On the Large Deviations Behaviour of Acyclic

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Consider a single class, acyclic network of G/G/1 queues. Customers arrive to the network in a number of independent streams and are treated uniformly by the ...
On the Large Deviations Behaviour of Acyclic Networks of G/G/1 Queues1 Dimitris Bertsimas2 [email protected]

Ioannis Ch. Paschalidis3 ;4 [email protected]

John N. Tsitsiklis5 [email protected]

December 1994 Revised: November 1997

LIDS Report: LIDS-P-2278

The Annals of Applied Probability, Vol. 8 (1998), No. 4, pages 1027{1069.

1 Research

supported by a Presidential Young Investigator award DDM-9158118 with matching funds from Draper Laboratory, by the ARO under grant DAAL-03-92-G-0115, and by the NSF under grant NCR-9706148. 2 Sloan School of Management, M.I.T., Cambridge MA 02139. 3 Department of Manufacturing Engineering, Boston University, 15 St. Mary's Str., Boston, MA 02215. 4 Please address all correspondence to the second author. 5 Department of EECS, M.I.T., Cambridge, MA 02139.

Abstract We consider a single class, acyclic network of G/G/1 queues. We impose some mild assumptions on the service and external arrival processes and we characterize the large deviations behaviour of all the processes resulting from various operations in the network. For the network model that we are considering, these operations are passing-through-asingle-server-queue (the process resulting from this operation being the departure process), superposition of independent processes, and deterministic splitting of a process to a number of processes. We also characterize the large deviations behaviour of the waiting time and the queue length observed by a typical customer in a single server queue. We prove that the assumptions imposed on the external arrival processes are preserved by these operations, and we show how to inductively apply these results to obtain the large deviations behaviour of the waiting time and the queue length in all the queues of the network. Our results indicate how these large deviations occur, by concretely characterizing the most likely path that leads to them.

Keywords: Communication Networks, Large Deviations, Queueing Networks. AMS Subject Classi cations (MSC 1995): 90B12, 60K30, 60F10, 60K25.

1 Introduction Consider a single class, acyclic network of G/G/1 queues. Customers arrive to the network in a number of independent streams and are treated uniformly by the network. Di erent streams may share a queue and the rst-come rst-serve (FCFS) policy is implemented. A constant fraction pij of customers departing a queue i, is routed to queue j and a fraction pi0 leaves the network. The aim of this paper is to derive large deviations results for the waiting time and the queue length observed by an arbitrary customer at di erent queues of the network. The main application area that motivates the study of such systems is the design and the operation of high speed, packet-switched communication networks. These networks will accommodate various types of trac, namely, digitized voice, encoded video, and data. The interesting problem arising is how to estimate and prevent congestion, which may cause long delays and packet losses. It is desirable to operate the network in a regime where packet loss probabilities are very small, e.g., in the order of 10?9 . Moreover, large delays should also have a correspondingly small probability. Thus, the need for understanding the large deviations behaviour of such a network arises. In this paper, we consider single class networks, which from the application point of view means that we are dealing only with one type of trac in the network. For this reason, the FCFS assumption can be made without loss of generality. The problem of estimating tail probabilities of rare events in a single queue has received extensive attention in the literature and has been approached by two main methodologies. The rst one is to use large deviations theory, as we do in this paper. This approach is used in [dVW92] to estimate the tail probability of the queue length in a G/G/1 queue. In that paper, a discrete time model was used in contrast to the continuous time model that we use in this paper. Similar results are obtained in [CW93]. The second approach is to use spectral decomposition techniques. This second approach is used in [EM93] to estimate the tail probability of the queue length in a queue with a deterministic server and Markov modulated arrival process. Results for the single queue case were rst obtained in [Hui88], [Kel91], [GH91] and later in [KWC93], [Whi93] and [GW94]. In all of these papers, the large deviations results obtained are used to derive appropriate admission control schemes for networks. The extension of these ideas to networks appears to be a rather challenging problem. Researchers have been able to obtain some bounds on the tail probabilities for delays and queue lengths in various networks models (see [Cha94, Cru91a, Cru91b, YS93]), but it is not clear whether these bounds are tight. Recently, large deviations results for two queues in tandem, with renewal arrivals and exponential servers, were reported in [GA94]. In [dVCW93], a very interesting approach is used to obtain results for networks with deterministic servers. The departure process from a single G/D/1 queue is characterized in the large deviations regime, using a discrete time model, in an attempt to treat the whole 1

network inductively. The main focus of [dVCW93] is to apply the large deviations results obtained to resource management for networks. A large deviations upper bound for the departure process appears also in [CHJS94]. It is important to point out that the departure process is a very dicult process to obtain exact results for (see for example [BN90]). However, we should note that it is not very clear to us how the large deviations result for the departure process in [dVCW93] can be applied inductively. The crux of the matter is that [dVCW93] uses a technical result from [DZ93a] in order to obtain the large deviations behaviour of the departure process. The latter result holds under certain technical assumptions on the arrival process. Since the departure process from a queue is the arrival process in another downstream queue in the network, one would need at this point to verify that the same technical assumptions hold for the departure process. This is not done in [dVCW93] and appears to be rather dicult. In the present paper, we consider a continuous time model and we extend the work in [dVCW93] to a network of G/G/1 queues. The objective is to obtain the large deviations behaviour of waiting times and queue lengths in all the nodes of the network. To this end, we initially seek to characterize the large deviations behaviour of the aggregate arrival process in each node. Our results are self-contained in the sense that we do not need the technical results of [DZ93a]. Instead, we impose certain assumptions on the external arrival processes and we characterize the large deviations behaviour of all the processes resulting from various operations in the network. For the network model that we are considering, these operations are passing-through-a-queue (the process resulting from this operation being the departure process), superposition of independent processes, and deterministic splitting of a process to a number of processes. We prove that the assumptions imposed on the external arrival processes are preserved by these operations, and thus we are able to apply these results inductively to obtain large deviations results for the aggregate arrival process in each node. As a by-product of our analysis we also obtain large deviations results for the internal trac in the network. For a single queue, in isolation, we characterize the large deviations behaviour of the waiting time incurred by a typical customer and, by using ideas from distributional laws (see [BN95, BM96]), the large deviations behaviour of the queue length observed by a typical customer. Finally, we compose the large deviations behaviour of the aggregate arrival process in each node of the network with the results for a single queue to obtain the large deviations behaviour of the waiting time and queue length in each node. Our approach provides particular insight on how these large deviations occur, by concretely characterizing the most likely path that leads to them. Characterizations of most likely paths were obtained for the single queue case in [Asm82], [Ana88] and [DZ93a]. After the submission of the present paper the work in [Cha95] and [CZ95] was brought to our attention. In [Cha95] the author independently obtained the large deviations behaviour for a network model of G/D/1 queues similar to ours, when the external arrival processes are bounded. In [CZ95] the authors obtain the large deviations behaviour of the departure 2

