Distribution-valued heavy-traffic limits for the G/GI/Infty queue

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Distribution-valued heavy-traffic limits for the G/GI/ ∞ queue

Josh Reed Rishi TalrejaStern School of Business Knight Capital Group

New York University Jersey City, NJNew York, NY

June 16, 2012

Abstract

We study the G/GI/ ∞ queue in heavy-traffic using tempered distribution-valued processeswhich track the age and residual service time of each customer in the system. In both cases, weuse the continuous mapping theorem together with functional central limit theorem results inorder to obtain uid and diffusion limits for these processes in the space of tempered distribution-valued processes. We nd that our diffusion limits are tempered distribution-valued Ornstein-Uhlenbeck processes.

1 Introduction

Limit theorems for the innite-server queue in heavy-traffic have a rich history starting with theseminal work of Iglehart [ 17] on the M/M/ ∞ queue. This work then inspired a line of research

aimed at extending the results of [17] to additional classes of service time distributions. Whitt [ 32]studies the GI/PH/ ∞ queue, having phase-type service-time distributions, and Glynn and Whitt[11] consider the GI/GI/ ∞ queue with service times taking values in a nite set. In [ 5], [25] and[31], the G/GI/ ∞ queue is studied with general service time distributions. [29] gives a survey of these results.

In this paper, we study two Markov processes associated with the G/GI/ ∞ queue. The rstprocess which we study is a tempered distribution-valued process which tracks the age of eachcustomer in the system. We refer to this process as the age process. The second process which westudy is also a tempered distribution-valued process and it tracks the residual service time of eachcustomer in the system as well as the amount of time since departure for each customer who hasleft the system. We refer to this process as the residual service time process. Although analyzing

either of these processes might at rst appear to be a difficult task, one of the key themes thatruns throughout the present paper is that techniques originally developed for establishing heavy-traffic limits in the nite-dimensional setting may also be successfully applied in the more abstractinnite-dimensional setting.

Our main results in this paper are to obtain uid and diffusion limits for both the age and resid-ual service time processes. In particular, for both the age and the residual service time process, weuse the continuous mapping theorem together with functional central limit theorem results in order

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p ∈ N and which also induce the same the topology on S as the seminorms ·α,β : α, β ∈ N given above. In particular, by Lemma 1.3.3 of [ 21], one has that for each p ∈ N , there exists ak ∈N and C > 0 such that

ϕ p ≤ C max0≤ α,β ≤ k+1

ϕ α,β for all ϕ ∈ S , (3)

and, by Lemma 1.3.4 of [ 21], one has that for each α, β ∈N , there exists a p ∈N and M > 0 suchthat

max0≤ α,β ≤ k+1

ϕ α,β ≤ M ϕ p for all ϕ ∈ S . (4)

For a precise construction of · p for each p ∈N , one may consult page 24 of [21] .The set of all linear maps from S to S is denoted by L(S , S ) and the strong topology on L(S , S )is dened in the following manner. A subset B of S is said to be bounded if for any neighborhood

U of ϕ ≡ 0 ∈ S , there exists a constant α > 0 such that α− 1B ⊂ U (see Denition 1.1.7 of [21]).The strong topology on L(S , S ) is then given by the following denition (see Theorem 1.2.1 of [21]).

Denition 1.2. For each bounded subset B of

S and p

∈N , let

q B,p (T ) ≡ supϕ∈B

Tϕ p for all T ∈ L(S , S ).

Then, q B,p constitutes a family of seminorms on L(S , S ) and the topology given by these semi-norms is referred to as the strong topology on L(S , S ).

1.1.3 The Space of Tempered Distributions

Many of the processes studied in this paper take values in the topological dual of S , which we denoteby S . Recall that S is the space of all continuous linear functionals on S . Elements of S arereferred to as tempered distributions and we now review some relevant facts concerning tempered

distributions as well as tempered distribution-valued processes.For each µ ∈ S and ϕ ∈ S , we denote the duality product of µ and ϕ by µ,ϕ ≡ µ(ϕ). Thedistributional derivative of µ ∈ S is denoted by µ and is dened to be the unique element of S such that

µ ,ϕ = −µ,ϕ for all ϕ ∈ S .

It is clear by the denition of S that µ is well-dened . For each µ ∈ S and t ∈R , we also deneτ t µ as the unique element of S such that

τ t µ,ϕ = µ, τ tϕ for all ϕ ∈ S ,

where τ tϕ ∈ S is the function dened by τ tϕ(·) ≡ ϕ(· −t).All statements in this paper regarding convergence in

S are with respect to the strong topology

on S , which we now dene. One may consult Section 1.1 of [21] for further details.Denition 1.3. For each bounded subset B ⊂ S , let

q B (µ) ≡ supϕ∈B | µ,ϕ | for all µ ∈ S . (5)

Then, the strong topology on S is the topology induced by the family of seminorms q B .

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Unfortunately, the space S is not metrizable with respect to the strong topology (see Section 2of [21]). Nevertheless, as we discuss below, one may still usefully speak of weak convergence of

S -valued random elements and processes taking values in S .Let D ([0, T ], S ) denote the space of functions from [0 , T ] to S that are right-continuous on[0, T ) with left limits everywhere on (0 , T ]. If (µt )t≥ 0 ∈ D ([0, T ], S ) and t ∈ [0, T ], we then dene

the tempered distribution t

0 µs ds to be the unique element of S (see Section 2 of [19]) such that

t

0µs ds,ϕ =

t

0µs ,ϕ ds for all t ≥ 0, ϕ ∈ S .

As noted immediately following Denition 1.3 above, the space S equipped with the strongtopology is not metrizable and so the Skorokhod metric and ensuing Skorokhod J 1-topology maynot dened on D ([0, T ], S ) in the usual manner. We therefore follow the approach of [21, 26] indening an appropriate topology on D ([0, T ], S ). Let Λ be the set of strictly increasing continuousmaps from [0, T ] onto itself such that for each λ ∈ Λ,

γ (λ) = sup0≤ s<t ≤ T

lnλ t −λ s

t −s<

∞.

We then have the following denition (see [ 21, 26]).

Denition 1.4. For each seminorm q B dening the strong topology on S , let

doqB (µ, ν ) = inf

λ∈Λ sup0≤ t≤ T |q B (µt −ν λ t ) + γ (λ)| for all µ, ν ∈ S .

The topology on D ([0, T ], S ) is then dened by the family of pseudometrics doqB .

By part (c) of Theorem 2.4.1 of [21], the topology given in Denition 1.4 above is equivalent to the

topology dened by the family of pseudometrics dqB , where

dqB (µ, ν ) = inf λ∈Λ

sup0≤ t≤ T |q B (µt −ν λ t )|+ sup

0≤ t≤ T |λ t −t| for all µ, ν ∈ S .

We also note that under this topology, D ([0, T ], S ) is a completely regular topological space [ 26].The following result is an important consequence of Proposition 5.2 of [26] regarding weak con-

vergence of processes taking values in the dual of a nuclear Frechet space. It provides a convenientcharacterization of weak convergence of processes taking values in S .Theorem 1.5 (Mitoma’s Theorem) . Let (µn )n≥ 1 be a sequence of random elements of D ([0, T ], S ).Then,

µn⇒ µ in D ([0, T ], S )

if the following two statements hold:

1. For each ϕ ∈ S , the sequence µn ,ϕ n ≥ 1 is tight in D ([0, T ], R ).

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2. For ϕ1, . . . ,ϕm ∈ S and t1, . . . , t m ∈ [0, T ],

µnt 1 ,ϕ1 , . . . , µn

tm ,ϕm ⇒ ( µt 1 ,ϕ1 , . . . , µt m ,ϕm ) in R m .

We now conclude the technical background section with some comments regarding martingalesand, in particular,

S -valued martingales. Let (

F t)

t≥ 0 be a ltration on an underlying probability

space (Ω, F , P ) and let M and N be two R -valued F t -matingales. The quadratic covariation of M and N is denoted by ( < M, N > t )t≥ 0 and the quadratic variation of M is denoted by ( << M >> t)t≥ 0 ≡ (< M,M > t ) t≥ 0. An S -valued process M is said to be an S -valued F t -martingale if for allϕ ∈ S , ( M t ,ϕ )t≥ 0 is an R -valued F t -martingale. For two S -valued martingales M and N , theirtensor quadratic covariation (< M,N > t )t≥ 0 is given for all t ≥ 0 and all ϕ, ψ ∈ S by

< M, N > t (ϕ, ψ) ≡< M ·,ϕ , N ·, ψ > t ,

and the tensor quadratic variation (<< M >> t )t≥ 0 of an S -valued martingale is given by ( <<M >> t ) t≥ 0 ≡ (< M, M > t )t≥ 0. Two S -valued martingales, M and N , are said to be orthogonal if < M, N > = 0 identically. Corresponding notions for the optional quadratic variation process [ M ]

are dened analogously.

2 System Equations

In this section, we obtain semi-martingale decompositions of the tempered distribution-valued ageprocess A ≡ (At )t≥ 0 and the tempered distribution-valued residual service time process R ≡(Rt )t≥ 0. We begin in §2.1 by treating the age process A and then move on in §2.2 to treating theresidual service time process R.

2.1 Age Process

We consider a G/GI/ ∞ queue with general arrival process ( E t )t≥ 0 ∈D

([0, ∞),R

). We assume thatE 0 = 0, P -a.s., and, for convenience in our proofs, we also dene E t = 0 for t < 0. We also makethe assumption that for each t ≥ 0, we have that E [E 2t ] < ∞. Next, for each i ≥ 1, we denote by

τ i = inf t ≥ 0 : E t ≥ ithe time of the arrival of the ith customer to the system after time t = 0. We assume that E [τ i ] < ∞for each i = 1 , 2,... We denote by ηi the service time of the ith customer to arrive to the system aftertime t = 0 and we assume that ηi , i ≥ 1 is an i.i.d. sequence of non-negative, mean 1 randomvariables with cumulative distribution function (cdf) F , complementary cumulative distributionfunction (ccdf) F = 1 − F , and probability density function (pdf) f . We also assume that thehazard rate function h of F satises the following assumption.

Assumption 2.1. The function h ∈ C ∞b (R + ).

Now let (At ) t≥ 0 ∈ D ([0, ∞), D ) be such that for each t ≥ 0 and y ≥ 0, the quantity At (y)represents the number of customers in the system at time t ≥ 0 that have been in the system forless than or equal to y units of time at time t. For y < 0, we set At (y) = 0. At time t = 0, weassume that there are A0(y) customers present who have been in the system for less than or equal

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to y ≥ 0 units of time and that there are a total of A0(∞) customers present. We assume thatE [A2

0(∞)] < ∞. For each i = 1 ,...,A 0(∞), we denote by

τ i = −inf y ≥ 0 : A0(y) ≥ ithe “arrival” time of the ith initial customer to the system. We denote by ˜ ηi the remaining service

time at time t = 0 of the ith initial customer in the system. The distribution of ˜ ηi , conditional onthe arrival time ˜τ i , is given by

P (ηi > x |τ i ) = 1−F (−τ i + x)

1 −F (−τ i ) , x ≥ 0. (6)

We denote by f τ i the conditional pdf associated with this distribution and we set h τ i (·) = h(· −τ i ).We now derive a convenient representation for the system equations for ( At )t≥ 0 and its tempered

distribution-valued counterpart, A, which we dene shortly. We begin by noting that by rstprinciples we have that for each t ≥ 0 for y ≥ 0,

At (y) =A0 (∞ )

i=1

1 t− τ i ≤ y1 t< ηi +E t

i=1

1 t− τ i ≤ y1 t− τ i <η i . (7)

Our rst result provides an alternative way to write ( 7). In the following, we set 0i=1 = 0.

Proposition 2.2. For each t ≥ 0 and y ≥ 0,

At (y) = A0(y) −A0 (∞ )

i=1

1 ηi ≤ t∧(y+ τ i ) −A0 (y)

i= A0 (y− t )+1

1 ηi >y + τ i

+ E t −E t

i=1

1 ηi ≤ (t− τ i )∧y −E ( t − y ) −

i=1

1 ηi >y . (8)

Proof. By (7), we have that

At (y) =A0 (y)

i=1

1 t− τ i ≤ y1 t< ηi +E t

i=1

1 t− τ i ≤ y1 t− τ i <η i

= A0(y) +A0 (y)

i=1

(1 t− τ i ≤ y1 t< ηi −1) + E t +E t

i=1

(1 t− τ i ≤ y1 t− τ i <η i −1). (9)

However,

1 −1 t− τ i ≤ y1 t< ηi = 1 t− τ i ≤ y1 ηi ≤ t + 1 t− τ i >y (10)= 1 t− τ i ≤ y1 ηi ≤ t + 1 t− τ i >y 1 − τ i +˜ηi ≤ y + 1 t− τ i >y 1 − τ i + ηi >y ,

and, similarly,

1 −1 t− τ i ≤ y1 t− τ i <η i = 1 t− τ i ≤ y1 ηi ≤ t− τ i + 1 t− τ i >y (11)= 1 t− τ i ≤ y1 ηi ≤ t− τ i + 1 t− τ i >y 1 ηi ≤ y + 1 t− τ i >y 1 ηi >y .

