Chang, C.-S. Sample path large deviations and intree networks. Queueing Systems Theory Appl., 20 (1995), 7--36.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... of large deviations techniques in communications (see [31] for a survey) The single class queue case has received extensiveattention [20,18,23,19,16,22,28] The extension of these ideas to single class networks, although much harder, has been treated in various versions and degrees of rigor in [4,17,6,10]. In [11] and [32] the authors obtain the asymptotic tails of the overflow probabilities for the GPS policy with deterministic service capacity. The analysis there is based on a large deviations result for the departure process from a G D 1 queue [10] Tail overflow probabilities for the GPS ....

.... 0 and a function ; Delta) with 0 ; y) 1, for all y 0, such that for all n M and all k 0 #: #k m with 1=k 0 k 1 Delta Delta Delta km = n, DeltaZ ] expf m X j=1 [ k j ; k j;1 j) j g# (5) where = 1 #: # m) and Z = S k 0 #S k 2 ; S k 1 #: #S km ; S km;1) In [6] a uniform bounding condition is given under which Assumptions B and Caresatisfied. It is verified that the set of processes satisfying these assumptions is large enough to include renewal, Markov modulated, and stationary processes with mild mixing conditions. Such processes can model ....

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C.S. Chang. Sample path large deviations and intree networks. Queueing Systems, 20:7--36, 1995.

....in a wide class of source models: PfX xg ffe ;jx as x 1, where X denotes a random variable presenting queue length, ff is called the asymptotic decay constantandj is called the asymptotic decay rate. This asymptotic property can be obtained by large deviation techniques (see, e.g. [4,6,15] and references therein) or analytical Ishizaki and Takine Bounds for the tail distribution 3 arguments (see, e.g. 1,7,21] and references therein) With these results, wecan efficiently find the asymptotic decayratej. Compared with the asymptotic decay rate, the asymptotic decay constant ff ....

C. -S. Chang, Sample path large deviations and intree networks, Queueing Systems 20 (1995) 7--36.

....probably satisfactory in terms of accuracy. Thus, it is desired to develop a method which has a proper tradeoff between efficiency and accuracy. The most powerful analytical tools which can be somewhat satisfactory in terms of both efficiency and accuracy are large deviation technique (see, e.g. [3, 5, 9] and references therein) and asymptotic analysis (see, e.g. 1, 6, 15] and references therein) Sohraby [15] has considered BMS s (binary Markov sources) and GBMS s (generalized binary Markov sources) as source models. He has considered a discrete time queueing model with an infinite buffer where ....

C.-S.Chang, "Sample path large deviations and intree networks," Queueing Systems, Vol.20, pp.7--36, 1995.

....queueing models: PfX xgfle ;jx as x 1, where X denotes a random variable representing the number of cells in the systems, fl is called the asymptotic decay constant and j is called the asymptotic decay rate. This asymptotic property can be obtained by large deviation techniques (see, e.g. [3, 6, 21] and references therein) or analytical arguments (see, e.g. 1, 8, 29] and references therein) With these results, we can efficiently find the asymptotic decay rate j. We first review the distribution of the number of cells in an infinite buffer queue where the arrival process is governed by a ....

C.-S. Chang, "Sample path large deviations and intree networks," Queueing Systems,Vol.20, pp.7--36, 1995.

....provides a new tool for looking at large deviations for queueing systems in equilibrium. Equilibrium systems have generally been treated on a case by case basis, with much work and or additional hypotheses necessary to prove large deviation principles (see, for example, Bertsimas et al. 1] Chang [6], Chang and Zajic [7] Ganesh and Anantharam [16] Majewski [18] and Ramanan and Dupuis [23] We provide a simple sucient condition for the usual sample path LDP (as in Mogulskii s theorem) to be strengthened to a topology for which the re ection mappings appearing in many queueing applications ....

C. S. Chang. Sample path large deviations and intree networks. Queueing Systems 20:7-36, 1995.

.... = 1: Assuming a preemptive resume model, large deviations results of the form (2) can also be obtained for statistical multiplexers with priorities [10, 6] based on earlier work characterizing the large deviations rate function of the output process of a single class, single server queue [15, 9]. A more general case has been considered in [7, 29] where large deviation asymptotics have been obtained for queue lengths and virtual waiting times in two queues operating under the generalized processor sharing (GPS) discipline. One of the contributions of our paper is to point out that for ....

