches a branch point. Also, ws→0 as R→∞. It is possible for the pole and saddle to coalesce. Setting ws(R)=wp=(ρ1+ρ2)-2 and solving this equation for R we find, again after some algebra, that
(112)ws(R)=wpwhenR=Rc,
where Rc is given by (26). If R>Rc(>1), we have ws<wp and if R∈(1,Rc), we have ws>wp. From (28) we can show that ws′(R)<0 so that the saddle location decreases with the queue length ratio m/n=R. If R>Rc we deform the contour in (10), which is a small counterclockwise loop about z=0, into a saddle point contour, on which |w|=ws(R). Then the new contour traverses the saddle in the direction(s) of steepest descent and the standard Laplace estimate of (10) is
(113)p(m,n)~1-ρ1-ρ22π2πnF′′(ws)f(ws)wsexp[nF(ws)].
But enF(ws)=ws-n[Z+(ws)]n-m and, in view of (111),
(114)F′′(w)=1w2+(1-R)[Z+′′(w)Z+(w)-(Z+′(w)Z+(w))2].
Then using (11) we commute ws2F′′(ws) to find that ws2(R)F′′(ws(R)) is the same as M(R) in (30). Thus (113) is the same as the right side of (27), and we have established the asymptotics of p(m,n) for R>Rc.If1<R<Rc,wp>ws and in deforming the loop in (10) to the saddle point contour |w|=ws(R) we must take into account the contribution from the residue at the pole wp. But we have F(wp)⩾F(ws) with equality only if R=Rc, when wp=ws. Thus the pole contribution dominates the saddle contribution and we have
(115)p(m,n)~-(1-ρ1-ρ2)wp-n-1[Z+(wp)]n-mnumf(wp)den′(wp).
From (13) we find that
(116)numf(wp)=-2ρ2(ρ1+ρ2)2[(ρ1+ρ2)2(1-ν)+ρ1]×(1-ρ1-ρ2)
and from (11)
(117)Z+(wp)=12ρ1[ρ1+ρ2+1+(ρ1+ρ2+1)2-4ρ1-4ρ1ρ2(ρ1+ρ2)2]=(ρ1+ρ2)2+ρ2ρ1(ρ1+ρ2),
when (ρ1+ρ2)2+ρ2-ρ1>0. With (115)–(117) and (108) we have established (34) for R∈(1,Rc), where enF(ws;R) gives the product form expression and (1-ρ1-ρ2)|numf(wp)|/[wpden′(wp)] corresponds to the multiplicative constant C+.AsR→∞ we have ws→0 and we must then reconsider the asymptotics of (10). From (28) ws=O(R-1) as R→∞ so we scale w=u/m in the integral in (10) and consider the limit m→∞ with n=O(1). Then
(118)[Z+(w)]n-m=[Z+(0)+Z+′(0)um+O(m-2)]n-m~[Z+(0)]n-mexp(-Z+′(0)Z+(0)u)
and (10) becomes asymptotically
(119)p(m,n)~(1-ρ1-ρ2)2πif(0)[Z+(0)]n-m×∮‍mnun+1exp[-Z+′(0)Z+(0)u]du=(1-ρ1-ρ2)f(0)[Z+(0)]n-m(-Z+′(0)Z+(0))nmnn!.
Since Z+′(0)=-ρ2/(ρ1+ρ2+1)2-4ρ1 we see that (119) is the same as (25).Now considerR=m/n≈Rc, where the saddle and pole are close to each other. This is a standard problem that is discussed, for example, in [17, 18]. We now expand the integrand in (10) about w=wp, ultimately scaling w-wp=O(n-1/2), and then the integrand will approach a limiting value as n→∞. We have, by Taylor/Laurent series,
(120)f(w)=numf(wp)den′(wp)1w-wp+O(1),w⟶wp,(121)nF(w)=-nlogwp-(m-n)log[Z+(wp)]-[nwp+m-nZ+(wp)Z+′(wp)](w-wp)+12{nwp2+(m-n)[Z+′(wp)Z+(wp)]2-(m-n)Z+′′(wp)Z+(wp){[Z+′(wp)Z+(wp)]2}}(w-wp)2+O(n(w-wp)3).