process of a G/G/1 queue, in isolation. It is interesting to note, that in order to obtain the large deviations behaviour of the superposition operation we prove a general result that connects the stationary distribution (i.e., as it is seen at a random time) and the Palm distribution (i.e., as it is seen by a typical customer) of a point process in the large deviations regime. This result could be of independent interest. Regarding the structure of the paper, we start in Section 2 by reviewing some results from the theory of large deviations that we use in the sequel. In Section 3 we present the network model that we are considering and establish our notation. In Section 4 we treat the single queue case. This section is comprised by two subsections. In Subsection 4.1 we review the existing result for the large deviations behaviour of the waiting time and we completely characterize the most likely path along which the waiting time takes large values. In Subsection 4.2, using an idea from distributional laws we obtain the tail probability of the queue length. In Section 5 we derive the large deviations behaviour of the departure process from a G/G/1 queue. Particular attention is given to the way that such a deviation occurs. In Subsection 5.1, some special cases are studied. Namely, we apply the result for the departure process of a G/G/1 queue to a G/D/1 queue and an M/M/1 queue. For the latter case, Burke's Theorem is veri ed in the large deviations regime. In Sections 6 and 7 we study the large deviations behaviour of the processes resulting from the following operations: superposition of independent processes, and deterministic splitting of a process to a number of processes, respectively. In Subsection 6.1 we prove a result that connects the Palm and the stationary distribution of a point process in the large deviations regime. This result is used in the rest of Section 6 to derive the large deviations behaviour of the superposition process. In Section 8, we treat, as an example, a network consisting of two queues in tandem. We characterize the way that the waiting time in the second queue reaches large values and we include some numerical results. Finally, in Section 9 we provide some concluding remarks and discuss some open problems.

2 Preliminaries In this section we review some basic results from Large Deviations Theory that will be used in the sequel. We rst state the Gartner-Ellis Theorem (see [Buc90] and [DZ93b]) which establishes a Large Deviations Principle (LDP) for random variables. It is a generalization of Cramer's theorem which applies to independent and identically distributed (iid) random variables. Consider a sequence fS1 ; S2 ; : : : g of random variables, with values in R and de ne 41 Sn n () = n log E[e ]:

3

(1)

For the applications that we have in mind, Sn is a partial sum process. Namely, Sn = i=1 Xi , where Xi ; i  1, are identically distributed, possibly dependent random variables.

Pn

Assumption A 1. The limit

1 4 Sn () = nlim !1 n () = nlim !1 n log E[e ]

(2)

exists for all , where 1 are allowed both as elements of the sequence n () and as limit points. 4 2. The origin is in the interior of the domain D = f j () < 1g of ().

3. () is di erentiable in the interior of D and the derivative tends to in nity as  approaches the boundary of D . 4. () is lower semicontinuous, i.e., lim inf n ! (n )  (), for all .

Theorem 2.1 (Gartner-Ellis) Under Assumption A, the following inequalities hold Upper Bound: For every closed set F   1 S n lim sup n log P n 2 F  ? ainf  (a): 2F

n!1

(3)

Lower Bound: For every open set G 1 log P  Sn 2 G  ? inf  (a); lim inf n!1 n a2G n

(4)

where 4 sup(a ? ()):  (a) = 

(5)

We say that fSn g satis es a LDP with good rate function  (). The term \good" refers to the fact that the level sets fa j  (a)  kg are compact for all k < 1, which is a consequence of Assumption A (see [DZ93b] for a proof). It is important to note that () and  () are convex duals (Legendre transforms of each other). Namely, along with (5), it also holds () = sup(a ?  (a)): a

(6)

The Gartner-Ellis Theorem intuitively asserts that for large enough n and for small  > 0, P[Sn 2 (na ? n; na + n)]  e?n(a) : 4

However, in this paper, we are mostly estimating tail probabilities of the form P[Sn  na] or P[Sn  na]. We therefore de ne large deviations rate functions associated with such tail probabilities. P Consider the case where Sn = ni=1 Xi , the random variables Xi ; i  1, being identically distributed, and let m = E[X1 ]. It is easily shown (see [DZ93b]) that  (m) = 0. Let us now de ne (

 (a) if a > m 0 if a  m

(7)

(

 (a) if a < m 0 if a  m:

(8)

4 + (a) =

and 4 ? (a) =

Notice that + (a) is non-decreasing and ? (a) is a non-increasing function of a, respectively. The convex duals of these functions are 4 + () =

(

() if   0 +1 if  < 0

(9)

(

() if   0 +1 if  > 0

(10)

and 4 ? () =

respectively. In particular, ? (a) = sup (a ? ? ()) and + (a) = sup (a ? + ()). Using the Gartner-Ellis Theorem it can be shown that for all 1 ; 2 > 0 there exists n0 such that for all n  n0

e?n(?(a)+2 )  P[Sn  na]  e?n(? (a)?1 );

(11)

e?n(+(a)+2 )  P[Sn  na]  e?n(+ (a)?1 ) :

(12)

and More speci cally, the lower bound in (11) can be obtained by noting that P[Sn  na]  P[Sn < na], and using the lower bound of the Gartner-Ellis Theorem for the open set (?1; a). The upper bound can be obtained by an argument similar to the one we use in the proof of Lemma 4.2. A similar argument can be used for (12).

5

3 The Network Model In this section, we formally de ne the network model of which we will derive the large deviations behaviour. Moreover, we establish the notation that we will be using and state a set of assumptions on the arrival and service processes. Consider a directed acyclic graph (dag) with J nodes. For reasons that will become soon apparent, we assume that any two directed paths do not meet in more than one node. Each node of the graph is equipped with an in nite bu er and a single server. Customers enter the network in a number of independent streams A1 ; A2 ; : : : ; AJ . In particular, Ai is the stream of customers that enter the network at node i. Customers are treated uniformly by the network, i.e., the network is assumed to be single class. Let Z denote the set of integers. By Aji ; i 2 Z, we denote the interarrival time of the ith customer in the j th stream (the interval between the arrival epochs of the (i ? 1)st and the ith customer). By Bij ; i 2 Z, we denote the service time of the ith customer in the j th node. We assume that for each arriving stream j the process fAji ; i 2 Zg, is stationary, and Aji ; i 2 Z, are possibly dependent random variables. Moreover, for each node j , the service times Bij ; i 2 Z, are iid random variables. We also assume that interarrival and service times at a speci c node are mutually independent and that service times at di erent nodes are independent. Independent streams may share a queue and the FCFS policy is implemented. A fraction pij1 ; pij2 ; : : : of customers departing node i, which is connected to nodes j1 ; j2 ; : : : , are routed to these nodes, respectively, and a fraction pi0 leaves the network. The exact way that the routing is performed is not of importance in the large deviations regime. Roughly, out of every 1=pij customers leaving node i the routing mechanism sends one to node j . Figure 1 depicts an example of the class of networks considered. Such a network is intended to model packet-switched communication networks. We denote by W 1 ; W 2 ; : : : ; W J and L1 ; L2 ; : : : ; LJ the steady-state waiting times and queue lengths, incurred by a typical customer at nodes 1; 2; : : : ; J of the network, respectively. For each node j , Wnj (resp. Ljn ) denotes the waiting time incurred (resp. queue length observed) by the nth customer. We assume that the process f(Wnj ; Ljn ); n 2 Z; j = 1; : : : ; J g is stationary. (The existence and stationarity of this process contains an implicit stability assumption.) In this paper, we derive large deviations results for the steady-state waiting times 1 W ; W 2 ; : : : ; W J , and the corresponding queue lengths L1 ; L2 ; : : : ; LJ , incurred at nodes 1; 2; : : : ; J of the network, respectively (as these random variables are seen by a typical customer). Our strategy is rst to obtain large deviations results for the steady-state waiting time and the corresponding queue length in a single G/G/1 queue. Then it suces to derive a LDP for the partial sum of the aggregate arrival process in each queue of the network and apply the result for the single queue case. It is important to note that by the de nition of the network all the streams sharing the same queue are independent. Therefore, from the 6

