A. Dembo and T. Zajic. Large deviations: from empirical mean and measure to partial sums. Stochastic Processes and their Applications, 57:191--224, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[8] Glynn and Whitt [16] 3 Using the sample path large deviation not only provides a proof for the lower bound, but also yields a physical insight on how congestion occurs. This permits us to extend the conditional limit theorem for a single queue (see e.g. Anantharam [1] and Dembo and Zajic [10]) to intree networks. Our main sources for sample path large deviation are Dembo and Zeitouni [9] and Dembo and Zajic [10] Also, we note that the concept of sample path large deviation is also recently used in Glynn and Whitt [17] for inverting counting processes. To state our results precisely, ....

....yields a physical insight on how congestion occurs. This permits us to extend the conditional limit theorem for a single queue (see e.g. Anantharam [1] and Dembo and Zajic [10] to intree networks. Our main sources for sample path large deviation are Dembo and Zeitouni [9] and Dembo and Zajic [10]. Also, we note that the concept of sample path large deviation is also recently used in Glynn and Whitt [17] for inverting counting processes. To state our results precisely, we make the following assumptions on the external arrival processes and routing processes: A1) All the external arrival ....

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A. Dembo and Tim Zajic, "Large deviations from empirical mean and measure to partial sums processes," preprint, 1993.

.... the LDP for the partial sum random variable S n 2 R, is the LDP for the partial sum process (Sample path LDP) S n (t) bntc X i=1 X i # t 2 [0# 1] Note that the random variable S n = i=1 X i corresponds to the terminal value (at t =1) oftheprocess S n # t 2 [0# 1] In a key paper [12], under certain mild mixing conditions on the stationary sequence fX i # i 1g, the authors establish an LDP for the process S n in D[0# 1] the space of right continuous functions with leftlimits) equipped with the supremum norm topology. In the spirit of the sample path LDP in [12] ....

....a key paper [12] under certain mild mixing conditions on the stationary sequence fX i # i 1g, the authors establish an LDP for the process S n in D[0# 1] the space of right continuous functions with leftlimits) equipped with the supremum norm topology. In the spirit of the sample path LDP in [12] wewillbeassuming the following. Assumption B For all m 2 N, for every ffl 1 , for every ffl 2 0 sufficiently small, and for every scalars a 0 #: #a m;1 , there exists M 0 such that for all n M and all k 0 #: #k m with 1=k 0 k 1 Delta Delta Deltak m = n, nffl 2 P[jS k i 1 ; ....

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A. Dembo and T. Zajic. Large deviations: From empirical mean and measure to partial sums processes. Stochastic Processes and Applications, 57:191--224, 1995.

....processes X it is possible to choose a normalising function f(L) such that Condition 1 is satisfied. Often, the normalising function is just f(L) L, and the limit t has the simple linear form t ( P t i=1 1 ( i ) For an account of conditions under which this occurs, see Dembo and Zajic [14]. In Example 2.5 below, the normalising function is not linear and t has a more complicated form. Suppose for now that t has the simple linear form: this gives as the rate function I(x) P t 1 (x t ) Then Condition 2 is satisfied. To see this, choose v(t) t, so that ( 1 ( ....

....level, f(L) t. In the latter case, the time taken to fill the buffer tends to infinity and so the rate function I(b) depends only on the infinite time characteristics of the source. For Markov modulated fluid sources (and many other sources which satisfy conditions described by Dembo and Zajic [14]) it is appropriate to take f(L) L and so t ( 1) t lim L 1 L Gamma1 log E exp( X(0; L] Then the rate function I(b) simplifies to I(b) sup b, where the supremum is taken over all such that 1 ( C. By contrast, under the many sources asymptotic the rate function depends on the ....

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Amir Dembo and Tim Zajic. Large deviations: From empirical mean and measure to partial sums process. Stochastic processes and their applications, 57:191--224, 1995.