Substituting ( 11) and ( 10) into ( 9) and summing over A0(y) and E t completes the proof.

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We now provide an intuitive description of each of the terms appearing in ( 8). The rst termrepresents the number of customers in the system at time t = 0 that have been in the system forless than or equal to y units of time, the second term represents the number of departures by timet ≥ 0 of those initial customers that had total service less than or equal to y units of time at timet = 0, and the third term represents the number of initial customers whose total service time is

greater than y units of time and had been in the system for less than or equal to y units of time attime t = 0 but have been been in the system for greater than y units of time at time t ≥ 0. Thefourth, fth and sixth terms represent similar quantities but for those customers that arrived tothe system after time t = 0.

Now let D 0 = ( D 0t )t≥ 0 ∈D ([0, ∞), D ) be dened by setting

D 0t (y) =

A0 (∞ )

i=1

1 ηi ≤ t∧(y+ τ i ) − ηi∧t∧(y+ τ i )

0h τ i (u) du , t ≥ 0, y ≥ 0, (12)

and set D0t (y) = 0 for y < 0, and let D = ( D t )t≥ 0 ∈D ([0, ∞), D ) be dened by setting

D t (y) =

E t

i=1

1ηi ≤ (t− τ i )∧y −

ηi∧(t− τ i )∧y

0 h(u) du , t ≥ 0, y ≥ 0, (13)

and set Dt (y) = 0 for y < 0. It then follows from ( 8) that for each t ≥ 0 and y ≥ 0, we may write

At (y) = A0(y) + E t −D 0t (y) −D t (y) −

A0 (∞ )

i=1 ηi∧t∧(y+ τ i )

0h τ i (u) du

−E t

i=1 ηi∧(t− τ i )∧y

0h(u) du −

A0 (y)

i= A0 (y− t )+1

1 − τ i + ηi >y −E ( t − y ) −

i=1

1 ηi >y . (14)

The above expression for At (y) will become useful in a moment. However, we next move on to

expressing the age process as a tempered distribution-valued process using the Schwartz space S dened in ( 1). In particular, we associate with the process A dened in (7) the S -valued process

A = (At )t≥ 0 such that for each t ≥ 0 and ϕ ∈ S we set

At ,ϕ = Rϕ(y) dAt (y). (15)

In a similar manner, we associate the S -valued processes D0 = ( D0t )t≥ 0 and D = (Dt )t≥ 0 with D 0

and D , respectively. That is, for each t ≥ 0 and ϕ ∈ S we set

D0t ,ϕ = Rϕ(y) dD 0

t (y) and Dt ,ϕ = Rϕ(y) dD t (y).

We also associate the S -valued random variable A0 with A0 by setting

A0,ϕ = Rϕ(y) dA0(y), ϕ ∈ S .

It is straightforward to see that for each t ≥ 0, the quantities At , D0t and Dt are well dened

elements of S . Moreover, since for each xed ϕ ∈ S the sample paths of ( At ,ϕ ) t≥ 0, ( D0t ,ϕ )t≥ 0and ( Dt ,ϕ )t≥ 0 all lie in D ([0, ∞), R ), P -a.s., it follows that A, D0, D ∈D ([0, ∞), S ), P -a.s.

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For the remainder of the paper, we now replace the hazard rate function h : R + → R with afunction h : R → R such that h(x) = h(x) for x ≥ 0 and h ∈ C ∞b (R ). In a similar manner, wereplace the cdf F and ccdf F with corresponding functions F and ˜F such that F , ˜F ∈ C ∞b (R ). Forease of notation, we continue to refer to h, F and ˜F as h, F and F , respectively.

Our next step is to use the expression ( 14) in order to provide a convenient expression for the

tempered distribution-valued process A. We begin by noting that integrating test functions ϕ ∈ S term-by-term in ( 14) it follows that for each t ≥ 0 one has that

At ,ϕ = A0,ϕ − D0t + Dt ,ϕ −

A0 (∞ )

i=1 − τ i +(˜ηi∧t )

− τ iϕ(y)h(y) dy

−E t

i=1 ηi∧(t− τ i )

0ϕ(y)h(y) dy − R +

ϕ(y) dA0 (y)

i= A0 (y− t )+1

1 − τ i +˜ηi >y +E ( t − y ) −

i=1

1 ηi >y . (16)

The following two propositions now allow us to further simplify the expression in ( 16). We rsthave the following.

Proposition 2.3. For each t ≥ 0,

A0 (∞ )

i=1 − τ i +(˜ηi∧t )

− τ iϕ(y)h(y) dy +

E t

i=1 ηi∧(t− τ i )

0ϕ(y)h(y) dy =

t

0 As ,ϕh ds.

Proof. For each t ≥ 0,

A0 (∞ )

i=1 − τ i +(˜ηi∧t )

− τ iϕ(y)h(y) dy +

E t

i=1 ηi∧(t− τ i )

0ϕ(y)h(y) dy

=A0 (∞ )

i=1 t

01 0≤ s≤ ηi ϕ(s − τ i )h(s − τ i ) ds +

E t

i=1 t

01 0≤ s− τ i ≤ ηi ϕ(s −τ i )h(s −τ i) ds

= t

0

A0 (∞ )

i=1

1 0≤ s≤ ηi ϕ(s − τ i )h(s − τ i ) +E t

i=1

1 0≤ s− τ i ≤ ηi ϕ(s −τ i )h(s −τ i ) ds

= t

0 As ,ϕh ds.

This completes the proof.

Next, we have the following.

Proposition 2.4. For each t ≥ 0,

− R +

ϕ(y) dA0 (y)

i= A0 (y− t )+1

1 − τ i + ηi >y +E ( t − y ) −

i=1

1 ηi >y = E tϕ(0) + t

0 As ,ϕ ds.

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Proof. Let t ≥ 0. Then, integrating by parts we have that

−E tϕ(0) − R +

ϕ(y) dA0 (y)

i= A0 (y− t )+1

1 − τ i +˜ηi >y +E ( t − y ) −

i=1

1 ηi >y

= R +

A0 (y)

i= A0 (y− t )+1

1 − τ i +˜ηi >y +E ( t − y ) −

i=1

1 ηi >y ϕ (y) dy

= R +

A0 (∞ )

i=1

1 τ i ≥− y, − τ i +˜ηi >y, τ i + y<t +E t

i=1

1 ηi >y,τ i + y<t ϕ (y) dy

=A0 (∞ )

i=1 R +

1 τ i ≥− y, − τ i +˜ηi >y, τ i + y<t ϕ (y) dy +E t

i=1 R +

1 ηi >y,τ i + y<t ϕ (y) dy

=A0 (∞ )

i=1 t

01 0≤ s− τ i ≤− τ i +˜ηi ϕ (s − τ i ) ds +

E t

i=1 t

01 0≤ s− τ i ≤ ηi ϕ (s −τ i ) ds

= t

0

A0 (∞ )

i=1

1 0≤ s− τ i ≤− τ i + ηi ϕ (s − τ i ) +E t

i=1

1 0≤ s− τ i ≤ ηi ϕ (s −τ i ) ds

= t

0 R +

ϕ (u)dAs (u) ds

= t

0 As ,ϕ ds.


Now note that combining Propositions 2.3 and 2.4 with system equation ( 16), one nds thatfor each t ≥ 0 and ϕ ∈ S ,

At ,ϕ = A0,ϕ + E t − D0t − Dt ,ϕ −

t

0 As , hϕ ds + t

0 As ,ϕ ds, (17)

where we dene the S -valued process E = (E t )t≥ 0 to be such that E t ,ϕ = E tϕ(0) for each ϕ ∈ S and t ≥ 0. In general, we refer to ( 17) as the semi-martingale decomposition of A. This will becomeclear in §4 where we show that the process ( D0

t + Dt )t≥ 0 is a martingale.

2.2 Residual Service Time Process

We next move on to analyzing the residual service time process

R. As in

§2.1, we assume that

we have a G/GI/ ∞ queue in which customers arrive to the system according to a general arrivalprocess ( E t )t≥ 0 ∈D ([0, ∞), R ), where we assume that E 0 = 0, P -a.s. For each i ≥ 1, we denote by

τ i = inf t ≥ 0 : E t ≥ ithe time of the ith customer arrival to the system after time t = 0 and we let ηi be the service timeof the ith customer to arrive to the system after time t = 0. We assume that ηi , i ≥ 1 is an i.i.d.

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=R 0 (∞ )

i=1 R1 ηi − t≤ y< ηi ϕ (y) dy +

E t

i=1 R1 (τ i + ηi )− t≤ y<η i ϕ (y) dy

=R 0 (∞ )

i=1 t

0ϕ (ηi −s) ds +

E t

i=1 t

01 τ i ≤ sϕ (τ i + ηi −s) ds

= t

0

R 0 (∞ )

i=1ϕ (ηi −s) +

E t

i=1

1 τ i ≤ sϕ (τ i + ηi −s) ds

= t

0 R +

ϕ dR s (u) ds

= t

0 Rs ,ϕ ds.


Substituting ( 24) into ( 23), one now obtains that for each t ≥ 0 and ϕ ∈ S ,

Rt ,ϕ = R0,ϕ + Gt ,ϕ + E t F ,ϕ − t

0 Rs ,ϕ ds. (25)

We refer to ( 25) as the semi-martingale decomposition of R. This is due to the fact that in §4 itwill be shown that the process G in (25) is a martingale. We also point out the similarity of ( 25)with (4) of [8].

3 Regulator Map Result

Let B : S → S be a continuous linear operator and for each µ ∈D ([0, T ], S ), consider the solutionν

∈D ([0, T ],

S ) to the integral equation

ν t ,ϕ = µt ,ϕ + t

0ν s , Bϕ ds, t ∈ [0, T ], ϕ ∈ S . (26)

In this section, we rst show that under some mild restrictions on B, (26) denes a continuousfunction Ψ B : D ([0, T ], S ) → D ([0, T ], S ) mapping µ to ν . We then proceed in §3.1 and §3.2 tostudy particular continuous linear operators associated with the age process and the residual servicetime process, respectively.

We begin with the following denition from [ 19].

Denition 3.1. A family (S t ) t≥ 0 of linear operators on S is said to be a strongly-continuous(C 0, 1) semi-group if the following three conditions are satised:

1. S 0 = I , where I is the identity operator, and, for all s, t ≥ 0, S s S t = S s+ t .

2. The map t → S t is continuous in the strong topology of L(S , S ). That is, if tn → t in R + ,then for any bounded subset K ⊂ S and p ≥ 1,

supϕ∈K

S t n ϕ−S tϕ p → 0.

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3. For each q ≥ 0, there exist numbers M q, σq and p ≥ q such that for all ϕ ∈ S and t ≥ 0,

S tϕ q ≤ M qeσq tϕ p.

Remark 3.2. Note that Condition 2 of Denition 3.1 is stronger than the corresponding condition for a (C 0, 1) semi-group as given, for example, in [ 19 ] . The weaker denition in [ 19 ] only requires that the map t → S tϕ be continuous in the weak topology of L(S , S ). That is, if tn → t in R + , then for all ϕ ∈ S and m ≥ 1, S tn ϕ−S tϕ m → 0.

Recall now that for a family ( S t ) t≥ 0 of linear operators on S , the innitesimal generator B of (S t )t≥ 0 is dened to be such that

Bϕ = limt→0

S tϕ−ϕt

in S ,for all such ϕ ∈ S that the limit on the righthand side above exits. We refer to such ϕ ∈ S as

D(B ), the domain of B. We now have the following result, which is our main result regarding the

integral equation ( 26).Theorem 3.3. Let B ∈ L(S , S ) be the innitesimal generator of a strongly-continuous (C 0, 1)semi-group (S t )t≥ 0. Then, for each µ ∈D ([0, T ], S ), the equation (26) has a unique solution given by

ν t ,ϕ = µt ,ϕ + t

0µs , S t− s Bϕ ds, t ∈ [0, T ], ϕ ∈ S . (27)

Furthermore, (27) denes a continuous function ΨB : D ([0, T ], S ) →D ([0, T ], S ) mapping µ to ν .