A1 1

p21

7

p24

A2

2

4

p25 5

8

p35 A3

3

p67

p36

6

1 ? p67

A8

Figure 1: A network example. model description, it is apparent that it suces to obtain LDP's for the processes resulting from the following operations 1. Passing-through-a-queue (the process resulting from this operation being the departure process). 2. Superposition of independent streams. 3. Deterministic splitting of a stream to a number of streams. Let fAi ; i 2 Zg be an arbitrary external arrival process and fBi ; i 2 Zg be an arbitrary 4 Pj X = service process. Hereafter, we will be using the notation Si;j k=i Xk ; i  j for the 4 X = 0; i > j . partial sums of the random sequence fXi ; i 2 Zg along with the convention Si;j

Assumption B

1. The sequence of partial sums fS1A;n ; n  1g satis es

1 ? A nlim !1 n log P[S1;n  na] = ?A (a); 7

(13)

where 8
0

:

and 

The limit in the upper branch of (14) exists for all , where 1 are allowed both as elements of the sequence and as limit points. We will say that fS1A;n ; n  1g satis es a one sided LDP. Moreover, we assume that ? A (a) is a strictly convex function of a in the intersection of the interior of its domain and the interval (?1; E[A1 ]), and that A1 has moments of all orders, i.e., E[Ap1 ] < 1 for all p  0. 2. The sequence of partial sums fS1B;n ; n  1g satis es the requirements of the GartnerEllis theorem with limiting log-moment generating function

1 4 S1B;n ]; B () = nlim !1 n log E[e

(16)

and large deviations rate function 4 B (a) = sup(a ? B ()): 

(17)

Moreover, we assume that B (a) is a strictly convex function of a in the interior of its domain and that B1 has moments of all orders, i.e., E[B1p ] < 1 for all p  0.

Assumption C

1. For every 1 ; 2 ; a > 0, there exists MA such that for all n  MA ?

e?n(A (a)+2 )  P[S1A;i ? ia  1 n; i = 1; : : : ; n]:

(18)

2. For every 1 ; 2 ; a > 0, there exists MB such that for all n  MB ?

B ? (j ? i + 1)a  1 n; 1  i  j  n]; e?n(B (a)+2 )  P[Si;j

(19)

and +

B ? (j ? i + 1)a  ?1 n; 1  i  j  n]: e?n(B (a)+2 )  P[Si;j

(20)

We consider external arrival and service processes that satisfy Assumptions B and C. We will show that these assumptions are satis ed by the processes resulting from the three operations mentioned above. In this way, our approach provides a calculus of acyclic net8

works since we will be able to determine the large deviations behaviour of each individual queue inductively. Assumption B provides a LDP for the arrival and service processes. Based on these LDP's we will derive LDP's for all the processes of interest in the network. Note that only the tail probability of the external arrival processes corresponding to \many arrivals" is characterized by Assumption B. We will prove that in order to estimate probabilities of large waiting times and long delays, as we do in this paper, only such a tail probability of the aggregate arrival process in each queue of the network is needed. The strict convexity assumption on the large deviations rate functions of interest is needed to avoid some technical issues that have to do with the di erentiability of the corresponding limiting log-moment generating functions. Assumption C is needed in order to derive a LDP for the departure process of a G/G/1 queue. It intuitively asserts that besides the LDP for the partial sum random variable S1;n , we also have a LDP for the partial sum process fS1;i ; i = 1; : : : ; ng for the arrivals and fSi;j ; 1  i  j  ng for the service times. In other words, (18) and (19) guarantee that, with high probability, the partial sum process follows a path that never overshoots the straight line of slope a, in order to reach an improbable level S1;n  na. A similar interpretation can be given to (20). Mild mixing conditions on the arrival and service processes suce to guarantee Assumption C. A thorough treatment is given in [DZ93a]. In the Appendix we provide some conditions under which Assumption C is satis ed based on the results of [DZ93a]. In [Cha95] a uniform bounding condition is given under which the above assumption is true. We should note here that we do not need the full power of the sample path large deviations results in [DZ93a] and [Cha95] to establish our results. We only need Assumptions B and C which, as we will show, are preserved by the internal trac in the network. Assumptions B and C are satis ed by processes that are used to model external arrival and services in communications networks, such as renewal processes, stationary processes with mild mixing conditions, as well as Markov-modulated processes with some uniformity assumptions on the stationary distribution (see [DZ93a, Section 4]).

4 Large Deviations of a G/G/1 Queue In this section, we establish a LDP for the Palm distributions of the steady-state waiting time and queue length (i.e., as these random variables are seen by a typical customer), in a G/G/1 queue with stationary arrivals and service times. The setting is the same as in Section 3. We denote by fAi ; i 2 Zg the stationary aggregate arrival process to the queue and we assume that it satis es Assumption B.1. We also denote by fBi ; i 2 Zg the stationary service process and we assume that it satis es Assumption B.2. For this section, the independence assumption for the service times can 9

be relaxed. For stability purposes, we further assume E[A] > E[B ], where A (resp. B ) denotes a typical interarrival (resp. service) time.

