# Two Coupled Queues with Vastly Different  Arrival Rates: Critical Loading Case

Authors: Charles Knessl; John A. Morrison
Journal: Advances in Operations Research
 (2011)

---

## Abstract

We consider two coupled queues with a generalized processor sharing service discipline. The second queue has a much smaller Poisson arrival rate than the first queue, while the customer service times are of comparable magnitude. The processor sharing server devotes most of its resources to the first queue, except when it is empty. The fraction of resources devoted to the second queue is small, of the same order as the ratio of the arrival rates. We assume that the primary queue is heavily loaded and that the secondary queue is critically loaded. If we let the small arrival rate to the secondary queue beO(ε), where 0≤ε≪1, then in this asymptotic limit the number of customers in the first queue will be large, of order
O(ε-1), while that in the second queue will be somewhat smaller, of order O(ε-1/2). We obtain a two-dimensional diffusion approximation for this model and explicitly solve for the joint steady state probability distribution of the numbers of customers in the two queues. This work complements that in (Morrison, 2010), which the second queue was assumed to be heavily or lightly loaded, leading to mean queue lengths that were
O(ε-1) or O(1), respectively.


---

## Body

## 1. Introduction

The study of two coupled queues is a fundamental problem in queueing theory and applied probability. Classic examples include the shortest queue problem [1–3], the longer queue problem [4], fork-join models [5–7] and two coupled queues with generalized processor sharing [8–10], which is the subject of the present investigation. Computing the joint probability distribution for these models typically leads to functional equations that may sometimes be recast as boundary value problems [11], such as Dirichlet and Riemann-Hilbert problems.Generalized processor sharing (GPS) models have become quite popular in recent years, as they provide scheduling algorithms that yield both service differentiation among different customer classes and also gains from statistical multiplexing. Some recent investigations and applications of such models appear in [12–15], where they are used, for example, for flow control in integrated service networks.We consider here two parallel queues with respective Poisson arrival ratesλ and εσ, where 0<ε≪1. Thus, the arrivals to the second queue are much less frequent than those to the first, and we immediately scale the second arrival rate by ε, thus introducing σ. The service times are assumed to be exponentially distributed in both queues, with respective means 1/μ and 1. Thus, we are taking the unit of time as the mean service time in the second queue. There is a single processor sharing server that works at unit rate and devotes 1-εκ=1-O(ε) of its capacity to the first queue and the remaining εκ to the second queue, provided both queues are nonempty. If one queue is empty, the processor devotes all of its capacity to the other queue. The total load is given by λ/μ+εσ, and we assume that the system is in heavy traffic so that this quantity will be close to 1. Hence, we define ω from the relation λ/μ+εσ=1-εω and assume that ω>0, so that the system is stable. This also means that the first queue is heavily loaded. The second queue has traffic intensity εσ/(εκ)=σ/κ, and we say it is underloaded if σ/κ<1, overloaded if σ/κ>1 and critically loaded if σ/κ≈1 (more precisely σ/κ=1+O(ε)). The underloaded and overloaded cases were analyzed in [16]. We denote by N1 (resp., N2) the number of customers in the first (resp., second) queue and the joint steady state probability distribution will be denoted by p(m,n)=Prob[N1=m,N2=n].For the underloaded case most of the mass occurs on the scalem=O(ε-1) and n=O(1), so there will tend to be only a few customers in the second queue. Asymptotically, p(m,n) has a product form behavior, with an exponential distribution in εN1 and a geometric distribution in N2 (see [16]). This analysis was recently extended to an arbitrary number of parallel queues by Morrison and Borst [17], as long as one queue is heavily loaded and all the others are underloaded (with similar assumptions about arrival rates and processor-sharing factors as above). For the overloaded case most of the probability mass occurs for both m,n=O(ε-1), and in [16] a diffusion limit of the form p(m,n)~ε2𝒫*(ζ,τ)=ε2𝒫*(εm,εn) is obtained. Here 𝒫* may be characterized as the solution to a parabolic PDE, in the variables ζ and τ. Here, we will analyze the critically loaded case, which also leads to a diffusion limit with now p(m,n)~ε3/2ϕ0(ζ,w)=ε3/2ϕ0(εm,εn), where ϕ0 will satisfy an elliptic PDE. Thus the critically loaded case has N2=O(ε-1/2) and leads to a somewhat more difficult problem than either the underloaded or overloaded cases.We will obtain the PDE forϕ0(ζ,w) as a limiting case of the difference equation(s) satisfied by p(m,n), and explicitly solve the PDE by transform methods. We will obtain detailed results for the marginal distributions Prob[N1=m]=∑n=0∞p(m,n) and Prob[N2=n]=∑m=0∞p(m,n), as well as the mean queue lengths. We will also obtain other approximations to p(m,n) that are valid on scales where m=o(ε-1) and/or n=o(ε-1/2). In particular, we shall show that p(m,n) is O(ε) on the scale m=O(ε-1/2) and n=O(1).Previous work on this model includes Fayolle and Iasnogorodski [8] (see also [10]) and the more recent study of Guillemin and Pinchon [9]. There the authors consider the double generating function F(x,y)=∑m,nxmynp(m,n) and obtain a functional equation for the boundary values F(x,0) and F(0,y). This is ultimately converted to a Dirichlet problem, which is solved to yield the boundary values of F in terms of elliptic integrals. One of the authors (J. A. Morrison) has verified that by analyzing the results in [8] and [9], in the asymptotic limit we consider and with the scaling x=1-O(ε) and y=1-O(ε), and then inverting the double transform, we also obtain our main approximation p(m,n)~ε3/2ϕ0(ζ,w). However, this involves a lengthy calculation that takes far more work than our direct approach, which consists of deriving a limiting PDE and solving it, along with appropriate boundary conditions. Also, this direct approach should work for other models of this type, including ones with finite capacities of customers, and with 3 or more coupled queues.Other recent work on diffusion approximations for generalized processor sharing models includes Ramanan and Reiman [18] (see also references therein). This work, however, is more concerned with theoretical aspects of the diffusion approximations, such as convergence of the discrete problem to a certain diffusion process. Here, our focus is on obtaining the explicit solution to the limiting diffusion equation that arises from the balance equations. It is highly likely that this equation can be interpreted as the Kolmogorov forward equation for some appropriate diffusion process, but we do not consider such “process level” aspects here, as our approach is largely analytical.Our approach has the merit that it can be used to compute correction terms to the diffusion approximations, and we do this in some cases here. Also, we treat scales other than the basic diffusion scale, where, for example, some of the variables remain discrete, which we then relate to the diffusion scale by asymptotic matching. This type of analysis is needed, for example, to accurately compute boundary probabilities.From a mathematical viewpoint, the diffusion approximation we obtain (i.e.,ϕ0(ζ,w)) is somewhat nonstandard in that the density vanishes as ζ→0 and the approximation breaks down for ζ=O(ε). Also, the corner behavior of the problem is much different than what is typical. In [19] we analyzed a more general version of this model in another heavy traffic limit, assuming that the arrival rates and processor-sharing factors were of comparable magnitude. That analysis led to an elliptic PDE that was more complicated than the one obtained here, but probably more representative of typical diffusion approximations to two coupled queues, such as those considered in [20–22].The present scaling limit leads to a separable, elliptic PDE in the variablesεm and εn. Since the boundary conditions are somewhat simpler than those for the diffusion model in [19], we are able to obtain a more explicit solution to this equation, using classical transform theory [23]. We then evaluate the solution in various limiting cases, to obtain even simpler results that yield more insight into model behavior.Yet another analysis of the model considered here is done in [24], but there it was assumed that both the arrival and service rates of the secondary customers are small, while the server devotes comparable resources to each queue.The paper is organized as follows. In Section2, we state the problem more precisely and give the balance equations satisfied by p(m,n). In Section 3, we summarize all of our main results. The derivations are given in Section 4 for the scale m=O(ε-1),n=O(ε-1/2) and in Section 5 for the other ranges of m,n.Throughout the paper we will use the notationf(x)~g(x) to mean limx→x0[f(x)/g(x)]=1, f(x)=o(g(x)) to mean limx→x0[f(x)/g(x)]=0, and f(x)=O(g(x)) to mean that |f(x)/g(x)| is bounded for x sufficiently close to x0.