Proof. That Ψ B is a well-dened function from D ([0, T ], S ) to D ([0, T ], S ) and the form of thesolution ( 27) follows from Theorem 2.1 of [19] (see also Corollary 2.2 of [19]).

We now show that Ψ B is continuous. By ( 27), it suffices to show that the function mappingD ([0, T ], S ) to D ([0, T ], S ) dened by µ → ·

0 B∗S ∗·− s µs ds, where B∗ and S ∗t denote the adjointoperators of B and S t , respectively, is continuous. Let ( µn )n ≥ 1 be a sequence converging to µ inD ([0, T ], S ) . Then, by Denition 1.4 and the comment below it (see also Theorem 2.4.1 of [21]),there exists a sequence ( λn )n ≥ 1 of strictly increasing homeomorphisms of the interval [0 , T ] suchthat for each bounded set K ⊂ S ,

sup0≤ t≤ T

supϕ∈K

µnt −µλ n

t ,ϕ → 0 and sup

0≤ t≤ T |λnt −t| → 0 as n → ∞. (28)

Moreover, it suffices to consider homeomorphisms ( λn )n ≥ 1 that are absolutely continuous withrespect to Lebesgue measure on [0 , T ] having corresponding derivatives ( λn )n ≥ 1 satisfying λn −1 T

→ 0 as n

→ ∞ (see pages 112-114 of [1]). It then follows that for each t

∈ [0, T ] and ϕ

∈ S we

14





as n → ∞, which implies (30).We next prove the limit ( 31). First note that by Lemma 2.2 of [18], there exist θ ≥ 0 and q ≥ 1

such that

supt∈[0,T ]

supϕ∈K

t

0µλ n

s , (S t− s −S λ nt − λ n

s )Bϕ ds (36)

≤ T sups∈[0,T ]

sup0≤ w≤ v≤ T

supϕ∈K

µs , (S v− w −S λ nv − λ n

w )Bϕ

≤ T θ sup0≤ w≤ v≤ T

supϕ∈K

(S v− w −S λ nv − λ n

w )Bϕ q.

Next, note that for w ≤ v we may write

S v− w −S λ nv − λ n

w = 1(v− w)− (λ nv − λ n

w )≥ 0(S (v− w)− (λ nv − λ n

w ) −I )S λ nv − λ n

w

+ 1 (λ nv − λ n

w )− (v− w)≥ 0(I −S (λ nv − λ n

w )− (v− w) )S v− w .

Thus, recalling the denition of the set K = S u Bϕ, u ∈ [0, T ],ϕ ∈ K , it follows that

sup0≤ w≤ v≤ T

supϕ∈K

(S v− w −S λ nv − λ n

w )Bϕ q (37)

≤ sup0≤ w≤ v≤ T

supϕ∈K

1 (v− w)− (λ nv − λ n

w )≥ 0(S (v− w)− (λ nv − λ n

w ) −I )S λ nv − λ n

w Bϕ q

+ sup0≤ w≤ v≤ T

supϕ∈K

1 (λ nv − λ n

w )− (v− w)≥ 0(I −S (λ nv − λ n

w )− (v− w) )S v− wBϕ q

≤ sup0≤ w≤ v≤ T

supϕ∈K

1 (v− w)− (λ nv − λ n

w )≥ 0(S (v− w)− (λ nv − λ n

w ) −I )ϕ q


supϕ∈K

1 (λ nv − λ n

w )− (v− w)≥ 0(I −S (λ nv − λ n

w )− (v− w) )ϕ q.

However, note that since λn −1 T → 0 as n → ∞, it follows that

sup0≤ w≤ v≤ T |(v −w) −(λn

v −λnw)| → 0 as n → ∞.

Thus, since by ( 34) the set K ⊂ S is bounded, it follows by Part 2 of Denition 3.1 that

sup0≤ w≤ v≤ T

supϕ∈K

1 (v− w)− (λ nv − λ n

w )≥ 0(S (v− w)− (λ nv − λ n

w ) −I )ϕ q


supϕ∈K

1 (λ nv − λ n

w )− (v− w)≥ 0(I −S (λ nv − λ n

w )− (v− w) )ϕ q

→ 0 as n → ∞,which, by ( 36) and ( 37), implies (31).

Finally, ( 32) follows from the fact that

supt∈[0,T ]

supϕ∈K

t

0µn

s −µλ ns , S t− s Bϕ ds ≤ T sup

s∈[0,T ]sup

u∈[0,T ],ϕ∈K µn

s −µλ ns , S u Bϕ → 0, (38)

as n → ∞, where the nal convegence follows from ( 28) and ( 34). This completes the proof.

16



3.1 Age Process

Now let B A be the linear operator dened on S such that

B Aϕ = ϕ −hϕ for all ϕ ∈ S . (39)

Our main result in this subsection is to verify that BA

generates a strongly-continuous ( C 0, 1)semi-group. This will then be useful in §5.1 and §6.1, where we prove our uid and diffusion limits,respectively, for the age process. In particular, by Theorem 3.3 of the preceding subsection and(17) of §2.1, this will then allow us to write

A = Ψ B A (A0 + E −(D0 + D)) , (40)

where the map Ψ B A : D ([0, T ], S ) →D ([0, T ], S ) is a continuous map. We begin with the followinglemma. Its proof may be found in the Appendix.

Lemma 3.4.

1. For each n ≥ 1 and t ≥ 0,

supx≥ 0

F (x + t)F (x)

(n )

< ∞. (41)

2. For each T > 0, there exists a sequence (M n )n ≥ 0 such that for each s, t ∈ [0, T ],

supx≥ 0

F (x + t)F (x) −

F (x + s)F (x)

(n )

≤ M n |t −s|. (42)

Proof. See Appendix.

Next, we have the following.Lemma 3.5. For each ϕ ∈ S , t ∈R and α, β ∈N , we have

τ tϕ α,β ≤ (1∨ |t|α )2α max0≤ i≤ α

ϕ i,β . (43)

Proof. For each ϕ ∈ S , t ∈R and α, β ∈N , we have

τ tϕ α,β = ϕ(· −t) α,β = supx∈R

xαϕ

(β ) (x −t)

= supx∈R

[t + ( x −t )]αϕ

(β ) (x −t)

= supx∈R

α

i=0

αi

tα − i (x −t)iϕ(β ) (x −t)

≤α

i=0

αi |t|α − i

ϕ i,β ≤ (1∨ |t|α )2α max0≤ i≤ α

ϕ i,β .


17



The following is now our main result of this subsection.

Proposition 3.6. The linear operator BA dened by (39) generates a strongly-continuous (C 0, 1)semi-group (S At )t≥ 0 given by

S At ϕ = (1 / F )τ − t Fϕ for all ϕ

∈ S . (44)

Proof. We rst check that B A is indeed the innitesimal generator of the semi-group given by ( 44).In order to do so, it suffices to check that for each α, β ∈N ,

limt→0

S At ϕ−ϕt −(ϕ −hϕ)

α,β = 0 . (45)

We begin by noting that for each t ≥ 0, x ∈R and ϕ ∈ S , we have that

S At ϕ(x) =1 −F (x −t)

1 −F (x) ·τ tϕ(x) ≤ e h ∞ t · |τ tϕ(x)|,

and so it follows that S At ϕ ∈ S . Now let x ∈R be xed and note that for each t ≥ 0, we may write

S At ϕ(x) = exp − x+ t

xh(v)dv ϕ(x + t).

Hence, by Taylor’s theorem, expanding in terms of t we obtain that we may write

S At ϕ(x) = ϕ(x) + ( ϕ (x) −h(x)ϕ(x)) t + R(x, t ), (46)

where the remainder term R(x, t ) has the from

R(x, t ) = 1

2

t

0

d2

du2 exp −

x+ u

x

h(v)dv ϕ(x + u) udu.

Now differentiating with respect to x in (46), we obtain that for each α, β ∈N ,

limt→0

S At ϕ−ϕt −(ϕ −hϕ)

α,β = lim

t→0supx∈R

xα R (β ) (x, t )t

,

where R(β ) (x, t ) denotes the β th derivative of R(x, t ) with respect to x. Hence, in order to verify(45) it now suffices to show that for each α, β ∈N ,

limt→0

supx∈R

xα R (β ) (x, t )t

= 0 .

First note that for each x ∈R xed, we may write

xα R (β ) (x, t ) = 1

2 t

0xα d2

du2 exp − x+ u

xh(v)dv ϕ(x + u)

(β )

udu. (47)

18



However, since h ∈ C b(R + ) and ϕ ∈ S , it is straightforward to show that for each α, β ∈ N , thereexists a constant C α,β < ∞ such that for u sufficiently small,

supx∈R

xα d2

du2 exp − x+ u

xh(v)dv ϕ(x + u)

(β )

< C α,β .

Hence, by ( 47) we obtain that

limt→0

supx∈R

xα R (β ) (x, t )t

< limt→0

C α,β

2t t

0udu = lim

t→0

C α,β

4 t = 0 .

This now completes the verication of the fact that BA is the innitesimal generator of the semi-group ( S At )t≥ 0 given by (44).

We next verify that the semi-group ( S At )t≥ 0 is a strongly-continuous ( C 0, 1) semi-group. Itis straightforward to see that Part 1 of Denition 3.1 is satised. We next check that Part 2 of Denition 3.1 is satised. Consider 0 ≤ s < t , a bounded set K ⊂ S , and α, β ∈ N . We then havethat we may write

supϕ∈K

S As ϕ−S At ϕ α,β = supϕ∈K

(1/ F )τ − s ( Fϕ) −(1/ F )τ − t ( Fϕ) α,β

= supϕ∈K

supx∈R

xα F (x + s)

F (x) ϕ(x + s) −

F (x + t)F (x)

ϕ(x + t)(β )

≤ supϕ∈K

supx∈R

xα F (x + s)

F (x) −F (x + t)

F (x)ϕ(x + s)

(β )

+ supϕ∈K

supx∈R

xα F (x + t)

F (x) (ϕ(x + s) −ϕ(x + t))

(β )

. (48)

We now handle each of the terms in ( 48) separately.For the rst term in ( 48), note that by Lemmas 3.4 and 3.5 we have that

supϕ∈K

supx∈R

xα F (x + s)

F (x) −F (x + t)

F (x)ϕ(x + s)

(β )

= supϕ∈K

supx∈R

β

i=0

β i

xα F (x + s)

F (x) −F (x + t)

F (x)

(β − i)

ϕ(i) (x + s)

= |t −s|β

i=0

M β − iβ i

supϕ∈K

supx∈R

xαϕ

(i) (x + s)

≤ |t −s|(1∨ |t|α )2αβ

i=0M β − i

β i

supϕ∈K

max0≤ j ≤ α

ϕ j,i . (49)

We next focus on the second term in ( 48). For each n ≥ 1, denote the left-hand-side of ( 41) by

19



Ln . By the mean value theorem, for each x ∈R there exists an rx ∈ [s, t ] such that

supϕ∈K

supx∈R

xα F (x + t)

F (x) (ϕ(x + s) −ϕ(x + t))

(β )

= supϕ∈K supx∈R

β

i=0

β i x

α F (x + t)

F (x)

(β − i)

ϕ

(i)(x + s) −ϕ

(i)(x + t)

= supϕ∈K

supx∈R

β

i=0

β i

Lβ − i xαϕ

(i) (x + s) −ϕ(i) (x + t)

≤ |t −s| supϕ∈K

supx∈R

β

i=0

β i

Lβ − i xαϕ

(i+1) (x + r x )

≤ |t −s|(1∨ |t|α )2αβ

i=0

β i

Lβ − i supϕ∈K

max0≤ j ≤ α

ϕ j,i +1 , (50)

where the nal inequality follows as a consequence of Lemma 3.5. Combining ( 49) and ( 50) with(48) and taking the limit as s → t now yields Part 2 of Denition 3.1.We now complete the proof by verifying that Part 3 of Denition 3.1 is satised. First note

that for each ϕ ∈ S , t ≥ 0 and α, β ∈N , we may write

S At ϕ α,β = supx∈R

xα F (x + t)

F (x) ϕ(x + t)

(β )

= supx∈R

xαβ

i=0

β i

F (x + t)F (x)

(β − i)

ϕ(i) (x + t)

≤β

i=0

β i Lβ − i supx∈R x

α

ϕ

(i)

(x + t)

≤ (1∨ |t|α )2αβ

i=0

β i

Lβ − i max0≤ j ≤ α

ϕ j,i

≤ (1∨ |t|α )2α + β max0≤ i≤ β

L i max0≤ j ≤ α

max0≤ i≤ β

ϕ j,i ,

where the nal inequality above follows from Lemma 3.5. Part 3 of Denition 3.1 now follows from(3) and ( 4) of §1.1 and the fact that · p ≤ · p+1 for each p ∈N . This completes the proof.