....In this section, we will present a very quick summary of the relevant background. Denote by S T n the restriction of S n to the interval [0; T ] and by A T the space of absolutely continuous functions on [0; T ] with (0) 0, eqquipped with the uniform topology. Dembo and Zajic [12] establish quite general conditions for which S T n satis es the LDP in A T with good convex rate function given by I( Z T 0 ( ds: For the LDP to hold in the iid case, it is sucient that the moment generating function Ee X 1 exists and is nite everywhere; this is a ....

Amir Dembo and Tim Zajic (1995). Large deviations: from empirical mean and measure to partial sums process. Stoch. Proc. Appl. 57:191{ 224.

....principle with the good rate function . Now for n = 1; 2; define a sequence of scaled partial sum processes of fY n ; n = 1; 2; g on [0; 1] Z (n) t) 1 n bntc X i=1 Y i ; 0 t 1: Clearly, Z (n) 2 D( 0; 1] IR) Let (n) be the probability measure of Z (n) In [13], conditions are established under which f (n) n = 1; 2; g satisfies the sample large deviation deviation principle and the good rate function I(OE) is identified with 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; ....

A. Dembo and T. Zajic, "Large Deviations:from empirical mean and measure to partial sums process Techniques ", Preprint, 1994.

.... from this behavior and look like some other function a(t) 1 To keep the discussion simple, let us assume that t 2 [0; t f ] t f 1: We will now informally argue that, for very large n; P (S n (t) a(t) e Gamman R t f 0 I( a(s) ds : A precise version of the above result is in [14]. Let us divide the interval [0; t f ] into M small subintervals, each of length Deltat: P (S n (t) a(t) P (S n (0) a(0) S n ( Deltat) a( Deltat) S n (2 Deltat) a(2 Deltat) S n (M Deltat) a(M Deltat) P (S n (M Deltat) a(M Deltat)jS n ( M Gamma 1) Deltat) a( M ....

A. Dembo and T. Zajic. Large deviations: from empirical mean and measure to partial sums process. Stochastic Processes and Applications, 67:195--211, 1995.

.... principle with the good rate function (x) Now for n = 1; 2; 3; define a sequence of scaled partial sum processes of fYn ; n 2 INg on [0; 1] Z (n) t) 1 n bntc X i=1 Y i ; 0 t 1: Clearly, Z (n) 2 D( 0; 1] IR) Let (n) be the probability measure of Z (n) In [12], conditions are established under which f (n) n = 1; 2; g satisfies the sample path large deviation deviation principle and the good rate function I(OE) is identified 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) ....

A. Dembo and T. Zajic, "Large Deviations:from empirical mean and measure to partial sums process Techniques", Preprint, 1994.

.... for small ffl 0, P[S n 2 (na Gamma nffl; na nffl) e Gamman (a) A stronger concept than the LDP for the partial sum random variable S n 2 R, is the LDP for the partial sum process (to be referred as Sample path LDP) S n (t) 1 n bntc X i=1 X i ; t 2 [0; 1] In a key paper [16], under certain mild mixing conditions on the stationary sequence fX i ; i 1g, the authors establish an LDP for the process S n ( Delta) in D[0; 1] right continuous functions with left limits) equipped with the supremum norm topology. In the spirit of the sample path LDP, we will be assuming ....

.... km = n, e Gamma ( nffl 2 P m Gamma1 i=0 (k i 1 Gammak i ) a i ) P[jS k i 1 Gamma S k i Gamma (k i 1 Gamma k i )a i j ffl 1 n; i = 0; m Gamma 1] 3) A detailed discussion of this Assumption, and the technical conditions under which it is satisfied can be found in [16]. Intuitively, Assumption B deals with the probability of sample paths that are constrained to be within a tube around a polygonal path made up with linear segments of slopes a 0 ; am Gamma1 . We will also be making the following assumption, which can be viewed as the convex dual ....

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A. Dembo and T. Zajic. Large deviations: From empirical mean and measure to partial sums processes. Stochastic Processes and Applications, 57:191--224, 1995.