## 2. Formulation

We consider two parallel infinite capacity queues for different traffic classes. The jobs arrive as Poisson processes with ratesλ for the primary class, and εσ for the secondary class, where 0<ε≪1. Hence, the secondary jobs arrive much less frequently than the primary ones. Moreover, it is assumed that the primary and secondary jobs have exponentially distributed service requirements with mean service times 1/μ and 1, respectively, where μ=O(1). The server works at unit rate, and if neither queue is empty devotes fractions 1-εκ and εκ of its effort to the primary and secondary queues, respectively, where κ=O(1). The corresponding service rates are (1-εκ)μ and εκ. If one queue is empty the server works at unit rate on the other queue, so that this model is work conserving. It is assumed that the primary queue is heavily loaded, with(2.1)λμ+εσ=1-εω,0<ω=O(1).
Moreover, we assume that the secondary queue is critically loaded, with(2.2)σ=κ+δε,δ=O(1),
where δ may have either sign. Thus, the asymptotic limit we consider has ε→0+ with ω,μ,κ and δ fixed, and then σ and λ vary with ε so that (2.1) and (2.2) hold.Sinceω>0 the system is stable. Let p(m,n) denote the stationary probability that there are m jobs in the primary queue and n jobs in the secondary queue. Then, the balance equations satisfied by p(m,n) are(2.3)[λ+εσ+(1-εκ)μ+εκ]p(m,n)=λp(m-1,n)+εσp(m,n-1)+(1-εκ)μp(m+1,n)+εκp(m,n+1),m⩾1,n⩾1,(2.4)(λ+εσ+μ)p(m,0)=λp(m-1,0)+μp(m+1,0)+εκp(m,1),m⩾1,(2.5)(λ+εσ+1)p(0,n)=εσp(0,n-1)+(1-εκ)μp(1,n)+p(0,n+1),n⩾1,(2.6)(λ+εσ)p(0,0)=μp(1,0)+p(0,1).
The normalization condition is(2.7)∑m=0∞∑n=0∞p(m,n)=1.The mean number of jobs in the primary and secondary queues are(2.8)E(N1)=∑m=0∞∑n=0∞mp(m,n),E(N2)=∑m=0∞∑n=0∞np(m,n),
From Little’s Law, the corresponding mean waiting times are E(N1)/λ and E(N2)/(εσ). From [8], the conservation law of Kleinrock implies, using (2.1) and (2.2), that(2.9)E(N1)μ+E(N2)=1εω(λμ2+εσ)=1εμω[1+ε(μκ-ω-κ)+ε3/2(μ-1)δ].