C. S. Chang. Sample path large deviations and intree networks. Queueing Systems, 1995.

....of some key concepts and results from large deviation theory and its application to the study of discrete time G D 1 are presented. To save space, in this section we only list a few results that are to be used later in the paper 2 The following presentation follows closely the formulation in [4, 5]. We describe the arrival process to a discrete time G D 1 1 queueing system by a sequence of bounded, nonnegative random variables on IR, fa(t) t 2 INg, where IN is the set of nonnegative integers. In other words, at time t 3 , the amount of arrivals to the queue is a(t) For any = 0; 1; ....

....is a(t) For any = 0; 1; 2; and any t 2 IN , t , define A( t) P t Gamma1 s= a(s) the number of arrivals during the time interval [ t) Also let A( 0. We call A the cumulative arrival process. We make the following assumptions on the arrival process fa(t) t = 0; 1; 2; g [4, 5]. A1) The arrival process fa(t) t = 0; 1; 2; g is ergodic and stationary. A2) For any 2 IR, A ( lim t 1 1 t log Ee A(0;t) 1 (1) and is differentiable. A3) fa(t) t = 0; 1; 2; g is adapted to a filtration fF A t ; t 2 INg with the following property: for any 2 IR, ....

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C. S. Chang, "Sample Path Large Deviation and Intree Network", To appear in Queueing Systems, 1994.

....ours. However, due to their resort to a Loynes type argument [28] 3 , their lower bound argument is somewhat less convincing and rigorous. In contrast, we argue directly with the stationary version of the processes. To obtain the lower bound, we apply the sample path large deviation principle [14, 7] which requires stronger assumptions on the arrival processes. Our results for the GPS system with more than two queues are more general than theirs, as we exploit the bandwidth sharing dynamics in more details. The rest of Part I of the paper is organized as follows. Section 2 briefly reviews the ....

....the following form: for any OE 2 D( 0; 1] IR) I(OE) R 1 0 (OE 0 (t) dt; if OE 2 AC 0 ( 0; 1] IR) 1; otherwise 4 where AC 0 ( 0; 1] IR) is the space of absolutely continuous functions from [0; 1] to IR with OE(0) 0, and OE 0 (t) is the derivative of OE(t) at t. Following [7], we say the sequence of probability measures f (n) n = 1; 2; g (or fZ n ; n = 1; 2; g) satisfies the sample path large deviation principle with respect to . Large deviation theory has been widely applied in queueing theory to study the tail probabilities of various queueing ....

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C. S. Chang, "Sample Path Large Deviation and Intree Network", To appear in Queueing Systems, 1994.

....a lot of bursty sources even with a zero buffer. Large deviations asymptotics obtained by simultaneously scaling both the buffer and bandwidth are provided in [6] The results presented here are only valid for a single node. Extensions to certain special types of networks can be found in [10]. Obtaining results for general networks is still an open issue. In [13] a different approach is explored. There, the bandwidth needed to decouple the network into a collection of non interacting nodes has been studied. A different direction of research is to extend the results to multiple ....

C.S. Chang. Sample path large deviations and intree networks. Queueing Systems, 20:7--36, 1995.

....O Connell [7] These results include queues in which the input process can exhibit complex dependency structure. Under certain conditions, these results extend from a single node to an intree network, which is a useful model of real world high speed ATM networks; the reader is referred to Chang [5] for details. The use of extremal statistics is one means of taking advantage of A1 To illustrate this point, consider a discrete time real valued stationary sequence (Xn : n 0) for which the tail probability satis es A1: If Mn = maxfX j : 0 j n 1g and Fn ( is the empirical distribution ....

Chang, C.-S. Sample path large deviations and intree networks. Queueing Systems Theory Appl., 20 (1995), 7-36.