4.1 Large Deviations of the Waiting Time Let us rst characterize the steady-state waiting time, W , incurred by a typical customer. By Wn we denote the waiting time of the nth customer. The condition E[A] > E[B ] is necessary 1 for the existence and the uniqueness of a stationary process (see [Wal88]). From the Lindley equation, the waiting time of the 0th customer, at steady-state, is given by 4 max[W + B ? A ; 0] = max[S B W0 = [W?1 + B?1 ? A0 ]+ = ? S?Ai;0; 0]: ?1 ?1 0 i0 ?i?1;?1

(21)

The intuitive meaning of this relation is the following: For a particular sample path, if i is the optimum i, then the customer with label ?i ? 1 is the one who initializes the busy period in which the 0th customer is served. The next theorem establishes a LDP for W0 . This result is not new. The proof is almost identical with the proof in [dVW93, Thm 3.1], where a discrete time model is used, and is therefore omitted. A similar argument is also given in [Cha94]. An upper bound on the tail probability of the steady-state waiting time, for renewal arrival and service processes, was rst obtained by Kingman [Kin70].

Theorem 4.1 The tail of the Palm distribution of the steady-state waiting time, W , in a

FCFS G/G/1 queue with arrivals and service times satisfying Assumption B is characterized by

1 log P[W  U ] =  ; lim U !1 U

(22)

?A () + B (?) = 0:

(23)

where  < 0 is the smallest root of the equation

Remarks : Intuitively, Theorem 4.1 asserts that for large enough U , we can state P[W  U ]  e U ;

where  < 0 is such that

?A ( ) + B (?) = 0:

(24)

Note that  exists as an extended real number since E[A] > E[B ] and the functions ?A (); B () are convex 2 . Figure 2 depicts the function ?A ()+B (?) and the root . If 1 for suciency ergodicity is also needed. 2 This is proven under the conditions of Assumption B in [DZ93b, Lemma 2.3.9]

10

?A ()+B (?) < 0 for all  < 0 we use the convention  = 1. In Figure 2 we make use of the di erentiability of ?A (); B (), which is guaranteed by the strict convexity assumption  that we have imposed on ? A () and B () (see Assumption B). More speci cally, convex duality arguments (see [Roc70, Thm. 26.3]) guarantee that the convex dual of a strictly convex function is di erentiable in the interior of its domain. Regarding ?A () we are only interested in its di erentiability for  < 0, which can be established using the strict ? convexity of ? A (a) for a < E[A1 ]. (As we have de ned A () it has only a left derivative at zero, which is positive and equal to E[A] ? E[B ].

?A () + B (?)



 B (?)

slope E[A] ? E[B ]

?A ()

Figure 2: The root of ?A () + B (?) = 0. It is instructive to characterize the most likely \path" along which the large deviation of the waiting time occurs. Such a characterization can also provide an alternative proof of Thm. 4.1. Let a > 0 and x1 ; x2 2 R+ , such that x2 ? x1 = a. Using Eq. (21), we have

P[W0  (i + 1)a]  P[S?Bi?1;?1 ? S?Ai;0  (i + 1)a]  P[S?Ai;0  (i + 1)x1 ]P[S?Bi?1;?1  (i + 1)x2 ] + A (x1 )+B (x2 )+] ;  e?(i+1)[?

(25)

where the last inequality makes use of Assumption B and holds for any  > 0 and for large i. Setting U = (i + 1)a, we obtain 

 1 ?  + P[W0  U ]  exp ?U a> inf0 a x2 ?inf [ (x ) + B (x2 )] ? U : x1 =a A 1

(26)

Let a > 0 be a solution to the above optimization problem. Thus, for large U , and by 11

taking  ! 0 in (26), we obtain (

)

+ ? P[W0  U ]  exp ?U inf x2?x1=a [Aa(x1) + B (x2 )] :

(27)

The tightness of this bound can be proven by obtaining a matching (i.e., with the same exponent) upper bound; the proof is omitted. Let i be de ned by the equation i +1 = U=a . Let also x1 and x2 solve the optimization problem in (27). Consider a scenario where customers ?i ; : : : ; ?1; 0 arrive at an empirical arrival rate of x11 and customers (?i ? 1); : : : ; ?1 are served with an empirical service rate of x12 . Such a scenario, which is depicted in Figure 3, has probability comparable to the right hand side of (27) and is therefore a most likely way for the large deviation of the waiting time to occur. cust. (?i ? 1) arrives

cust. 0 arrives

S?Ai ;0 ; rate x11

cust. (?1) arrives

W0

S?Bi ?1;?1 ; rate x12

cust. (?1) departs

Figure 3: The optimal path for large deviations in the waiting time.

4.2 Large Deviations of the Queue Length In this subsection, we present a LDP for the steady-state queue length in a G/G/1 queue, as seen by a typical customer (Palm distribution). To accomplish that we use the main argument used in deriving distributional laws; that is, a probabilistic relation between the waiting time and the queue length. A detailed discussion of distributional laws and their applications can be found in [BN95, BM96]. It is important to note that distributional laws have been proven there only for renewal arrival and service processes. However in the large deviations setting, we are able to relax the renewal assumption and state a result that holds even for correlated arrival and service processes. Let us now characterize the steady-state queue length L seen by a typical customer (not 12

including herself) upon arrival (this is sometimes denoted by L? in the literature). The goal is to estimate P[L  n]. Let us denote by Ln the queue length observed by the nth customer. As in Section 3, we assume that the process f(Ln ; Wn ); n 2 Zg is stationary. The main idea, in order to establish a relation between the waiting time and the queue length, is to look backwards in time from the arrival epoch of the nth customer. Figure 4 depicts the situation. We denote with T0 ; T1 ; : : : the arrival epochs of customers 0; 1; : : : , S1A;n T0

T1 T2

Tn?1

Tn

t

Figure 4: The system at time Tn. respectively. Recall that Wn and Bn denote the waiting and the service time of the nth customer, respectively. The main observation is the following: In order for the queue length right before Tn to be at least n, the 0th customer should be in the system at that time. Namely,

P[Ln  n] = P[W0 + B0  S1A;n]

(28)

? S?Ai;n; ?S1A;n]  0] = P[Ln  n] = P[max [S B i0 ?i?1;0 [S B = P[max ? S?Ai;n]  0]: i?1 ?i?1;0

(29)

and by using (21) we obtain

The next theorem establishes a LDP for Ln . We will need a technical lemma which we prove next. (This lemma is also used in the next section.)