From (11) and (122) we find that
(122)Z+′(wp)Z+(wp)=1wpnn-m|R=Rc=1wp11-Rc,
since, at R=Rc, wp=ws. It follows that
(123)-nwp-m-nZ+(wp)Z+′(wp)=nwpR-RcRc-1=nξwp(Rc-1),
where ξ was defined above (31). Also, from (11) we find that
(124)Z+′′(wp)=-2ρ1ρ22(ρ1+ρ2)3[(ρ1+ρ2)2+ρ2-ρ1]3.
Then setting
(125)w-wp=1nwp[RcRc-1+2ρ1ρ2((ρ1+ρ2)2+ρ2-ρ1)2]-1/2v
and scaling ξ as in (32), (121) becomes
(126)nF(w)=-nlogwp-(m-n)log[Z+(wp)]+ξ1v+12v2+O(n-1/2).
Then we use (120) and (126) to get
(127)p(m,n)=wp-n[Z+(wp)]n-m×(-C+)12πi∫Br-eξ1vev2/21vdv×[1+O(n-1/2)],
where Br- is a vertical contour with Re(v)<0. To obtain (127) we shifted the original contour in (10) into the circle |w|=wp-ε1, ε1>0, and note that |w|<wp implies to leading order that Re(v)<0. The integral in (127) can be expressed in terms of a parabolic cylinder function of order p=-1 (see (61)), which can be expressed in terms of the standard error function, using the identity
(128)-12πi∫Br-1veξ1vev2/2dv=12π∫ξ1∞e-u2/2du.
With (128) and (117), (127) becomes the same as (31), so we have derived the leading term for the range R-Rc=O(n-1/2). We have thus covered all of the necessary ranges that have R⩾1, and established items (i)–(v) in Theorem 2.Now we consider ranges withR<1(m<n) where p(m,n) is given by the integral in (15). The analysis is completely analogous to the expansion of (10), so we merely sketch the details. The function g(w) in (16) has a simple pole at w=wp=(ρ1+ρ2)-2>1 if (ρ1+ρ2)2>ρ1-ρ2, with residue numg(wp)/den′(wp) where
(129)numg(wp)=-2ρ1(ρ1+ρ2)2[(ρ1+ρ2)2ν+ρ2](1-ρ1-ρ2).
We write w-m[Z+*(w)]m-n=exp[nF*(w;R)] where
(130)F*(w;R)=-Rlogw+(R-1)log[Z+*(w)].
Thus for m,n simultaneously large F* will have a saddle where F*′(w)=0, and this leads to w=ws*=ws*(R), as in (43). The saddle and pole coalesce where R=Rc*(<1) in (37). For R∈(Rc*,1) we have wp<ws*(R) and the pole determines the asymptotic behavior of p(m,n). Then
(131)p(m,n)~-(1-ρ1-ρ2)numg(wp)den′(wp)wp-m-1[Z+*(wp)]m-n,
and this leads to (38) with (39), as now Z+*(wp)=[(ρ1+ρ2)2+ρ1]/[ρ2(ρ1+ρ2)]. For R∈(0,Rc*) the saddle determines the asymptotics, and the estimate
(132)p(m,n)~1-ρ1-ρ22πn1F*′′(ws*)g(ws*)ws*exp[nF*(ws*)]
leads to (42), with (43)–(45). As R→0 the saddle ws*→0 and the result in (46) can be obtained by approximating the integrand in (15) for w=O(n-1), similarly as in (118) and (119) (with Z+ replaced by Z+*, f by g, and m by n). Thus we obtain (46). Finally, when R≈Rc* the pole and saddle are close. Then we scale R-Rc*=η/n=O(n-1/2) and expand the integrand in (15) near w=wp, with w-wp=O(n-1/2). This leads ultimately to (40) with (41). We have thus sketched the derivation of items (vi)–(ix) in Theorem 2.Next we take(ρ1+ρ2)<ρ1-ρ2(>0) and establish Theorem 3. Now the pole at w=wp is absent. We first consider the diagonal probabilities in (19). The only singularities of h(w) as the branch points at wb and wb*, and the former determines the large n asymptotics, as wb<wb* in view of (105). Expanding h(w) about w=wb will transfer immediately to an asymptotic series for p(n,n), but here we focus on the leading term. From (14) and (21) we have
(133)num(w)=A+Bwb-w+O(w-wb),den(w)=C+Dwb-w+O(w-wb),
where
(134)A=-2ρ1ρ2wb+(ρ1+ρ2+1)[ρ1ν+ρ2(1-ν)]-2νρ1ρ1-ρ2,(135)B=-2ρ1ρ2ρ2(1-ν),(136)C=(ρ1+ρ2+1)[ρ1ν+ρ2(1-ν)]-2ρ1(ρ2wb+ν)ρ1-ρ2,(137)D=-2ρ1ρ2[ρ1ρ2wb+ρ2(1-ν)].