....found in [26] The result for the two queue GPS system was first stated in [14] under weaker assumptions than ours. However, they provide a non rigorous heuristic proof based on a Loynes type argument [17] In contrast, we establish the result by applying the samplepath large deviation principle [13, 6] which requires stronger assumptions on the arrival processes. The rest of this paper is organized as follows. Section 2 briefly reviews the large deviation principle and states several results regarding discrete time G D 1 1 queueing systems which will be used later. In section 3 we state and ....

.... with the following form: for any OE 2 D( 0; 1] IR) I(OE) ae R 1 0 (OE 0 (t) dt; if OE 2 AC 0 ( 0; 1] IR) 1; otherwise where AC 0 ( 0; 1] IR) is the space of absolutely continuous functions from [0; 1] to IR with OE(0) 0, and OE 0 (t) is the derivative of OE(t) Following [6], we say the sequence of probability measures f (n) n 2 IN; n 0g (or fZn ; n = 1; 2; g) satisfies the sample path large deviation principle with respect to . Large deviation theory has been widely applied in queueing theory to study the tail probabilities of various queueing systems ....

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C. S. Chang, "Sample Path Large Deviation and Intree Network", Queueing Systems, Vol. 20, No.I-II, 1995.

....approach devised in [17] We introduce a key concept partial feasible sets which provides an avenue to capture the asymptotic bandwidth sharing dynamics among the sessions in the GPS system. With the help of this concept, we apply the sample path large deviation principle (see, e.g. [6, 2]) to the analysis of the GPS system and obtain upper and lower bounds on the asymptotic decay rates of the queue length tail distributions for the sessions. Although the upper and lower bounds do not match exactly in general, in the special case of a two queue GPS server, the lower and upper ....

....of notation, we study the steady state system at time 0 and look backward in time. Therefore, we describe the arrival process to the system by a sequence of bounded, nonnegative random variables on the real line IR, fa( Gammat) t 2 INg and make the following assumptions on the arrival process [1, 2]. A1) The arrival process fa( Gammat) t 2 INg is ergodic and stationary. A2) For any 2 IR, the following limit A ( lim t 1 1 t log Ee A( Gammat;0) 1 (1) exists as a finite number. Moreover, A ( is strictly convex and differentiable. A3) fa( Gammat) t 2 INg is adapted to a ....

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C. S. Chang, "Sample Path Large Deviation and Intree Network", Queueing Systems, Vol. 20, No.I-II, 1995.

.... for all n M and all k 0 ; km with 1 = k 0 k 1 Delta Delta Delta km = n, E[e DeltaZ ] exp 8 : m X j=1 [ k j Gamma k j Gamma1 ) j ) Gamma( j ) 9 = 4) where = 1 ; m ) and Z = S k0 ; S k2 Gamma S k1 ; S km Gamma S km Gamma1 ) In [7] a uniform bounding condition is given under which Assumptions B and C are satisfied. It is verified that the set of processes satisfying these assumptions is large enough to include renewal, Markov modulated, and stationary processes with mild mixing conditions. Such processes can model ....

....practical problems with this approach. On the technical side, the network problem appears to be particularly hard, since in essence it is needed to obtain distributions of queue lengths and delays in a multiclass network of G G 1 queues. This has been accomplished in singleclass acyclic networks [4,7]. Related work is reported in [13,25] The multiclass case, however, appears much harder and no LDP results exist (in fact, some negative results have been reported in [26] On the practical side, a network mechanism has to scale to the full range of speeds and administrative domains that ....

C.S. Chang. Sample path large deviations and intree networks. Queueing Systems, 20:7--36, 1995.

....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 ....

....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 process of a G G 1 queue, in isolation. It is ....

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C.S. Chang, Sample path large deviations and intree networks, Queueing Systems 20 (1995), 7--36.

.... of buffered networks without delay jitter control is beyond the scope of this paper, our techniques can be extended to rate controlled servers [18] using techniques such as in [11] or to more general classes of networks using other techniques for end to end performance evaluation, e.g. [1,13]. 2.3 Experimental Workload Throughout this paper we use two sources for admission control and simulation experiments: a periodic on off source and a 30 minute trace of an MPEG compressed video of an action movie. The periodic on off source can be characterized by three parameters, i.e. the on ....