## 3. Summary of Results

We considerm=ζ/ε=O(1/ε) and n=w/ε=O(1/ε), with(3.1)p(ζε,wε)=ε3/2[ϕ0(ζ,w)+εϕ1(ζ,w)+O(ε)],0<ζ=O(1),0<w=O(1).
We then haveProposition 3.1.
(3.2)
ϕ
0
(
ζ
,
w
)
+
I
(
δ
<
0
)
δ
ω
κ
exp
⁡
(
-
ω
ζ
+
δ
w
κ
)
=
2
ω
π
exp
⁡
(
-
ω
ζ
2
+
δ
ω
2
κ
)
∫
0
∞
β
[
β
cos
⁡
(
w
β
)
+
(
δ
/
2
k
)
sin
(
w
β
)
]
(
β
2
+
δ
2
/
(
4
κ
2
)
)
exp
⁡
[
-
ω
2
+
δ
2
κ
μ
+
4
κ
β
2
μ
ζ
2
]
d
β
.This gives the limiting density of(N1,N2), which applies for fixed values of ζ and w. We next evaluate this density in various limiting cases, such as w→∞ and δ→±∞, to gain more insight into its structure, and to verify consistency with results in [16].Corollary 3.2.
Ifδ=O(1), w≫1 and 0<ζ/w≪1, then
(3.3)ϕ0(ζ,w)~ω(δ2+κμω2)3/4ζπμ(δ+δ2+κμω2)w3/2×exp⁡(-ωζ2)exp⁡[-w2κ(δ2+κμω2-δ)].Corollary 3.3.
If-δ≫1,w/|δ|>0 and 0<ζ=O(1), then
(3.4)ϕ0(ζ,w)~ω|δ|2κexp⁡(-ωζ2)exp⁡[-w2κ(δ2+κμω2+|δ|)]exp⁡(μω24|δ|w)×{exp⁡(-ωζ2)Erfc⁡[(μωw-|δ|ζ)2|δ|μw]-exp⁡(ωζ2)Erfc⁡[(μωw+|δ|ζ)2|δ|μw]},
where the complementary error function is given by [25]
(3.5)Erfc⁡(z)=2π∫z∞e-u2du.Corollary 3.4.
Ifw≫-δ≫1 and 0<ζ=O(1), then
(3.6)ϕ0(ζ,w)~2|δ|5/2ζπμκμωw3/2exp⁡(-ωζ2)exp⁡[-w2κ(δ2+κμω2+|δ|)].Corollary 3.5.
If-δ≫1,0<w=O(1) and 0<ζ=O(1), then
(3.7)ϕ0(ζ,w)~ω|δ|κexp⁡(-ωζ-|δ|wκ).Remark 3.6.
This matches with [16, Result 7], with n=w/ε,0<ε≪1 and |δ|ε≪1.Corollary 3.7.
Ifδ≫1,0<w/δ=O(1) and 0<ζ=O(1), then
(3.8)ϕ0(ζ,w)~ωδζ2πμw3/2exp⁡[-δ4μw(ζ+μωwδ)2].Remark 3.8.
This matches with [16, Result 4], with τ=wε,0<ε≪1 and δε≪1.In Propositions3.9 and 3.16 below, we give the limiting marginals of N2 and N1, which apply for w and ζ fixed, respectively. Then, we simplify these marginal densities in various limiting cases.Proposition 3.9.
The scaled lowest order asymptotic approximation to the stationary distribution of the number of jobs in the secondary queue is(3.9)∑m=0∞p(m,wε)~ε∫0∞ϕ0(ζ,w)dζ,
where
(3.10)∫0∞ϕ0(ζ,w)dζ+I(δ<0)δκexp⁡(δwκ)=4ωπexp⁡(δw2κ)×∫0∞β[βcos⁡(wβ)+(δ/2κ)sin(wβ)](β2+δ2/4κ2)[ω+ω2+δ2/(κμ)+4κβ2/μ]dβ.Corollary 3.10.
Ifδ=O(1) and w≫1, then
(3.11)∫0∞ϕ0(ζ,w)dζ~4(δ2+κμω2)3/4πμω(δ+δ2+κμω2)w3/2exp⁡[-w2κ(δ2+κμω2-δ)],
which is consistent with (3.3).Corollary 3.11.
If-δ≫1 and 0<w/|δ|=O(1), then
(3.12)∫0∞ϕ0(ζ,w)dζ~|δ|κexp⁡[-w2κ(δ2+κμω2+|δ|)]×[(1+μω2w2|δ|)exp⁡(μω2w4|δ|)Erfc⁡(ω2μw|δ|)-ωμwπ|δ|].Corollary 3.12.
Ifw≫-δ≫1, then
(3.13)∫0∞ϕ0(ζ,w)dζ~8|δ|5/2πμκμω3w3/2exp⁡[-w2κ(δ2+κμω2+|δ|)],
which is consistent with (3.6).Corollary 3.13.
If-δ≫1 and 0<w=O(1), then
(3.14)∫0∞ϕ0(ζ,w)dζ~|δ|κexp⁡(-|δ|wκ),
which is consistent with (3.7).Corollary 3.14.
Ifδ≫1 and 0<w/δ=O(1), then
(3.15)∫0∞ϕ0(ζ,w)dζ~ωμδ[1πwexp⁡(-μω24δw)-ω2μδErfc⁡(ω2μwδ)],
which is consistent with (3.8).Remark 3.15.
This matches with [16, Result 3], with τ=wε,0<ε≪1 and δε≪1.Proposition 3.16.
The scaled asymptotic approximation to the stationary distribution of the number of jobs in the primary queue is(3.16)∑n=0∞p(ζε,n)=εωexp⁡(-ωζ)+ε3/2Q1(ζ)+O(ε2),
where
(3.17)Q1(ζ)-I(δ>0)δ(1-ωζ)exp⁡(-ωζ)=ωμπexp⁡(-ωζ2)∫0∞β2(β2+δ2/4κ2)2×{[ω+ω2+δ2κμ+4κβ2μ]exp⁡[-ω2+δ2κμ+4κβ2μζ2]-2ωexp⁡(-ωζ2)}dβ.Corollary 3.17.
If|δ|≫1, then
(3.18)∑n=0∞p(ζε,n)=εexp⁡(-ωζ)[ω+εI(δ>0)δ(1-ωζ)]+O(ε2).Remark 3.18.
This matches with [16, Result 7] for δ<0, and with [16, Result 2] for δ>0 since, for δε≪1,
(3.19)(ω+σ-κ)exp⁡[-(ω+σ-κ)ζ]=exp⁡(-ωζ)[ω+εδ(1-ωζ)]+O(ε).We next give expansions for the mean queue lengths, forε→0 (with ω,κ,μ,δ fixed).Proposition 3.19.
The lowest order asymptotic approximation to the stationary mean secondary queue length is(3.20)E(N2)~sε,s=δμω2{1+12πc4[(1-c4)cos⁡-1(-c)-(1+c4)cos⁡-1c-2c1-c2(1-2c2)]},