3.2 Residual Service Time Process

Now let B R be the linear operator dened on S such that

B Rϕ = −ϕ for all ϕ ∈ S . (51)

In this subsection, we verify that BR generates a strongly-continuous ( C 0, 1) semi-group. Thiswill be useful in §5.2 and §6.2, where we prove our uid and diffusion limits, respectively, for the

20



residual service time process. In particular, by ( 25) of §2.2 and Theorem 3.3, this then implies thatwe may write

R = Ψ B R (R0 + E F + G), (52)

where the map Ψ B R : D ([0, T ], S ) →D ([0, T ], S ) is a continuous map.Our main result of this subsection is the following.

Proposition 3.7. The linear operator B R dened by (51) generates the strongly-continuous (C 0, 1)semi-group (τ t )t≥ 0.

Proof. First note that it is clear by Lemma 3.5 that for each t ≥ 0 and ϕ ∈ S , we have that τ tϕ ∈ S .We next check that for each α, β ∈N , we have the convergence

limt→0

τ tϕ−ϕt −(−ϕ )

α,β = 0 . (53)

This will then be sufficient to verify that BR as dened by ( 51) generates the semi-group ( τ t ) t≥ 0.Let x ∈ R and ϕ ∈ S be xed. It then follows by Taylor’s theorem, expanding in terms of t, thatwe may write

τ tϕ(x) = ϕ(x) −ϕ (x)t + R(x, t ), (54)

where the remainder term R(x, t ) is given by

R(x, t ) = 1

2 − t

0ϕ (x + u)(−t −u)du. (55)

Now differentiating in ( 54) with respect to x, we obtain that

limt→0

τ tϕ−ϕt −(−ϕ )

α,β = lim

t→0supx∈R

xα R (β ) (t, x )t

,

where R (β ) (x, t ) denotes the β th derivative of R with respect to x. Thus, in order to prove ( 53), itnow suffices to check that

limt→0

supx∈R

xα R (β ) (t, x )t

= 0 . (56)

First note that by ( 55) we may write

xα R (β ) (x, t ) = xα

2 − t

0ϕ

(β +2) (x + u)(−t −u)du = 12

− t

0xαϕ

(β +2) (x + u)(−t −u)du.

However, by Lemma 3.5 there exists a constant C α,β < ∞ such that for sufficiently small u,

supx∈R

xαϕ

(β +2) (x + u) < C α,β .

We therefore obtain that

limt→0

supx∈R

xα R (β ) (t, x )t

< limt→0

C α,β

2t − t

0(−t −u)du = lim

t→0

C α,β

4 t = 0 ,

21



thus proving ( 56). This completes our verication of the fact that B R as dened by ( 51) generatesthe semi-group ( τ t )t≥ 0.

We next proceed to verify that ( τ t )t≥ 0 is a strongly-continuous ( C 0, 1) semi-group. In orderto check this fact, we will verify that ( τ t )t≥ 0 satises Parts 1 through 3 of Denition 3.1. It isclear that ( τ t ) t≥ 0 satises Part 1 of Denition 3.1. We next check that ( τ t )t≥ 0 satises Part 2 of

Denition 3.1. Let s < t , K ⊂ S be a bounded set, and let α, β ∈ N

. Then, by the mean valuetheorem, for each x ∈R there exists an rx ∈ [x −t, x −s] such that

supϕ∈K

τ sϕ−τ tϕ α,β = supϕ∈K

supx∈R

xαϕ

(β ) (x −s) −ϕ(β ) (x −t)

= supϕ∈K

supx∈R

(r x + ( x −r x ))αϕ

(β )(x −s) −ϕ(β ) (x −t)

= supϕ∈K

supx∈R

α

i=0

αi

r α − ix (x −r x )i

ϕ(β ) (x −s) −ϕ(β ) (x −t)

= |t −s| supϕ∈K

supx∈R

α

i=0

αi

r α − ix (x −r x )i

ϕ(β +1) (x −r x )

≤ |t −s|α

i=0

αi

(t + 1) α − i supϕ∈K

ϕ i,β +1 .

Part 2 of Denition 3.1 now follows from the bound ( 3) and the fact that K is a bounded set.We now complete the proof by verifying that ( τ t )t≥ 0 satises Part 3 of Denition 3.1. Note that

for each ϕ ∈ S , t ≥ 0 and α, β ∈N , we have that

τ tϕ α,β = ϕ(· −t) α,β

= supx∈R

xαϕ

(β ) (x −t)

= supx∈R

[t + ( x −t)]α ϕ(β ) (x −t )

= supx∈R

α

i=0

αi

tα − i(x −t)iϕ

(β ) (x −t)

≤α

i=0

αi

tα − iϕ i,β

≤ (1∧tα )2α max0≤ i≤ α

ϕ i,β .

Part 3 of Denition 3.1 now follows as a result of the bounds ( 3) and ( 4).

4 Martingale Results

In this section, we show that the process D0 + D dened in §2.1 and the process G dened in §2.2are both S -valued martingales. The fact that D0 + D is an S -valued martingale will ultimately beused together with the martingale functional central limit theorem [10] and the continuous mappingtheorem [ 1] in §5.1 and §6.1 in order to prove our uid and diffusion limits, respectively, for the age

22



process. The fact that G is an S -valued martingale is not necessarily needed in order to prove limittheorems for the residual service time process but may be used to show that the residual servicetime process is in fact a Markov process. We begin by studying D0 + D.

4.1 Age Process

In this subsection, we show that the process D0 + D dened in §2.1 is an S -valued martingale withrespect to the ltration ( F At )t≥ 0 dened by

F At = σ1 ηi ≤ s− τ i , s ≤ t, i = 1 , 2,...,A 0(∞)∨στ i , i = 1 ,...,A 0(∞)∨σ1 ηi ≤ s− τ i , s ≤ t, i = 1 , 2,...,E t∨σE s , s ≤ t∨N .

Moreover, we explicitly identify the tensor quadratic variation of D0 + D. The following is our mainresult of this subsection.

Proposition 4.1. The process D0 + D is an S -valued F At -martingale with tensor quadratic vari-ation process given for all ϕ, ψ ∈ S by

<< D0 + D >> t (ϕ, ψ) =A0 (∞ )

i=1 ηi∧t

0ϕ(x − τ i )ψ(x − τ i )h τ i (x) dx

+E t

i=1 ηi∧(t− τ i )+

0ϕ(x)ψ(x)h(x) dx. (57)

Proof. Let ϕ ∈ S . We claim that ( (D0 + D)t ,ϕ ) t≥ 0 is an R -valued F At -martingale, which issufficient to show that D0 + D is an S -valued F At -martingale. Let t ≥ 0. We rst show thatE [| (D0 + D)t ,ϕ |] < ∞. Note that by ( 12) and ( 13) we may write

E (D0 + D) t ,ϕ ≤E

A0 (∞ )

i=1 t

0ϕ(x − τ i ) d 1 ηi ≤ x −

ηi∧x

0h τ i (u) du (58)

+ EE t

i=1 (t− τ i )+

0ϕ(x) d 1 ηi ≤ x −

ηi∧x

0h(u) du

≤E sup0≤ s< ∞ |ϕ(s)|

A0 (∞ )

i=1

1 ηi ≤ t + ηi∧t

0h τ i (u) du

+ E sup0≤ s< ∞ |ϕ(s)|

E t

i=1

1 ηi ≤ (t− τ i )+ + ηi∧(t− τ i )+

0h(u) du

≤ sup0≤ s< ∞ |ϕ(s)|(1 + t h ∞ )(E [A0(∞)] + E [E t ])

< ∞,

where the nal inequality follows from the assumptions made on E [A0(∞)] and E [E t ] in §2.1. Thus,E [| (D0 + D)t ,ϕ |] < ∞ as desired.

23



Next, we show that ( (D0 + D) t ,ϕ ) t≥ 0 possesses the martingale property with respect to theltration ( F At )t≥ 0. That is, we show that for each 0 ≤ s ≤ t,

E [ (D0 + D) t ,ϕ |F As ] = (D0 + D)s ,ϕ . (59)

First note that by ( 12) and ( 13), we may write

(D0 + D)t ,ϕ =∞

i=1

1 i≤ A0 (∞ ) D0,it ,ϕ +

∞

i=1Dit ,ϕ , (60)

where, for each i ≥ 1, we set

1 i≤ A0 (∞ ) D0,it ,ϕ = 1i≤ A0 (∞ )

t

0ϕ(x − τ i ) d 1 ηi ≤ x −

ηi∧x

0h τ i (u) du

and

Dit ,ϕ =

(t− τ i )+

0

ϕ(x) d 1 ηi ≤ x −

ηi∧x

0

h(u) du . (61)

We now show that for each i ≥ 1,

E [ Dit ,ϕ |F As ] = Dis ,ϕ . (62)

The proof that

E [1 i≤ A0 (∞ ) D0,it ,ϕ |F A

s ] = 1i≤ A0 (∞ )E [ D0,it ,ϕ |F A

s ] = 1 i≤ A0 (∞ ) D0,is ,ϕ , (63)

for each i ≥ 1, is similar and will not be included. For each i ≥ 1 and y ≥ 0, set

D it (y) = 1 ηi ≤ (t− τ i )+ ∧y −

ηi∧(t− τ i )+

∧y

0

h(u) du. (64)

We now claim that in order to show ( 62), it suffices to show that E [D it (y)|F As ] = D i

s (y) for eachy ≥ 0. This is true since it will then follow that

E Dit ,ϕ |F As = −E R +

D it (y)ϕ (y) dy|F As

= − R +

E D it (y)|F As ϕ (y) dy

= − R +

D is (y)ϕ (y) dy,

=

Di

s,ϕ .

First note that since y∧(t −τ i )+ = ( t∧(τ i + y) −τ i )+ , we may write

D it (y) = D i

t∧(τ i + y) (∞).

Next note that since by the assumptions in §2.1, we have that E [τ i ] < ∞, it is straightforwardto verify that τ i + y is an F At -stopping time for each y ≥ 0. Thus, by Problem 3.2.4 of [ 22], it

24





Therefore, as on page 259 of [25], it follows that

<< Di ,ϕ >> t = (t− τ i )+

0ϕ(x)2 d 1 ηi ≤ x −

ηi∧x

0h(u) du =

ηi∧(t− τ i )+

0ϕ(x)2h(x) dx, (67)

which implies that

<< Di >> t (ϕ, ψ) = < Di ,ϕ , Di , ψ > t

= 1

4<< Di ,ϕ + ψ >> t − << Di ,ϕ−ψ >> t

= 1

4 ηi∧(t− τ i )+

0(ϕ(x) + ψ(x)) 2 h(x) dx

− ηi∧(t− τ i )+

0(ϕ(x) −ψ(x))2h(x) dx

=

ηi∧(t− τ i )+

0

ϕ(x)ψ(x)h(x) dx,

where the second equality in the above follows from the polarization identity and the third equalityfollows from (67). In a similar manner, one may also show that for all i ≥ 1,

<< 1 i≤ A0 (∞ )D0,i >> t (ϕ, ψ) = 1 i≤ A0 (∞ ) ηi∧t

0ϕ(x − τ i )ψ(x − τ i )h τ i (x) dx,

for all ϕ, ψ ∈ S .We now claim that in order to show that the tensor quadratic variation of D0 + D is given by

(57), it suffices to show the following three facts:

1.

Di is orthogonal to

D j for i

= j ,

2. 1i≤ A0 (∞ )D0,i is orthogonal to 1 j ≤ A0 (∞ )D0,j for i = j ,

3. 1i≤ A0 (∞ )D0,i is orthogonal to D j for all i, j ≥ 1.

The fact that the tensor quadratic variation of D0 + D is given by (57) can then be shown in thefollowing manner. For each k ≥ 1,ϕ ∈ S and t ≥ 0, let

(D0 + D)kt ,ϕ =

k

i=1

1 i≤ A0 (∞ ) D0,it ,ϕ +

k

i=1Dit ,ϕ ,

and set ( D0 + D)k = (( D0 + D)kt , t ≥ 0). It is then clear that Claims 1 through 3 above imply that

<< (D0 + D)k >> t (ϕ, ψ) =k

i=1

1 i≤ A0 (∞ ) ηi∧t

0ϕ(x − τ i )ψ(x − τ i )h τ i (x) dx

+k

i=1 ηi∧(t− τ i )+

0ϕ(x)ψ(x)h(x) dx. (68)

26



Moreover, using similar arguments as above and the simple inequality ( x1 + x2)2 ≤ 4(x21 + x2

2), itis straightforward to show that for each k ≥ 1, one has P -a.s. the bound

(D0 + D)kt ,ϕ (D0 + D)k

t , ψ

− k

i=1 1 i≤ A0 (∞ ) ηi∧t

0 ϕ(x − τ i)ψ(x − τ i)h τ i (x) dx +

k

i=1 ηi∧(t− τ i )+

0 ϕ(x)ψ(x)h(x) dx

≤ 4 sup0≤ s< ∞

(|ϕ(s)|+ |ψ(s)|)2

(1 + t h ∞ )2(E 2t + A20(∞))

+ sup0≤ s< ∞

(|ϕ(s)||ψ(s)|)(1 + t h ∞ )(E t + A0(∞)) .