C. Chang. Sample path large deviations and intree networks. Queueing Systems, Theory and Applications, 20(1-2):7--36, 1995.

....But, first, we discuss some special cases when the departure process does have linear geodesics. Suppose there is a single input stream. If the service process is deterministic, then the departure process has linear geodesics. So, a recursive analysis of networks of such queues is possible, as in [3]. Even if the service process is stochastic, we show that, conditional on the departure rate from a queue exceeding its mean, the departure process has linear geodesics. We are typically interested in the probability of queue lengths exceeding some large threshold, and in well designed networks ....

....of the arrival and service processes intersecting at ff. If the service process is deterministic, the latter cannot happen, and in this case it can be shown that the departure process has linear geodesics. This makes it possible to analyze networks of deterministic server queues, as in Chang [3]. Likewise, if we consider only ff EX, then too it can be shown that the departure process conditioned on having mean rate ff is linear. Since we are typically interested in the problem of queue lengths exceeding some large threshold, and since in well designed networks this requires departure ....

C.S. Chang (1995). Sample path large deviations and intree networks. Queueing Systems, 20(1-2): 7-36.

.... of large deviations techniques in communications (see [31] for a survey) The single class queue case has received extensive attention [20,18,23,19,16,22,28] The extension of these ideas to single class networks, although much harder, has been treated in various versions and degrees of rigor in [4,17,6,10]. In [11] and [32] the authors obtain the asymptotic tails of the overflow probabilities for the GPS policy with deterministic service capacity. The analysis there is based on a large deviations result for the departure process from a G D 1 queue [10] Tail overflow probabilities for the GPS ....

.... 0, such that for all n M and all k 0 ; km with 1 = k 0 k 1 Delta Delta Delta km = n, E[e DeltaZ ] expf m X j=1 [ k j Gamma k j Gamma1 ) j ) Gamma( j ) g; 5) where = 1 ; m ) and Z = S k 0 ; S k 2 Gamma S k 1 ; S km Gamma S km Gamma1 ) In [6] a uniform bounding condition is given under which Assumptions B and C are satisfied. It is verified that the set of processes satisfying these assumptions is large enough to include renewal, Markov modulated, and stationary processes with mild mixing conditions. Such processes can model ....

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C.S. Chang. Sample path large deviations and intree networks. Queueing Systems, 20:7--36, 1995.

....An end to end overflow (or delay) probability is simply the largest such probability among all nodes in the route from source to destination. The latter probabilities can be calculated by applying the single node results. A more rigorous decomposition approach has been developed in [15] and [16]. Related work is reported in [17, 18] Among the main contributions of the work in this paper we consider: ffl The multiclass character of the analytical results and the admission control algorithm. One significant advantage of our scheme, over other priority schemes proposed in the literature ....

C. Chang, "Sample path large deviations and intree networks," Queueing Systems, vol. 20, pp. 7--36, 1995.

....approach devised in [17] We introduce a key concept partial feasible sets which provides an avenue to capture the asymptotic bandwidth sharing dynamics among the sessions in the GPS system. With the help of this concept, we apply the sample path large deviation principle (see, e.g. [6, 2]) to the analysis of the GPS system and obtain upper and lower bounds on the asymptotic decay rates of the queue length tail distributions for the sessions. Although the upper and lower bounds do not match exactly in general, in the special case of a two queue GPS server, the lower and upper ....

....of notation, we study the steady state system at time 0 and look backward in time. Therefore, we describe the arrival process to the system by a sequence of bounded, nonnegative random variables on the real line IR, fa( Gammat) t 2 INg and make the following assumptions on the arrival process [1, 2]. A1) The arrival process fa( Gammat) t 2 INg is ergodic and stationary. A2) For any 2 IR, the following limit A ( lim t 1 1 t log Ee A( Gammat;0) 1 (1) exists as a finite number. Moreover, A ( is strictly convex and differentiable. A3) fa( Gammat) t 2 INg is adapted to a ....

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C. S. Chang, "Sample Path Large Deviation and Intree Network", Queueing Systems, Vol. 20, No.I-II, 1995.

