a fluid control problem in queueing networks with general service...

A Fluid Control Problem in Queueing Networks with

General Service Times

Gennady Shaikhet

Department of Mathematical Sciences

Carnegie Mellon University

Pittsburgh, PA, 15213, USA

E-mail: [email protected]

January 20, 2010

Abstract

We consider a queueing network with several customer classes and several service

pools, each consisting of many statistically identical servers. The service requirement of

each customer is assumed to be generally distributed and is allowed to depend on both

the customer class and the service pool. Customers may also abandon while waiting in

queue, with exponentially distributed patience time. For each class, the holding cost

per unit time is given as a convex function of the corresponding queue length. It is

shown that, under an overload condition and under a certain fluid scaling, the problem

of identifying an optimal dynamical scheduling that minimizes the long average holding

cost, can be reduced to solving a simple deterministic problem.

1 Introduction

Many-server queues with general service times have recently received considerable attentionin the literature. The papers of Kaspi and Ramanan [6], [7] and Reed [11], [12] study fluidand diffusion approximations of G/G/N queues. The recent paper of Kang and Ramananstudies fluid approximation of the G/G/N+G queues. The before mentioned works addressperformance measure of single queues, while the issue of asymptotic control of many-serverqueueing networks with general service times still remains a challenging research direction.We cite the work of Whitt and Talreja [15], that surveys control problems in overloadedfluid queueing models with general service and patience times.

1

In this paper we consider a particular control problem in a network setting and proposean asymptotic solution. Our queueing network consists of I customer classes and J many-server service pools. Each customer can be served only once, in either one of the pools,and then leaves the system. The service requirement of customer i in pool j is generallydistributed. If not routed upon arrival, a customer joins the queue, from which he canabandon according to his exponential clock. There is an accumulated penalty on the numberof waiting customers, and the goal is to minimize long-run average total cost. The abovecost can also be extended to include abandonment penalties. We consider the problem ina many-server overloaded fluid regime, when both the arrival rates and number of seversin pools are proportional to N → ∞. In particular we are interested in the case when thesystem is overloaded, i.e., in the case when the queues of order O(N) are expected underany routing policy. In this regime we seek to minimize

1T

∫ T0

∑i

Ci

(Y

(N)

i (t))dt,

where Y(N)

i =Y

(N)i

N is a scaled queue length of type i and Ci some convex function. We areable to solve this problem asymptotically, when first N and then T increase to infinity.

This paper is motivated by the work of Atar, Giat and Shimkin [1], which considersthe same problem in the fully exponential setting, in a network with just one service pooland linear penalties on queue lengths. The authors identify the so called cµ/θ-policy - therouting that assigns fixed priorities on classes according to the value of ciµi/θi - where ciis a holding cost, and µi and θi are the corresponding service and abandonment rates forclass i. In particular, each time a server in the pool becomes idle - he admits to service acustomer with the highest value of ciµi/θi.

In the long run, the policy achieves the following effect: there are classes of customersthat are fully served, with no remaining queue; there are classes of customers that are notserved at all, and customers remain in queue until they abandon; and there is one classfor which only a fixed proportion of customers is served. The latter behavior is stationary,i.e., does not depend on time, with all the quantities derived from a simple deterministiclinear optimization problem. Moreover, it turns out that any policy with a similar largetime behavior will asymptotically minimize the cost. Therefore, showing the equivalenceto a deterministic optimization problem plays the central role in our attempt to expandthe results to a setting with general service times. This in a contrast to [1], where theequivalence is deduced easily from the exponential nature of service, namely, from the fact,that asymptotically, the number of served (resp., abandoned) customers is proportional totheir total service (resp., waiting) time, with service (resp., abandonment) rate being theproportionality coefficient. A similar equivalence for the number served holds in our settingas well, although, in contrast to [1], the averaging over a long interval of length T , is essentialfor the treatment. Together with the asymptotic optimality result, the ”exponential” likeproportionality relation for the number of served are the main contributions of the currentpaper. We believe, however, that a similar ”exponential” like relation does not hold for thenumber of abandoned customers. See also [4] and [9].

2

While [1] concentrate on showing the asymptotic optimality of the cµ/θ policy, no suchattempt is made here and we introduce a separation policy, that completely separates pools’capacity, creating dedicated servers - servers, that can only serve a certain class, or properlydefined sub-class of incoming customers. The use of a separation policy allows us to treat thesystem as a composition of several disjoint G/G/N + M systems which has been alreadywell treated in the literature (e.g., [6]). In addition, this allows an effortless transitionfrom the one-pool model to a several pools’ model. An application of Jensen’s inequalityallows to include convex costs. In the context of a one-pool setting, the cµ/θ policy seemsadvantageous over the separation in that it does not require the knowledge of the incomingarrival rates. We comment, however, that even in the exponential setting of [1], cµ/θ rule istreated under an additional assumption on initial system conditions - a system must startclose to an optimal servers allocation, and for that one still needs to know the arrival rates.For our results we do not need any particular assumption on initial conditions.

Notations. We write R+ for [0,∞). For x ∈ Rn we use ‖x‖ =∑ni=1 |xi|. The symbol I{A}

denotes the indicator function of a given set A. For a collection A of random variables,σ{A} denotes the sigma-field generated by this collection. For a finite set B of elementswe write |B| for the number of elements in the set. By DR+ [0,∞) we define the space offunctions from R+ to R+ that are right continuous on R+ and have finite left limits on(0,∞) (RCLL), endowed with the usual Skorohod topology J1 - topology [10]; a.s. meanalmost surely; unless otherwise specified, convergence → of random variables means a.s.convergence in the appropriate probability space. Since the limit processes we consider havehas continuous sample paths, so the notion of convergence on the underlying function spaceDR+ [0,∞) coincides with uniform convergence on closed bounded intervals [14].

2 Model

2.1 Description of the queueing system

A precise definition of the model is as follows. A complete probability space (Ω,F ,P) isgiven, supporting all the stochastic processes defined below. Expectation with respect to Pis denoted by E. The queueing model is parameterized by N ∈ N, where N represents thescale of the system, has I ≥ 2 customer classes and J ≥ 2 service stations (as illustrated inthe picture below).

The classes are labeled as 1, . . . , I and the stations as I + 1, . . . , I + J : the correspondingindex sets are denoted

I = {1, . . . , I}, J = {I + 1, . . . , I + J}.

Each station j ∈ J has N (N)j statistically identical servers working independently.

Service. Class i customers may be served in some of the stations from J . Service,once started with a certain server in a certain station, can not be interrupted or switched

3

Figure 1: The model of queueing network with 3 customer classes and 4 stations.

to another server. Let E denote the set of all pairs (i, j) ∈ I × J such that the class i canbe served in station j. The service requirements for each class i ∈ I are given by the i.i.d.sequence of vectors {vki }, k ∈ Z, where the components of the vector vki = (vki,j , j ∈ J ) areindependent with common cumulative distribution function Gij for each (i, j) ∈ E . Thus,for a given i and k, a vector {vkij , j ∈ J } represents service requirements of the kth arrivingcustomer of class i for each station j ∈ J . The non-positive values of k correspond toservice requirements of customers, initially presented in the system (see the description ofthe initial conditions below).