Note that (ρ1+ρ2+1)2-4ρ2(ρ1wb+1)=2ρ1-ρ2. Using (133)–(137) we have
(138)h(w)=AC+BC-ADC2wb-w+O(w-wb).The expansion ofh(w) will be in powers of wb-w, but only the odd powers will contribute to the asymptotics. Now,
(139)12πi∮‍wb-wwn+1dw=wbwb-n12πi∮‍1-UUn+1dU=-wbwb-n12n-1(2n)!4n(n!)2~-12wbπn-3/2wb-n,n⟶∞,
where we used the binomial expansion of 1-U and Stirling's formula. It follows that the leading term for p(n,n) is, in view of (138) and (139),
(140)p(n,n)~-12wbπBC-ADC2n-3/2wb-n,
with a correction that is O(n-1) relative to the leading term, which may be computed from the O((wb-w)3/2) term in (138), and a refined Stirling approximation of the factorials in (139). Some of the algebra in our calculations is simplified by introducing a=ρ1+ρ2 and b=ρ1-ρ2. Then C in (136) factors as
(141)C=(a-b)(b-1)[b(ν-12)-12(a+b+1)]=-[ρ1-ρ2-(ρ1+ρ2)]×[1-ρ1-ρ2][12(ρ1+ρ2+1)+ρ1-ρ2(1-ν)],
and C<0 if (ρ1+ρ2)2<ρ1-ρ2. After some calculation we find that
(142)BC-AD=2(ρ1ρ2)3/2(1-ρ1-ρ2)2wb×[(ρ1+ρ2)(ν-12)-12-ρ1-ρ2]
so that BC-AD<0 and then (AD-BC)/(2C2) is the same as the constant C¯ in (53). We have thus established (52).Next we considerR=m/n>1. The saddle point calculation that led to (27) does not depend on whether (ρ1+ρ2)2≷ρ1-ρ2. If the pole at wp is absent, then (27) will apply for all R∈(1,∞). For n=O(1) and m→∞ (25) again holds, since this calculation is independent of whether wp exists or not. However, now a different analysis is needed for R≈1, and the appropriate scale is to fix m-n and let n→∞. As R↓1 we have ws(R)→ws(1)=wb and then
(143)Z+(ws)⟶Z+(wb)=ρ1+ρ2+12ρ1.
The expansion in (27) breaks down as R→1 (if (ρ1+ρ2)2<ρ1-ρ2) as (30) shows that
(144)M(R)~2(R-1)2(ρ1+ρ2+1)2(ρ1+ρ2+1)2-4ρ1,R↓1
and thus the factor 1/M(R) in (27) vanishes linearly as R↓1, which indicates a problem in the asymptotics.We thus reexamine (10) for m-n=O(1) and n large. We again employ singularity analysis and expand the integrand about the branch point wb. We let
(145)numf(w)=Af+Bfwb-w+O(w-wb)=Af+O(w-wb),
since Bf=0 in view of (13). Then
(146)f(w)=AfC-AfDC2wb-w+O(w-wb),
where
(147)Af=-2ρ1ρ2wb+ρ12(1-ν)+ρ1ρ2-ρ2(1-ν)+(ρ12+ρ1)ν+2[ρ2-(ρ1+ρ2)ν]ρ1-ρ2=(1-b)2[νa-12(a+1)-b]=(1-ρ1-ρ2)2×[(ν-12)(ρ1+ρ2)-12-ρ1-ρ2]
and we again used some algebraic factorization in the a,b variables. From (11), Z+(w)=[ρ1+ρ2+1+2ρ1ρ2wb-w]/(2ρ1) and hence
(148)[Z+(w)]m-n=(ρ1+ρ2+12ρ1)m-n×[1+2ρ1ρ2(ρ1+ρ2)+1(m-n)wb-w+O(w-wb){2ρ1ρ2(ρ1+ρ2)+1}].