....of this paper is to establish functional (or sample path) large deviation principles (FLDPs) for stochastic processes arising in queues and acyclic networks of queues. A distinguishing feature from previous work in this direction, notably by de Veciana, Courcoubetis and Walrand [37] and Chang [7], is our focus on FLDPs in the function space D with the (non uniform) Skorohod [33] topologies, where the rate functions may be finite on some discontinuous functions. Establishing such general FLDPs is challenging and interesting mathematically, but there also is substantial practical ....

....can be approached by establishing LDPs for departure processes. If we can establish an LDP for a departure process, then we can extend the effective bandwidth concept to acyclic networks of queues. Significant progress on that program was carried out by de Veciana, Courcoubetis [37] and Chang [7]. They found, again under regularity conditions, that the departure process (of completed work) D(t) has the almgf ffi( 8 : ff( ff( c( Gamma ) 1.4) where c is the constant output rate from the queue, ff( j P ff i ( is the almgf for the aggregate ....

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Chang, C. S. (1995). Sample Path Large Deviations and Intree Networks. Queueing Systems 20, 7--36.

.... = 1: Assuming a preemptive resume model, large deviations results of the form (2) can also be obtained for statistical multiplexers with priorities [10, 6] based on earlier work characterizing the large deviations rate function of the output process of a single class, single server queue [15, 9]. A more general case has been considered in [7, 30] where large deviation asymptotics have been obtained for queue lengths and virtual waiting times in two queues operating under the generalized processor sharing (GPS) discipline. One of the contributions of our paper is to point out that for ....

C. S. Chang. Sample path large deviations and intree networks. Queueing Systems, 1995.

.... of rare events in a single class queue has received extensive attention in the literature [Hui88, GH91, Kel91, KWC93, GW94, EM93, TGT95] The extension of these ideas to single class networks, although much harder, has been treated in various versions and degrees of rigor in [BPT97a, GA96, Cha95, O C95a, dVCW93] Closer to the subject of this paper, the asymptotic tails of the overflow probabilities for the GPS policy with deterministic service capacity are obtained in [dVK95] and [Zha97] Both papers use a large deviations result for the departure process from a G D 1 queue [dVCW93] ....

.... for all n M and all k 0 ; km with 1 = k 0 k 1 Delta Delta Delta km = n, E[e DeltaZ ] exp ae m X j=1 [ k j Gamma k j Gamma1 ) j ) Gamma( j ) oe ; 8) where = 1 ; m ) and Z = S k 0 ; S k 2 Gamma S k 1 ; S km Gamma S km Gamma1 ) Chang [Cha95] provides a uniform bounding condition under which Assumption B is true, and verifies that the condition is satisfied by renewal, Markov modulated, and stationary processes with mild mixing conditions. Using his uniform bounding condition it can be verified (see [Cha95] for a proof) that ....

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C.S. Chang, Sample path large deviations and intree networks, Queueing Systems 20 (1995), 7--36.

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C.S. Chang, \Sample path large deviations and intree networks," Queueing Systems, Vol. 20, pp. 7-36, 1995.

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C.S. Chang, "Sample path large deviations and intree networks," to appear in Queueing Systems.

....achieved by choosing 1 = p( in the family of UEPs of a(t) Also, note that Z 1;i ( st p( Z 1;i ( p( i = 1; k, since Z 1;i , i = 1; k, are exponentially distributed. Now consider a discrete time intree network with J queues and multiple classes of customers as in [8, 7]. In such a network, there is a natural partial ordering among the J queues. If part of the departure process of queue i is an input process of queue j, we say queue j is a successor of queue i and queue i is a predecessor of queue j. Let Suc(j) resp. P re(j) be the set of queues that are ....

....contributes to the J queue, then Z J ( is simply an L stage Erlang random variable, where L = J 1 P J j=1 k j . Thus, P r(q J x) e J ( J ) L 1 =0 J (x ( J ) J x : 45) Also, the recursive equation in (41) is di erent from that in Chang [7], where the exact decay rate of q J is computed. It can be shown that J computed by this algorithm is not greater than the exact decay rate of q J . The gap is due to the input output relation 7 obtained in this paper, where we simply choose a k ( Comparing this with the corresponding ....

