A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Queues, Communication and Computing. London, U.K.: Chapman & Hall, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....6 we prove the matching upper bound. Section 7 treats the special case of prioritypolicies and provides an alternativewayofcalculating the large deviations exponent. Conclusions are in Section 8. 2. Preliminaries In this section wereview some basic results on the theory of Large Deviations [5,13,30] that will be used in the sequel. Consider a sequence fS 1 #S 2 #: g of random variables, with values in R and define n ] 1) For the applications that wehaveinmind, S n is a partial sum process. Namely, S n = i=1 X i , where X i # i 1, are identically distributed, possibly ....

A. Shwartz and A. Weiss. Large deviations for performance analysis. Chapman and Hall, New York, 1995.

....of sums of i.i.d. random vectors with values in a separable Banach space. Our results are stated as functional large deviations with a Gaussian rate function. General references on (functional) large deviations are Bahadur [4] Varadhan [23] Deuschel and Stroock [9] and Shwartz and Weiss [21]. We consider stochastic processes as elements of l (T ) where T is an index set. l (T ) is the Banach space consisting of the bounded functions defined in T with the norm #x## x(t) We will use the following definition: Definition 1.1. Given a sequence of stochastic processes T , ....

A. Shwartz and A. Weiss, Large Deviations for Performance Analysis. Queues, Communications, and Computing. Chapman & Hall, London, 1995.

....the stationary probability of being in state #vn# is asymptotically exp( L(v)n) for large n. For results on large deviations for specific types of constrained random walks in see [20] There are numerous works on large deviation in the context of queueing systems, see Shwartz and Weiss [36] for a survey. Specifically, Kurkova and Suhov [26] study large deviation rates for a two dimensional random walk corresponding to join the shortest queue. The analysis is quite intricate and uses complex analytic techniques developed by Malyshev [28] 27] 29] back in 70 s. To the best of our ....

A. Shwartz and A. Weiss, Large deviations for performance analysis, Chapman and Hall, 1995.

....framework so that losses can be distributed more evenly among flows subscribing to the same class of statistical service. The results are reported in [28] which also contains more discussions of related work. We will also look into using other statistical tools (such as the large deviation theory [23]) to improve the admission control accuracy. APPENDIX Proof of Theorem 1: Define as the set of packets in the arrival sequence whose priority values are less than or equal to that of packet and as the total work (in bits) done by the FC server for packets in in the time interval . We will carry ....

A. Schwarz and A. Weiss, Large Deviations for Performance Analysis. London, U.K.: Chapman and Hall, 1995.

....phase transition takes place. In their assessment of distributed control of packetized voice, Gibbens and Kelly [9] rely on the approximations of [19] Contribution. The main contribution of the present paper is the analysis of the combined packet burst model applying large deviations asymptotics [5, 23]. In order to do that, we apply the so called many sources asymptotic : we scale link rate and buffer with the number of sources n (such that we get C nc and B rib) and then we let n grow large. We are interested in the probability of the buffer level exceeding rib. We call this probability ....

A. SHWARTZ AND A. WEISS. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, 1995.

....positive function of b and c. In section 2 we return to these results. Notice that there are several concepts of loss probability : fraction of fluid lost, fraction of time the buffer is full, etc. In [4] it is argued that all these performance measures have the same exponential decay rate. Weiss [20, 23] however gives only for Markov fluid a method to calculate the decay rate in an alternative manner. Consider the case of one type of sources; then a path f is a function of time with values in a probability measure space: f i (s) is the fraction of modulating Markov chains that is in state i ....

....path for general Markov fluid sources and general buffer size. Then we prove that our conjecture path must be optimal because it optimizes the above mentioned functional J( Delta) To our knowledge, this it the first successful attempt to unify the two different approaches of [2, 4, 21] and [20, 23]. However, we notice that there is a striking analogy with the relations between first and second level large deviations theorems and the contraction principle [7] For periodic on off sources, the notion of most likely path to overflow was not introduced in the literature yet. It can be ....

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A. Shwartz and A. Weiss. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, 1995.