For t ≥ 0 and i ∈ I let X(N)i (t) be the number of class i customers in the system, (whichincludes those in queue and those being served), at time t. We also let Y (N)i (t) be thenumber of class i customers in the queue at time t and let Ψ(N)ij (t), (i, j) ∈ E , be the numberof class i customers being served at station j at time t. The above quantities satisfy, foreach t ≥ 0,

Y(N)i (t) +

∑j∈J

Ψ(N)ij (t) = X(N)i (t), i ∈ I, (1)∑

i∈IΨ(N)ij (t) ≤ N

(N)j , j ∈ J . (2)

Also, the following non-negativity conditions hold by definition

X(N)i (t) ≥ 0, Y

(N)i (t) ≥ 0, Ψ

(N)ij (t) ≥ 0, i ∈ I, j ∈ J , t ≥ 0. (3)

We will write X(N) for the vector (X(N)i , i ∈ I) and similarly Y (N) = (Y(N)i , i ∈ I).

Arrivals. For each i ∈ I introduce a counting process A(N)i , representing the totalnumber of class i customers that arrive to the system in the time interval [0, t]. We willintroduce some additional assumptions on A(N) as paper progresses.

4

Abandonments. For each i ∈ I introduce a Poisson process R̃(N)i , with rate θi. Thecumulative number of class i abandonments up to time t, which we denote by R(N)i (t), is

assumed to be equal R̃(N)i( ∫ t

0Y

(N)i (s)ds

). Namely, each time when R̃(N)i

( ∫ t0Y

(N)i (s)ds

)jumps, one customer from the queue is blocked. We make a convention that it is alwaysthe end-of-the-line customer, which is being blocked, although it is not important for ouranalysis. Throughout the paper we assume that the processes ({A(N)i (t), t ≥ 0}, {R̃i(t), t ≥0}, {vki , k ∈ Z})i∈I are mutually independent.

Service completions. The ”exponential” modeling of abandonment process can nolonger be applied for the departure processes. Since the service is not exponential, it is notenough to know how many customers are currently being served in each station. We needto know the routing destination for each customer, as well as his ”age”, which is defined tobe the time spent in service from the moment of routing (a precise representation is givenin (5) below). Introduce routing index as follows. For k ∈ Z and i ∈ I let αk,(N)i be thearrival time of the kth customer of class i. For such customer define for t ≥ 0

σk,(N)i (t) =

0 if t < αk,(N)i1 if the customer is still in queue at t−1 if the customer abandoned in interval [0, t]I + j if the customer is routed to station j in interval [0, t]

(4)

At any time t ≥ 0, if there are waiting customers of some class i ∈ I and idle servers at somestation j with (i, j) ∈ E , the policy maker can route a waiting class i customer to servicein station j. The later will result in changing the corresponding σk,(N)i (t) from 1 to I + j.Within a class i, we assume, it’s always the head of the line customer (the one with the leastk among those waiting), which is routed. Note that we do not assume work conservation,therefore allowing situations where there are both idle servers and waiting customers.

Let βk,(N)i be the routing time of customer k of class i to service, to either one of theavailable stations. For (i, j) ∈ E , t ≥ 0 and k ∈ Z, satisfying σk,(N)i (t) = I + j, define theage processes

ak,(N)ij (t) =

{(t− βk,(N)i ) ∧ 0 if t− β

k,(N)i < v

kij

vkij otherwise(5)

We need to define ages ak,(N)ij only is for such k and (i, j), so that kth customer of class i

was routed to station j. The departure processes will be introduced according to measure-theoretic representation of [6]. With (5) in hand, we define D(N)ij (t) - the cumulative numberof class i departures from the station j in the interval [0, t]

D(N)ij (t) =

∑s∈[0,t]

A(N)i (t)∑

k=−X(N)i (0)+1

I {σ

k,(N)i (s)=I+j

}I {da

k,(N)ijdt (s−)>0 , a

k,(N)ij (s)=v

kij

}. (6)The last term requires a short explanation. For the customer in service, its age grows linearlyuntil it reaches its service requirement, and remains at that value afterwards. Thus, the last

5

indicator on the r.h.s. being 1 is equivalent to the fact that the customer finishes his serviceat time s. We will also need the routing process K(N)ij (t), which represents the cumulativenumber of class i customers, that are routed to the service in station j in the interval [0, t].

K(N)ij (t) =

A(N)i (t)∑

k=−Y (N)i (0)+1

I {σ

k,(N)i (t)=I+j

}. (7)

Initial conditions. We assume that the system initially has {Y (N)i (0)}i∈I customersin each queue, and {Ψ(N)ij (0)}(i,j)∈E customers in each type of service. For convenience,we assign numbers k = −X(N)i (0) + 1, ...,−Y

(N)i (0) for all class i customers initially in the

system, as well as define their routing indices σk,(N)i (0). We repeat the procedure for eachi ∈ I. Note, that the alternative way of introducing Grij was to assume the initial agesak,(N)ij (0) are random and the residual service time has distribution G

rij :

Grij(x) =Gij

(x+ ak,(N)ij (0)

)−Gij

(ak,(N)ij (0)

)1−Gij

(ak,(N)ij (0)

) , x ≥ 0. (8)Non-anticipating controls. Naturally, we would like to assume that the policy makerdoes not know the service requirements of customers, which are either waiting in queue orhave not arrived to the system. To formulate that, for any i ∈ I and t ≥ 0 define

τ(N)i (t) = inf{s > t : K

(N)i (s) > K

(N)i (s−)}

to be the time of the first routing event from class i, after t. For each t ≥ 0 and i ∈ I set

k(N)i (t) = max{k ≥ 0 : σ

k,(N)i (t) > I}

to be the order number of the last class i customer, routed to service in the interval [0, t].Let i(N)∗ (t) = arg mini{τ (N)i (t)} be the index of the first class to be routed after time t. Weare ready to state the non-anticipation assumption (for clarity we omit the superscript (N)).

Assumption 2.1 For any t ≥ 0, both the next routing time τi∗(t) and the routing decisionat that time σk∗(t)i∗(t) (τi∗(t)) are independent of the collection

∪iσ{vk0i , k0 > ki(t)

}.

In particular, this will imply that for any pair (i, j) ∈ E , the service requirements of allcustomers, routed from class i to station j in the interval [0, t], are mutually independent,with the distribution Gij . The latter property will be used in the paper. Also note thatAssumption 2.1 allows policies that observe the remaining service time - turns out this doesnot influence our result.

Throughout we impose the following assumption on service distribution.

6

Assumption 2.2 Each distribution from the set G = {Gij , Grij , (i, j) ∈ E} has a densityand satisfy either one of the following conditions

1. Has a finite support,

2. has a finite second moment and a mean residual life of at most linear growth - thereexist constants l0, l1 ≥ 0, such that for any X ∼ G ∈ G, E[X − t | X > t] ≤ l0 + l1tfor all t > 0.

Remark 2.3 The part of Assumption 2.2 (2) about the residual life is indeed a mild assump-tion. It includes, for example, all the distributions, which decay faster than 1−G(t) ∼ ct−β,for β > 2 (see Proposition 11(b) in [5]).

Assumption 2.2 implies that any distribution from G has a finite mean. For any pair(i, j) ∈ E define the service rates µij and µrijas

(µij)−1 =∫ ∞

0

[1−Gij(u)]du, (µrij)−1 =∫ ∞

0

[1−Grij(u)]du. (9)

We end the section with the following mass balance equations: for t ≥ 0

X(N)i (t) =X

(N)i (0) +A

(N)i (t)−

∑j∈J

D(N)ij (t)−R

(N)i (t), i ∈ I, (10)

Ψ(N)ij (t) =Ψ(N)ij (0) +K

(N)ij (t)−D

(N)ij (t), (i, j) ∈ E . (11)

Any process π(N) =(σ(N), D(N), R(N), X(N), Y (N),Ψ(N)

)will be regarded as a policy,

provided the relations (1)–(11) hold and Assumption 2.1 is satisfied. Let Π(N) denote theset of all policies.