However, by the assumptions in §2.1, one has that E [E 2t + A20(∞)] < ∞ and E [E t + A0(∞)] < ∞.

Hence, using the dominated convergence theorem for conditional expectations [ 6], it follows thatfor 0 ≤ s ≤ t,

E [ (D0 + D)t ,ϕ (D0 + D) t ,ϕ − << (D0 + D) >> t (ϕ, ψ)|F As ]

= E [ limk→∞ ( (D0

+ D)kt ,ϕ (D

0+ D)

kt , ψ − << (D

0+ D)

k>> t (ϕ, ψ)) |F

As ]

= limk→∞

E [ (D0 + D)kt ,ϕ (D0 + D)k

t , ψ − << (D0 + D)k >> t (ϕ, ψ)|F As ]

= limk→∞

( (D0 + D)ks ,ϕ (D0 + D)k

s , ψ − << (D0 + D)k >> s (ϕ, ψ))

= (D0 + D)s ,ϕ (D0 + D)s , ψ − << (D0 + D) >> s (ϕ, ψ).

This then implies that the tensor quadratic variation of D0 + D is given by (57). We now proceedto prove Claims 1 through 3, which is sufficient to complete the proof.

We begin with Claim 1. Let ϕ, ψ ∈ S and i = j . We show that ( Dit ,ϕ D jt , ψ ) t≥ 0 is an

R -valued F At -martingale, which is sufficient to show that Di is orthogonal to D j . First note that itis clear as in (58) that for each t

≥ 0, we have that E [

| Dit ,ϕ

D jt , ψ

|] <

∞. Next, let 0

≤ s

≤ t. By

the independence of ηi from A0, ηk , k = 1 ,...,A 0(∞), E = ( E t )t≥ 0 and ηk , k = i, and, similarly,the independence of η j from A0, ηk , k = 1 ,...,A 0(∞), E = ( E t )t≥ 0 and ηk , k = j , it follows that

E [ Dit ,ϕ D jt , ψ |F A

s ] = E [ Dit ,ϕ D jt , ψ |1 τ i ≤ s, 1 τ j ≤ s, 1 ηi ≤ s− τ i , 1 ηj ≤ s− τ j ].

However, by the independence of ηi from η j and τ j , and, similarly, the independence of η j from ηiand τ i , we have that

E [ Dit ,ϕ D jt , ψ |1 τ i ≤ s, 1 τ j ≤ s, 1 ηi ≤ s− τ i , 1 ηj ≤ s− τ j ]

= E [ Dit ,ϕ |1 τ i ≤ s, 1 ηi ≤ s− τ i ]E [ D jt , ψ |1 τ j ≤ s, 1 ηj ≤ s− τ j ]

= Dis ,ϕ D js , ψ ,

where the nal equality follows from ( 62) and the fact that for k ≥ 1,E [ Dkt ,ϕ |1 τ k ≤ s, 1 ηk ≤ s− τ k ] = E [ Dkt ,ϕ |F A

s ]. (69)

Thus, it is clear that ( Dit ,ϕ D jt , ψ )t≥ 0 possesses the martingale property and so ( Dit ,ϕ D j

t , ψ ) t≥ 0is an R -valued F At -martingale, and hence Di is orthogonal to D j . The proof of Claim 2 above fol-lows similarly. The proof of Claim 3 above follows in a similar manner as well. In particular,

27



let i, j ≥ 1. We show that ( 1 i≤ A0 (∞ ) D0,it ,ϕ D j

t , ψ )t≥ 0 is an R -valued F At -martingale for eachϕ, ψ ∈ S , which is sufficient to show that 1i≤ A0 (∞ )D0,i is orthogonal to D j . For each t ≥ 0, it isclear as in ( 58) that E [|1 i≤ A0 (∞ ) D0,i

t ,ϕ D jt , ψ |] < ∞. Next, note that since ˜ ηi is independent

of A0, ηk , k = 1 ,...,A 0(∞); k = i, E = ( E t ) t≥ 0 and ηk , k ≥ 1, and, similarly, η j is independent of A0, ηk , k = 1 ,...,A 0(∞), E = ( E t )t≥ 0 and ηk , k = j , we have that

E [1 i≤ A0 (∞ ) D0,it ,ϕ D j

t , ψ |F As ]

= E [1 i≤ A0 (∞ ) D0,it ,ϕ D j

t , ψ |τ i , 1 τ j ≤ s, 1 ηi ≤ s− τ i , 1 ηj ≤ s− τ j ].

However, by the independence of ˜ηi from η j and τ j , and, similarly, the independence of η j from ηiand τ i , we have that

E [1 i≤ A0 (∞ ) D0,it ,ϕ D j

t , ψ |τ i , 1 τ j ≤ s, 1 ηi ≤ s− τ i , 1 ηj ≤ s− τ j ]

= E [1 i≤ A0 (∞ ) D0,it ,ϕ |τ i , 1 ηi ≤ s− τ i ]E [ D jt , ψ |1 τ j ≤ s, 1 ηj ≤ s− τ j ]

= 1i≤ A0 (∞ ) D0,is ,ϕ D j

s , ψ ,

where the nal equality follows by ( 62), (63), (69) and the fact that for k ≥ 1,E [1 k≤ A0 (∞ ) D0,k

t ,ϕ |τ k , 1 ηk ≤ s− τ k ] = E [1 k≤ A0 (∞ ) D0,kt ,ϕ |F A

s ].

Thus, it is clear that ( 1 i≤ A0 (∞ ) D0,it ,ϕ D j

t , ψ )t≥ 0 possesses the martingale property and so(1 i≤ A0 (∞ ) D0,i

t ,ϕ D jt , ψ ) t≥ 0 is an R -valued F At -martingale, and hence D0,i is orthogonal to D j .

This proves Claim 3, which completes the proof.

4.2 Residuals

In this subsection, we show that the process G dened in §2.2 is a martingale. This fact may beuseful in future work where one wishes to show that the residual service time process is a Markov

process. Let ( F Gt )t≥ 0 be the natural ltration generated by G. We then have the following result.Proposition 4.2. The process G is an S -valued F Gt -martingale with tensor optional quadratic variation process given for all ϕ, ψ ∈ S by

[G]t (ϕ, ψ) =E t

i=1(ϕ(ηi ) − F ,ϕ )(ψ(ηi ) − F , ψ ). (70)

Proof. Let ϕ ∈ S . We rst show that ( Gt ,ϕ )t≥ 0 is an R -valued F Gt -martingale. Dene theltration ( Hk )k≥ 1 by setting Hk = σE t , t ≥ 0∨ση1, η2,...,ηk∨N for each k ≥ 1. Next, denethe discrete-time D -valued process ( Gk )k≥ 1 by

Gk (y) =k

i=1

1 ηi ≤ y −F (y) , y ≥ 0, (71)

and, for convenience, let Gk (y) = 0 for y < 0. Then, let (Gk )k≥ 1 be the S -valued process associatedwith Gk . Since ϕ ∈ S is bounded, it is clear that E [| Gk ,ϕ |] < ∞ for each k ≥ 1. Moreover, by theindependence of the service times from the arrival process, one has that ( Gk ,ϕ )k≥ 1 possesses the

28



martingale property with respect to ( Hk )k≥ 1. Hence, ( Gk ,ϕ )k≥ 1 is an R -valued Hk-martingale.However, since for each t ≥ 0 we have by the assumptions in §2.1 that E [E t ] < ∞, it is straightfor-ward to see that E t is a stopping time with respect to the ltration ( Hk )k≥ 1. Thus, the ltration(HE t ) t≥ 0 is well-dened and, furthermore, it follows by the optional sampling theorem [22] that( Gt ,ϕ )t≥ 0 = ( GE t ,ϕ ) t≥ 0 is an HE t -martingale. The result now follows since any martingale is a

martingale relative to its natural ltration.The form of the tensor optional quadratic variation ( 70) is immediate by Theorem 3.3 of [29].

5 Fluid Limits

In this section, we provide our main uid limit results. We begin in §5.1 by studying the age processand in §5.2 we study the residual service time process. Our setup in both subsections is the same.In particular, we consider a sequence of G/GI/ ∞ queues indexed by n ≥ 1, where the arrival rateto the system grows large with n while the service time distribution does not change with n .

5.1 Ages

We begin by studying the age process A dened in §2.1. For each n ≥ 1, dene the uid scaledquantities

¯An0 ≡ An

0n

, E n ≡ E n

n , ¯D0,n ≡ D0,n

n , ¯Dn ≡ Dn

n , ¯An ≡ An

n , (72)

and set ¯E n ≡ E n δ 0. Using (17), Theorem 3.3 and Proposition 3.6, it is straightforward to showthat one may write

¯An = Ψ B A ( ¯An0 + ¯E n −( ¯D0,n + ¯Dn )) , (73)

where the map Ψ B A : D ([0, T ], S ) →D ([0, T ], S ) is continuous. We now prove that if ( ¯An0 + ¯E n )n≥ 1

weakly converges, then so too does ( ¯An0 + ¯E n −( ¯D0,n + ¯Dn )) n ≥ 1.

Proposition 5.1. If ¯An0 + ¯E n

⇒ ¯A0 + ¯E in D ([0, T ], S ) as n → ∞, then

¯An0 + ¯E n −( ¯D0,n + ¯Dn ) ⇒ ¯A0 + ¯E in D ([0, T ], S ) as n → ∞.

Proof. We rst note that by Theorem 1.5, it is sufficient to show that if ¯An0 + ¯E n

⇒ ¯A0 + ¯E as

n → ∞, then¯D0,n + ¯Dn

⇒ 0 in D ([0, T ], S ) as n → ∞. (74)

Let T > 0 and 0 ≤ t ≤ T . Then, for each ϕ ∈ S , we have by Proposition 4.1 that

| << ¯D0,n + ¯Dn >> t (ϕ,ϕ)|= 1

n2

An0 (∞ )

i=1 ηi∧t

0ϕ

2(x − τ ni )h τ ni (x) dx +E nt

i=1 ηi∧(t− τ n

i )+

0ϕ

2(x)h(x) dx

≤ h ∞

n2

An0 (∞ )

i=1 t

0ϕ

2(x − τ ni ) dx + ϕ2h ∞

nE nT . (75)

29



Thus, from ( 75) we obtain that for each 0 ≤ t ≤ T ,

| << ¯D0,n + ¯Dn >> t (ϕ,ϕ)| (76)

≤ h ∞

n2

An0 (∞ )

i=1

t

0ϕ

2(x − τ ni ) dx + ϕ2h ∞

nE nT

= h ∞

n2 t

0

An0 (∞ )

i=1ϕ

2(x − τ ni ) dx + ϕ2h ∞

nE nT

= h ∞

n t

0 ¯An

0 , τ − xϕ2 dx + ϕ2h ∞

nE nT

≤ h ∞ tn

q K ( ¯An0 ) + ϕ2h ∞

nE nT ,

where the set K is given by K = τ − xϕ2, 0 ≤ x ≤ t. By Lemma 3.5, the set K is bounded in S and

hence q K is a semi-norm on S by Denition 1.3. This then implies that q K is a continuous functionon

S . Hence, since by assumption ¯

An0

⇒ ¯

A0 and E nT

⇒ E T , it follows by Slutsky’s Theorem that

h ∞ tn

q K ( ¯An0 ) + ϕ2h ∞

nE nT ⇒ 0 as n → ∞.