Lemma 4.2 Under Assumption B, and for  < 0, satisfying ?A () + B (?) < 0, it holds A B lim sup n1 log E[e? maxi?1 [S?i?1;0 ?S?i;n ]]  ?A ():

n!1

Proof : We have E[e? maxi?1[S?Bi?1;0 ?S?Ai;n]] 

13

X

i?1

E[e?S?Bi?1;0 ]E[eS?Ai;n ]:

(30)

From (16) it can be seen that for any  > 0 there exists j > 0 such that for all i > j it holds

E[e?S?Bi?1;0 ]  e(i+2)(B (?)+):

(31)

Also from (14), we have that for  < 0 and for any  > 0 there exists N such that for all n > N and all i  ?1

E[eS?Ai;n ]  e(n+i+1)(?A ()+) :

(32)

Fix now some  < 0 satisfying ?A () + B (?) < 0 and some  > 0 such that ?A () + B (?) + 2 < 0. Note that the existence of such a  is guaranteed by the condition E[A] > E[B ] (see Figure 2). We then have that for all n > N

E[e? maxi?1 [S?Bi?1;0 ?S?Ai;n]] 

j X i=?1

E[e?S?Bi?1;0 ]E[eS?Ai;n ] +

en(?A ()+)

j X

X

i>j

E[e?S?Bi?1;0 ]E[eS?Ai;n ]

E[e?S?Bi?1;0 ]e(i+1)(?A ()+)

i=?1 X ? ? ? n ( + e A ()+) e2B (?)+A ()+3 ei(B (?)+A ()+2) i>j ? K (; j; )en(A ()+); (33)

where K (; j; ) is some constant depending on , j and  but not on n. To see that, notice that in the last inequality above we use the fact that the rst sum is nite, and the in nite geometric series in the second sum converges to a constant independent of n. From Eq. (33) we obtain B A lim sup n1 log E[e? maxi?1 [S?i?1;0 ?S?i;n ]]  ?A () + :

n!1

(34)

Since this is true for all small enough  > 0, the result follows.

Theorem 4.3 The tail of the Palm distribution of the steady-state queue length, L, in a FCFS G/G/1 queue with arrivals and service times satisfying Assumption B is characterized by

1

?  );

nlim !1 n log P[L  n] = A (

(35)

where  < 0 is the smallest root of the equation

?A () + B (?) = 0: 14

(36)

Proof : Due to stationarity, it suces to characterize the tail distribution of Ln. For an upper bound de ne

4 Gn = max [S B ? S?Ai;n]: i?1 ?i?1;0

(37)

Using the Markov inequality and Eq. (29), we obtain

P[Ln  n] = P[Gn  0]  E[e?Gn ]; for  < 0. Taking the limit as n ! 1, using Lemma 4.2, and optimizing over  to get the best bound we obtain lim sup 1 log P[L  n]  (38) inf [? ()] = ? ( ); n!1

n

n

fj ?A ()+B (?) ( + 0 )f ((a + b)=( + 0 )):

(86)

Consider now the de nition of ? A1;2 (a), which we rewrite as [? (a=) + (1 ? )? ? A2 (a=(1 ? ))]: A1;2 (a) = 2inf [0;1] A1 Let  (respectively, 0 ) be the minimizer in the optimization problem corresponding to ? ? A1;2 (a) (respectively, A1;2 (b) ). We then have 1 ? 1 ? 2 A1;2 (a)+ 2 A1;2 (b) = 1 ?  ? 0 ? 0 1 ? 0 ? 0 = 2 ? A1 (a=) + 2 A2 (a=(1 ? )) + 2 A1 (b= ) + 2 A2 (b=(1 ?  )) 0 2 ?  ? 0 ? ((a + b)=(2 ?  ? 0 )) 0 (( a + b ) = (  +  )) +   +2  ? A1 A2 2     ( a + b ) = 2 ( a + b ) = 2 ? ? = A1 + (1 ?  )A2 1 ?        ( a + b ) = 2 ( a + b ) = 2 ? ? + (1 ?  )A2 1 ?    A1   2inf  [0;1] =? A1;2 ((a + b)=2): The rst inequality above is due to (86) and is strict unless a= = b=0 and a=(1 ? ) = b=(1 ? 0 ), which implies that  = 0 and a = b. Thus, as long as a 6= b, we have established the strict convexity of ? A1;2 (). The next Theorem establishes that the process resulting from the superposition satis es Assumption C1.

Theorem 6.3 Assume that the m independent processes A1i ; : : : ; Ami i 2 Z satisfy Assump30

tion C1. The aggregate process resulting from their superposition also satis es Assumption C1.

Proof : It suces to prove the result for m = 2 since by using induction we can prove it

for any m. We need to prove that for every 1 ; 2 ; a > 0, there exists MS such that for all

n  MS

?

e?n(A1;2 (a)+2 )  P[S1A;j1;2 ? ja  1n; j = 1; : : : ; n]:

(87)

Following the steps of the proof of Theorem 6.1 we consider the scenario that a fraction  of customers of the aggregate process originates from the A1 process. Again, H1 denotes the event that customer 1 of the aggregate process originates from the A1 process. We have

P[S1A;j1;2 ? ja  1n; j = 1; : : : ; n j H1]   sup P[S1A;j1 ? ja  1n; j = 1; : : : ; n]  2[0;1]

PR [S1A;j2(1?) ? ja  1n; j = 1; : : : ; n]:

(88)

Using Assumption C1 for the A1 stream we obtain for large enough n ?

P[S1A;j1 ? ja  1n; j = 1; : : : ; n]  e?n(A1 (a=)+0 ) :

(89)

In Subsection 6.1 (Lemma 6.6) it is shown that for large enough n ?

PR [S1A;j2(1?) ? ja  1n; j = 1; : : : ; n]  e?n(1?)(A2 (a=(1?))+00) :

(90)

To obtain (87) it suces to choose appropriate 0 and 00 such that for large enough n and given 2 ? ? ? 00

0

e?n inf 2[0;1] [(A1 (a=)+ )+(1?)(A2 (a=(1?))+ )]  e?n(A1;2 (a)+2 ) :

6.1 Connection between Palm and stationary distributions in the large deviations regime In this subsection we show that the stationary and the Palm distribution of the same point process have the same large deviations behaviour. Consider a stationary arrival process satisfying Assumptions B with interarrivals Ai ; i 2 Z. We have (91) lim 1 log P[S A  na] = ?? (a): n!1 n

1;n

31

A

As explained in the proof of Thm. 6.1, P[] denotes the probability distribution seen by a random customer (customer 1 in the case of Eq. (91)). Consider now a random time (say t = 0) and assume that customer 0 is the rst customer to arrive after t = 0. Let U; V denote the duration and the age, respectively, of A0 . The situation is depicted in Figure 10. By PR [] we denote the probability distribution seen at the random time t = 0 and we t=0 V t S1A;n

U

Figure 10: The arrival process seen at a random time. are interested in obtaining a LDP for S1A;n under PR []. The next theorem establishes the result. Moreover, we are also interested in obtaining a LDP result for the partial sum process fS1A;j ; j = 1; : : : ; ng under PR [] when Assumption C1 is satis ed. The latter result is obtained in Lemma 6.6.