....= P[Y (n) n ] it follows from Theorem 3.1 and 0 that X (D ) Y ( 1.12) On the other hand, from the LDP(sp) upper bound for X , we have X (B D ) inf 2B D I X ( 1.13) Combining (1.12) and (1.13) and making use of Lemma 2. 8 of Shwartz and Weiss [21], we have, for some 0, which clearly implies (1.11) ii) Using the continuity of the mapping appearing in (1.9) the desired weak convergence follows from (i) We now consider the second model, the lter based on (1.1) dX(s) dX(s) 1.14) Before going ....

A. Shwartz and A. Weiss, Large deviations for performance analysis. Chapman & Hall, London, 1995.

....where appropriate, at later parts of the dissertation. 1.3.1 Performance Analysis and Traffic Engineering The overflow probability in broadband networks will typically be less than 10 , hence the overflow probability is a rare event. For this reason, the theory of large deviations [Wei95, SW95] which is a collection of techniques for estimating properties of rare events, lends itself naturally to the study of the overflow probability in broadband networks. Large asymptotics is an application of large deviations which studies the tail of the overflow probability of a link when some ....

....supinf formula (2.3) is in good agreement with the value computed using (2. 17) Note that the steps in the value computed using the supinf formula are expected since the many sources asymptotic (and large deviations theory in general) captures only the most probable way overflow can occur [Wei95, SW95] On the other hand, the curve labeled simulation in Figure 2.8(a) includes all the events that contribute to buffer overflow. Recall from the discussion in Section 2.1.1 that the time parameter t can be interpreted as the most probable duration of the busy period prior to buffer overflow. ....

A. Shwartz and A. Weiss. Large Deviations for Performance Analysis. Chapman and Hall, NY, 1995.

....we first provide an intuitive interpretation. 1 t P V K2 (19) with # 1 : 1 # 1 1. To understand the above formula, it is useful to draw a comparison with the workload V 1 of class 1 when served in isolation at constant rate 1. Large deviations results for the M M 1 queue [26] suggest that the most likely way for V 1 to reach a large level x is that class 1 temporarily experiences abnormal activity. Specifically, class 1 must essentially behave as if its tra#c intensity were increased from the normal value # 1 to the value # 1 , causing a positive drift # 1 1 ....

Shwartz, A., Weiss, A. (1995). Large Deviations for Performance Analysis (Chapman & Hall, London).

....which is established using probabilistic techniques. 11 1 t P V # 1 1 (35) with # 1 : 1 # 1 1. To understand the above formula, it is useful to draw a comparison with the workload V 1 when Q 1 is served in isolation at constant rate 1. Large deviations results for the M M 1 queue [26] suggest that the most likely way for V 1 to reach a large level x is that class 1 temporarily experiences abnormal tra#c activity. Specifically, class 1 essentially behaves as if its tra#c intensity were increased from the normal value # 1 to the value # 1 , causing a positive drift # 1 1 ....

Shwartz, A., Weiss, A. (1995). Large Deviations for Performance Analysis (Chapman & Hall, London).

....we hope that the problem framework which we establish in the SRBM setting will provide motivation for further research into the interesting and challenging open problems beyond the 2 dimensional case. There is a large body of literature on LDPs for random walks and queueing networks. The book [33] by Shwartz and Weiss contains an excellent list of references. We can only provide a short survey of the latest works which are most closely related to our study. Recent work on LDPs for queueing networks include O Connell [30, 31] and Dupuis and Ramanan [13] on multi bu er single server systems, ....

A. Shwartz and A. Weiss. Large deviations for performance analysis. Chapman & Hall, New York, 1995.

....decay rate of the probability of a non empty buffer is given by IP (WN 0) I(0) When #p (# p) c(regime (A) I(0) is given by c(# p) p# r and when #r (# r) c# #p (# p) regime (B) I(0) is given by # . Proof. Directly from Theorem 11.15 of [29], the decay rate of the probability of a non empty buffer equals m x x #(1 x) dx. Here x is the (downward) probability flux per source, when the number of sources in the on state is Nx: x : p if xp c, x 1 otherwise. Direct calculation yields the stated expression. # ....