where
(3.21)-1<c=δδ2+κμω2<1,0<cos⁡-1(±c)<π.Remark 3.20.
We have verified from (3.17) that ∫0∞ζQ1(ζ)dζ=-μs so that from (3.16),
(3.22)E(N1)=1εω-μsε+O(1),
which is consistent with (2.9).Corollary 3.21.
If|δ|ε≪1, then
(3.23)s~δμω2,ifδ≫1;s~-κδ,if-δ≫1.
These match with [16, Result 1 and Corollary 10], respectively.We next give some asymptotic results forϕ0(ζ,w) that apply for δ fixed and ζ and/or w→∞. We also give the “corner” behavior as (ζ,w)→(0,0).Proposition 3.22.
(i)ζ,w→∞ with δ<0 and 0⩽w/ζ<|δ|/(μω),
(3.24)ϕ0(ζ,w)~ω|δ|κexp⁡(-ωζ-|δ|κw),
(ii)
ζ,w→∞ with 0<w/ζ<∞ for δ>0, or |δ|/(μω)<w/ζ<∞ for δ⩽0,
(3.25)
ϕ
0
(
ζ
,
w
)
~
K
(
ζ
,
w
)
exp
⁡
(
-
ω
2
ζ
+
δ
2
κ
w
-
1
2
ω
2
+
δ
2
μ
κ
μ
κ
w
2
+
ζ
2
)
,
(3.26)
K
(
ζ
,
w
)
=
μ
π
κ
ω
b
s
b
s
+
δ
/
(
2
κ
)
(
ω
2
+
δ
2
κ
μ
)
1
/
4
ζ
(
ζ
2
+
μ
κ
w
2
)
-
3
/
4
,
b
s
=
b
s
(
w
ζ
)
=
μ
2
κ
(
ω
2
+
δ
2
κ
μ
)
1
/
2
w
ζ
2
+
(
μ
/
κ
)
w
2
,
(iii)
ζ
→
∞
,
w
=
O
(
1
)
,
δ
>
0,
(3.27)
ϕ
0
(
ζ
,
w
)
~
ω
δ
μ
3
/
2
κ
(
ω
2
+
δ
2
κ
μ
)
3
/
4
1
π
ζ
3
/
2
(
w
+
2
κ
δ
)
exp
⁡
[
δ
w
2
κ
]
exp
⁡
[
-
1
2
(
ω
+
ω
2
+
δ
2
κ
μ
)
ζ
]
,
(iv)
w
,
ζ
→
∞
,
ζ
=
O
(
w
),
(3.28)
ϕ
0
(
ζ
,
w
)
~
2
b
+
κ
μ
b
+
ω
b
+
+
δ
/
(
2
κ
)
ζ
π
w
3
/
2
exp
⁡
[
(
δ
2
κ
-
b
+
)
w
-
ω
2
ζ
-
b
+
κ
2
μ
ζ
2
w
]
,
(3.29)
b
+
=
μ
2
κ
ω
2
+
δ
2
κ
μ
,
(v)
ζ
,
w
→
0 with 0<w/ζ<∞,
(3.30)
ϕ
0
(
ζ
,
w
)
~
2
ω
π
μ
κ
ζ
ζ
2
+
(
μ
/
κ
)
w
2
,
(vi)
ζ
→
0 with 0<w<∞,
(3.31)
ϕ
0
(
ζ
,
w
)
~
κ
μ
ω
π
ζ
a
(
w
)
exp
⁡
(
δ
w
2
κ
)
,
(3.32)
a
(
w
)
=
δ
2
κ
1
w
∫
-
∞
∞
e
i
w
β
(
μ
/
4
κ
)
(
ω
2
+
δ
2
/
μ
κ
)
-
i
(
β
δ
/
2
κ
)
(
β
+
i
δ
/
2
κ
)
2
[
β
2
+
(
μ
/
4
κ
)
(
ω
2
+
δ
2
/
κ
μ
)
]
1
/
2
d
β
+
μ
4
κ
(
ω
2
+
δ
2
μ
κ
)
1
w
2
∫
-
∞
∞
e
i
w
β
[
β
2
+
μ
4
κ
(
ω
2
+
δ
2
κ
μ
)
]
-
3
/
2
d
β
.Remark 3.23.
The results show that forδ<0 the density ϕ0 is asymptotically of product form in a sector, and distinctly nonproduct form in the complimentary sector. For δ>0 the product form behavior is absent. Item (v) shows that ϕ0 has an integrable singularity near the corner, while item (vi) shows that ϕ0=O(ζ) as ζ→0. The second integral in (3.32) may be expressed in terms of a modified Bessel function, using the identity
(3.33)∫-∞∞eiZU(Z2+1)-3/2dZ=2UK1(U).The approximationp(m,n)≈ε3/2ϕ0(ζ,w) is only valid for m=O(ε-1) and n=O(ε-1/2). For other ranges of (m,n), other expansions must be constructed, and these we summarize below.Proposition 3.24.
(
i
)   m=S/ε=O(ε-1/2), n=w/ε=O(ε-1/2),
(3.34)p(m,n)~ε2[SP+(w)+P-(w)],P+(w)=ωπκμexp⁡(δw2κ)a(w),P-(w)=μ∫w∞P+(u)du,
(
ii
)
m
=
S
/
ε
=
O
(
ε
-
1
/
2
)
,
n
=
O
(
1
),
(3.35)
p
(
m
,
n
)
~
ε
2
ω
π
∫
0
π
cos
⁡
(
Ω
2
)
cos
⁡
[
(
n
+
1
2
)
Ω
]
exp
⁡
[
-
2
κ
μ
S
sin
(
Ω
2
)
]
d
Ω
(
iii
)
m
,
n
=
O
(
1
),
(3.36)
p
(
m
,
n
)
~
ε
3
/
2
2
ω
π
κ
μ
[
4
(
2
n
+
3
)
(
2
n
-
1
)
m
+
4
n
4
n
2
-
1
μ
]
,
n
⩾
1
,
p
(
m
,
0
)
~
ε
ω
.Remark 3.25.
We comment that forn=O(1) and ζ>0(m=O(ε-1))p(m,n)~ε3/2ϕ0(ζ,0) so that the diffusion approximation still applies. For m=O(1) and w>0 item (i) still applies, and then p(m,n)~ε2𝒫-(w), which is independent of m and can be used to estimate the boundary probabilities p(0,n), which are O(ε2) for n=O(ε-1/2). Note that in item (ii) and for n=0 in item (iii), p(m,n) is O(ε), which is larger than the order of magnitude (O(ε3/2)) of the diffusion approximation on the (ζ,w) scale. But, the total mass in the range in (ii) is O(ε), while that in ranges (i) and (iii) is O(ε).