Theorem 6.4 Under Assumption B we have 1 ? A nlim !1 n log PR [S1;n  na] = ?A (a):

(92)

Proof : Let ER[] denote the expectation with respect to PR []. We use a standard procedure to relate ER [] to E[] (see [Wal88]). Consider an arbitrary function f () of S1A;n . It can be shown ([Wal88, ch. 7]) that

ER [f (S1A;n) j V = v; U = u] = E[f (S1A;n) j A0 = u]: Thus following the steps in [Wal88, ch. 7],

ER [f (S1A;n)]

Z 1 Z u 1 = E[A ] E[f (S1A;n) j A0 = u] dv dFA0 (u) 1 u=0 v=0

= E[

Z

A0

f (S1A;n) dv] v=0 = E[A0 f (S1A;n)];

(93)

where we have assumed without loss of generality that E[A1 ] = 1, and we have used the notation FA0 () for the distribution function of A0 . A To obtain an upper bound on ER [eS1;n ] we set f () = e and use Holder's inequality. 32

Namely, (p + q = 1)

ER[eS1A;n ] = E[A0eS1A;n ] ? A A  = E[(A10=p )p e(=q)S1;n q ]  E[A10=p ]p E[e(=q)S1;n ]q ; (94)

which for   0 implies 1=p A lim sup n1 log ER [eS1;n ]  lim sup p log En[A0 ] + q?A (=q) = q?A (=q);

n!1

n!1

(95)

since the rst term of the right hand side vanishes. Taking the now limit as q ! 1 in the above equation we obtain for   0 A lim sup n1 log ER [eS1;n ]  ?A ():

n!1

(96)

Therefore using Eq. (96) and the Markov inequality we obtain lim sup n1 log PR [S1A;n  na]  ?? A (a): n!1

(97)

To obtain now a lower bound on PR [S1A;n  na] set f (S1A;n) = 1fS1A;n  nag in Eq. (93), where 1fg denotes the indicator function. We have

PR [S1A;n  na]

=

1

Z

0

uP[S1A;n  na j A0 = u] dFA0 (u)

Z 1 1  n2 2 P[S1A;n  na j A0 = u] dFA0 (u) 1=n 1 P[S A  na; A  1 ]: =

n2

1;n

0

n2

(98)

We need the following lemma the proof of which is deferred until the end of the current proof.

Lemma 6.5 Under Assumption B and for every positive  and a, there exists Na; such that for every n  Na; it holds A (a)+) : P[S1A;n  na; A0  n12 ]  e?n(?

We now use Lemma 6.5 in Eq. (98) and take  ! 0 to obtain lim inf n1 log PR [S1A;n  na]  ?? A (a): n!1

33

(99)

0 0 Proof of Lemma 6.5: Eq. (91) implies that for every positive 0 and a there exists Na; 0 0 it holds such that for every n  Na; ?

?

e?n(A (a)+0 )  P[S1A;n  na]  e?n(A (a)?0 ):

(100)

Fix now a; 0 > 0, and let  = 0 . We have

P[S1A;n  na; A0  n12 ] = (by stationarity) (union bound)

n X 1 = n P[SiA+1;i+n  na; Ai  n12 ] i=1

 n1 P[9i 2 [1; n] s.t. SiA+1;i+n  na; Ai  n12 ]  1 P[S A  na; 9i 2 [1; n] s.t. A  1 ] i

1;n(1+)

n

n X 1 A  n P[S1;n(1+)  na; Ai  nn2 ]

(P[A \ B ]  P[A] ? P[B C ])

n2

i=1

 n1 P[S1A;n(1+)  na] ? n1 P[S1A;n  n ] 0 ? a  n1 e?n(1+)(A ( 1+ ) +  ) ? n1 P[S1A;n  n ];(101)

where the last inequality holds for all n  N 0 a ; 0 . Note that we have used the notation 1+ B C to denote the complement of B . We next show that for n ! 1 (keeping a; ; 0 xed) we can neglect the second term in the right hand side of (101). To see that note that for all positive there exists N ;0 such that for all n  N ;0 it holds 0 A ( )? ) : P[S1A;n  n ]  P[S1A;n  n ]  e?n(?

(102)

By taking ,  and 0 small enough and n  N ;0 we can achieve ? a 0 0 ? A ( ) ?  > (1 + )(A ( 1+ ) +  ):

(103)

? a ? Here we are using the fact that for suciently small , ? A ( ) > A ( 1+ ) since A ( ) is monotonically increasing as # 0. Observe now that the value of which satis es Eq. (103) is a function of a,  and 0 . Therefore, using Eq. (102), there exists Na;;0 such that for all n  Na;;0 it holds 0

? n1 P[S1;n  n ]  ? 2n1 e?n(1+)(A ( 1+ ) +  ) : ? a

(104)

Combining Eqs. (104) and (101) we conclude that there exists N^a;;0 such that for all

34

n  N^a;;0 it holds ? a

0

P[S1A;n  na; A0  n12 ]  2n1 e?n(1+)(A ( 1+ ) +  ):

(105)

We now choose 0 such that (recall  = 0 ) a 0 1 e?n(1+0 )(? A ( 1+0 ) +  )  e?n(? A (a)+) ;

2n0

for all n  Na; . This can be done due to the lower semicontinuity of ? A () (see the argument in proof of Lemma 2.2.5 in [DZ93b]).

Lemma 6.6 Under Assumptions B and C1 we have that for every 1; 2 ; a > 0 there exists Na;1 ;2 such that for all n  Na;1 ;2 it holds A (a)+2 ) : PR [S1A;j  ja + 1n; j = 1; : : : ; n]  e?n(?

(106)

Proof : Following the proof of the lower bound in Thm. 6.4 (using the argument used to derive Eq. (98) but applied to the sample path S1A;j  ja + 1 n; j = 1; : : : ; n) we have PR [S1A;j  ja + 1 n; j = 1; : : : ; n]  n12 P[S1A;j  ja + 1 n; j = 1; : : : ; n; A0  n12 ]:

(107)

Now, as in the proof of Lemma 6.5, xing a; 1 ; 2 > 0 we obtain

P[S1A;j  ja + 1 n; j = 1; : : : ; n; A0  n12 ] =

n 1 X A 1 = n P[S1+ k;j +k  ja + 1 n; j = 1; : : : ; n; Ak  n2 ] k=1 1 A 1  n P[9k 2 [1; n] s.t. S1+ k;j +k  ja + 1 n; j = 1; : : : ; n; Ak  n2 ] A 1  n1 P[8k 2 [1; n] S1+ k;j +k  ja + 1 n; j = 1; : : : ; n; 9k 2 [1; n] s.t. Ak  n2 ] 1 A A   n1 P[8k 2 [1; n] S1+ k;j +k  ja + 1 n; j = 1; : : : ; n] ? n P[S1;n  n ]: (108)

Now notice that A P[8k 2 [1; n] S1+ k;j +k  ja + 1 n; j = 1; : : : ; n] = P[8k 2 [1; n] S1A;j+k ? S1A;k  (j + k)a ? ka + 1n; j = 1; : : : ; n]  P[S1A;j+k  (j + k)a + 12n ; 8k 2 [1; n]; j = 1; : : : ; n; S1A;k  ka ? 12n ; 8k 2 [1; n]] = 0 A (a)+ ) ; (109) P[S1A;j+k  (j + k)a + 12n ; 8k 2 [1; n]; j = 1; : : : ; n]  e?n(1+)(?