A. Shwartz and A. Weiss. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, NY, USA, 1995.

....Let us now de ne J(B) lim N 1 1 N log Pr(B N ) for any sequence of events B N when such a limit exists. Now consider the sequence of random variables fA (m) 1 ( T ; 0)g; m 0. For each T , these form a sequence of i.i.d. random variables. Let I A1 ;T (x) be its rate function (see [16]) Formally, for a sequence of random variables Zm ; m 0, we de ne the rate function as follows: De ne Z m ( 1 m log E e Zm . Assume that the limit Z ( lim N 1 Z m ( exists pointwise. Then, I Z (x) sup x Z ( Similarly de ne I A2 ;T (x) ....

.... N 2 ( T ; 0) NC(T m 1) i We have that the most likely way that E T would occur is for A N 1 ( T ; m) x and A N 2 ( T ; 0) NC(T m 1) x, for some x such that x E(A N 1 ( T ; m) T m 1) and x C(T m 14 1) T m 1) Formalizing this through the contraction principle ([16]) we have J(E T ) inf 0 x e C(T m 1) fI A1 ;T m (x) I A2 ;T (C(T m 1) x)g = inf (T m 1) x e C(T m 1) fI A 1 ;T m (x) I A 2 ;T (C(T m 1) x)g (9) Using Assumption 4.1, we show that the event E T identi ed above in asymptotically dominates the delay asymptote. The rest of this ....

A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Chapman and Hall, 1995. 22

....convolution of F (1) x,0 (t) for all positive x and positive integers n, that the mean of F (1) x0 (t) is x (1 # 1 ) e.g. see Theorem 7 of [6] and that the Laplace transform of f (1) x0 (t) is given in (2.7) Hence, we can apply Cherno# s bound, e.g. 1.6a) on p. 14 of Shwartz and Weiss [48], to obtain (7.26) and (7.27) By (29) and (30) of [6] and (7.23) and (7.24) we see that the root of z # 1 ( s) 0 is # 1 , which implies that #(#) # # 1 as # # 0. We apply Lemma 7.1 to obtain (7.29) and (7.30) We now obtain control of the asymptotics of the second term of F c (t) in ....

A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Chapman and Hall, London, 1995.

....[a; b] equipped with the norm of uniform convergence. We next recall the definition of the large deviations principle on a metric space S equipped with the topology generated by the metric and the corresponding Borel oe field. Refer to, e.g. Dembo and Zeitouni [11] and Schwartz and Weiss [26] for more details. Definition 2.1 A sequence of probability measures (n) on a metric space S satisfies the large deviations principle, with speed i n 1 and good rate function I( Delta) S [0; 1] if (i) the level sets fx 2 S : I(x) ffg, ff 2 IR, are compact, ii) for any closed set F , ....

....flg: We have from the proof of Corollary 3.2 that inf OE2 Gamma Z 1 Gamma1 1 2oe 2 (OE 0 (s) 2 ds = Z 1 Gamma1 1 2oe 2 (OE 0 fl (s) 2 ds = Y (fl) Furthermore, observe that the set B ffl Gamma fOE : I X (OE) 2 Y (fl)g is closed. An application of Lemma 2. 8 of [26] yields inf OE2B ffl Gamma Z 1 Gamma1 1 2oe 2 (OE 0 (s) 2 ds Y (fl) Applying Theorem 3.1 yields (11) ii) Since the map F m : D C[0; 1] defined in (8) is continuous, the probability law of f F m (n H Gamma1=2 X (n) Delta) jY (n) 1) flg converges in ....

A. Shwartz and A. Weiss, Large deviations for performance analysis. Chapman & Hall, London, 1995.

....In particular, we wonder how the MBAC algorithm suffers from the consequences of distributions with heavy tails. Existing literature and contribution. Our analysis of the probabilities of extreme input rate fluctuations is of an asymptotic nature. We derive a large deviations approximation [27] of the transient overflow probability asymptotically in the number of flows. In earlier work [22] we restricted ourselves to the case where the sojourn times in the states of the flow had an exponential distribution; the analysis provided here holds for generally distributed sojourn times. We ....

....that n flows are present at time 0, of which nff are in the on state. The link rate C is denoted by nfi (corresponding with the notation introduced in Section 2) Without loss of generality, we assume the flows peak rates to equal 1. Our reasoning will be of a heuristic nature. As justified in [27], we will extensively use the so called Laplace s principle [12] stating that the decay rate of an integral equals the decay rate of the maximum of the integrand. We denote by A n (t) the number of flows (out of n) in the on state at time t, assuming that they do not leave the system. Notice that ....