By multiplying (146) by (148) and using (139), the leading term for p(m,n) becomes
(149)p(m,n)~(1-ρ1-ρ2)×wbπn-3/2wb-n(2ρ1ρ1+ρ2+1)m-n×Af2C[2ρ1ρ2ρ1+ρ2+1(m-n)+DC].
But (149) is the same as (48), with C¯+=(1-ρ1-ρ2)Af/(2C) and Δ+=D/C, since D in (137) may also be written as
(150)D=-2ρ1ρ2[ρ2(1-ν)-ρ1+14(ρ1+ρ2+1)2].
We have thus established (48).Now considerR∈[0,1) with (ρ1+ρ2)2<ρ1-ρ2. The integrand in (15) again has a saddle at ws*(R), and ws*→0 as R→0. For m=O(1) and n→∞ we again obtain (46). As R↑1 the saddle ws*(R)→ws*(1)=wb*>wb. Thus there is a critical value of R, which we call Rd, such that
(151)ws*(Rd)=wb.
Using (130), and the facts that F*′(ws)=0 and (ρ1+ρ2+1)2-4ρ2(ρ1wb+1)=ρ1-ρ2, we obtain
(152)Rdwb[ρ1+ρ2+1+2ρ1-ρ2]=ρ1ρ2ρ1-ρ2(1-Rd),
which is a linear equation for Rd whose solution is given by (54). If 0<R<Rd then ws*(R)<wb and the saddle point approximation in (42) holds. If Rd<R<1 the branch point at w=wb determines the asymptotics of p(m,n), and we again use singularity analysis. Now we expand g(w) about w=wb, writing (17) as
(153)numg(w)=Ag+Bgwb-w+O(w-wb),
where
(154)Ag=-2ρ1ρ2wb+ρ22(1-ν)+ρ1ρ2+ρ2(1-ν)+(ρ12-ρ1)ν,Bg=[(ρ1+ρ2)ν-ρ2]2ρ1ρ2.
If we furthermore scale w-wb=U/n=O(n-1), then
(155)[Z+*(w)]m-n=[Z+*(wb)+UnZ+*′(wb)+O(n-2)]m-n~[Z+*(wb)]m-nexp[(R-1)Z+*′(wb)Z+*(wb)U],(156)w-m-1~wb-m-1exp(-RUwb),(157)g(w)=AgC+BgC-AgDC2-Un+O(1n).
It follows, by using (155)–(157) in (15), that
(158)p(m,n)~(1-ρ1-ρ2)BgC-AgDC2n-3/2wb-m-1[Z+*(wb)]m-n×12πi∫Br-exp{[-Rwb+(R-1)Z+*′(wb)Z+*(wb)]U}×-UdU,
where Re(U)<0 on Br-. After some calculation we find that, again using a=ρ1+ρ2 and b=ρ1-ρ2,
(159)Ag=(b+1)(b-1)[12+(ν-12)a]<0,(160)BgC-AgD=-14ρ1ρ2(1-b)2(-2νa+1+a+2b)×[2ν(b2+a)+a2+1-2b2]=-2ρ1ρ2(1-ρ1-ρ2)2×[12+ρ1-ρ2+(ρ1+ρ2)(12-ν)]×[14(ρ1+ρ2+1)2-ρ1(1-ν)],(161)-Rwb+(R-1)Z+*′(wb)Z+*(wb)=-1wbR-Rd1-Rd.
The integral in (158) can then be evaluated as
(162)12πi∫Br--Uexp[-1wbR-Rd1-RdU]dU=wb3/2(1-RdR-Rd)3/2(-12π).
Combining (141) with (158)–(162) we obtain precisely the result in (54)–(56).Now consider the caseR≈Rd, where the branch point at w=wb is close to the saddle at w=ws*(R). The standard scaling for such situations with coalescing singular points (see [18]) is w-wb=O(n-1/2) and R-Rd=O(n-1/2). By expanding in Taylor series about w=wb, setting (R-Rd)n=S and scaling S as in (59), and letting
(163)w-wb=wbn{Rd[11-Rd+ρ1ρ2wb2(ρ1-ρ2)]}-1/2V,
we find that
(164)exp{-mlogw+(m-n)log[Z+*(w)]}=wb-m[Z+*(wb)]m-nexp[-S1V+12V2]×[1+O(n-1/2)].