2.2 Many-server fluid scaling and optimization problem

Consider the sequence of such systems, indexed by N . We assume that the number ofservers and the arrival rates increases by the factor of N - the fact, which is summarized inthe following assumption.

Assumption 2.4 1. There exist constants λ1 > 0, ..., λI > 0, such that for each i ∈ I,the sequence of processes {A(N)i , N ≥ 1} satisfies A

(N)i /N → λit, in DR+ [0,∞) a.s.

for N →∞.

2. There exist constants nj > 0, j ∈ J , such that for each j, N (N)j /N → nj, as N →∞.

3. There exists a constant 1 < m < ∞, independent of N so that the scaled initialconditions ‖X(N)(0)/N‖ ≤ m for all N ≥ 1.

7

Introduce scaled quantities

A(N)

i =1NA

(N)i , R

(N)

i =1NR

(N)i , (12)

X(N)

i =1NX

(N)i , Y

(N)

i =1NY

(N)i , , D

(N)

ij =1ND

(N)ij . (13)

Let C1,...,CI be continuously differentiable increasing convex functions. For a given T > 0,N ∈ N and a routing policy π(N) ∈ Π(N) consider the cost

C(N)(T ) = C(N)(π(N), T ) =1T

E[ ∫ T

0

∑i

Ci

(Y

(N)

i (t))dt]

(14)

and define

V (N)(T ) = infπ(N)∈Π(N)

C(N)(π(N), T

). (15)

We are interested in asymptotically solving the problem. The asymptotic considered istaking N →∞ for fixed T and then taking T →∞. Namely, we are looking for

v = lim infT→∞

lim infN→∞

V (N)(T ). (16)

and for a policy to achieve v asymptotically. The structure if the cost suggests that the onlycases of interest are those when the system is overloaded on the order of O(N) (fluid) level,otherwise the optimal value will trivially be zero.

For a given time T introduce y(N)i = y(N)i (T ), i ∈ I and ψ

(N)ij = ψ

(N)ij (T ), (i, j) ∈ E , as

y(N)i (T ) =

1T

∫ T0

Y(N)

i (s)ds , ψ(N)ij (T ) =

1T

∫ T0

Ψ(N)

ij (s)ds (17)

The above quantities represent the averages over time of the scaled queue length of eachclass i ∈ I and each serving population for each pair (i, j) ∈ E . Theorem 2.5 below discoversthe asymptotic connection of θiy

(N)i (T ) and µijψ

(N)ij (t), namely

e(N)i (T ) := λi −

∑j∈J

µijψ(N)ij (T )− θiy

(N)i (T ) ≈ 0. (18)

Theorem 2.5 There exist a continuous function γ : [0,∞)→ [0,∞), with γ(0) = 0, suchthat for any ε > 0 small enough, T ≥ ε−1 and for any sequence of routing policies π(N)

lim infN→∞

P(‖e(N)(T )‖ < γ(ε)

)≥ 1− γ(ε).

Before proving the theorem we will discuss it’s role in finding v from (16). Define S to bethe set of all pairs (y, ψ) = ({yi}, {ψij} , i ∈ I, (i, j) ∈ E), satisfying

∑j∈J µijψij + θiyi = λi,∑i∈I ψij ≤ nj ,

ψij ≥ 0, yi ≥ 0.(19)

8

Consider the following deterministic optimization problem:

Minimize∑i

Ci(yi) over all pairs (y, ψ) ∈ S (20)

Define V ∗ to be the minimal value of (20). Observe that V ∗ is attained and finite as aminimum of a continuous function over a closed set. From (2), (18) and Assumption 2.4(2)we see that (y(N)(T ), ψ(N)(T )) asymptotically belong to S, which gives us a hint of finding vvia solving the deterministic optimization problem. And we justify that claim by introducingthe main result of the paper

There exists an asymptotically optimal policy, and v = V ∗. (21)

The result will be formalized as a theorem in Section 4. We start with proving Theorem2.5.

3 Proof of Theorem 2.5

From (1) and (10)-(11) we have, for any i ∈ I and T > 0

Y(N)

i (T ) = Y(N)

i (0) +A(N)

i (T )−R(N)

i (T )−∑j∈J

K(N)

ij (T ). (22)

Dividing by T , and regrouping terms using the definition of e(N)i from (18), we get

e(N)i (T ) =

1T

(Y

(N)

i (T )− Y(N)

i (0))−(

1TA

(N)

i (T )− λi)

(23)

+(

1TR

(N)

i (T )− θiy(N)i (T )

)+∑j∈J

(1TK

(N)

ij (T )− µijψ(N)ij (T )

)To show (18) we need to prove that the r.h.s. of (23) tends to zero as N → ∞ for T largeenough. The rest of the proof will be divided into four steps:

Step 1: we estimate the second and the third terms of the r.h.s. of (23);Step 2: we estimate the first term of the r.h.s. of (23);Step 3: we estimate the fourth term of the r.h.s. of (23);Step 4: we finalize the proof using the estimates obtained in Steps 1-3.

Step 1: For arbitrary T > 0 and δ ∈ (0, 1), define E(N)A = E(N)A (T, δ) and E

(N)R = E

(N)R (T, δ)

E(N)A =

{maxi

supt∈[0,T ]

∣∣∣A(N)i (t)− λit∣∣∣ < δ}; (24)E

(N)R =

{maxi

supt∈[0,T ]

∣∣∣R(N)i (t)− θi ∫ t0

Y(N)

i (s)ds∣∣∣ < δ}. (25)

9

Lemma 3.1 P(E

(N)A ∩ E

(N)R

)→ 1 as N →∞.

Proof: The convergence P(E(N)A )→ 1 as N →∞ follows from Assumption 2.4(1), and wemove to the E(N)R . From the assumed representation R

(N)i (t) = R̃i

(∫ t0Y

(N)i (s)ds

)we have

R(N)

i (t) =1NR̃i

(N

∫ t0

Y(N)

i (s)ds). (26)

Moreover, from (22), we have Y (N)i (t) ≤ Y(N)i (0)+A

(N)i (t). Using Assumption 2.4(3), on the

event E(N)A , we have Y(N)

i (t) ≤ m+λit+δ, hence∫ t

0Y

(N)

i (s)ds ≤ (m+δ)T+λ2iT 2 for t ≤ T .Recall that T is fixed and set M = M(T ) = (m+ δ)T +λ2iT

2. Since R̃i is a Poisson processwith rate θi, we have (see Chpt. 5 of [3]) the a.s. convergence supt∈[0,M ] | 1N R̃i(Nt)−θit| → 0as N →∞. Together with

∫ t0Y

(N)

i (s)ds ≤M this concludes the lemma. 2

Step 2: We now proceed to the first term of the r.h.s. of (23).

Lemma 3.2 There exist constant c0 and n0, such that for every δ ∈ (0, 1), T ≥ 1, N ≥ n0and π(N) ∈ Π(N), we have ‖Y (N)(T )‖ ≤ c0 on E(N)A (T, δ) ∩ E

(N)R (T, δ).