A. Shwartz and A. Weiss. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, USA, 1995.

....of two independent M M 1 queues where Q 1 has arrival rate # 1 and service rate 1 while Q 2 has arrival rate # 2 # and service rate 2 . During this large deviation there will be some time t when Q 1 (t) exceeds a# while Q 2 (t) exceeds (1 a)# where 0 a 1. By Section 11.2 in [15], the cost or action associated with a trajectory of the first queue from 0 to a# in time t plus the action associated 5 with a trajectory of the second queue from 0 to (1 a)# in time t is t[L 1 (a# t) L 2 ( 1 a)# t) where L 1 (c) # c log( c # c 2 4# 1 1 2# 1 ) # 1 ....

....as an infinite sum of product measures in the asymmetric case when 1 #= 2 but # 1 = # 2 = 0. # 1 2 was required for stability. Knessl, Matkowsky, Schuss and Tier [10] used a heuristic technique to give the stationary distribution for the model in this paper. Shwartz and Weiss [15] developed a large deviation theory for Markov jump processes with a boundary and applied it (heuristically) to deviations of the number of customers in a join the shortest queue network like the one studied here. The existence of a large deviation principle for this system is given in Dupuis and ....

Shwartz, A. and Weiss, A. (1994). Large deviations for performance analysis, Chapman and Hall.

.... as well as to queueing problems [CH91] Recently these techniques have resurged in the random graph community, initiated by the work of Wormald [Wor95] The text by Shwartz and Weiss on large deviations provides a solid introduction into the entire area of large deviations, including Kurtz s work [SW95] There are by now many examples of works that use large deviation bounds and di erential equations for a variety of problems, including but in no way limited to [AM97, AH90, AFP98, KMPS95, LMS 97] 38 Given this framework, it is easy to nd the limiting fraction of empty bins after m = cn ....

A. Shwartz and A. Weiss. Large Deviations for Performance Analysis. Chapman & Hall, 1995.

....equilibrium and the joint large deviations behaviour of the outputs are analysed, and queues with dedicated bu ers. Most of the problems we have discussed, including those studied in [44, 45, 46] along with variants and related problems, have been analysed using di erent methods in (for example) [1, 2, 4, 6, 7, 9, 16, 17, 20, 21, 22, 23, 31, 40, 50, 53]. 2.7 Application: A problem in stochastic control Consider the following queueing system. We have a stationary and ergodic arrivals process X k , and the queue evolves as follows: Q n = minf(Q n 1 X n c(Q n 1 ) Bg; 29) 30 where c is some function which we are allowed to choose. The ....

Adam Shwartz and Alan Weiss (1995). Large Deviations for Performance Analysis. Chapman and Hall. 51

....to be computed. With the ODEs, fewer ODEs need to be considered. Finally, asymptotic formulas can serve as alternatives to exact numerical algorithms in the appropriate asymptotic regimes. Such asymptotic formulas are given in Mitra and Weiss [22] Knessl [18] and Chapter 12 of Shwartz and Weiss [25]. These asymptotic formulas are very attractive when they are both simple and su#ciently accurate. If the asymptotic formulas are not simple, then they properly should be viewed as alternatives to numerical algorithms. For example, the direct numerical inversion here would seem to be preferable to ....

Shwartz, A. and Weiss, A. (1995) Large Deviations for Performance Analysis, Chapman and Hall, London.

.... as well as to queueing problems [CH91] Recently these techniques have resurged in the random graph community, initiated by the work of Wormald [Wor95] The text by Shwartz and Weiss on large deviations provides a solid introduction into the entire area of large deviations, including Kurtz s work [SW95] There are by now many examples of works that use large deviation bounds and differential equations for a variety of problems, including but in no way limited to [AM97, AH90, AFP98, KMPS95, LMS 97] 38 Given this framework, it is easy to find the limiting fraction of empty bins after m = ....