Using (163) and (164) in (15) and again expanding g(w) as in (157) we obtain
(165)p(m,n)=(1-ρ1-ρ2)Md-1/2wb-m[Z+*(wb)]m-nn-1/2×12πi∫Br-e-S1VeV2/2×[AgC+BgC-AgDC2wbMd-1/4-Vn1/4+O(n-1/2){-Vn1/4}]dV,
where Br- is to the left of all singularities. Here we also used the definition of Md in (58). The integral(s) in (165) may be easily evaluated using
(166)12πi∫Br-e-S1VeV2/2dV=e-S12/22π,12πi∫Br-e-S1VeV2/2-VdV=e-S12/42πD1/2(-S1),
where the latter follows from (61), after replacing -V by V. But g(wb)=Ag/C by (157), and then (165) with (166) give precisely the approximation in (57)–(60). Note that K is the same as (BgC-AgD)/C2.We can easily compute higher order terms in the expansion, and our analysis shows that the asymptotic series will now involve powers ofn-1/4. Actually, the leading term in (57), which has a Gaussian dependence on S1 (hence on S), can be obtained by simply expanding the saddle point approximation in (42), for R↑Rd. However, the O(n-1/4) correction term is necessary to see the transition to the range R>Rd, where (55) applies. In view of (62) the correction term becomes comparable to the leading term when
(167)e-S12/2~n-1/4S1-3/2,S1>0
or, roughly, when |S1|~(1/2)logn. Thus for S1=O(1) or S1≫1 with |S1|≪(1/2)logn the Gaussian dominates while for |S1|≫(1/2)logn the O(n-1/4) term dominates. Note that K>0 and D1/2(-S1)<0 for S1 sufficiently negative, by (62). Thus for S1~(1/2)logn both the O(1) and O(n-1/4) terms in (57) are positive. The asymptotic matching of (57) for S1→+∞ (with S1≫(1/2)logn) with (55) as R↓Rd is then easily verified, as the approximation in (55) has an O[(R-Rd)-3/2] algebraic singularity as R↓Rd, which agrees with the second formula in (62). Also, up to a multiplicative constant n-3/2(R-Rd)-3/2 agrees with N-3/4S1-3/2. This completes the derivation of Theorem 3, where (ρ1+ρ2)2<ρ1-ρ2.Next we analyze how the results in Theorem2 transition to those in Theorem 3, as (ρ1+ρ2)2 decreases through ρ1-ρ2(>0). We could simply assume that (ρ1+ρ2)2=ρ1-ρ2 and then obtain the necessary asymptotic results. But to see the transition it is necessary to also analyze cases where (ρ1+ρ2)2≈ρ1-ρ2. To make this more precise we write
(168)(ρ1+ρ2)2-(ρ1-ρ2)=δβ,
as in (63), and assume that δ→0+. Then β≷0 according as (ρ1+ρ2)2≷ρ1-ρ2. Since only the product δβ is important, we can set β=+1,0,-1, according to the cases (ρ1+ρ2)2>ρ1-ρ2,(ρ1+ρ2)2=ρ1-ρ2,(ρ1+ρ2)2<ρ1-ρ2. Then we must relate the small parameter δ to the large parameters m,n, and we show below that a natural scaling is to take m,n=O(δ-2) as δ→0+.The asymptotic results form→∞ and n=O(1) are the same in Theorems 2 and 3, and thus no transition is needed here. We can write these results in terms of, say, (ρ1,δβ) rather than (ρ1,ρ2) and expand for small δ to somewhat simplify the expression in (25), but we will not do so here. If (ρ1+ρ2)2<ρ1-ρ2 the saddle point approximation in (27) applies for all R>1, while if (ρ1+ρ2)2>ρ1-ρ2 it applies only for R>Rc. But Rc→1 (cf. (26)) as (ρ1+ρ2)2→ρ1-ρ2 so it will apply for any fixed R>1. But R≳1 will require a separate analysis. We also note that the sector 1<R<Rc, where the product form solution in (34) applies, shrinks to zero. Thus if such an approximation will play a role here, it must be contained near R=1.We begin by considering the diagonal probabilitiesp(n,n), using the scaling in (168). We will approximate h(w) in (19) in such a way that the integrand approaches a nondegenerate limit. The pole at wp=(ρ1+ρ2)-2, if it exists, is now close to the branch point wb=[(ρ1+ρ2+1)2-4ρ1]/(4ρ1ρ2), since (ρ1+ρ2)2≈ρ1-ρ2. Note that a-2=[(a+1)2-2(a+b2)]/(a2-b4) if a(=ρ1+ρ2)=b(=ρ1-ρ2). Introducing the scaling w=wb(1+U/n), so that w-wb=O(n-1), we have w-n-1dw~n-1wb-ne-UdU. Now we write h(w) near wb as
(169)h(w)=A+O(wb-w)C+Dwb-w+O(w-wb).