Proof: From Assumption 2.4(2) there exists a constant n0 such that N(N)j /N ≤ nj + 1

for all N ≥ n0. Fix such a constant n0 and fix δ ∈ (0, 1), T ≥ 1, N ≥ n0 and π(N) ∈Π(N). For each (i, j) ∈ E the path of D(N)ij is an increasing step function, with steps ofat most nj + 1. Let D̃

(N)ij to be an interpolation of D

(N)

ij on the interval [0, T ]. Then

supt∈[0,T ]∣∣∣D̃(N)ij (t)−D(N)ij (t)∣∣∣ ≤ nj + 1 and, using (22) and (24)–(25), we have for t ≤ TX

(N)

i (t) = X(N)

i (0) + η(N)i (t) + λit− θi

∫ t0

Y(N)

i (s)ds−∑j∈J

D̃(N)ij (t), (27)

for some function ηi with η(N)i (0) = 0 and satisfying η

(N)i (t) ≤ 2δ + ‖n‖ + 1 ≤ 3 + ‖n‖ on

E(N)A (δ, T )∩E

(N)R (δ, T ). The interpolated function D̃

(N)ij is piece-wise linear non-decreasing,

and can be represented as an integral over some nonnegative step function. Let ξ be aunique solution to

ξ(t) = X(N)

i (0) + η(N)i (t) + (θi(‖n‖+ 1) + λi)t− θi

∫ t0

ξi(s)ds. (28)

Subtracting (28) from (27), and taking a derivative, we have

d

dt

(X

(N)

i − ξ)

(t) = − θi(Y

(N)

i (t) + ‖n‖+ 1)

+ θiξ(t)−d

dtD̃

(N)ij (t) (29)

≤ − θi(X

(N)

i − ξ(t))

10

Now we can apply the standard comparison theorem for ODE (Theorem 7, p. 22, [2]), toget X

(N)

i ≤ ξ. In turn, a solution to (28) is given by

ξ(t) =X(N)

i (0)e−θit + η(N)i (t)− θi

∫ t0

η(N)i (u)e

θi(u−t)du+θi(1 + ‖n‖) + λi

θi(1− e−θit).

(30)

And, by Assumption 2.4(3), is bounded by

ξ(t) ≤ m+ 2‖η(N)i ‖∗T + (1 + ‖n‖) +

λiθi. (31)

And the lemma follows from (see (1)) Y (N)i ≤ X(N)i , by taking c0 = |I|(7+3‖n‖+m)+

∑iλiθi

.2

Step 3: We now proceed to the fourth term of the r.h.s. of (23). For any (i, j) ∈ E wehave (

1tK

(N)

ij (t)− µijψ(N)ij (t)

)=

1t

(K

(N)

ij (t)− µijSK(N)ij (t))

+µijt

(SK

(N)ij

(t)− tψ(N)ij (t))

(32)

Here SK

(N)ij

(t) is the scaled total amount of the work routed from class i to station j in the

interval [0, t].

SK

(N)ij

(t) =1N

A(N)i (t)∑

k=−Y (N)i (0)+1

I{σk,(N)i (t) = I + j

}vkij . (33)

In what follows we estimate each term of (32) separately. The first term vanishes asymp-totically due to the Law of Large Numbers, while to estimate the second term we will needAssumption 2.2 about the residual service time.

Estimating the first term in (32): For fixed T > 0, i ∈ I and (i, j) ∈ E consider thefollowing sequence of i.i.d. service requirements

{vkij , k = −Y(N)i (0) + 1, ...,−Y

(N)i (0) + 2λiNT}. (34)

For a natural d ≤ 2λiNT and a subsequence ξ = {ξ1, .., ξd} ⊂ {−Y (N)i (0)+1, ...,−Y(N)i (0)+

2λiNT} of integers, chosen independently of random variables(34), define a partial sumS

(N)ij (ξ, d) =

∑dr=1 v

ξrij . For arbitrary δ ∈ (0, 1) introduce an event E

(N)1,K = E

(N)1,K (T, δ)

E(N)1,K =

{max

(i,j)∈Esupd, ξ

∣∣∣1dS

(N)ij (ξ, d)−

1µij

∣∣∣µij dN T

< δ}. (35)

Lemma 3.3 For every δ ∈ (0, 1), T ≥ 1, N ≥ (m + 1)(mini λi)−1 and π(N) ∈ Π(N), wehave

max(i,j)

∣∣∣∣∣∣ 1µij −SK

(N)ij

(t)

K(N)

ij (t)

∣∣∣∣∣∣ µijK(N)

ij (t)t

< δ (36)

on the event E(N)A (T, δ) ∩ E(N)1,K (T, δ). Moreover, P

(E

(N)A ∩ E

(N)1,K

)→ 1 as N →∞.

11

Proof: Using Assumption 2.4, on the event E(N)A ∩ E(N)1,K we have Y

(N)i (0) + A

(N)i (t) ≤

λiNT + δ + m ≤ 2λiNT for N from the statement of the lemma. Together with the non-anticipation of the policies, this implies that the event (36) is a subset of E(N)1,K from (35).

Let us prove P(E

(N)1,K

)→ 1. We have P

((E(N)1,K )

c)≤ P

(B(N)1

)+ P

(B(N)2

), where

B(N)1 ={∃(i, j) ∈ E and {vξkij }

dk=1, d ≤ N1/2 such that

∣∣∣∣∣1dd∑k=1

vξkij −1µij

∣∣∣∣∣ µij dN T ≥ δ }

B(N)2 ={∃(i, j) ∈ E and {vξkij }

dk=1, N

1/2 < d ≤ 2λiNT such that∣∣∣∣∣1dd∑k=1

vξkij −1µij

∣∣∣∣∣ µij dN T ≥ δ }.Recall that, for a given (i, j) ∈ E , the random variables {vξkij }dk=1 are independent. Hence

P(B(N)1

)≤

∑(i,j)∈E

P

µijN T

√N∑

k=1

vξkij ≥ δ/2

+ ∑(i,j)∈E

P

(√N

N T≥ δ/2

)→ 0, (37)

as N → ∞, where the convergence of the first term is due to Markov inequality and thesecond term convergence is trivial. As for B(N)2 , we have, as N →∞

P(B(N)2

)≤∑

(i,j)∈E

P

( ∣∣∣∣∣1dd∑k=1

vξkij −1µij

∣∣∣∣∣ d µijN T ≥ δ for some N1/2 ≤ d ≤ 2λiNT)

≤∑

(i,j)∈E

P

( ∣∣∣∣∣1dd∑k=1

vξkij −1µij

∣∣∣∣∣ ≥ δ2λiµij for some d ≥ N1/2)→ 0

where the convergence is due to a strong Law of Large Numbers. Lemma 3.4 below completesthe proof. 2

Lemma 3.4 Consider two sequences {An}n≥1 and {Bn}n≥1 , and assume P(An) → 1 asn→∞, Then

lim infn→∞

P(An ∩Bn) ≥ 1− lim supn→∞

P(Bcn). (38)

In particular, P(An ∩Bn)→ 1 if P(Bn)→ 1.

Proof: Indeed, since P(An ∩Bn) = 1− P(Acn ∪Bcn), we have

lim infn

P(An ∩Bn) ≥ 1− lim supn→∞

P(Acn ∪Bcn) (39)

≥ 1− lim supn→∞

P(Acn)− lim supn→∞

P(Bcn)

= 1− lim supn→∞

P(Bcn).