A. Shwartz and A. Weiss. Large Deviations for Performance Analysis. Chapman & Hall, 1995.

....Abstract This paper deals with the transient behavior of the Erlang loss model. After scaling both arrival rate and number of trunks, an asymptotic analysis of the blocking probability is given. Apart from that, the most likely path to blocking is given. Compared to Shwartz and Weiss [14], more explicit results are obtained by using probabilistic arguments. The computation method is applied to the problem of (real time) dimensioning of virtual paths in ATM networks, and to the problem of integrating scheduled and switched connections in a single network. Keywords: Erlang loss ....

....integrals appeared to allow for explicit calculation. The paper is organized as follows. In section 2, we consider the model with infinitely many trunks, and find an expression for the decay rate of the probability of exceeding level n in t units of time, starting at level nx. As argued in [14], this decay rate equals the decay rate of the transient blocking probability of our interest. In section 3, we focus on the most likely trajectory to blocking. With probabilistic arguments, e.g. Laplace s method, we recover the results of [10] In section 4, a number of examples are considered. ....

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A. Shwartz and A. Weiss. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, 1995.

....S of absolutely continuous functions on the interval [0, t] lim n## 1 n log P (F n (s) # S, s # [0, t] F n (0) f 0 ) inf f#S:f(0) f0 J t (f) 4) Here J t (f) R t 0 I f(s) f # (s) ds is called the action functional. Notice that in Theorem 5. 1 of Shwartz and Weiss [29] this theorem is proved for general dimension d, but under the assumption that the logarithm of the transition rates is bounded. In our model, the rate of jumping from i to j is m# ij if m sources are in state i. As m can attain value zero, the logarithm of this rate is not bounded, and Theorem ....

....f , say f # , of variational problem (4) has an interesting interpretation. Given that the rare event under consideration occurs, with overwhelming probability it does so with F n ( following a path that lies close to f # (where n ##) A formal treatment of this concept is found in Chapter 6 of [29], in particular Theorem 6.15. Clearly this optimal path gives much insight into the system conditional on overflow, and appears to be useful in developing e#cient simulation methods, see Section 4. 5 3 Analysis This section finds the decay rate (2) for the case of exponential on o# sources, and ....

[Article contains additional citation context not shown here]

A. Shwartz and A. Weiss. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, 1995.

....we hope that the problem framework which we establish in the SRBM setting will provide motivation for further research into the interesting and challenging open problems beyond the 2 dimensional case. There is a large body of literature on LDPs for random walks and queueing networks. The book [33] by Shwartz and Weiss contains an excellent list of references. We can only provide a short survey of the latest works which are most closely related to our study. Recent work on LDPs for queueing networks include O Connell [30,31] and Dupuis and Ramanan [13] on multi bu#er single server systems, ....

A. Shwartz and A. Weiss. Large deviations for performance analysis. Chapman & Hall, New York, 1995.

....that there is a particular path f # such that lim n## 1 n log H n = lim n## 1 n log P L n (s) # f # (s) s # [0, t] Therefore, f # is the most likely path. A formal treatment lies in the theory of large deviations for sample paths and is described in detail in Shwartz Weiss [8]. However, their method gives an implicit representation of the optimal path, viz. the solution of a 7 variational problem in function space. In [6] we determined the optimal path explicitly in the Markov fluid model with a problem which is slightly di#erent from the one we consider here. Namely, ....

A. Shwartz and A. Weiss. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, 1995.

....Chernoff Hoeffding bounds are large deviation estimates for sums S = P n i=1 X i of independent, identically distributed random variables X i , that is, they provide bounds of the form Pr(S an) inf t 0 e Gammaant (E[exp(tX 1 ) n for a E[X 1 ] E[S] n ; 6. 1) see, for example, [15, 10]. The independence assumption can be replaced by the requirement that E[exp(t X i X i ) Y i E[exp(tX i ) Because of Proposition 4, this is easily seen to be fulfilled if the X i s are negatively associated. Theorem 14 Let X 1 ; X n be identically distributed random variables ....