By (76), this then implies that

<< ¯D0,n + ¯Dn >> (ϕ,ϕ) ⇒ 0 in D ([0, T ], R ) as n → ∞. (77)

We now verify that Parts 1 and 2 of Theorem 1.5 are satised for the sequence ( ¯D0,n + ¯Dn )n ≥ 1,with the limit point being the function which is identically 0. We begin with Part 1. Using thefact that the maximum jump of both ¯D0,n + ¯Dn ,ϕ and << ¯D0,n + ¯Dn >> (ϕ,ϕ) is bounded overthe interval [0 , T ] uniformly in n, we obtain by ( 77) and the martingale FCLT (see Theorem 7.1.4

of [10] or [34]) that¯D0,n + ¯Dn ,ϕ ⇒ 0 in D ([0, T ], R ) as n → ∞. (78)

Thus, Part 1 of Theorem 1.5 holds. We next check that Condition 2 holds. Let m ≥ 1 and lett1,...,t m ∈ [0, T ] and ϕ1, . . . ,ϕm ∈ S . By (78), we have that for each 1 ≤ i ≤ m,

( ¯D0,n + ¯Dn ) t i ,ϕi ⇒ 0 in R as n → ∞.However, by Theorem 3.9 of [ 1] this now implies that

( ¯D0,n + ¯Dn ) t1 ,ϕ1 , . . . ( ¯D0,n + ¯Dn )t m ,ϕm ⇒ (0, . . . , 0) in R m as n → ∞.Thus, we have shown that Part 2 of Theorem 1.5 holds and so (74) is proven. This completes theproof.

We are now in a position to prove the main result of this subsection. We have the following.

30



Theorem 5.2. If ¯An0 + ¯E n

⇒ ¯A0 + ¯E in D ([0, T ], S ) as n → ∞, then

¯An⇒

¯A in D ([0, T ], S ) as n → ∞,where ¯A is the unique solution to the integral equation

¯At ,ϕ = ¯A0,ϕ + ¯E t ,ϕ − t

0 ¯As , hϕ ds + t

0 ¯As ,ϕ ds, t ∈ [0, T ], (79)

for all ϕ ∈ S .

Proof. By the assumption that ¯An0 + ¯E n

⇒ ¯A0 + ¯E , it follows immediately by Proposition 5.1 that

¯An0 + ¯E n −( ¯D0,n + ¯Dn ) ⇒ ¯A0 + ¯E in D ([0, T ], S ) as n → ∞. (80)

Next, recall that by ( 73) we have that ¯An = Ψ B A ( ¯An0 + ¯E n −( ¯D0,n + ¯Dn )), where the map Ψ B A :

D ([0, T ], S ) → D ([0, T ], S ) is continuous. The result now follows by ( 80) and Proposition 1.1applied to Ψ B A .

Remark 5.3. Note that one may now use Theorem 5.2 along with Theorem 3.3 in order to obtain an explicit expression for ¯A. Similarly, one may obtain explicit expressions for ¯R, Â and ˆR in Theorems 5.6, 6.5 and 6.9, respectively, below.

We also note that at a heuristic level, one may attempt to substitute the function 1x≥ 0 intothe explicit formula provided by Theorem 3.3 for ¯Ain order to obtain an expression for the limiting,uid scaled total number of customers in the system. For instance, suppose that ¯A0 = 0 so thatthe system is initially empty and that ¯E ,ϕ = λϕ(0)e for each ϕ ∈ S . Then, using the form of the generator BA from (39) and the semi-group ( S At )t≥ 0 from Proposition 3.6, one obtains aftersubstituting into Theorem 3.3 that heuristically the total number of customers in the system attime t ≥ 0 is given by

λt −λ t

0sf (t −s)ds = λ

t

0F (t −s)ds.

We now conclude this subsection by providing an additional condition on the arrival processunder which a stationary solution to the uid limit equation ( 79) may be explicitly found. Notealso that our condition in Proposition 5.4 below holds, for example, if the arrival process to thenth system is a renewal process which has been sped up by a factor of n (as will be the case forthe GI/GI/ ∞ queue).

Proposition 5.4. If ¯E ,ϕ = λϕ(0)e for each ϕ ∈ S , then ¯A = λF e is a stationary solution to the uid limit equation (79).

Proof. Substituting A = λF e and E ,ϕ = λϕ(0)e into (79), we see that it suffices to verify that

λ R +

ϕ(y) dF e(y) = λ R +

ϕ(y) dF e(y) + λtϕ (0) −λt R +

(h(y)ϕ(y) −ϕ (y)) dF e(y).

31



However, this follows since

λt R +

(h(y)ϕ(y) −ϕ (y)) dF e(y) = λt R +

(h(y)ϕ(y) −ϕ (y)) F (y) dy

= λt

R

+

(f (y)ϕ(y)

− F (y)ϕ (y)) dy

= −λt R +

F (y)ϕ(y) dy

= λtϕ (0) .


5.2 Residuals

We next proceed to analyze the residual service time process Rof §2.2. Our setup in this subsectionis the same as that in the previous subsection. However, in addition to the uid scaled quantitiesalready dened in

§5.1, we also now dene for each n

≥ 1 the new uid scaled quantities

¯Rn ≡ Rn

n , ¯Rn

0 ≡ Rn0

n and ¯G ≡ G

n

n .

Using (25), Theorem 3.3 and Proposition 3.7, it is now straightforward to show that

¯Rn = Ψ B R ( ¯Rn0 + E nF + ¯Gn ), (81)

where the map Ψ B R : D ([0, T ], S ) →D ([0, T ], S ) is continuous. In our rst result of this subsection,we prove that if ( ¯Rn

0 + E n F )n ≥ 1 weakly converges, then so too does ( ¯Rn0 + E nF + ¯Gn )n ≥ 1.

Proposition 5.5. If ¯Rn0 + E nF ⇒ ¯R0 + E F in D ([0, T ], S ) as n → ∞, then

Rn0 + E nF + Gn

⇒ R0 + E F in D ([0, T ], S ) as n → ∞. (82)

Proof. We rst note that by Theorem 1.5, it is sufficient to show that if ¯Rn0 + E nF ⇒ ¯R0 + E F as

n → ∞, then

¯Gn⇒ 0 in D ([0, T ], S ) as n → ∞. (83)

Let T > 0 and 0 ≤ t ≤ T . We then have by Proposition 4.2 and the assumption that ¯Rn0 + E nF ⇒¯R0 + E F , that for each ϕ, ψ ∈ S ,

|[ ¯

Gn ]t (ϕ, ψ)

| =

1

n2

E nt

i=1

ϕ(ηi )ψ(ηi )

≤ 1

n2 E nT sup

0≤ s< ∞ |ϕ(s)ψ(s)

| ⇒ 0 in R as n

→ ∞. (84)

The remainder of the proof now proceeds in a similar manner to the proof of Proposition 5.1. Weomit the details.

The following is now our main result of this subsection.

32



Theorem 5.6. If ¯Rn0 + E nF ⇒ ¯R0 + E F in D ([0, T ], S ) as n → ∞, then

¯Rn⇒

¯R in D ([0, T ], S ) as n → ∞,where ¯R is the unique solution to the integral equation

¯

Rt ,ϕ = ¯

R0,ϕ + ¯E t F ,ϕ −

t

0¯

Rs ,ϕ ds, t ∈ [0, T ], (85) for all ϕ ∈ S .

Proof. By the assumption that ¯Rn0 + E nF ⇒ ¯R0 + E F , it follows immediately by Proposition 5.5

that¯Rn

0 + E nF + ¯Gn⇒

¯R0 + E F in D ([0, T ], S ) as n → ∞. (86)

Next, recall that by ( 81) we have that ¯Rn = Ψ B R ( ¯Rn0 + E nF + ¯Gn ), where the map Ψ B R :

D ([0, T ], S ) → D ([0, T ], S ) is continuous. The result now follows by ( 86) and Proposition 1.1applied to Ψ B R .

6 Diffusion Limits

In this section, we prove our main diffusion limit results. In §6.1, we study the age process andin §6.2 we study the residual service time process. Before we provide our main results, however,we rst must provide the denition of an S -valued Wiener process and a generalized S -valuedOrnstein-Uhlenbeck process. Our denitions are the same as those in [ 2].

Denition 6.1. A continuous S -valued Gaussian process W = ( W t )t≥ 0 is called a generalized

S -valued Wiener process with covariance functional

K (s,ϕ ; t, ψ ) = E [ W s ,ϕ W t , ψ ] , s, t ≥ 0 and ϕ, ψ ∈ S ,

if it has continuous trajectories and, for each s, t ≥ 0 and ϕ, ψ ∈ S , K (s,ϕ ; t, ψ ) is of the form

K (s,ϕ ; t, ψ ) = s∧t

0Quϕ, ψ du,

where the operators Qu : S → S , u ≥ 0, possess the following two properties:

1. Qu is linear, continuous, symmetric and positive for each u ≥ 0,

2. the function u → Quϕ, ψ is in D ([0, ∞), R ) for each ϕ, ψ ∈ S .

If Qu does not depend on u ≥ 0, then the process W is called an S -valued Wiener process .

Now, using the above denition of a generalized S -valued Wiener process, we may provide thefollowing denition of a generalized S -valued Ornstein-Uhlenbeck process.

Denition 6.2. An

S -valued process X = ( X t )t≥ 0 is called a (generalized)

S -valued Ornstein-

Uhlenbeck process if for each ϕ ∈ S and t ≥ 0,

X t ,ϕ = X 0,ϕ + t

0X u , Aϕ du + W t ,ϕ ,

where W ≡ (W t )t≥ 0 is a (generalized) S -valued Wiener process and A : S → S is a continuous operator.

33





Now note that clearly ˇA0+ ˇE ,ϕ has the same distribution as Â0+ Ê ,ϕ and, similarly, ( ˇD0+ ˇD),ϕhas the same distribution as ( ˆD0 + ˆD),ϕ . We now verify that ˇA0 + ˇE ,ϕ and ( ˇD0 + ˇD),ϕ areindependent of one another. This will then imply the convergence Ân

0 + Ê n ,ϕ −( ˆD0,n + ˆDn ),ϕ ⇒ Â0 + Ê ,ϕ −( ˆD0 + ˆD),ϕ in D ([0, T ], R ) as n → ∞,along the given subsequence. However, since the subsequence was arbitrary, this will then implyconvergence along the entire sequence, thus verifying Part 1 of Theorem 1.5.

Let t1, t2 ∈ [0, T ] with t1 ≤ t2 and let a1, a2, b1, b2 ∈R and let x, y ∈R . We will show that

P (a1 Ân0 + Ê n

t1 ,ϕ + a2 Ân0 + Ê n

t 2 ,ϕ ≤ x, b1 ( ˆD0,nt1 + ˆDn

t 1 ),ϕ + b2 ( ˆD0,nt2 + ˆDn

t2 ),ϕ ≤ y)

→ P (a1 Â0 + Ê t 1 ,ϕ + a2 Â0 + Ê t 2 ,ϕ ≤ x) (91)

× P (b1 ( ˆD0t 1 + ˆDt 1 ),ϕ + b2 ( ˆD0

t2 + ˆDt 2 ),ϕ ≤ y)

as n → ∞. The analogous proof for t1,...,t m ∈ [0, T ] with m > 2 follows similarly. This will thenbe sufficient to show that ˇA0 + ˇE ,ϕ and ( ˇD0 + ˇD),ϕ are independent of one another. First notethat we may write

P (a1 ˆ

An0 + ˆ

E nt1 ,ϕ + a2 ˆ

An0 + ˆ

E nt 2 ,ϕ

≤ x, b1 ( ˆ

D0,nt1 + ˆ

Dnt 1 ),ϕ + b2 ( ˆ

D0,nt 2 + ˆ

Dnt2 ),ϕ

≤ y)

= E 1 a 1 An0 + E nt 1

,ϕ + a 2 An0 + E nt 2

,ϕ ≤x1 b1 ( D0 ,nt 1

+ Dnt 1

) ,ϕ + b2 ( D0 ,nt 2

+ Dnt 2

),ϕ ≤y .

However, by the tower property of conditional expectations [ 6], we have that

E 1 a 1 An0 + E nt 1

,ϕ + a 2 An0 + E nt 2

,ϕ ≤x1 b1 ( D0 ,nt 1

+ Dnt 1

) ,ϕ + b2 ( D0 ,nt 2

+ Dnt 2

),ϕ ≤y

= E 1 a 1 An0 + E nt 1

,ϕ + a 2 An0 + E nt 2

,ϕ ≤xE 1 b1 ( D0 ,n

t 1 + Dn

t 1) ,ϕ + b2 ( D0 ,n

t 2 + Dn

t 2) ,ϕ ≤y|An , E n .