From (134) we find that
(170)A⟶-12(1-a)2(1+a)asb⟶a(orδ⟶0+),
while (136) or (141) shows that
(171)C~(b-a)(1-a)(32a+12-aν)~-δβ4a(1-a)(3a+1-2aν),
since a2-b2=δβ and thus a-b=δβ/(a+b)~δβ/(2a). From (137) or (151) we also find that as b→a(172)D⟶-14a(1-a)1-a2(3a+1-2aν).
Then from (169)–(172) we see that C becomes comparable to Dwb-w if δ=O(n-1/2), or n=O(δ-2). Then setting n=y/δ2 and using (169)–(172), we can approximate (19) to obtain
(173)p(m,n)~(1-a)2δn2(1+a)3a+1-2aνwb-n×12πi∫Br-e-U-(β/a)+1-a2(-U/y)dU.
Thus it is useful to introduce β1=β/(a1-a2) and then we evaluate the contour integral in (173) as, changing U→-U,
(174)12πi∫Br+eUyU-β1ydU=2y2πi∫Br+eZ2(Z-β1y+β1y)Z-β1ydZ=yπ+yπiβ1y∫Br+eZ2Z-β1ydZ=yπ[1+β1yπieβ12y∫Br+eV2e2β1yVVdV]=yπ[1+2β1yeβ12y∫-∞β1ye-u2du].
Here we used a conformal map U=Z2, some contour deformations, (61) with p=-1, and an identity that relates D-1(·) to the standard error function, which yields the last equality in (174). Using (174) in (173) and noting that y/(δn)=n-1/2, we obtain (66).Now considerR≳1 and we already discussed the case R>1 with the transition range scaling in (168). We expand now (10) similarly as we expanded (15). For w-wb=O(n-1), the factor [Z+(w)]n-m can be approximated by
(175)[Z+(w)]n-m~(2ρ1ρ1+ρ2+1)m-nexp[2ρ1ρ2wbρ1+ρ2+1-Un(n-m)]
for m-n=O(n). We thus let z=δ(m-n) so that (m-n)/n=z/y. Also, wb→a-2 as δ→0+, and then
(176)2ρ1ρ2wbρ1+ρ2+1⟶1-a1+b,b⟶a.
Hence (10) becomes
(177)p(m,n)~(1-ρ1-ρ2)wb-n(2ρ1ρ1+ρ2+1)m-n1n×12πi∫Br-AfC+Dwb-U/ne-U×exp[-1-a1+a-UZy]dU.
Now C and D can be evaluated as in (171) and (172), and, as b→a, Af in (147) becomes
(178)Af~-12(1-a)2(3a+1-2νa).
Then (177) can be evaluated similarly to (173), and we ultimately obtain (68), with z~=(1-a)/(1+a)zy as in (69).ForR<1(n>m) we need only consider the ranges R∈(Rd0,1) and R≈Rd0, where Rd0 is in (64), with Rd0 being the limit of both Rc* and Rd, as b→a or δ→0+. For R>Rd0 the saddle at ws*(R) exceeds the branch point wb and hence the latter determines the asymptotic behavior of p(m,n). For a fixed R, we scale w=wb+U/n and use
(179)w-m-1~wb-m-1exp(-RwbU),(180)[Z+*(w)]m-n~[Z+*(wb)]m-nexp[(R-1)Z+*′(wb)Z+*(wb)U].
We again expand g(w) in the form in (169), with A now replaced by Ag, where
(181)Ag~-12(1-a2)(1-a+2νa),b⟶a.
Using (161) with Rd replaced by Rd0, along with (180) and (181), the integral in (7) becomes
(182)(1-a)(1+a)(1-a+2νa)3a+1-2νa2nδwb-m-1[Z+*(wb)]m-n×12πi∫Br+exp((V/wb)((R-Rd0)/(1-Rd0)))-(β/a)+1-a2a(V/y)dV.