12

2

Estimating the second term in (32): Using (33) and (17), rewrite (32) asµijt N

(V

(N)ij (t)− U

(N)ij (0)

), where

U(N)ij (0) =

−Y (N)i (0)∑k=−X(N)i (0)+1

I {σ

k,(N)i (0)=I+j

} (vkij − ak,(N)ij (0)) (40)and

V(N)ij (t) =

A(N)i (t)∑

k=−Y (N)i (0)+1

I {σ

k,(N)i (t)=I+j

}vkij + U (N)ij (0)− ∫ t0

Ψ(N)ij (s)ds. (41)

Here U (N)ij (0) is the total initial service requirements of type i in station j, whereas V(N)ij (t)

is the total residual amount of work of type i that is left at station j at time t. For t > 0and ε > 0 introduce an event E(N)2,K = E

(N)2,K (t, ε) as

E(N)2,K =

{max

(i,j)∈E

{ µijt N

∣∣∣V (N)ij (t)− U (N)ij (0)∣∣∣} < ε1/8 } . (42)Lemma 3.5 For any ε > 0 small enough, t > ε−1, N ≥ n0 (from Lemma 3.2) and π(N) ∈Π(N)

P(E

(N)2,K (t, ε)

)≥ 1− ε1/16. (43)

Proof: For arbitrary t > 0, π(N) ∈ Π(N) and (i, j) ∈ E let A(N)ij = A(N)ij (t, π

(N)) be the set ofagents in station j, that are currently (i.e., at time t) busy with class i customers. Formally,{a ∈ A(N)ij } ⇔ {an agent a in station j is serving class i customer at time t}. For an agenta ∈ A(N)ij we let R

(N)a (t) be the residual service time of his current job. By this we have

V(N)ij (t) =

N(N)j∑a=1

Ia∈A(N)ij

Ra(t), (44)

Next we consider two cases according to Assumption 2.2:

(Case 1.) If the distribution Gij has a finite support Gij(Mij) = 1 for some Mij ε−1

µijt N

∣∣∣V (N)ij (t)− U (N)ij (0)∣∣∣ ≤ 2µij Mij N (N)jt N ≤ 2µijMij(nj + 1)ε ≤ ε1/8, (45)for ε small enough and (43) is trivially satisfied.

13

(Case 2.) Decompose A(N)ij as a union of two disjoint A1,(N)ij,ε and A

2,(N)ij,ε , such that

{a ∈ A1,(N)ij,ε } ⇔ {a ∈ A(N)ij and a started his current job less then ε

−1/2 time units ago};

{a ∈ A2,(N)ij,ε } ⇔ {a ∈ A(N)ij and a started his current job more then ε

−1/2 time units ago}.

For the group A1,(N)ij,ε of agents we have

E

µijt N

N(N)j∑a=1

I{a∈A1,(N)ij,ε

}Ra(t) = µij

tE

1N

N(N)j∑a=1

I{a∈A1,(N)ij,ε

}E(Ra(t) ∣∣∣ A1,(N)ij,ε ) .

(46)

By Assumption 2.2 we have E(Ra(t)

∣∣∣ A1,(N)ij,ε ) = E(Ra(t) ∣∣∣ a ∈ A1,(N)ij,ε ) ≤ l0 + l1 ε−1/2.Therefore

E

µijt N

N(N)j∑a=1

I{a∈A1,(N)ij,ε

}Ra(t) ≤ µij (nj + 1)

ε−1

(l0 + l1 ε−1/2

)≤ ε1/4 (47)

for ε small enough. Similarly, for the group A2,(N)ij,ε of agents we have

E

µijt N

N(N)j∑a=1

I{a∈A2,(N)ij,ε

}Ra(t) = µij

tE

1N

N(N)j∑a=1

I{a∈A2,(N)ij,ε

}E(Ra(t) ∣∣∣ A2,(N)ij,ε )(48)

≤ (l0ε+ l1)µij E

1N

N(N)j∑a=1

I{a∈A2,(N)ij,ε

} = (l0ε+ l1)µij 1

N

N(N)j∑a=1

P(a ∈ A2,(N)ij,ε

).

Observe that P(a ∈ A2,(N)i,ε

)≤ 1 −min{Gij(ε−1/2), Grij(ε−1/2)}. By Assumption 2.2 both

Gij and Grij have a second moment, i.e.,∫∞

0t (1−Gij(t)) dt < ∞. Hence 1 − Gij(t) =

o(1/t) as t → ∞. The same holds for Grij and we may assume ε small enough to satisfy1−min{Gij(ε−1/2), Grij(ε−1/2)} ≤ ε1/2. Thus, continuing (48)

E

µijt N

njN∑a=1

I{a∈A2,(N)ij,ε

}Ra(t) ≤ µij (nj + 1) (l0ε+ l1) ε1/2 ≤ ε1/4, (49)

for ε small enough. Now apply Markov inequality to (44), (47) and (49) to get

P(µijt N

V(N)ij (t) >

ε1/8

2

)≤ 2ε1/8

E

µijt N

njN∑a=1

I{a∈A(N)ij,ε

}Ra(t) ≤ 4ε1/4

ε1/8. (50)

14

As for the U (N)ij (0) from (40), we can unconditionally bound it by Markov inequality usingµrij as a mean of G

rij

P(µijt N

U(N)ij (0) >

ε1/8

2

)≤ 2ε1/8

µijtµrij(nj + 1) ≤ ε3/4, (51)

for ε small enough. Therefore

P((E

(N)2,K (t, ε)

)c)≤∑

(i,j)∈E

P( µijt N

∣∣∣V (N)ij (t)− U (N)ij (0)∣∣∣ > ε1/8)≤∑

(i,j)∈E

P(µijt N

V(N)ij (t) >

ε1/8

2

)+∑

(i,j)∈E

P(µijt N

U(N)ij (0) >

ε1/8

2

)

≤∑

(i,j)∈E

(4ε1/4

ε1/8+ ε3/4

)≤ ε1/16

where for the last line we used (50) and (51). This concludes the lemma. 2

Step 4: Finalizing proof of Theorem 2.5: We use (23), (24), (25), Lemma 3.2,(36), (35) and (42). For T > 0, ε > 0 and N ≥ n0 + (m + 1)(mini λi)−1, on the eventE

(N)A (T, ε) ∩ E

(N)R (T, ε) we have∣∣∣ 1

TA

(N)

i (T )− λi∣∣∣ < ε

T,∣∣∣ 1TR

(N)

i (T )− θiy(N)i (T )

∣∣∣ < εT, Y

(N)

i (T ) ≤c0T. (52)

Moreover, on the event E(N)A (T, ε) ∩ E(N)1,K (T, ε) ∩ E

(N)2,K (T, ε), for T > ε

−1,∣∣∣∣ 1T K(N)ij (T )− µijψ(N)ij (T )∣∣∣∣ ≤ ε+ ε1/8, (53)

Therefore, from (23), on E(N)A (T, ε) ∩ E(N)R (T, ε) ∩ E

(N)1,K (T, ε) ∩ E

(N)2,K (T, ε), for T > ε

−1,

‖e(N)(T )‖ =∑i∈I|e(N)i (T )| ≤ |I|(2ε+ c0 +m)ε+ |I||J |(ε+ ε

1/8) ≤ ε1/16, (54)

for ε small enough. As a result, for T > ε−1

P(‖e(N)(T )‖ ≤ ε1/16

)≥ P

(E

(N)A (T, ε) ∩ E

(N)R (T, ε) ∩ E

(N)1,K (T, ε) ∩ E

(N)2,K (T, ε)

)Lemmas 3.1, 3.3 and Lemma 3.4 below imply P

(E

(N)A (T, ε) ∩ E

(N)R (T, ε) ∩ E

(N)1,K (T, ε)

)→ 1

as N →∞. Therefore, by using Lemma 3.4, we have

lim infN→∞

P(‖e(N)(T )‖ ≤ ε1/16

)≥ 1− lim sup

N→∞P((E

(N)2,K (t, ε)

)c)≥ 1− ε1/16, (55)

with the last inequality following from (43). And the theorem follows for γ(ε) = ε1/16. 2

15

4 Solving Optimization Problem

We will proceed as follows. First we show that v ≥ V ∗, and Theorem 2.5 will be the keyfor the proof. After that we will prove that, under an additional assumption on the arrivalprocesses, the opposite inequality takes place, thus v = V ∗. By showing that, we alsointroduce a sequence of routing policies on which the limit V ∗ is achieved.