## 4. Analysis of the Main Diffusion Approximation

If we letm=ζ/ε,n=w/ε and use (3.1), (2.1), and (2.2), in (2.3) and (2.4), we obtain to lowest order(4.1)μ∂2ϕ0∂ζ2+μω∂ϕ0∂ζ+κ∂2ϕ0∂w2-δ∂ϕ0∂w=0,ζ>0,w>0,
and the boundary condition(4.2)κ∂ϕ0∂w(ζ,0)=δϕ0(ζ,0),ζ>0.
We will discuss the second boundary condition, along ζ=0, after (4.16).We let(4.3)ϕ0(ζ,w)=exp⁡(-ωζ2+δw2κ)Φ0(ζ,w).
It follows that(4.4)μ∂2Φ0∂ζ2+κ∂2Φ0∂w2-14(μω2+δ2κ)Φ0=0,ζ>0,w>0,(4.5)∂Φ0∂w(ζ,0)=δ2κΦ0(ζ,0).To solve (4.4) and (4.5) we apply a transform in the w variable. Using the theory of distributions and Green’s functions for ordinary differential equations (see [23, p. 294, exercise 4.24]), we have the following transform pair:(4.6)G(B)=∫0∞[Bcos⁡(Bx)+Asin(Bx)]F(x)dx,(4.7)F(x)=-I(A<0)CAeAx+2π∫0∞Bcos⁡(Bx)+Asin(Bx)A2+B2G(B)dB.
Here, the constant C appears only when A<0 and the term AeAx corresponds to a single discrete eigenvalue in the spectral theory. By multiplying (4.7) by eAx and integrating from x=0 and x=∞, we find that C=2∫0∞F(x)eAxdx. Applying (4.6) with B=β and x=w, we let(4.8)Ω0(ζ,β)=∫0∞[βcos⁡(wβ)+δ2κsin(βw)]Φ0(ζ,w)dw,
then integration by parts and the boundary condition (4.5) leads to(4.9)∫0∞[βcos⁡(βw)+δ2κsin(βw)]∂2Φ0∂w2(ζ,w)dw=-β2Ω0(ζ,β).
Hence, from (4.4),(4.10)∂2Ω0∂ζ2-14(ω2+δ2κμ+4κβ2μ)Ω0=0,
so that(4.11)Ω0(ζ,β)=Ω0(0,β)exp⁡[-ω2+δ2κμ+4κβ2μζ2].
Applying the inversion formula (4.7)(4.12)Φ0(ζ,u)+I(δ<0)δκexp⁡(δu2κ)∫0∞exp⁡(δw2κ)Φ0(ζ,w)dw=2π∫0∞[βcos⁡(uβ)+(δ/2κ)sin(uβ)](β2+δ2/(4κ2))Ω0(ζ,β)dβ.
Let(4.13)P0(ζ)=∫0∞ϕ0(ζ,w)dw.
Then, from (4.1) and (4.2),(4.14)d2P0dζ2+ωdP0dζ=0,ζ>0.
But, from the normalization condition (2.7), (3.1) and the Euler-Maclaurin summation formula [26], ∫0∞P0(ζ)dζ=1, so that(4.15)P0(ζ)=ωe-ωζ.
Hence, from (4.3) and (4.13),(4.16)∫0∞exp⁡(δw2κ)Φ0(ζ,w)dw=ωe-ωζ/2.
Also, if we use (3.1),  (2.1), and (2.2), in (2.5), we obtain the lowest order boundary condition ∂ϕ0/∂w(0,w)=0, for w>0. We conclude that ϕ0(0,w) and Φ0(0,w) are proportional to a delta function at w=0+ and hence, from (4.8) and (4.16), that Ω0(0,β)=ωβ. Proposition 3.1 follows from (4.3), (4.11), (4.12), and (4.15).We may rewrite the integral in (3.2) as(4.17)12∫-∞∞βeiwβ(β+iδ/(2κ))exp⁡[-ω2+δ2κμ+4κβ2μζ2]dβ.
We then deform the contour of integration to one around a cut in the β-plane from (i/2κ)δ2+κμω2 to i∞, and let(4.18)β=i(12κδ2+κμω2+y).
For δ<0 there is a contribution from the pole at β=-iδ/(2κ), and we obtain(4.19)ϕ0(ζ,w)=2ωπexp⁡(-ωζ2+δw2κ)exp⁡(-w2κδ2+κμω2)×∫0∞(δ2+κμω2+2κy)e-wy(δ+δ2+κμω2+2κy)sin(yκy+δ2+κμω2ζμ)dy.
For δ=O(1),w≫1 and 0<ζ/w≪1, the main contribution to the integral comes from y=O(1/w), and Corollary 3.2 follows.If-δ≫1,w>0 and 0<ζ=O(1), then the integral in (4.19) is approximated by(4.20)|δ|2κ∫0∞e-wy(y+μω2/4|δ|)sin(ζ|δ|yμ)dy=|δ|κ∫0∞xexp⁡(-(μw/|δ|)x2)(x2+ω2/4)sin(ζx)dx,
which leads [27] to Corollary 3.3. Corollaries 3.4 and 3.5 follow from the asymptotic approximation [25] Erfc(z)~exp(-z2)/(πz),z≫1, and the limiting value Erfc(-∞)=2. If δ≫1,0<w/δ=O(1) and 0<ζ=O(1), then from (4.19), we obtain the approximation(4.21)ϕ0(ζ,w)~ωπexp⁡(-ωζ2-μω24δw)∫0∞e-wysin(δyμζ)dy,
and hence Corollary 3.7.Proposition3.9 follows from Proposition 3.1, (3.1) and the Euler-Maclaurin summation formula [26]. Next, from (4.19)(4.22)∫0∞ϕ0(ζ,w)dζ=8ωπμexp⁡(δw2k)exp⁡(-w2κδ2+κμω2)×∫0∞(δ2+κμω2+2κy)yκy+δ2+κμω2e-wy(δ+δ2+κμω2+2κy)[μω2+4y(κy+δ2+κμω2)]dy.
For δ=O(1) and w≫1, the main contribution to the integral comes from y=O(1/w), and Corollary 3.10 follows. If -δ≫1 and 0<w/|δ|=O(1), then(4.23)∫0∞ϕ0(ζ,w)dζ~ωμ|δ|πκexp⁡[-w2κ(δ2+κμω2+|δ|)]∫0∞ye-wy(y+μω2/(4|δ|))2dy,
which leads [27] to Corollary 3.11. Corollaries 3.12, 3.13, and 3.14 follow directly from Corollaries 3.4, 3.5, and 3.7, respectively.We now consider the first order correction termϕ1(ζ,w) in (3.1). If we use (3.1), (2.1), and (2.2), in (2.3) and (2.4), we obtain(4.24)μ∂2ϕ1∂ζ2+μω∂ϕ1∂ζ+κ∂2ϕ1∂w2-δ∂ϕ1∂w+δ(μ∂ϕ0∂ζ+12∂2ϕ0∂w2)=0,
for ζ>0 and w>0, and the boundary condition(4.25)κ∂ϕ1∂w(ζ,0)-δϕ1(ζ,0)+μ∂2ϕ0∂ζ2(ζ,0)+μ(ω+κ)∂ϕ0∂ζ(ζ,0)+κ2∂2ϕ0∂w2(ζ,0)=0.
We let(4.26)P1(ζ)=∫0∞ϕ1(ζ,w)dw.
It follows, from (4.15) and (4.24)–(4.26), that(4.27)μd2P1dζ2+μωdP1dζ+μ∂2ϕ0∂ζ2(ζ,0)+μ(ω+κ)∂ϕ0∂ζ(ζ,0)+κ2∂2ϕ0∂w2(ζ,0)-δ2∂ϕ0∂w(ζ,0)=δμω2e-ωζ.
Hence, from (4.1) at w=0+,(4.28)P1(ζ)=Q1(ζ)-12ϕ0(ζ,0),