Here we let U=-V and used (171) and (172) to approximate C and D. Scaling β as in (178) and evaluating the integral in (182) similarly to (173) leads to (70). If n-m=O(1), the same analysis applies, as then we can simply replace R by 1 and β2 by β1 in (70) and (71), but must maintain the factor [Z+*(wb)]m-n.WhenR≈Rd0 we let S0=(R-Rd0)n=O(1). Now the saddle ws*(R) will be close to the branch point wb (with also wb~a-2 by (168)). We scale w-wb=O(n-1/2) with
(183)w-wb=-Vnwb{Rd[11-Rd+ρ1ρ2wb2(ρ1-ρ2)]}-1/2
and introduce S10 as in (73). For b~a we also have
(184)11-Rd+ρ1ρ2wb2(ρ1-ρ2)~1+5a+7a2+19a34a(1+3a)
and then
(185)S10S0=(1+4a+11a2)3/22a(1+3a)11-a211+5a+7a2+19a3.
Since C=O(δ) and Dwb-w=O(n-1/4)=O(δ) we have
(186)g(w)=Ag+O(wb-w)C+Dwb-w+O(wb-w)~AgD1wb-w~2a1+a1-a1-a+2νa1+3a-2νa1wb-w
and then
(187)p(m,n)~AgD(1-a)1n1/4wb[Rd0(11-Rd0+1-a28a2)]-1/4×wb-m[Z+*(wb)]m-n12πi∫Br+eS10VeV2/2VdV.
Expressing the integral as a parabolic cylinder function of order p=-1/2 (see (61)) we obtain the expression in (72), with (73) or (185).If we letδ→0+ but consider even larger values of n, with n=y1/δ4=O(δ-4), a slightly different expansion applies. Now both C and wb-w in (186) become O(δ). Setting w-wb=-V¯/n, we use
(188)g(w)~2(1+a)(1-a+2νa)3a+1-2νa[-βy11/4a+a1-a2V¯]-1
in (15) to ultimately obtain the expression in (74), which involves the contour integral in (76). The function ℒ(Z;γ) can be expressed as an infinite sum of parabolic cylinder functions, as
(189)ℒ(Z;γ)=[∑j=0∞‍γjD-(j+1)/2(-Z)]e-Z2/42π.This completes the analysis of the transition range where(ρ1+ρ2)2≈ρ1-ρ2, and we have thus established Theorem 4.


## 5. Light and Heavy Traffic

We establish Theorem5. First consider the light traffic limit, where ρ1,ρ1→0. We now scale
(190)w=uρ1ρ2
and then, from (11), (12), (16), and (20), obtain
(191)Z+(w)~12ρ1[1+1-4u],Z+*(w)~12ρ2[1+1-4u],f(w)~g(w)~h(w)~11-4u.
Hence the diagonal probabilities have the limiting form
(192)p(n,n)~(ρ1ρ2)n12πi∮‍u-n-11-4udu=(ρ1ρ2)n[un](1-4u)-1/2=ρ1nρ2n(2nn),
where we used the binomial theorem, and [un] denotes the coefficient of un in the Taylor expansion of what follows.Form>n we similarly obtain
(193)p(m,n)~(ρ1ρ2)n(2ρ1)m-n×12πi∮‍1un+1(11+1-4u)m-ndu1-4u=ρ1mρ2n(m+nn),
and this holds also for m⩽n. By refining (191) with (190) we can obtain higher order terms, as an expansion involving powers of ρ1 and ρ2. We have thus established (78).For the first heavy traffic limit (HTL1) we letε=1-ρ1-ρ2→0+, replace ρ2 by 1-ρ1-ε and also set
(194)n=Yε,m=n+ℓ,w=1-εξ.