Shwartz, A. and Weiss, A. (1995) Large Deviations for Performance Analysis. Chapman & Hall, London. 10

....a state z 2 S x . Given a sequence oe of length n, let b p z be the number of times oe reaches state z, divided by n, and let b p z;i be the number of times oe takes transition i out of state z, divided by the total number of transitions taken by oe out of state z. From large deviation theory [ShW], we know that Pr(jb p z Gamma p z j ffi) is exponentially small in n for ffi 0, where n is the length of the sequence. Similarly, from [ShW] we see that Pr(jb p z;i Gamma p z;i j ffi) is exponentially small in n. The exponentially small probability is of the form O(e Gammaaffi 2 n ....

....number of times oe takes transition i out of state z, divided by the total number of transitions taken by oe out of state z. From large deviation theory [ShW] we know that Pr(jb p z Gamma p z j ffi) is exponentially small in n for ffi 0, where n is the length of the sequence. Similarly, from [ShW], we see that Pr(jb p z;i Gamma p z;i j ffi) is exponentially small in n. The exponentially small probability is of the form O(e Gammaaffi 2 n ) where a depends on the Markov source. Using Lemmas 5 and 6, we have E(Fault X;n ) Gamma F oe p 2ffi (18) with exponentially small ....

A. Shwartz & A. Weiss, Large Deviations for Performance Analysis, Chapman & Hall, 1995.

....approach, in which we develop a deterministic model corresponding to the limiting system as n # #. We call this system the infinite system, and also refer to the method as the infinite system approach. This approach has successfully been applied previously to study load balancing problems in [1, 12, 13, 14, 15, 22] (see also [1] for more references, or [21] for the use of this approach in a different setting) and it can be seen as a generalization of the previous Markov chain analysis. Using this technique, we examine several new models of load balancing in the presence of old information. In conjunction ....

....modeled by an infinite system. The infinite system consists of a set of differential equations, which we shall describe below, that describe the expected behavior of the system. This corresponds to the exact behavior of the system as n # #. More information on this approach can be found in [6, 8, 12, 13, 14, 15, 22]. We note, however, that this approach works only because the systems for finite n have an appropriate form as a Markov chain; indeed, we initially require exponential service times and Poisson arrivals to ensure this form. Previous experience suggests that using the infinite system to estimate ....

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A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, 1995, Chapman & Hall.

....and time scales respectively. Also, its longterm effective bandwidth is given by ff j (s) lim t 1 ff j (s; t) Effective bandwidths have been computed in literature for a wide variety of source models, and large deviations estimates of the queue length tail probabilities have been studied [21]. For our purposes, we use the following form for the queue length tail probabilities [7] which is found to work well for markovian source models : PfQ qg e Gammaffiq (10) where ffi is the queue length decay rate computed as follows: ffi = maxfs : J X j=1 k j ff j (s) Cg (11) where C ....

A. Shwartz, A. Weiss, Large Deviations for Performance Analysis, Chapman & Hall, 1995. 17

....11 Let us now de ne J(B) lim N 1 1 N log Pr(B N ) for any sequence of events B N when such a limit exists. Now consider the sequence of random variables fA (m) 1 ( T ; 0)g; m 0. For each T , these form a sequence of i.i.d. random variables. Let I A1 ;T (x) be its rate function (see [15]) Formally, for a sequence of random variables Zm ; m 0, we de ne the rate function as follows: De ne Z m ( 1 m log E e Zm . Assume that the limit Z ( lim N 1 Z m ( exists pointwise. Then, I Z (x) sup x Z ( Similarly de ne I A2 ;T (x) ....

A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Chapman and Hall, 1995.

....2] may also be used to establish results for the continuous time GPS system. The paper deals only with the large buffer asymptotics under the GPS scheduling. Another future direction is to study the asymptotical behavior of the GPS scheduling with a large number of sources a la the methods of [29, 3]. Acknowledgement I am indebted to Prof. Richard Ellis for teaching me Probability Theory and Large Deviation Theory and to my advisor Don Towsley for encouragement and many helpful discussions. A Proof of Sample Path Lower Bound on Output Processes Proof of Lemma 6: First note that (25) follows ....