35

where the last equality is obtained by choosing suciently small  such that na ? 12n < 0 which implies that P[S1A;k  ka ? 12n ; 8k 2 [1; n]] = 1. The last inequality holds, due to 0 0 . Now, as in Lemma 6.5 it can be shown that there exists Assumption C1, for all n  Na; 1 ; 00 0 such that for all n  N 00 0 it holds Na;; a;; 0 A (a)+ ) : ? n1 P[S1;n  n ]  ? 2n1 e?n(1+)(?

(110)

Combining Eqs. (107), (108), (109) and (110) we conclude that there exists N^a;1 ;;0 such that for all n  N^a;1 ;;0 it holds 0 A (a)+ ) : PR [S1A;j  ja + 1n; j = 1; : : : ; n]  2n13  e?n(1+)(?

(111)

We now choose 0 and if necessary  smaller than the one chosen above for the purposes of (109), such that ? 1 e?n(1+)(? 0 A (a)+ )  e?n(A (a)+2 ) ; 3 2n  for n  Na;1 ;2 .

7 Deterministic splitting of a stream In this section we treat the splitting operation of our network model. In particular, we derive a LDP for the process resulting from the splitting of a stream to a number of streams and we show that splitting preserves Assumptions B and C1. Consider a stream with stationary interarrival times Ai ; i 2 Z, which is split to 2 substreams. In particular, a fraction p of arrivals of the \master" stream is directed to substream 1 and a fraction 1 ? p to substream 2. The next theorem provides a LDP for stream 1. Since stream 1 is chosen arbitrarily, by relabelling the streams one can obtain a LDP for stream 2. The more general case in which the master stream is split to more than two substreams can be handled by successive splitting to two substreams. Let us denote by A1i ; A2i ; i 2 Z, the interarrival times of substreams 1; 2, respectively. A () and A () denote the large deviations rate function and the limiting log-moment generating function of the master stream.

Theorem 7.1 Under Assumption B, the partial sum S1A;n1 of substream 1 satis es 1 1 ? A1 nlim !1 n log P[S1;n  na] = ? p A (ap):

(112)

Proof : To have n arrivals in substream 1 we need n=p arrivals of the master stream.

Since we are interested in large values of n we will ignore integrality issues (i.e., it holds 36

bn=pc=n ! 1=p, as n ! 1). Thus, A (ap)?) : P[S1A;n1  na] = P[S1A;n=p  na]  e?(n=p)(?

Similarly for the lower bound. We now argue that splitting preserves Assumptions B, and C1. It is clear that the process resulting from splitting satis es Assumption B, since we have proven a one sided LDP for this process with large deviations rate function expressed as a function of the large deviations rate function of the master process. Moreover, the process A1 has moments of ? all orders since A has, and p1 ? A (ap) is strictly convex in (?1; E[Ai ]=p) since A (a) is in the interval (?1; E[Ai ]). The next theorem establishes that the process resulting from splitting satis es Assumption C1.

Theorem 7.2 Assume that the process fAi ; i 2 Zg, satis es Assumptions B and C1. Then

the A1 process satis es Assumption C1.

Proof : The proof is very similar to the proof of Theorem 7.1. A (ap)+) ; P[S1A;j1 ? ja  1n; j = 1; : : : ; n]  P[S1A;j=p ? ja  1n; j = 1; : : : ; n]  e?(n=p)(?

for n large enough and all 1 ;  > 0 by using Assumption C1 for the master process.

8 An Example: Queues in Tandem In this section we apply the results derived so far to obtain LDP's for two G/GI/1 queues in tandem. Moreover we work out a numerical example in order to get a qualitative understanding of the results. Large deviations results for tandem queues with renewal arrivals and exponential servers have been reported in [GA94]. Consider two G/GI/1 queues in tandem. Let Ai ; i 2 Z, denote the interarrival times in the rst queue and Bi1 ; Bi2 ; i 2 Z, the service times in the rst and second queue respectively. These processes are mutually independent, stationary and satisfy Assumptions B and C. According to Corollary 5.5 the limiting log-moment generating function of the departure process from the rst queue is given by ?D () =

(

inf x+y= f?B1 (x) + ?A (y)g if   ^ if  < ^ ?B1 ( ? ) + A ( )

37

(113)

where

i 4 d h ? ? ^ = : ( a ) +  ( a )  A a=0 ( ) da B1

A 1

Applying Theorem 4.1 we obtain that the tail probability of the stationary waiting time, W2, seen by a customer in the second queue, is characterized by

P[W2  U ]  e2U ;

(114)

where U is large enough and 2 < 0 is the smallest root of the equation D ()+B2 (?) = 0. Since for   0 the equation D ()+B2 (?) = 0 has exactly the same roots as the equation ?D () + +B2 (?) = 0, it turns out that 2 is the smallest root of the equation inf x+y= f?B1 (x) + ?A (y)g + +B2 (?) = 0 if   ^ if  < ^ ?B1 ( ? ) + A ( ) + +B2 (?) = 0 It is instructive to characterize a most likely path along which the LDP for the waiting time occurs in the second queue. The remarks after the proof of Theorem 4.1, suggest that a most likely path for the waiting time in the second queue is characterized by

P[W02  (i + 1)a]  x ?sup P [S?Di;0  (i + 1)x1 ]P[S?Bi2?1;?1  (i + 1)x2 ]; x =a 2

1

(115)

where W02 denotes the waiting time of the 0th customer in the second queue and i is large enough. Setting U = (i + 1)a, we obtain for large enough U 



[? (x ) + B+2 (x2 )] : P[W02  U ]  exp ?U a> inf0 a1 x2 ?inf x1 =a D 1

(116)

Let (a ; x1 ; x2 ) be an optimal solution of the optimization problem appearing in (116). Eq. (116) suggests that the waiting time in the second queue builds up by maintaining an empirical rate of 1=x1 for the process D (departure from rst queue) and an empirical service rate (process B 2 ) of 1=x2 . We use the remarks after Theorem 5.4 to characterize a most likely path for the process D to maintain an empirical rate of 1=x1 . Let i be de ned by the equation i + 1 = U=a . From (115), it can be seen that it suces to characterize a most likely path along which the event fS?Di ;0  (i + 1)x1 g occurs. As shown in Theorem 5.4, this most likely path is characterized by

P[S?Di ;0  (i + 1)x1 ]



(







y1 ? exp (i + 1) sup sup ?(1 +  )? A 1+  0 y1 ?y2 =a    ( a ) : (117) ? B+1 y2 ? (i + 1)? B1 