We now claim that

E 1 b1 ( D0 ,nt 1

+ Dnt 1

),ϕ + b2 ( D0 ,nt 2

+ Dnt 2

) ,ϕ ≤y|An , E n (92)P

→ P (b1 ( D0t 1 + Dt 1 ),ϕ + b2 ( D

0t2 + Dt2 ),ϕ ≤ y)

as n → ∞. Then, since by assumption

E 1 a 1 An0 + E nt 1

,ϕ + a 2 An0 + E nt 2

,ϕ ≤x → P (a1 Â0 + Ê t1 ,ϕ + a2 Â0 + Ê t2 ,ϕ ≤ x)

as n → ∞, this will then imply ( 91), thus verifying Part 1 of Theorem 1.5.In order to see that ( 92) holds, rst note that

E 1 b1 ( D0 ,nt 1

+ Dnt 1

),ϕ + b2 ( D0 ,nt 2

+ Dnt 2

),ϕ ≤y|An , E n

= P b1 ( ˆD0,nt 1 + ˆDn

t1 ),ϕ + b2 ( ˆD0,nt 2 + ˆDn

t 2 ),ϕ ≤ y|An , E n .

Now recall from (60) that we may write

b1 ( ˆD0,nt 1 + ˆDn

t1 ),ϕ + b2 ( ˆD0,nt2 + ˆDn

t 2 ),ϕ =An

0 (∞ )

i=1

( ˆD0,n,it 1 , b1ϕ + ˆD0,n,i

t 2 , b2ϕ )

+E nt 2

i=1( ˆDn,i

t1 , b1ϕ + ˆDn,it2 , b2ϕ ).

35



Moreover, given An and E n , we have that the random variables ( ˆD0,n,it1 , b1ϕ + ˆD0,n,i

t 2 , b2ϕ ), i =1,...,A n

0 (∞), and ( ˆDn,it 1 , b1ϕ + ˆDn,i

t 2 , b2ϕ ), i = 1 ,...,E nt2 , are mutually independent, with meanzero. In addition, it is straightforward to calculate that

E An

0 (∞ )

i=1( ˆ

D0,n,it1 , b1ϕ + ˆ

D0,n,it 2 , b2ϕ )2

|An0 ,

E n (93)

+ E E nt 2

i=1

( ˆDn,it 1 , b1ϕ + ˆDn,i

t2 , b2ϕ )2|An0 , E n

= E t1

0 ¯An

u , (b1ϕ + b2ϕ)2h du|An0 , E n .

Also, note that for each i = 1 ,...,An0 (∞),

E ( ˆD0,n,it1 , b1ϕ + ˆD0,n,i

t2 , b2ϕ )3|An0 , E n (94)

= E ( D0,n,it1 , b1ϕ + D

0,n,it2 , b2ϕ )

2( D

0,n,it 1 , b1ϕ + D

0,n,it2 , b2ϕ )|A

n0 , E

n

≤ (|b1|+ |b2|) ϕ ∞ (1 + t2 h ∞ )

√ n E ( ˆD0,n,it1 , b1ϕ + ˆD0,n,i

t 2 , b2ϕ )2|An0 , E n ,

and, similarly, for each i = 1 ,...,E nt 2 ,

E ( ˆDn,it 1 , b1ϕ + ˆD0,n,i

t2 , b2ϕ )3|An0 , E n (95)

≤ (|b1|+ |b2|) ϕ ∞ (1 + t2 h ∞ )

√ n E ( ˆDn,it 1 , b1ϕ + ˆDn,i

t 2 , b2ϕ )2|An0 , E n .

Now let Φ denote the CDF of a standard, normal random variable. It then follows by ( 93), (94), (95)and an application of the Berry-Esseen Theorem [ 6] for independent (but not necessarily identicallydistributed) random variables that

Pb1 ( ˆD0,n

t1 + ˆDnt 1 ),ϕ + b2 ( ˆD0,n

t 2 + ˆDnt2 ),ϕ

E t1

0 ¯Anu , (b1ϕ + b2ϕ)2h du|An

0 , E n1/ 2 ≤ y|An , E n −Φ(y)

≤ 1√ n

(|b1|+ |b2|) ϕ ∞ (1 + t2 h ∞ )

E t 1

0 ¯Anu , (b1ϕ + b2ϕ)2h du|An

0 , E n1/ 2 .

Hence, in order to complete the proof of ( 92) and hence verify Part 1 of Theorem 1.5, it suffices toshow that

E t1

0 ¯An

u , (b1ϕ + b2ϕ)2h du|An0 , E n P

→ t1

0 ¯Au , (b1ϕ + b2ϕ)2h du as n → ∞. (96)

However, note that by Theorem 5.2 and the continuity of the integral map [ 1], we have that

t1

0 ¯An

u , (b1ϕ + b2ϕ)2h du P

→ t1

0 ¯Au , (b1ϕ + b2ϕ)2h du as n → ∞. (97)

36



Next, note that the uniform integrability of An0 (∞), n ≥ 1 and E nT , n ≥ 1 implies the uniform

integrability of

t1

0 ¯An

u , (b1ϕ + b2ϕ)2h du,n ≥ 1 . (98)

It is then straightforward to show that ( 97) implies (96), thus completing the verication of Part 1of Theorem 1.5. The proof of the verication of Part 2 of Theorem 1.5 follows in a similar mannerto the above and has been omitted for the sake of brevity. This completes the proof.

The following is now our main result of this section. Its proof is a straightforward consequenceof Theorem 1.1, (87) and Proposition 6.4.

Theorem 6.5. If Ân0 + Ê n

⇒ Â0 + Ê in D ([0, T ], S ) as n → ∞, then

Ân⇒

Â in D ([0, T ], S ) as n → ∞, (99)

where Â is the solution to the stochastic integral equation

ˆ

At,ϕ =

ˆ

A0,ϕ + ˆ

E t,ϕ

−ˆ

D0 + ˆ

D,ϕ

− t

0 ˆ

As, hϕ ds +

t

0 ˆ

As,ϕ ds, t

∈ [0, T ], ϕ

∈ S . (100)

In addition, if Ê is an S -valued Wiener process with covariance functional K E (s,ϕ ; t, ψ ) = σ2(s∧t)ϕ(0)ψ(0) , then Â is a generalized S -valued Ornstein-Uhlenbeck process driven by a generalized

S -valued Wiener process with covariance functional

K E− ( D0 + D ) (s,ϕ ; t, ψ ) = σ2(s∧t)ϕ(0)ψ(0) + s∧t

0 ¯Au h,ϕψ du. (101)

Proof. First note that by ( 87) we have that Ân = Ψ B A ( Ân0 + Ê n −( ˆD0,n + ˆDn )), where the map

ΨB A : D ([0, T ], S ) → D ([0, T ], S ) is a continuous map. The convergence ( 99) now follows byTheorem 1.1 and Proposition 6.4.

Next, suppose that ˆ

E is an S -valued Wiener process with covariance functional K E (s,ϕ ; t, ψ ) =σ2(s ∧ t)ϕ(0)ψ(0). Then, combining this with ( 89) and the fact that ˆD0 + ˆD and Â0 + Ê areindependent from Proposition 6.4, yields (101). Thus, by Denition 6.2, Âis an S -valued Ornstein-Uhlenbeck process.

Recall now from Proposition 5.4 of §5.1 that if ¯E ,ϕ = λϕ(0)e for each ϕ ∈ S and some λ ≥ 0,then a stationary solution to the uid limit equation ( 79) is given by ¯A = λF e. We now show thatunder the additional condition that Ê is an S -valued Wiener process with covariance functionalK E (s,ϕ ; t, ψ ) = σ2(s ∧ t)ϕ(0)ψ(0), then the resulting limiting diffusion scaled age process Â of Theorem 6.5 is a time-homogeneous Markov process. Our result is the following. Note also that asimilar approach may be used to analyze the diffusion limit of the residual service time process inTheorem 6.9 in the following subsection.

Proposition 6.6. If ¯E ,ϕ = λϕ(0)e for each ϕ ∈ S , ¯A0 = λF e and Ê is an S -valued Wiener pro-cess with covariance functional K E (s,ϕ ; t, ψ ) = σ2(s∧t)ϕ(0)ψ(0) , then Â is an S -valued Ornstein-Uhlenbeck process driven by an S -valued Wiener process with covariance functional given for each ϕ, ψ ∈ S and s, t ≥ 0 by

K E− ( D0 + D ) (s,ϕ ; t, ψ ) = ( s∧t ) σ2δ 0 + λF ,ϕψ . (102)

37



Proof. It is clear that the covariance functional of Ê is given by

K E (s,ϕ ; t, ψ ) = ( s∧t) σ2δ 0,ϕψ . (103)

We now show that the covariance functional of ˆD0 + ˆD is given by

K D0 + D (s,ϕ ; t, ψ ) = λ(s∧

t) F ,ϕψ . (104)

Then, since Ê and ˆD0 + ˆD are independent, summing ( 103) and ( 104) will prove (102), which willcomplete the proof.

Note that by Proposition 5.4 we have that since by assumption ¯A0 = λF e , it follows that¯A = λF e is the unique solution to the uid limit equation ( 79). Therefore, by Lemma 6.3 we have

that for each ϕ, ψ ∈ S and s, t ≥ 0,

K D0 + D (s,ϕ ; t, ψ ) = s∧t

0 ¯Au ,ϕψh du

=

s∧t

0 R +

ϕ(x)ψ(x)h(x) d Au (x) du

= λ(s∧t) R +

ϕ(x)ψ(x)h(x) F (x) dx

= λ(s∧t) R +

ϕ(x)ψ(x)f (x) dx.

This proves ( 104), which completes the proof.

We now note that using ( 27) of Theorem 3.3 and ( 44) of Proposition 3.6, one may obtain anexplicit representation of Âin terms of the S -valued Wiener process given in Proposition 6.6 above.Direct calculations may then be used in order to obtain the transient and limiting distribution of ˆ

A. In particular, assuming that ˆ

A0 is a Gaussian random variable, one may then show that foreach t ∈ [0, T ], Ât is a Gaussian random variable with mean

E [ Ât ,ϕ ] = A0, F − 1τ − t ϕF , ϕ ∈ S , (105)

and covariance functional given for each ϕ, ψ ∈ S and t ∈ [0, T ] by

E [ Ât ,ϕ Ât , ψ ] = λ F e , F − 1τ − t ϕψ F 1 − F − 1τ − t F

+ t

0ϕ(u)ψ(u) λF (u) + σ2 F (u) F (u)du. (106)

In addition, taking limits as t → ∞, one also nds that Ât weakly converges as t → ∞to a Gaussian

random variable A∞ with mean zero and covariance functional given for each ϕ, ψ ∈ S byE [ Â∞ ,ϕ Â∞ , ψ ] = F e , λF + σ2 F ϕψ .

We now conclude this subsection by noting that one may heuristically attempt to substitutethe test function 1x≥ 0 into the formula for Â provided by Theorem 3.3 in order to obtain anexpression for the limiting diffusion scaled total number of customers in the system. For instance,

38



suppose that Â = 0 and that, as in the statement of Proposition 6.6, we have that ¯E ,ϕ = λϕ(0)efor each ϕ ∈ S and ¯A0 = λF e and Ê is an S -valued Wiener process with covariance functionalK E (s,ϕ ; t, ψ ) = σ2(s∧t)ϕ(0)ψ(0). Then, using the form of the generator B A from (39), the semi-group ( S At )t≥ 0 from Proposition 3.6 and ( 102) of Proposition 6.6, one obtains after a substitutioninto Theorem 3.3 that the limiting diffusion scaled number of customers in the system at time t ≥ 0

is heuristically given by

B t − t

0B s f (t −s)ds =

t

0F (t −s)d B s ,

where B = ( B t )t≥ 0 is a Brownian motion with innitesimal variance σ2 + λ.

6.2 Residuals

We next proceed to study the residual service time process R dened in §2.2. Our setup is the sameas in §5.2. That is, we consider a sequence of G/GI/ ∞ queues indexed by n, where the arrivalrate to the nth system is of order n and the service time distribution does not change with n . For

the remainder of this subsection, we also assume that ¯

Rn0 +

¯E

n

F ⇒ ¯

R0 + ¯E F as n → ∞, where¯R0 + E F is a non-random quantity. By Theorem 5.6 of §5.2, this implies that ¯R is non-random as

well.Now, for each n ≥ 1, in addition to the diffusion scaled quantities dened in §6.1, let us also

now dene the diffusion scaled quantities

ˆRn ≡ √ n ¯Rn − ¯R , ˆRn0 ≡ √ n ¯Rn

0 − ¯R0 and ˆGn ≡ √ n ¯Gn .

Then, after recalling the form of the uid limit ¯R from Theorem 5.6, note that using systemequation ( 25) in conjunction with Theorem 3.3 and Proposition 3.7, one has that

ˆRn = Ψ B R ( ˆRn0 + E nF + ˆGn ), (107)

where the map Ψ B R : D ([0, T ], S ) → D ([0, T ], S ) is a continuous map. Our strategy now is toproceed similar to as in §6.1. That is, we rst prove a weak convergence result for the sequence( ˆRn

0 + E nF + ˆGn )n ≥ 1 and then we apply Theorem 1.1 together with ( 107) in order to obtain adiffusion limit result for the sequence ( ˆRn )n ≥ 1.