A. Shwartz and A. Weiss, Large Deviations for Performance Analysis. New York, Chapman and Hall, 1995.

....regime, great caution should be exercised. The first equality involving lim inf B 1 1 B log in the lower bound proof (p. 11) of [16] where Q 1 and B are simultaneously replaced by Q 1;m and mfi, is dubious. 4 For simplicity, we do not state the results in their most general form. [14, 20, 37] are good sources for reference on the subject. For application of large deviation theory in communication networks, see the excellent survey paper [38] 3 Upper Bound: For every closed set F , lim sup n 1 1 n log n (F ) Gamma inf x2F I(x) 1) Lower Bound: For every open set G, lim ....

....4] may also be used to establish results for the continuous time GPS system. The paper deals only with the large buffer asymptotics under the GPS scheduling. Another future direction is to study the asymptotical behavior of the GPS scheduling with a large number of sources a la the methods of [37, 5]. Acknowledgement I am indebted to Prof. Richard Ellis for teaching me Probability Theory and Large Deviation Theory and to my advisor Don Towsley for encouragement and many helpful discussions. A Proofs of the Lemmas in Section 2 Proof of Lemma 2: E[e (Q( Gammat) A( Gammat;0) E h ....

A. Shwartz and A. Weiss, Large Deviations for Performance Analysis. New York, Chapman and Hall, 1995.

....where such an analysis is attempted on the distribution of a busy period of fractional Brownian storage. Our technical framework here is the theory of large deviations in path space of a Gaussian process. The inspiration for doing this came from reading the recent book by A. Shwartz and A. Weiss [10]. It deals, however, only with Markovian systems, and the general theory must in our case be taken from the world of general Gaussian processes. The problem of describing long busy periods of the fractional Brownian storage was raised in [6] in the following context. There is lot of evidence that ....

A. Shwartz and A. Weiss. Large Deviations for Performance Analysis. Chapman & Hall, 1995. 19

.... Technion Israel Institute of Technology Haifa 32000, Israel Alan Weiss Bell Laboratories, Lucent Technologies Murray Hill, NJ March 2002 Abstract The theory of large deviations for jump Markov processes has been generally proved only when jump rates are bounded below, away from zero [2, 5, 7]. Yet various applications of interest do not satisfy this condition. We describe several classes of models where jump rates diminish to zero in a Lipschitz continuous way. Under appropriate conditions, we prove that the sample path large deviations principle continues to hold. Under our ....

....our conditions, the rate function remains an integral over a local rate function, which retains its standard representation. 1 Introduction One of the common technical assumptions in existing large deviations theory for jump Markov processes is that jump rates are bounded below, away from zero [2, 5, 7]. This is not merely a technical assumption: if the rates may go down to zero, the process may get stuck at a point, or it may or may not be possible to reach certain regions. We illustrate these issues below through # Research supported in part by the United States Israel Binational Science ....

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A. Shwartz and A. Weiss, Large Deviations for Performance Analysis, Queues, Communication and Computing. London, U.K.: Chapman & Hall, 1995.

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A. Shwartz and A. Weiss (1995). Large Deviations for Performance Analysis. Chapman and Hall, New York.

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A. Schwarz and A. Weiss, Large Deviations for Performance Analysis. London, U.K.: Chapman and Hall, 1995.

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A. Shwartz and A. Weiss, Large deviations for performance analysis. Chapman & Hall, London, 1995.

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A. Shwartz and A. Weiss. Large deviations for performance analysis, queues, communication, and computing. Chapman and Hall, New York, NY, USA, 1995.

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A. Shwartz and A. Weiss. Large deviations for performance analysis. Chapman and Hall, 1995.

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A. Shwartz and A. Weiss. Large deviations for performance analysis. Chapman & Hall, New York, 1995.

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Adam Shwartz and Alan Weiss. Large Deviations for Performance Analysis. Chapman and Hall, 1995. ISBN 0-412-06311-5.

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A. Weiss. Large Deviations for Performance Analysis. Chapman & Hall, London, 1995.

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A. Shwartz and A. Weiss. Large deviations for performance analysis. Chapman and Hall, 1995.

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Schwartz, A. & Weiss, A. (1995) Large Deviations For Performance Analysis. Chapman & Chall, London.

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Schwartz, A. and Weiss, A. (1995) Large Deviations For Performance Analysis. Chapman and Chall, London.

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A. Shwartz, A. Weiss, Large Deviations for Performance Analysis, 1995, Chapman & Hall.

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