where(4.29)d2Q1dζ2+ωdQ1dζ=δω2e-ωζ-κ∂ϕ0∂ζ(ζ,0).From Proposition3.1,(4.30)ϕ0(ζ,0)+I(δ<0)δωκe-ωζ=2ωπexp⁡(-ωζ2)∫0∞β2(β2+δ2/(4κ2))exp⁡[-ω2+δ2κμ+4κβ2μζ2]dβ.
It follows that(4.31)d2Q1dζ2+ωdQ1dζ-I(δ>0)δω2e-ωζ=ωκπe-ωζ/2∫0∞β2[ω+ω2+δ2/(κμ)+4κβ2/μ](β2+δ2/(4κ2))exp⁡[-ω2+δ2κμ+4κβ2μζ2]dβ.
From (3.1), (4.13), (4.26), (4.28), and the Euler-Maclaurin summation formula [26],(4.32)∑n=0∞p(ζε,n)=εP0(ζ)+ε3/2Q1(ζ)+O(ε2).
Hence,(4.33)∫0∞Q1(ζ)dζ=0.
Proposition 3.16 follows in an elementary manner from (4.31) and (4.33).If we let2κy=δ2+κμω2(coshu-1) in (4.22), we obtain(4.34)∫0∞ϕ0(ζ,w)dζ=2ωπμκ∫0∞sinh⁡2ucosh⁡uexp⁡[-(w/2κ)δ2+κμω2(cosh⁡u-c)](cosh⁡u+c)(cosh⁡2u-c2)du,
where c is given by (3.21). Hence,(4.35)∫0∞∫0∞wϕ0(ζ,w)dζdw=8δ(1-c2)3/2πμω2c∫0∞sinh⁡2ucosh⁡u(cosh⁡u+c)2(cosh⁡u-c)3du.
The evaluation of the integral in (4.35) is routine, but tedious, and Proposition 3.19 follows from (3.1), since(4.36)E(N2)~1ε∫0∞∫0∞wϕ0(ζ,w)dζdw.We next establish the various asymptotic formulas in Proposition3.22. We first note that the integrand in Proposition 3.1 is an even function of β, and if β is viewed as complex it has a simple pole at β=-iδ/(2κ) and branch points at(4.37)β=±iμ2κω2+δ2κμ≡ib±.
Then, we can represent, for any δ, ϕ0 as the contour integral(4.38)ϕ0(ζ,w)=exp⁡[-ω2ζ+δ2κw]1π∫Cβωβ+iδ/2κ×exp⁡[iwβ-12ζω2+δ2κμ+4κμβ2]dβ.
Here, 𝒞 is a horizontal contour in the β-plane, on which(4.39)max⁡{-δ2κ,0}<Im(β)<b+.
The condition in (4.39) insures that if δ<0 the pole at β=i|δ|/(2κ) lies below the contour 𝒞.Ifδ>0 we can shift the contour to the real β-axis, and then (4.38) becomes the same as (3.2). If δ<0 the pole must be taken into account in making this shift, and the residue from this pole yields the exponential terms in the left hand side of (3.2).To evaluate (4.38) for ζ,w→∞ we employ the saddle point method. There is a saddle point where(4.40)ddβ[iwβ-12ζω2+δ2κμ+4κμβ2]=0,
so that(4.41)β=ibs=ibs(wζ)≡iμ2κω2+δ2κμw(μ/κ)w2+ζ2.
The saddle is on the imaginary axis and the directions of steepest descent are arg(β-ibs)=0,π. Then shifting the contour 𝒞 into another horizontal contour through ibs leads to (3.24). Such a shift is always permissible if δ>0, but if δ<0 we need the saddle to lie above the pole, that is, bs>|δ|/(2κ), and this occurs precisely when w/ζ>|δ|/(μω). We thus obtain the condition in item (i) of Proposition 3.22. If δ<0 and w/ζ<|δ|/(μω) the pole dominates the saddle point contribution, and we obtain (3.23).Forζ→∞ and w=O(1) a different analysis is needed, as now bs→0 so the saddle approaches the real axis, where the integrand in (4.38) has as simple zero. In this case (which applies only if δ>0), we shift 𝒞 back to the real axis and expand the integrand for β→0. Using(4.42)βωβ+iδ/(2κ)eiwβ=2βκωiδ[1+iβ(w+2κδ)+O(β2)],β⟶0,
we thus obtain(4.43)ϕ0(ζ,w)~exp⁡[-(ω2+12ω2+δ2κμ)ζ]exp⁡(δ2κw)×2κωπδ∫-∞∞[βi+β2(w+2κδ)+O(β3)]exp⁡[-κζμΔβ2]dβ,
where Δ=ω2+δ2/(κμ). Evaluating the integral(s) in (4.43) leads to (3.25). If we consider the opposite limit, where ζ=O(1) and w→∞, then the saddle point approaches the upper branch point at ib+. But by deforming 𝒞 to an integral about the branch cut we can show that the final result coincides with the expansion of (3.25) for w/ζ≫1, which is given by (3.28).Next, we consider the corner behavior ofϕ0 as ζ,w→0. Now, the main contribution to the integral will come from where |β| is large. From (3.2) for δ>0, we then obtain (4.44)ϕ0(ζ,w)~ωπ∫-∞∞eiwβe-(κ/μ)ζ|β|dβ=ωπμκ2ζ[1+μκw2ζ2]-1,
which yields (3.30), and this can be shown to remain valid for δ<0.Finally, we fixw and let ζ→0. Simply, setting ζ=0 in (3.2) leads to a divergent integral. However, an integration by parts leads to, for δ>0, (4.45)ϕ0(ζ,w)=ωπexp⁡(δ2κw)∫-∞∞exp⁡[iwβ-κμβ2+μ4κΔ2ζ][-g0′(β)]dβ,
where Δ=ω2+δ2/κμ and(4.46)g0(β)=ββ+iδ/2κ[iw-ββ2+(μ/4κ)Δ2κμζ]-1.
Expanding (4.46) for ζ→0 and noting that, by contour integration (if δ>0),(4.47)∫-∞∞eiwβddβ[ββ+iδ/(2κ)]dβ=0,
we write the integrand as(4.48)eiwβ[1-κμβ2+μ4κΔ2ζ+O(ζ2)]×{iwddβ(ββ+iδ/(2κ))+ζw2ddβ[ββ+iδ/(2κ)ββ2+(μ/(4κ))Δ2]+O(ζ2)}.
By using (4.47), identifying the O(ζ) terms in (4.48), and explicitly performing the differentiation with respect to β, we ultimately obtain (3.31) and (3.32). This completes the (sketched) derivation of Proposition 3.22.