....W up to time nT : S T n (t) W nt n ; t 2 [0; T ] 77) The sequence S T takes values in the space of absolutely continuous paths on [0; T ] which start at zero. Let P[0; T ] denote this space, equipped with the topology induced by the norm kOEk = sup 0tT jOE(t)j: 78) Dembo and Zajic [DZ95] have given general conditions under which the sequence of paths S T satisfies a large deviation principle in P[0; T ] with convex rate function L given by L T (OE) Z T 0 l( OE)dt; 79) for each finite T . This result would suffice for a study of the transient queue length distribution ....

A. Dembo and T. Zajic. Large deviations: from empirical mean and measure to partial sums process. Stoch. Proc. Appl., 57:191--224, 1995.

....difficult 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 ....

....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 ....

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A. Dembo and T. Zajic, Large deviations: From empirical mean and measure to partial sums processes, Preprint, 1993.

....n Sn [nt] 1 n Gamma Sn [nt] n : 2) For 2 lR 2 set ( lim n 1 1 n log Ee DeltaS n (1) 3) whenever this limit exists. Write for the convex dual of . Denote by A the space of absolutely continuous functions OE on [0; 1] with OE(0) 0. Dembo and Zajic [2] establish very general conditions for which Sn satisfies the LDP in A 2 with respect to the uniform topology, with good convex rate function given by I(OE) Z 1 0 ( OE)ds: 4) For such an LDP to hold in the i.i.d. case (see, for example, 3, Chapter 5] it is sufficient that the ....

Amir Dembo and Tim Zajic. Large deviations: from empirical mean and measure to partial sums process. Preprint.

....c) a (x) b (c) 7) We refer to the hypothesis (H2) as the linear geodesic property . It follows from (H2) the convexity of and Jensen s inequality, that the optimal path from point to point is a straight line. Such an LDP has been shown to hold quite generally by Dembo and Zajic [6]: roughly speaking, it holds provided the sequence is, in some sense, stationary and mixing. Under the above hypotheses, it was shown in [10] that the sequence R n satisfies the LDP in L d with good rate function given by I d ( inffI(OE) Delta(OE) g (8) where Delta : C d 1 C ....

A. Dembo and T. Zajic (1995). Large deviations: from empirical mean and measure to partial sums process. Stoch. Proc. Appl. 57:191--224.

....been established in a large class of queues serving increasingly many sources that at a given time, the most likely way for the queue length Q to exceed a given level b is for arrivals to build up over the previous interval whose duration (b) is asymptotically proportional to b for large b. See [2, 12] for some general results of this nature for sources obeying mixing conditions; see [15, 16] specifically for large superpositions) Now suppose that the arrival process takes a time t r to relax back to stationarity from its To be published in Performance Evaluation conditioned state. Then ....

A. Dembo and T. Zajic, Large deviations: from empirical mean and measure to partial sums process, Stoch. Proc. Appl. 57 (1995) 191--224.

.... for the partial sum random variable S n 2 R, is the LDP for the partial sum process (Sample path LDP) S n (t) 1 n bntc X i=1 X i ; t 2 [0; 1] Note that the random variable S n = P n i=1 X i corresponds to the terminal value (at t = 1) of the process S n (t) t 2 [0; 1] In a key paper [12], under certain mild mixing conditions on the stationary sequence fX i ; i 1g, the authors establish Bertsimas, Paschalidis, Tsitsiklis Large Deviations Analysis of the GPS Policy 7 an LDP for the process S n ( Delta) in D[0; 1] the space of right continuous functions with left limits) ....

....1g, the authors establish Bertsimas, Paschalidis, Tsitsiklis Large Deviations Analysis of the GPS Policy 7 an LDP for the process S n ( Delta) in D[0; 1] the space of right continuous functions with left limits) equipped with the supremum norm topology. In the spirit of the sample path LDP in [12] we will be assuming the following. Assumption B For all m 2 N, for every ffl 1 , for every ffl 2 0 sufficiently small, and for every scalars a 0 ; am Gamma1 , there exists M 0 such that for all n M and all k 0 ; km with 1 = k 0 k 1 Delta Delta Delta km = n, e Gamma ....