We rst show that for each n ≥ 1, the process ˆGn may be well approximated by a process whichis independent of ˆRn

0 + E nF . For each n ≥ 1, let ˇGn be the S -valued process dened for ϕ ∈ S by

ˇGnt , ϕ =

1√ n

n E t

i=1(ϕ(ηi ) − F ,ϕ ), t ≥ 0.

Note that it is clear that ˇ

Gn

is independent of ˆ

Rn0 +

Ê

n

F . We now have the following result. Itsproof may be found in the Appendix.

Lemma 6.7. If ˆRn0 + E nF ⇒ ˆR0 + E F in D ([0, T ], S ) as n → ∞, then

ˆGn − ˇGn⇒ 0 in D ([0, T ], S ) as n → ∞, (108)

39



and

ˇGn⇒

ˆG in D ([0, T ], S ) as n → ∞, (109)

where ˆG is an S -valued Wiener process with covariance functional given for ϕ, ψ ∈ S and s, t ≥ 0by

K G (s,ϕ ; t, ψ ) = ( E s ∧ E t )Cov(ϕ(η), ψ(η)) , (110)where η is a random variable with cdf F .

Proof. See Appendix.

Using Lemma 6.7, we may now prove the following result on the weak convergence of ( ˆRn0 +

E nF + ˆGn )n ≥ 1. We have the following.

Proposition 6.8. If ˆRn0 + E nF ⇒ ˆR0 + E F in D ([0, T ], S ) as n → ∞, then

ˆRn0 + E nF + ˆGn

⇒ ˆR0 + E F + ˆG in D ([0, T ], S ) as n → ∞, (111)

where G is as given in Lemma 6.7 and is independent of R0 + E F .Proof. Since Gn is independent of ˆRn

0 + E nF for each n ≥ 1, it follows Theorem 1.5 and (109) of Lemma 6.7 that we have the convergence

ˆRn0 + E nF + Gn

⇒ ˆR0 + E F + ˆG in D ([0, T ], S ) as n → ∞. (112)

The result now follows by ( 112), Theorem 1.5, (108) of Lemma 6.7 and the fact that we may write

ˆRn0 + E n F + Gn = ˆRn

0 + E nF + Gn + ( Gn − Gn ).

The following is now the main result of this subsection. It provides a weak limit for the sequence( ˆRn )n ≥ 1.

Theorem 6.9. If ˆRn0 + E nF ⇒ ˆR0 + E F in D ([0, T ], S ) as n → ∞, then

ˆRn⇒

ˆR in D ([0, T ], S ) as n → ∞, (113)

where ˆR is the solution to the the stochastic integral equation

ˆRt ,ϕ = ˆR0,ϕ + ˆGt ,ϕ + E t F ,ϕ − t

0ˆRs ,ϕ ds, t ∈ [0, T ], ϕ ∈ S . (114)

In addition, if E is a Brownian motion with diffusion coefficient σ, then R is a generalized S -valued Ornstein-Uhlenbeck process driven by a generalized S -valued Wiener process with covariance functional

K E F + G (s,ϕ ; t, ψ ) = σ2(s∧t)E [ϕ(η)]E [ψ(η)] + E s ∧ E t Cov(ϕ(η), ψ(η)) , (115)

where η is a random variable with cdf F .

40



Proof. First note that by ( 107) we have that ˆRn = Ψ B R ( ˆRn0 + E nF + ˆGn ), where the map Ψ B R :

D ([0, T ], S ) → D ([0, T ], S ) is a continuous map. The convergence ( 113) now follows by Theorem1.1 and Proposition 6.8.

Next, note that if E is a Brownian motion with diffusion coefficient σ, then it is easily checkedthat E F is an S -valued Wiener process with covariance functional

K E F (s,ϕ ; t, ψ ) = σ2(s∧t)E [ϕ(η)]E [ψ(η)]. (116)

Combining ( 110) with ( 116) and the fact that E F and ˆG are independent, yields ( 115).

Remark 6.10. Note that in the special case when the arrival process to the n th system is a Poisson process with rate λn , we then have that E = λe and so E turns out to be a Brownian motion with diffusion coefficient λ. It then follows that K E F (s,ϕ ; t, ψ ) = λ(s∧t)E [ϕ(η)ψ(η)] and so Theorem 6.9 gives us a version of Theorem 3 of [8].

7 Acknowledgements

The authors would like to thank the referees for their numerous helpful comments and suggestionswhich have helped to improve the clarity and overall exposition of the paper.

8 Appendix

In the Appendix, we provide the proofs of several supporting lemmas from the main body of thepaper. We begin with the proof of Lemma 3.4.

Proof of Lemma 3.4. We prove Part 1 by induction. For each n ≥ 0 and t ≥ 0 xed, denote thequantity on the lefthand side of ( 41) by Ln . For the base case of n = 0, it is straightforward to seethat L0 ≤ 1. Next, for the inductive step, suppose that ( 41) holds for n = 0 , . . . , k −1, and t ≥ 0.

Then, we have that

F (x + t)F (x)

(k)

= F (x + t)

F (x)

(1) (k− 1)

= (h(x) −h(x + t))F (x + t)

F (x)

(k− 1)

≤k− 1

i=0

k −1i

(h(x) −h(x + t)) (k− 1− i ) F (x + t)F (x)

(i)

≤ 2

k− 1

i=0

k

−1

i h(k− 1− i)

∞ L i

< ∞.

This completes the proof of Part 1.

41



We next prove Part 2 by induction as well. First recall that for s, t ≥ 0, we may write

F (x + t)F (x + s)

= exp − x+ t

x+ sh(u)du . (117)

Hence, for the base case of n = 0, we have that

supx≥ 0

F (x + t)F (x) −

F (x + s)F (x)

= supx≥ 0

F (x + s)F (x)

1 −F (x + t)F (x + s)

≤ supx≥ 0

F (x + s)F (x)

supx≥ 0

1 −F (x + t)F (x + s)

≤ supx≥ 0

1 −F (x + t)F (x + s)

≤ 1 −e− h ∞ | t− s |

≤ h ∞ |t −s|,

where the third inequality above follows from ( 117) and the nal inequality follows from the meanvalue theorem. Next, for the inductive step, suppose that ( 42) holds for n = 0 , 1,...,k −1. We thenhave that

supx≥ 0

F (x + t)F (x) −

F (x + s)F (x)

(k)

= supx≥ 0

(h(x) −h(x + s))F (x + s)

F (x) −(h(x) −h(x + t))F (x + t)

F (x)

(k− 1)

= supx≥ 0

(h(x + t) −h(x + s))F (x + s)

F (x) −(h(x) −h(x + t)) F (x + t)

F (x) −F (x + s)

F (x)

(k− 1)

= supx≥ 0

k− 1

i=0

k −1i

[(h(x + t) −h(x + s))](k− 1− i) F (x + s)F (x)

(i)

−k− 1

i=0

k −1i

(h(x) −h(x + t)) (k− 1− i) F (x + t)F (x) −

F (x + s)F (x)

(i)

. (118)

Now note that since by Assumption 2.1 we have that h ∈ C ∞b (R + ), it follows that all of thederivatives of h are bounded and hence uniformly continuous as well. Using this fact, Part 1 andthe inductive hypothesis it now follows that ( 118) is less than or equal to

2|t −s|k− 1

i=0

k

−1

i h(( k− 1− i)− 1)

∞ L i + h(k− 1− i )

∞ M i ≡ M k |t −s|.This proves Part 2 and completes the proof.

We next provide the proof of Lemma 6.3

42





In order to do so, we will verify that Parts 1 and 2 of Theorem 1.5 are satised. We begin withPart 1. Let ϕ, ψ ∈ S and note that by Proposition 4.2, the functional strong law of large numbers[33] and the random time change theorem [ 1], we have that

[ ˆ

Gn ](ϕ, ψ) =

1

n

E n

i=1

(ϕ(ηi )

− F ,ϕ )(ψ(ηi )

− F , ψ ) (123)

⇒ E Cov(ϕ(η), ψ(η)) in D ([0, T ], R ) as n → ∞,

where η is a random variable with CDF F . Now letting ϕ = ψ in (123) and using the fact thatthe maximum jump of ˆGn ,ϕ over the interval [0 , T ] is bounded uniformly in n, along with themartingale FCLT [10], yields the limit

ˆGn ,ϕ ⇒ ˆG,ϕ in D ([0, T ], R ) as n → ∞.Thus, Part 1 of Theorem 1.5 holds. We next prove that Part 2 of Theorem 1.5 holds. Let m ≥ 1and let ϕ1, . . . ,ϕm ∈ S . Then, using the limit ( 123) and the fact that the maximum jump of

ˆ

Gn ,ϕ1 , . . . , ˆ

Gn ,ϕm over the interval [0 , T ] is bounded uniformly in n , along with the martin-

gale FCLT [10], yields the limit

ˆGn ,ϕ1 , . . . , ˆGn ,ϕm ⇒ ˆG,ϕ1 , . . . , ˆG,ϕm in D m ([0, T ], R ) as n → ∞.This limit then provides convergence of the nite dimensional distributions of the random vectoron the left-hand side above, which shows that Part 2 of Theorem 1.5 holds. Thus, ( 122) it proven.

In order to complete the proof, it now suffices to show that

ˆGn − ˇGn⇒ 0 in D ([0, T ], S ) as n → ∞. (124)

However, in order to show ( 124), it suffices by Theorem 1.5 to show that for each ϕ ∈ S ,

Gn

,ϕ −Gn

,ϕ ⇒ 0 in D ([0, T ], R ) as n → ∞. (125)We proceed as follows. In a similar manner to the above, one may show using the martingale FCLTthat

ˇGn⇒

ˆG in D ([0, T ], S ) as n → ∞. (126)

Hence, for each ϕ ∈ S , the sequence ( ˆGn ,ϕ −ˇGn ,ϕ )n ≥ 1 is tight in D ([0, T ], R ) and so in orderto show (125), it suffices to show that for each 0 ≤ t ≤ T ,

ˆGnt , ϕ −ˇGn

t , ϕ ⇒ 0 as n → ∞. (127)

First note that we may write

ˆGnt , ϕ −ˇGn

t , ϕ = 1E nt ≥ n E t 1

√ nE nt

i= n E t

(ϕ(ηi ) − F ,ϕ )

−1 n E t ≥E nt 1

√ nn E t

i= E nt

(ϕ(ηi ) − F ,ϕ ) .

44



Now squaring both sides of the above and using the basic identity ( x1 + x2)2 ≤ 2(x21 + x2

2), it isstraightforward to show that one may write

( ˆGnt , ϕ −ˇGn

t , ϕ )2 ≤ 1 E nt ≤ 2 E t 2n

E nt ∨ n E t

i= E nt ∧ n E t

(ϕ(ηi ) − F ,ϕ )

2

(128)

+ 1 E nt ≥ 2 E t 2n

E nt ∨ n E t

i= E nt ∧ n E t

(ϕ(ηi ) − F ,ϕ )

2

. (129)

We now show that each of the terms ( 128) and ( 129) converges to 0 in probability as n tends to

∞, which implies ( 127) and completes the proof.We begin with ( 128). First note that by the independence of ηi , i ≥ 1 from the arrival process

E n , and the i.i.d. nature of the sequence ηi , i ≥ 1, we have that

E2

n1 E n

t ≤ 2 E t

E nt ∨ n E t

i= E nt ∧ n E t

(ϕ(ηi )

− F ,ϕ )

2

= 2

nE 1 E nt ≤ 2 E t

E nt ∨ n E t

i= E nt ∧ n E t

(ϕ(ηi ) − F ,ϕ )2

≤ 2

nE

(E nt ∧ 2n E t )∨ n E t

i=( E nt ∧ 2n E t )∧ n E t

(ϕ(ηi ) − F ,ϕ )2

≤ 8 ϕ2∞ E [|( E nt ∧2 E t ) − E t |].

However, since E nt ⇒ E t as n → ∞, it follows that

E [|( E nt ∧2 E t ) − E t |] → 0 as n → ∞.This then implies that ( 128) converges to 0 in probability as n tends to ∞, as desired.

We next proceed to ( 129). Note that since E nt ⇒ E t as n → ∞, it follows that for each ε > 0,

limn →∞

P 1 E nt ≥ 2 E t 2n

E nt ∨ n E t

i= E nt ∧ n E t

(ϕ(ηi ) − F ,ϕ )

2

> ε = 0 .

This shows that ( 129) converges to 0 in probability as n tends to ∞, which completes the proof.

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