4.1 Lower bound

Theorem 4.1 Let Assumptions 2.1–2.4 hold. Then v ≥ V ∗.

Proof : Fix an arbitrary ε > 0, T > ε−1 and let nε be such that |N (N)j /N − nj | ≤ ε forall N ≥ nε (see Assumption 2.4). Let γ(·) be a function from Theorem 2.5. Recall thedefinition (18) and consider the event

{‖e(N)(T )‖ ≤ γ(ε)

}. On this event the quantities

(y(N)(T ), ψ(N)(T )) satisfy

∑j∈J µijψ

(N)i (T ) + θiy

(N)i (T ) = λ̃

(N)i

∑i∈I ψ

(N)i (T ) ≤ ñ

(N)j

ψ(N)ij (T ) ≥ 0, y

(N)i (T ) ≥ 0,

(56)

for some λ̃(N) and ñ(N), satisfying ‖λ̃(N) − λ‖ + ‖ñ(N) − n‖ ≤ γ(ε) for ε small enough.Observe that the solution to (20) is continuous in parameters λ and n, therefore, on theevent

{‖e(N)(T )‖ ≤ γ(ε)

}, one has∑

i

Ci

(y

(N)i (T )

)≥ V ∗ − ρ(ε) (57)

for some continuous function ρ : [0,∞) → [0,∞), vanishing at zero. By Jensen inequality,we have for each i ∈ I

1T

∫ T0

Ci

(Y

(N)

i (t))dt ≥ Ci

(1T

∫ T0

Y(N)

i (t)dt

)= Ci

(y

(N)i (T )

), (58)

where we used the definition (17) of y(N)i (T ). Hence, for any sequence of policies π(N) ∈

Π(N), N ≥ nε

C(N)(π(N), T ) ≥ E

(∑i∈I

Ci

(y

(N)i (T )

))≥ E

(I{‖e(N)(T )‖ ≤ γ(ε)

}∑i∈I

Ci

(y

(N)i (T )

))≥ P

(‖e(N)(T )‖ ≤ γ(ε)

)(V ∗ − ρ(ε)) ,

16

where, for the last inequality we used (57). Using Theorem 2.5 and (57), for T ≥ ε−1 wehave

lim infN→∞

C(N)(π(N), T ) ≥ (V ∗ − ρ(ε))(1− γ(ε)) ≥ V ∗ − ρ0(ε), (59)

for ρ0(ε) = ρ(ε) + V ∗γ(ε). We finalize the proof as follows. From the definition (16) we canassume that T ≥ ε−1 is such that

lim infN→∞

V (N)(T ) ≤ v + ε.

Using (15), fix also a sequence of policies π(N) ∈ Π(N), N ≥ 1, for which

lim infN→∞

C(N)(π(N), T ) ≤ v + 2ε. (60)

We conclude the theorem by combining (59), (60) and the fact that ε > 0 is arbitrary. 2

4.2 Upper bound

Our remaining goal is to show that V ∗ is asymptotically achievable by a certain sequenceof policies. Let y∗ and ψ∗ be optimal quantities from the optimization problem (20). Com-paring (19) with (56) and using Definition (17) dictates to look for a policy with ψ∗ij as anaverage amount of type i work in station j, and y∗i as an average queue length of type i. Thissuggest to consider a sequence of policies, which for each pair (i, j) ∈ E dedicate [ψ∗ijN ] + 1agents in station j for serving type i - once assigned, those [ψ∗ijN ] + 1 agents can not serveany other class, even if it results in system being not work conserving. Intuitively such poli-cies imply that, for each class i the corresponding queue may be viewed as an infinite serversystem with exponential service, and with the arrival flow of (λi −

∑j µijψ

∗ij)N + o(N).

The fluid version of such system (i.e., divided by N) is known to converge, as N → ∞, toa deterministic process, which on the long run converges to y∗i = (λi −

∑j µijψ

∗ij)θ−1i , thus

leading the asymptotic cost of operating such system to∑i Ci(y

∗i ) = V

∗. In what followswe will make the above arguments mathematically precise.

Notice that we can not guarantee that∑i

([ψ∗ijN ] + 1

)≤ N (N)j for all N ≥ 1, (see

Assumption 2.4(2)), thus a correction is needed. This is analogous to a discussion preceding(57). Fix arbitrary ε > 0 and take γ(ε) = ε1/16, like in the proof of Theorem 2.5. ByAssumption 2.4(2), for each j ∈ J

|N (N)j − njN | ≤γ(ε)2|J |

N, (61)

for all N large enough. Therefore, for such N , we would have

njN −γ(ε)|J |

N ≤ N (N)j −γ(ε)2|J |

N. (62)

17

Consider an ε–optimal pair y∗(ε) and ψ∗(ε), which solve (19)-(20) with nj replaced bynj − γ(ε)/2|J |, such that

∑i Ci(y

∗i (δ)) ≤ V ∗ + ρ(ε) for some function ρ, vanishing at zero.

The inequality ∑i

ψ∗ij(ε) ≤ nj −γ(ε)2|J |

(63)

implies ∑i

Nψ∗ij(ε) ≤ N(nj −N

γ(ε)2|J |

)≤ N (N)j −

γ(ε)2|J |

N, (64)

and ∑i

([ψ∗ij(ε)N ] + 1

)≤ N (N)j −

γ(ε)2|J |

N + |I| ≤ N (N)j −γ(ε)3|J |

N, j ∈ J ; (65)

for a given ε and all N large enough. And now we can guarantee that∑i

([ψ∗ijN ] + 1

)≤

N(N)j .

With a slight abuse of notation, throughout the rest of the paper we will write ψ∗ andy∗ for ψ∗(ε) and y∗(ε). For each i ∈ I define{

λij = µijψ∗ij for j ∈ Jλi0 = λi −

∑j∈J λij for j = 0.

, αij =λijλi

, j ∈ J ∪ {0}. (66)

Ans recall for y∗i =(λi −

∑j µijψ

∗ij

)/θi, i ∈ I, that∑

i

Ci (y∗i ) ≤ V ∗ + ρ(ε). (67)

Arrival processes. To proceed we will assume that arrival processes are renewal processes.For each i ∈ I let {Ûi(1), Ûi(2), ...} be a sequence of i.i.d. random variables with E(Ûi(1)) =1. Assume we are given a sequence of deterministic rates {λ(N)i , N ≥ 1}, and define

U(N)i (k) =

Ûi(k)

λ(N)i

, k ≥ 1, i ∈ I

With the convention∑0

1 define

A(N)i (t) = sup

{m ≥ 1 :

m∑k=1

U(N)i (k) ≤ t

}, i ∈ I, t ≥ 0. (68)

Therefore the first class–i customer arrives at U (N)i (1), and the time between the (m− 1)st

and mth arrival of class–i customers is U (N)i (k), k = 2, 3, ..

Assumption 4.2 1. There exist constants {y0i ≥ 0, i ∈ I} and {λi ≥ 0, i ∈ I}, so thatfor each i ∈ I 1N Y

(N)i (0)→ y0 a.s. and 1N λ

(N)i → λi as N →∞;

18

2. There exist constants α > 0, β1 ≥ 2 and β < β1 so that, for each i ∈ I: E(Ûi(1))β1 <∞ and Ci(1 + x) ≤ α(1 + x)β for x ∈ R+.