## 5. Analysis of Boundary and Corner Regions

We analyze cases wherem=o(ε-1) and/or n=o(ε-1/2). While these carry mass that is asymptotically small, they must be considered to insure that p(m,n) is properly normalized to higher orders in ε, and to compute higher order approximations to the moments. Also, to determine ϕ0(ζ,w) we used the boundary condition ϕ0(0,w)=ωδ(w), and analysis of cases where ζ and w are small will allow us to examine this condition more carefully.First we observe from (3.31) that ϕ0(ζ,w) vanishes linearly as ζ→0, which indicates a nonuniformity in the asymptotics. We first consider the scale m=O(1) with w>0 and set(5.1)p(m,n)=εν1P̃(m,w;ε)~εν1P̃(m,w).
Here, ν1 is a constant that will be determined by asymptotic matching. From (2.3), in terms of the variables m and w, we have(5.2)[λ+μ+ε(σ+κ-κμ)]P̃(m,w;ε)=λP̃(m-1,w;ε)+εσP̃(m,w-ε;ε)+μ(1-εκ)P̃(m+1,w;ε)+εκP̃(m,w+ε;ε).
Thus, the leading term must satisfy, since λ=μ+O(ε),2μ𝒫̃(m,w)=μ𝒫̃(m-1,w)+μ𝒫̃(m+1,w), and we write 𝒫̃ as(5.3)P̃(m,w)=mP̃+(w)+P̃-(w),
which is a linear function of m. The expansion for m=O(1) must satisfy the boundary condition (2.5), which is(5.4)(λ+εσ+1)P̃(0,w;ε)=εσP̃(0,w-ε;ε)+μ(1-εκ)P̃(1,w;ε)+P̃(0,w+ε;ε).
To leading order, (5.4) implies that 𝒫̃(0,w)=𝒫̃(1,w) so that (5.3) becomes 𝒫̃(m,w)=𝒫̃-(w). But, as m→∞,εν1𝒫̃-(w) cannot match to ε3/2ϕ0(ζ,w), since ϕ0 vanishes as ζ=mε→0. This indicates that we must analyze another scale, which has m≫1 and m≪1/ε (with w fixed).We thus setm=S/ε and(5.5)p(m,n)=εν2P(S,w;ε)~εν2P(S,w).
By rewriting (2.3) on the (S,w) scale, we obtain(5.6)μ[1-ε(ω+κ)-ε3/2δ][P(S,w;ε)-P(S-ε,w;ε)]=εσ[P(S,w-ε;ε)-P(S,w;ε)]+εκ[P(S+ε,w;ε)-P(S,w;ε)]+μ(1-εκ)[P(S+ε,w;ε)-P(S,w;ε)].
The limiting form of (5.6) as ε→0 is 𝒫SS=0, so we write(5.7)P(S,w)=SP+(w)+P-(w).
On the (S,w) scale the boundary condition (2.5), using also λ=μ[1-ε(ω+σ)] and σ=κ+εδ, becomes(5.8)[μ+1+ε(σ-σμ-ωμ)]P(0,w;ε)=εσP(0,w-ε;ε)+P(0,w+ε;ε)+μ(1-εκ)P(ε,w;ε).
To leading order as ε→0, (5.8) implies that(5.9)Pw(0,w)+μPS(0,w)=0.
Combining (5.7) and (5.9), we thus write the approximation on the (S,w) scale as(5.10)p(m,n)~εν2[SP+(w)+μ∫w∞P+(u)du].
We then determine ν2 and 𝒫+(w) by asymptotically matching (5.10), as S→∞, to ε3/2ϕ0(ζ,w) as ζ→0. Noting that S=ζ/ε, we find, in view of (3.31), that(5.11)ν2=2,P+(w)=ωπκμexp⁡(δw2κ)a(w).
This establishes (3.34) in Proposition 3.24. Expression (5.10) remains valid for m=O(1) as then S=εm→0 and we obtain p(m,n)~ε2μ∫w∞𝒫+(u)du, which is independent of m and consistent with our previous analysis for m=O(1).We consider some limiting cases of the(S,w) scale result in (3.34). As δ→∞ from (3.28) we obtain b+~δ/(2κ) and then (3.32) yields, after some computations,(5.12)a(w)~exp⁡(-wδ2κ)2πw3/212δ2κ;δ⟶+∞,w=O(1).
It follows from (3.34) that(5.13)P+(w)~ω2πδκ1w3/2,P-(w)~ωπδμ1w;δ⟶+∞,w=O(1).
The above results are consistent with those of Morrison in [16] for the case σ>κ, where it was shown that(5.14)∑m=0∞p(m,n)~εωμπ(σ-κ)εn~p(m,n)ε(σ-κ);σ>κ,n≫1.
Hence, for δ=(σ-κ)/ε→∞ our approximation p(m,n)~ε2𝒫-(w) agrees with (5.14).We will next consider the scalem,n=O(1) and matching this to the (S,w) (or (m,w)) scale(s) will require the behavior of (3.34) as w→0. For w→0 the second integral in the definition of a(w) in (3.32) dominates, and we obtain(5.15)a(w)~2w2,w⟶0+,
so that(5.16)P+(w)~ωπκμ2w2,w⟶0+.
Also, 𝒫-(w) will be less singular (O(w-1)) than 𝒫+(w), and, since w=n/ε, for w→0+ the approximation ε2[εm𝒫+(w)+𝒫-(w)] becomes of order O(ε3/2) on the (m,n) scale.On the scalem,n=O(1) we will show that p(m,0) is asymptotically larger than p(m,n) for n⩾1, which we just inferred to be O(ε3/2). We thus set(5.17)p(m,n)={ε3/2Q(m,n)+O(ε2),n⩾1,εQ*(m)+ε3/2Q*(1)(m)+O(ε2),n=0.
On the (m,n) scale, the limiting form of (2.3), with (5.17), becomes(5.18)2μQ(m,n)=μQ(m-1,n)+μQ(m+1,n),
while the boundary condition (2.5) yields(5.19)(μ+1)Q(0,n)=μQ(1,n)+Q(0,n+1),n⩾2.
Hence, Q(m,n) is linear in m, and writing(5.20)Q(m,n)=Q+(n)⋅m+Q-(n),
we then obtain from (2.5), Q-(n)-Q-(n+1)=μQ+(n) which we sum to get(5.21)Q-(n)=μ∑j=n∞Q+(j).