Then after some calculation we find from (14) that
(195)den(w)=14ε2ξ(2+ξ)(2νρ2-ν+2-ρ2)+O(ε3),
and from (21), (13), and (17) that
(196)num(w)=εξρ2(1-ρ2)+O(ε2),(197)numf(w)=εξρ2(2-ν-ρ2)+O(ε2),(198)numg(w)=εξ(1-ρ2)(ν+ρ2)+O(ε2).Hence the limiting form of (19) is
(199)p(n,n)~ε4ρ2(1-ρ2)2νρ2-ν+2ρ212πi∫Br+eYξξ+2dξ=ε4ρ2(1-ρ2)2νρ2-ν+2-ρ2e-2Y,
which is the same as (80). With the scaling w=1-εξ the loop integral in (19) can be approximated by the vertical contour Br+, on which Re(ξ)>-2 in this case. Since w-n-1~w-m-1~eYξ and
(200)Z+(w)~1+ρ21-ρ2,Z+*(w)~2-ρ2ρ2
we use (200), (195), (197), and (198) to approximate the integrands in (10) and (15), and this leads to (79) and (81).We can also derive the HTL1 limits directly from the asymptotic formulas in (34), (35), and (38), as for ρ1=1-ρ2-ε(201)C0~4ρ2(1-ρ2)2-ρ2+(2ρ2-1)νε,C+~4ρ2(2-ν-ρ2)2-ρ2+(2ρ2-1)νε,C-~4(1-ρ2)(ρ2+ν)2-ρ2+(2ρ2-1)νε
and (ρ1+ρ2)2n=(1-ε)2Y/ε~e-2Y. This shows that it is permissible in this case to first let m,n→∞ with a fixed ρ1,ρ2 and then let ρ1+ρ2↑1. Note also that in HTL1 the condition (ρ1+ρ2)2>ρ1-ρ2 is certainly satisfied, so Theorem 2 applies.In the second heavy traffic limit, HTL2, we again letρ1+ρ2=1-ε and w=1-εξ, but now ρ2=γε=O(ε) is small. We now set m=X/ε and n=Y/ε, and we have
(202)(1+ρ1+ρ2)2-4ρ1-4ρ1ρ2w=ε2[(1+2γ)2+4γξ]+O(ε3),(203)Z+(w)=1+12[2γ+1+(2γ+1)2+4γξ]ε+O(ε2).
Then after some calculation we find that as ε→0(204)h(w)⟶h0(ξ)=(2-ν)ξ+(2γ+1)(1-ν)-(1-ν)(2γ+1)2+4γξ(2-ν)[ξ+2γ+1-(2γ+1)2+4γξ]
and thus p(n,n) is O(ε2) and given by the expression in (84) (again we have w-n-1~eYξ). Since h0(ξ)→1 as ξ→∞, the right side of (84) has probability mass along Y=0, and we can write
(205)12πi∫Br+eYξh0(ξ)dξ=δ(Y)+12πi∫Br+eYξ[(2γ+1)2+4γξ-(2γ+1)](2-ν)[ξ+2γ+1-(2γ+1)2+4γξ]dξ.
The presence of the δ(Y) term in (205) indicates that different approximations are needed for p(n,n) for Y>0 and for small Y=εn, such as n=O(1). But in HTL2 there is little mass along the diagonal m=n as a whole.In HTL2 we also obtain(206)numf(w)=ε2(2-ν)γξ+O(ε3),den(w)=ε2γ(2-ν)×[ξ+2γ+1-(2γ+1)2+4γξ]+O(ε3)
so that f(w)→f0(ξ), as in (83), and then p(m,n) is given by (82). We also note that, in view of (203) and the scaling (X,Y)=ε(m,n),
(207)[Z+(w)]n-m⟶exp{Y-X2[2γ+1+(2γ+1)2+4γξ]}.Ifℱ(X,Y) is the limit of ε-2p(m,n), for m>n, in HTL2, then by contour integration
(208)∫0∞∫Y∞ℱ(X,Y)dXdY≡∫0∞[12πi∫Br+2f0(ξ)eYξ2γ+1+(2γ+1)2+4γξdξ]dY=12πi∫Br-22γ+1+(2γ+1)2+4γξ(-1ξ)f0(ξ)dξ=12γ+1f0(0)=1
so that ℱ is a proper two-dimensional density function, with support in the range X>Y>0. In (208), on Br-,Re(ξ)∈(-(2γ+1)2/4γ,0).Form<n in HTL2 we have
(209)Z+*(w)~2-ρ2ρ2~2εγ
so that [Z+*(w)]m-n=O(εn-m). For n-m=O(1) we also have w-m-1~eXξ~eYξ and
(210)numg(w)=εν[(2γ+1)2+4γξ-(2γ+1)]+O(ε2).
Then g(w)~ε-1g¯(ξ) as below (86). Thus for n>m we have derived (86), and this completes the analysis of HTL2. Note that in HTL2 both p(n,n) and p(n-1,n) are O(ε2).



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*Source: 680539-2013-05-13.xml*