Note that under the above assumptions, for each i ∈ I, the scaled arrival process A(N)i → λitin DR+ [0,∞) a.s. for N →∞ (see chapter 5 of [3]). Therefore, the initial assumption 2.4(1),needed for the previous results, is satisfied.

We start with the description of the separation policy. First we decompose each arrivalprocess A(N)i ,i ∈ I, into |J | + 1 processes A

(N)ij , j ∈ 0 ∪ J , so that each A

(N)

ij → λijt inDR+ [0,∞) a.s. for N → ∞. This is done by a simple randomizing algorithm: with everyarrival of type i we generate an independent discrete random variable that attains the valuesin {0} ∪ J with probabilities {αij , j ∈ {0} ∪ J }. The resulting processes A(N)ij will remainrenewal processes with rate λijN + o(N) as N → ∞ and the convergence is a standardresult (see chapter 5 of [3]). Next, in each station j we separate the servers into |I| groups,with [ψ∗ijN ] + 1 servers in each group. For each (i, j) ∈ E the corresponding group willonly be serving customers from the arrival process A(N)ij . Therefore we have defined at most|I| × (|J |+ 1) independent (separated) queueing systems {Sij}. Systems Si0, i ∈ I will notserve customers but leave them in queue to abandon.

Initial conditions and initialization time. Based on the considerations above, wewould like to show that, in the given asymptotic regime, the systems {Sij , (i, j) ∈ E} donot accumulate any queues. This, however, is not immediately guaranteed, with the mainreason being non-zero initial conditions. We believe that the affirmative answer followsfrom the proof of Theorem 3.9 (2) in [6] (in particular, from lemmas 6.2 and 6.3) where theconsidered system does not allow its customers to abandon. However, we have decided notto exploit that fact due to obvious complication of exposition. Another way to deal with thatproblem is by introducing the so-called initialization time, i.e., the time, where the systemdoes not process any new customers and just waits for existing customers to complete theirservice and leave. If, for instance, the initial service distribution Grij had a compact support[0,Mij ] for each (i, j) ∈ E , then τ = max(i,j)∈EMij would be such time. If not, for a fixedε > 0 and function γ(·) from the proof of Theorem 2.5, set τ = τ(ε) = max(i,j)∈E τij(ε),where τij(ε) = inf{s , Grij(s) > 1− ε2γ(ε)}. Define

Υ(N)(ε) =

{∑i

Ψ(N)ij (τ(ε)) < Nγ(ε)6|J |

, ∀j ∈ J

}, (69)

Thus, if no customers were admitted to service in the interval [0, τ(ε)] we would have, byMarkov inequality,

P(

Υ(N)(ε))≥ 1− 6ε2|J |2 > 1− ε, (70)

for ε small enough. So that, on the event Υ(N)(ε), at time τ(ε), at each station j there willbe at least N (N)j −N

γ(ε)6|J | idle servers. Together with (65) this guarantees that we can define

systems {Sij , (i, j) ∈ E}, that would be empty at τ(ε) and each have [ψ∗ijN ] + 1 servers.We will need the following results:

19

Lemma 4.3 Consider an infinite server queueing system with initial condition Y (N)i (0),exponential (rate θi) service time and the arrival process being A

(N)i on the interval [0, τ(ε)]

and A(N)i0 afterwards. Let Z(N)i (t) be the number of customers in such system at time t.

Then Z(N)

i → zi, a.s., in D+[0,∞), where z satisfies

zi(t) =y0i + λit− θi∫ t

0

zi(s)ds, 0 ≤ t ≤ τ(ε), (71)

zi(t) =zi(τ(ε)) + λi0(t− τ(ε))− θi∫ tτ(ε)

zi(s)ds, t ≥ τ(ε) (72)

with the solution

zi(t) =y0i e−θit +

λiθi

(1− e−θit), 0 ≤ t ≤ τ(ε) (73)

zi(t) =zi(τ(ε))e−θi(t−τ(ε)) +λi0θi

(1− e−θi(t−τ(ε))), t ≥ τ(ε) (74)

Proof. Due to exponential nature of service of service, such convergence is a standardresult. Still we may cite Theorem 3.5 in [6], where in the equation ([6], (3.11)) one shouldtake f ≡ 1, 1−G(t) = e−θtt, dK(t) = λiI{t≤τ(ε)} + λi0I{t>τ(ε)} and

∫[0,M)

νt(dx) = z(t). 2

As an application of the above lemma, observe that zi(t) → λi0θi as t → ∞ and, as aresult, due to (67), (66), and since τ(ε) is finite and convex functions {Ci(·) , i ∈ I} arecontinuous, we have

limt→∞

1t

∫ ts=0

∑i

Ci(zi(s))ds =∑i

Ci(y∗i ). (75)

Lemma 4.4 Consider a queueing system, initially empty, with Gij as a service distribution,arrival process A(N)ij and [ψ

∗ijN ] + 1 servers. Let Q

(N)ij be the number of customers in such

system. Then, 1NQ(N)ij → qij a.s., in D+[0,∞), where qij converges monotonically to ψ∗ij,

in particular qij(t) ≤ ψ∗ij for all t ≥ 0.

Proof. The result is due to Theorem 3.9(1) from [6]. 2

The separation policy suggests that systems {Si0, i ∈ I} operate according to conditionsof Lemma 4.3. As for the systems {Sij , (i, j) ∈ E} -will satisfy the conditions of Lemma4.4 starting from time τ(ε). For each i ∈ I let X(N)i0 (t) be the total number of customersin system Si0 at time t. For and (i, j) ∈ E let X(N)ij (t) be the total number of customers insystem Sij at time τ(ε) + t.

For T > 0 introduce

Ω(N) = Ω(N)(ε, T ) ={

maxi∈I

supt∈[0,T ]

∣∣∣Y (N)i (t)− zi(t)∣∣∣ < ε}, (76)20

where, we recall, for each i ∈ I, Y (N) is the total queue length of the class i, (obtained byadding all queues in systems {Sij , j ∈ {0}∪J }). We are interested in the system behaviourafter the initialization, so assume T > τ(ε). Then

P(

Ω(N)(ε, T ))≥ P

(Ω(N)0 (ε, T )

), (77)

where

Ω(N)0 (ε, T ) =

{max

(i,j)∈Esup

t∈[0,T−τ(ε)]

(X

(N)

ij (t)− ψ∗ij)≤ 0

}∩

{maxi∈I

supt∈[0,T ]

∣∣∣X(N)i0 (t)− zi(t)∣∣∣ < ε}.

(78)

We have

P(

Ω(N)0 (ε, T ))≥ P

(Ω(N)0 (ε, T ) ∩Υ(N)(ε)

)= P

(Ω(N)0 (ε, T )

∣∣∣Υ(N)(ε))P(Υ(N)(ε)) . (79)But X(N)ij conditioned on Υ

(N)(ε) has the same distribution as Q(N)ij from Lemma 4.4, hence

P(

Ω(N)0 (ε, T )∣∣∣Υ(N)(ε)) (80)

=P

({max

(i,j)∈Esup

t∈[0,T−τ(ε)]

(Q

(N)

ij (t)− ψ∗ij)≤ 0

}∩

{maxi∈I

supt∈[0,T ]

∣∣∣Z(N)i (t)− zi(t)∣∣∣ < ε})

,

and, by Lemmae 4.3 and 4.4,

limN→∞

P(

Ω(N)0 (ε, T )∣∣∣Υ(N)(ε)) = 1. (81)

and, from (77)–(80) and (70)

lim infN→∞

P(

Ω(N)(ε, T ))≥ 1− ε. (82)

Hence

lim supN→∞

P(

Ω(N)(ε, T ))c

= 1− lim infN→∞

P(

Ω(N)(ε, T ))≤ ε. (83)

The sequence of separation policies from above will be denoted by π(N)sep (ε). We are nowready to complete the discussion.