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C.S. Chang, \Sample path large deviations and intree networks," to appear in Queueing Systems.

....input process Y . We characterize the transient queueing behavior; in particular, how large queues build up. In contrast to the large queue behavior in the case of queues with short range dependent input (refer to the conditional limit theorems in Anantharam [1] Dembo and Zajic [9] and Chang [4]) the most likely path to a large queue buildup here is nonlinear. Finally, we discuss steady state results, in terms of the maximum of the associated random walk, and compare them with the results of [11] We now brie y comment on the individual merits of the two models based on (1.1) and ....

.... (t) sup d ( d ( c(t s) Note that the conditional limit in Theorem 4. 2 (ii) which illustrates how the queue builds up when the input has long range dependence, is qualitatively di erent from the more standard case of input with shortrange dependence (e.g. [1, 4, 9]) where the most likely path is linear. Here, the most likely path is nonlinear. Using the continuity of the re ection mapping, t) sup f (t) s) c(t s)g; t 2 [0; 1] it follows from the contraction principle that the distribution of V (1) satis es the LDP with speed n V ( ....

C.S. Chang, \Sample path large deviations and intree networks," Queueing Systems, Vol. 20, pp. 7-, 1995.

....transform, the large deviation approach provided a probabilistic derivation of the general form of e ective bandwidths in [3] Also, it lled the gap for the lower bound needed for (1) in the single queue case. The lower bounds needed for intree networks with routing were justi ed by Chang [6] using the sample path large deviation principle and the contraction principle. With the exception of O Connell [24] it appears that all previous work on networks has assumed a constant capacity at each local node. In this paper, we consider a queue with a time varying capacity and identify the ....

....have the same e ective bandwidth when the queue has a constant capacity. ii) In order to have a large excursion of the stationary departure process, sometimes it is necessary to build up the queue rst at the rate characterized by (1) Our extension allows us to modify the algorithm in Chang [5, 6] for approximating the tail distributions of queue lengths in intree networks with time varying capacities. Since a multiple class queue under a headof line priority scheme can be modelled by a queue with a time varying capacity, our extension can also be applied for analyzing the performance of ....

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C.S. Chang, \Sample path large deviations and intree networks," to appear in Queueing Systems.

....[12] This phenomenon is qualitatively different from the exponential tail distribution in queues with renewal or Markov arrival processes inputs that have short range dependence. The exponential tail behavior is usually established via large deviations techniques; refer to, e.g. Chang [5, 6], Glynn and Whitt [14, 15] Whitt [28] and the references there. Our work is motivated in part by [12] Assuming that the netput input minus (potential) output satisfies a large deviations principle with a speed that is slower than n or, more precisely, an MDP, Duffield and O Connell ....

....) H Gamma1=2 d OE fl ( Gamma Z 0 Gamma1 [ s Gamma ) H Gamma1=2 Gamma ( Gamma ) H Gamma1=2 ]d OE fl ( Gamma c(t Gamma s) Remark. The most likely path to a large build up, fl , has been studied in the case of input with short range dependence by Anantharam [2] Chang [6] and Dembo and Zajic [10] where it has been found to be linear. Here, as in [7] the most likely path is nonlinear. In Figure 1, below, we have plotted the function fl for different values of H. Proof. Theorem 4.6] As in the proof of Theorem 3.3 (i) it suffices to show that inf OE2F B ....

C.S. Chang, "Sample path large deviations and intree networks," Queueing Systems, Vol. 20, pp. 7-36, 1995. 27

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Chang, C.-S. Sample path large deviations and intree networks. Queueing Systems Theory Appl., 20 (1995), 7--36.

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Chang, C.-S. (1995). "Sample path large deviations and intree networks". Queueing Systems, 20:7--36.

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C.S. Chang, Sample path large deviations and intree networks, Queueing Systems 20 (1995) 7-36.

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C.-S. Chang, Sample path large deviations and intree networks, Queueing Systems, 1995, 20, pp. 7-36.

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