38

Let (y1 ; y2 ;   ) be the solution of the optimization problem appearing in Eq. (117). We depict a most likely path in Figure 11. service rate in second queue arrival rate in rst queue service rate in rst queue (arrival in second)

rate 1

x1 1

x2 1+ 

y1

 y2

busy period of cust. (?i ? 1) starts in rst queue

cust. (?i ? 1) departs from cust. 0 departs from rst rst queue (arrives at second) queue (arrives at second)

Figure 11: A most likely path for the waiting time in the second queue. We now proceed with a numerical example. We chose the arrival process A to be a two-state Markov modulated deterministic process. More precisely, we consider a two-state Markov chain with transition probability matrix "

#

P = 0:8 0:2 ; 0:3 0:7

(118)

and we let the interarrival times be equal to 11 = 15 w.p.1 when the chain is at state 1, and equal to 12 = 101 w.p.1 when the chain is at state 2. The steady-state probability vector for this Markov chain is [1 2 ] = [0:6 0:4] and thus the mean inter-arrival is 11 1 + 12 2 = 0:16. We chose a deterministic server for both queues 1 and 2 with service times c = 0:13. Theorem 3.1.2 in [DZ93b] calculates the limiting log-moment generating function for 4 the arrival process as the largest eigenvalue of the matrix P = [pij e=j ] which in our case is #

"

=5 =10 P = 0:8e=5 0:2e=10 : 0:3e 0:7e

39

(119)

We performed several calculations using the software package Matlab. For the tail probability of the waiting time in the rst queue we found that 1 = ?9:47. We calculated the ? large deviations rate functions ? A (a) and D (a) for the arrival process and the departure process from the rst queue, respectively. The results appear in Figure 12. To calculate ? D (a) we used Eq. (72). It can be seen that the rst queue has a smoothing e ect on the 3.5

+1 3

? D (a)

2.5

? A (a)

2

1.5

1

0.5

0

−0.5 0

0.05

0.1

0.15

0.2

0.25

a ? Figure 12: ? A (a) and D (a) for the numerical example.

arrival process. In other words, the departure process deviates from its mean with smaller probability than the arrival process does. We also found that D () + B2 (?) is strictly negative for all  < 0, so as it can be seen from the proof of Theorem 4.1 that we have 2 = ?1, which means that a large queue does not built up in the second queue. Finally, we found that the departure process D2 from the second queue has large deviations rate ? function ? D2 (a) equal to D (a). This can also be seen analytically. Namely, observe that ? ? ? in Eq. (71) we have ? ? (a) = D (a) which implies D2 (a) = D (a).

40

9 Conclusions and Open Problems We considered a single class, acyclic network of G/G/1 queues, and characterized the large deviations behaviour of the waiting time and the queue length in all the queues of the network. We accomplished that by obtaining the large deviations behaviour of all the processes resulting from various operations in the network, which for the network model that we considered were passage-from-a-queue, superposition of independent processes, and deterministic splitting of a process to a number of processes. We concretely characterized the way that these large deviations occur. These results are to the best of our knowledge among the few that study large deviations in a network. It is clear that more work is needed in this area, especially in view of the important applications in high speed communication networks. It is an interesting open problem to derive similar results for network models that have feedback and accommodate more than one types of trac. It would also be interesting to study, in the large deviations regime, how di erent types of trac interact and how to choose scheduling policies in order to satisfy certain performance criteria. Work relevant to the latter problem, for the single queue case is reported in [Tse94] and [dVK95].

Acknowledgments

The authors wish to thank G. de Veciana, C. Courcoubetis, J. Walrand and T. Zajic for making available preprints of their work and D. Tse and O. Zeitouni for some helpful discussions. We would also like to thank three anonymous reviewers for their careful reading of the paper and their very detailed and useful comments.

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43

Appendix Here we consider an arbitrary process fXi ; i 2 Zg that satis es Assumption B and the following: For every 1 ; 2 ; ; a > 0, there exists MX such that for all n  MX ?

X ?(j ?i+1)a  1 n; 1  i  j  n s.t. (j ?i+1) > n]: e?n(X (a)+2 )  P[Si;j

(120)

Inequality (120) is implied by the results in [DZ93a], under some mild mixing assumptions on the process fXi ; i 2 Zg. We prove that the process fXi ; i 2 Zg satis es Assumption C for the service times (Eq. (19)), i.e., For every 1 ; 2 ; a > 0, there exists MX0 such that for all n  MX0 ?

X ? (j ? i + 1)a   n; 1  i  j  n]: e?n(X (a)+2 )  P[Si;j 1

(121)

Since Assumption C for the arrivals (Eq. (18)) is a weaker version of the above it is also satis ed by the process fXi ; i 2 Zg. Fix positive 1 ; 2 and a. We have

P[Si;jX ? (j ? i + 1)a  1n; 1  i  j  n] = X ? (j ? i + 1)a  1 n; 1  i  j  n s.t. (j ? i + 1) > n; = P[Si;j X ? (j ? i + 1)a   n; 1  i  j  n s.t. (j ? i + 1)  n] Si;j 1 X  P[Si;j ? (j ? i + 1)a  1n; 1  i  j  n s.t. (j ? i + 1) > n] ? P[9 i  j 2 [1; n] s.t. (j ? i + 1)  n and Si;jX ? (j ? i + 1)a  1n]: (122) where we have used the inequality P[A \ B ]  P[A] ? P[B C ]. Using the union bound and the Gartner-Ellis Thm. we obtain that for all 3 > 0 there exists N1 such that for all n  N1

P[9 i  j 2 [1; n] s.t. (j ? i + 1)  n and Si;jX ? (j ? i + 1)a  1n]  X  P[Si;jX ? (j ? i + 1)a  1 n] 

ij 2[1;n] (j ?i+1)n X

P[S1X;n  1 n]

ij 2[1;n] (j ?i+1)n +   n2e?n(X ( 1 ) ? 3) :

44

(123)

Now for given 02 > 0 choose 3 and  small enough in order for large n to have

n2e?n(X (  ) ? 3 )  21 e?n(X (a)+02 ) + 1

?

(124)

This can be done since X+ ( ) ! 1 as ! 1. Also, by using (120) we have that there exists N 00 such that for all n  N 00 0 X (a)+2 ) P[Si;jX ? (j ? i + 1)a  1n; 1  i  j  n s.t. (j ? i + 1) > n]  e?n(?

(125)

Combining (125), (124) and (123) with (122) we obtain that there exists N^ such that for all n  N^ 0 X (a)+2 ) P[Si;jX ? (j ? i + 1)a  1 n; 1  i  j  n]  21 e?n(?

Finally, to obtain (121) it suces to choose 02 such that for large enough n ? 1 e?n(? 0 X (a)+2 )  e?n(X (a)+2 ) 2

45

(126)