Next, we consider the boundary condition (2.4) and the corner condition (2.6). Since λ=μ+O(ε), we find that(5.22)2μQ*(m)=μQ*(m-1)+μQ*(m+1),
and Q*(1)(m) also satisfies (5.22). Hence, we have(5.23)Q*(m)=γ+γ*m,
and Q*(1)(m)=γ(1)+γ*(1)m. The corner condition (2.6), with (5.17), yields to leading order (O(ε))(5.24)μQ*(0)=μQ*(1),
and at the next order (O(ε3/2))(5.25)μQ*(1)(0)=μQ*(1)(1)+Q(0,1).
It follows from (5.24) that γ*=0. Also, (2.5) with n=1 becomes (μ+1)p(0,1)=μp(1,1)+p(0,2)+O(εp(0,0)) so that (5.19) holds to leading order even if n=1. To summarize, we have obtained(5.26)p(m,n)~{ε3/2[mQ+(n)+μ∑j=n∞Q+(j)],n⩾1,εγ+O(ε3/2),n=0.
It remains to determine Q+(n) and the constant γ. By letting m=S/ε and n→∞ and matching to the (S,w) scale, we conclude that as n→∞(5.27)Q+(n)~ωπκμ2n2,Q-(n)~ωπμκ2n.
While these relations are consistent with (5.21), they do not determine Q+(n) for n=O(1).We thus examine another scale, that hasm=S/ε=O(ε-1/2) and n=O(1). By matching to (5.26) as m→∞, we expect that p(m,n) will be O(ε) on this scale, so we set(5.28)p(m,n)~εQ¯(S,n).
Then, from (2.3), we obtain to leading order(5.29)μQ¯SS(S,n)+κ[Q¯(S,n+1)+Q¯(S,n-1)-2Q¯(S,n)]=0,n⩾1,
while the boundary condition (2.4) leads to(5.30)μQ¯SS(S,0)+κ[Q¯(S,1)-Q¯(S,0)]=0.Thus (5.29) corresponds to driftless diffusion in the S variable and a driftless random walk in the discrete n variable. We may replace (5.30) by the “artificial” boundary condition Q¯(S,-1)=Q¯(S,0), by extending (5.29) to hold also at n=0.To analyze (5.29) consider the discrete Green’s function problem(5.31)G(n+1)+G(n-1)+(ν-2)G(n)=-δ(n,n0),G(-1)=G(0),
where ν is a “spectral” parameter. This is easily solved to yield(5.32)G=G(n;n0)=1ν(ν-4)[A|n-n0|+An+n0+1],A=A(ν)=12[2-ν-ν(ν-4)].
By integrating (5.32) over a loop in the complex ν-plane that encircles the branch cut Re(ν)∈[0,4], we obtain(5.33)δ(n,n0)=12πi∮[A(ν)]|n-n0|+[A(ν)]n+n0+1ν(ν-4)dν=1π∫0π{cos⁡[(n-n0)Ω]+cos⁡[(n+n0+1)Ω]}dΩ,
where we evaluated the contour integral explicitly. Here, δ(n,n0) is the Kronecker delta symbol. From (5.33), we infer that for any sequence {f(n)}(5.34)f(n)=2π∑L=0∞∫0πf(L)cos⁡[(n+12)Ω]cos⁡[(L+12)Ω]dΩ,
which leads to the transform pair(5.35)g(Ω)=∑L=0∞f(L)cos⁡[(L+12)Ω]f(L)=2π∫0πg(Ω)cos⁡[(L+12)Ω]dΩ.
Returning to (5.29) we set(5.36)Q¯(S,n)=2π∫0πq(s,Ω)cos⁡[(n+12)Ω]dΩ,
and applying the transform in (5.35) to (5.29) and (5.30) leads to the ODE(5.37)qSS+2κμ(cos⁡Ω-1)q=0.
Using cosΩ-1=2sin2(Ω/2) and requiring also q to decay as S→∞, we thus have (5.38)q=f(Ω)exp⁡[-2κμsin(Ω2)s],
and hence(5.39)Q¯(S,n)=2π∫0πf(Ω)cos⁡[(n+12)Ω]exp⁡[-2κμsin(Ω2)S]dΩ.
We determine the function f(Ω) by asymptotic matching.First we consider the matching ofεQ¯(S,n) as S→∞ to ε3/2ϕ0(ζ,0) as ζ=εS→0. For S→∞ the major contribution to the integral in (5.39) will come from near Ω=0, and the Laplace method yields(5.40)Q¯(S,n)~2πμκf(0)S.
Setting w=0 in (3.2) and letting ζ→0, or, equivalently, using (3.30), we conclude that f(0)=ω. Now, we match the (S,n) and (m,n) scales.Forn⩾1,εQ¯(S,n) as S→0 should agree with ε3/2Q(m,n) as m→∞, which is possible only if Q¯(0,n)=0. Thus, f(Ω) in (5.39) must be orthogonal to cos[(n+1/2)Ω] for all n⩾1 and hence f(Ω)=kcos(Ω/2). The constant k must equal ω, since f(0)=ω.Having determinedf(Ω) we proceed to let S→0 in (5.39). If n=0,Q¯(S,0)~2ωπ-1∫0πcos2(Ω/2)dΩ=ω so that γ=ω in (5.26). If n⩾1(5.41)Q¯(S,n)~2ωπ(-2S)κμ∫0πcos⁡(Ω2)sin(Ω2)cos⁡[(n+12)Ω]dΩ=2ωπκμS(n+3/2)(n-1/2).
Thus, by matching ε3/2Q(m,n) as m→∞ to εQ¯(S,n) as S→0 we conclude that(5.42)Q+(n)=2ωπκμ4(2n+3)(2n-1).
Then, we easily obtain(5.43)∑j=n∞Q+(j)=2ωπκμ4n4n2-1.
Now, Q+,Q- in (5.20) are determined, as is γ in (5.26), and we have established (3.36) in Proposition 3.24. Also, using f(Ω)=ωcos(Ω/2) in (5.39) leads to (3.35) in Proposition 3.24.Thus, we have used a singular perturbation analysis to approximatep(m,n) on scales other than the (ζ,w) scale. While these carry mass that is o(1), the size of p(m,n) may actually be larger than O(ε3/2), as is evident from (3.35).



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*Source: 216790-2011-03-15.xml*