Theorem 4.5 (Upper bound.) Let Assumptions 2.1–4.2 hold. There exists a continuousfunction κ : [0,∞)→ [0,∞), with κ(0) = 0, such that, for arbitrary ε > 0

lim supT→∞

lim supN→∞

C(N),π(N)sep (ε)(T ) ≤ V ∗ + κ(ε).

Combined with (15), (16) and Theorem 4.1, since ε is arbitrary, this implies v = V ∗.

21

Proof of theorem 4.5 Fix ε > 0 and take Tε so that for all T ≥ Tε (see (75))

1T

∫ T0

∑i

Ci

(zi(s)

)ds ≤

∑i

Ci(y∗i ) + ε ≤ V ∗ + ρ(ε) + ε,

where the last inequality is due to (67). Fix arbitrary T > max{Tε, τ(ε)} and recall thedefinition (76). We have

C(N),π(N)sep (ε) ≤ (84)

E1T

[IΩ(N)

∫ T0

∑i

Ci

(Y

(N)

i (t))dt]

+ E1T

[I(Ω(N))c

∫ T0

∑i

Ci

(Y

(N)

i (t))dt]

For the first term on the r.h.s. of (84), we have

E1T

[IΩ(N)

∫ T0

∑i

Ci

(Y

(N)

i (t))dt]

(85)

≤

(1T

∫ T0

∑i

Ci (zi(t)) dt

)P(

Ω(N))

+ E1T

(IΩ(N)

∑i

∫ T0

∣∣∣Ci (Y (N)i (t))− Ci (zi(t)) ∣∣∣dt)

≤ V ∗ + ρ(ε) + ε+ E 1T

(IΩ(N)

∑i

∫ T0

ri

∣∣∣Y (N)i (t)− zi(t)∣∣∣ dt)≤ V ∗ + ρ(ε) + ε(1 + ‖r‖),

(86)

and the last inequality is due to the fact that convex increasing function Ci on finite interval[0, c0] (see Lemma 3.2) has derivatives bounded by some ri

r.h.s. on (88)can be bounded by (see Assumption 2.4(2) and (22))

E

[I“E

(N)A ∩E

(N)R

”c ∑i

Ci

(m+A

(N)

i (T ))]

(89)

≤ E

[I“E

(N)A ∩E

(N)R

”c ∑i∈I

α(m+A

(N)

i (T ))β]

≤∑i∈I

E[(L1 + L2

(A

(N)

i (T ))β)

I“E

(N)A ∩E

(N)R

”c],

for some constants L1, L2, independent of N and T . From the definition (68), for each i ∈ I,we have A(N)(t) = Ai(λ(N)t), where

Ai(t) = sup

{m ≥ 1 :

m∑k=1

Ui(k) ≤ t

}, i ∈ I, t ≥ 0. (90)

From Assumption 4.2(1), we have λ(N)i

N ≤ C for all N large enough, hence for all t ≥ 0

A(N)

i (t) =1NA

(N)i (t) =

1NAi

(λ

(N)i

NNt

)≤ 1NAi(N Ct).

Using Assumption 4.2(2), we can now apply Theorem 4 from [8] to get

E[(A

(N)

i (T ))β1]

≤ L3(CT + 1)β1 ,

for some L3 and C, independent of T and N . For a given T < ∞, together with β1 > β,this makes L1 +L2

(A

(N)

i (T ))β

uniformly integrable. Finally, using Lemma 3.1, the secondterm on the r.h.s. of (84) vanishes, as N → ∞. Therefore, for any ε > 0 and T > Tε wehave

lim supN→∞

C(N),π(N)sep (T ) ≤ V ∗ + ρ(ε) + ε(1 + ‖r‖+ ‖c‖),

where ci := Ci(c0). Since ε is arbitrary, this concludes the Theorem. 2

4.3 A connection to cµ/θ rule

For a particular case of a single pool model (i.e., |J | = 1, |I| ≥ 2) and linear costs (Ci(x) =cix, for x ≥ 0) the solution of the linear program (19), (20) assumes the following nicerepresentation (see [1]). Assume that customer classes in I are labeled in such a way that

c1µ1θ1≥ c2µ2

θ2≥ ... ≥ cIµI

θI. (91)

23

Denote by (y∗, ψ∗) a solution to the (19), (20) and let V ∗ = c · y∗ A solution is given as

ψ∗ =(λ1µ1, ...,

λi∗−1µi∗−1

, 1−i∗−1∑j=1

λjµj, 0, ..., 0

),

(92)

y∗ =(

0, ..., 0,λi∗ − µi∗zi∗

θi∗,λi∗+1θi∗+1

, ....,λIθI

),

where i∗ = max{i ∈ [1, ..., I + 1] :

∑i−1j=1

λjµj< 1}.

As an application to queueing model, this suggests the classes with higher value of cµ/θare given priority over the classes with the lower value. See the full exposition in the paper[1].

Acknowledgement. The author wants to thank Kavita Ramanan for valuable discussionsand comments.

References

[1] Atar R. , Giat Ch. and Shimkin N. (2009). The cµ/θ rule for many-server queueswith abandonment. Preprint.

[2] Birkhoff, G., G.-C. Rota. 1989. Ordinary Differential Equations, 4th ed. JohnWiley, New York.

[3] Chen H. and Yao D.D. (2009). Fundamentals of Queueing Networks: Performance,Asymptotics, and Optimization. Springer, New York, 2001.

[4] Dai J. G. and He S. (2009). Customer abandonment in many-server queues. Preprint.

[5] Hall, W. J. and Wellner, J. A. (1981). Mean residual life.Proceedings of the Inter-national Symposium on Statistics and Related Topics (eds. M. Csörgö, D. A. Dawson,J. N. K. Rao and A. K. M. E. Saleh),169-184, North Holland, Amsterdam.

[6] Kaspi H. and Ramanan K. (2009). Law of large numbers limits for many-server queues. To appear in Annals of Applied Probability. Available at http ://www.imstat.org/aap/future papers.html

[7] Kang W. and Ramanan K. (2008). Fluid limits of many-server queues with reneging.Preprint.

[8] Krichagina E. V. and Taksar, M. I. (1992). Diffusion approximation for GI/G/1controlled queues. Queueing Systems Theory Appl. 12 333-367.

[9] Mandelbaum A. and Momcilovic P. (2009). Queues with Many Servers and Im-patient Customers. Preprint.

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[10] Parthasarathy K.R. Probability measures on metric spaces. Academic Press (1967).

[11] Reed J. (2009). The G/GI/N Queue in the Halfin-Whitt Regime I: Infinite ServerQueue System Equations To appear in Annals of Applied Probability.

[12] Reed J. (2009). The G/GI/N Queue in the Halfin-Whitt Regime II: Idle Time SystemEquations. Preprint.

[13] Rogers L.C.G. and Williams D. Diffusions, Markov Processes and Martingales 2nd

ed. Cambridge University Press (2000)

[14] Whitt, W. Stochastic-Process Limits: An Introduction to Stochastic-Process Limitsand Their Application to Queues. (2002) Springer-Verlag, New York.

[15] Whitt W. and Talreja R. Fluid Models for Overloaded Multi-Class Many-ServerQueueing Systems with FCFS Routing. Management Science, vol. 54, No. 8, 2008, pp.1513-1527.

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