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A. Dembo and T. Zajic. Large deviations: From empirical mean and measure to partial sums processes. Stochastic Processes and Applications, 57:191--224, 1995. Bertsimas, Paschalidis, Tsitsiklis / Large Deviations Analysis of the GPS Policy 33

....case of X i which are real valued functions of a regular finite state Markov chain is treated. Schuette [18] analysed real valued, independent but not necessarily identical distributed X i under more restrictive conditions on the corresponding moment generating functions. Moreover Dembo and Zajic [4] proved an extension of Varadhan s result to cover the LDP for L n (t) 1 n [nt] X i=1 ffi X i ; t 2 [0; 1] 1.1) 1991 Mathematics Subject Classification. 60 F 10. Key words and phrases. Large deviations, partial sums, U processes. 2 PETER EICHELSBACHER AND MATTHIAS L OWE or for the ....

.... Gamma [nt] 1 Gamma n m Delta X C [nt] m Gamma1 h(X i 1 ; X i m Gamma1 ; X [nt] 1 ) is the polygonal approximation and is used in the proof of Theorem 2.4. The proof is based on the approach to Mogulskii s result taken in Dembo and Zeitouni [5, Section 5. 1] and in Dembo and Zajic [4]. The following two Lemmas are standard in the sense of this approach. Recall that the maps f e U n ( Delta) n 2 Ng and fU n ( Delta) n 2 Ng from( Omega ; S N ; P ) into L1 Gamma [0; 1] R d ; k Delta k) Delta are called exponentially equivalent if for each ffi 0 lim sup n 1 1 ....

Dembo, A. and T. Zajic, Large deviations: from empirical mean and measure to partial sums process, Stochastic Process. Appl. 57 (1995), 191--224.

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A. Dembo and T. Zajic, \Large deviations from empirical mean and measure to partial sums processes," Stoch. Proc. and Appl., Vol. 57, No. 2, pp. 191-224, 1995.

....dependent 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 ....

.... (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 ( ....

A. Dembo and T. Zajic, \Large deviations from empirical mean and measure to partial sums processes," Stoch. Proc. and Appl., Vol. 57, No. 2, pp. 191-224, 1995.

.... log E(e C(s;t s) jF C s ) C ( t C ( a:s: 8) where C ( is de ned in (A3) Moreover, C ( is di erentiable for all . These two conditions imply both a(t) and c(t) satisfy the sample path large deviation principle, a stronger concept than the large deviation principle (cf. [6, 10]) The following condition allows us to deal with the fact that in general, for t xed, a(t) and q(0) are dependent, as well as c(t) and q(0) A5) There exists 0 1 and a positive integer n such that sup j sup A2F j 1 ;B2F 1 j n f jP (A B) P (A)P (B)j P (A)P (B) g ; 9) where ....

.... 1 j n f jP (A B) P (A)P (B)j P (A)P (B) g ; 9) where F j 1 = a(t) 1 t j) and F 1 j n = a(t) j n t 1) Similarly (9) remains valid with a(t) replaced by c(t) We remark that our proof technique remains valid for other mixing conditions (for example, condition (S) of [10]) Indeed, a challenging problem is to nd an appropriate such condition which ensures nice mixing properties of the departure process. Finally, the following assumption is needed to justify applying the large deviation lower bound to closed sets in the proof of Theorem 2.2. A6) The following ....

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A. Dembo and T. Zajic, \Large deviations from empirical mean and measure to partial sums processes, " preprint, 1993.

....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 ffl Z 1 Gamma1 (OE 0 ....

A. Dembo and T. Zajic, "Large deviations from empirical mean and measure to partial sums processes," Stoch. Proc. and Appl., Vol. 57, No. 2, pp. 191-224, 1995.

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A. Dembo and T. Zajic. Large deviations: from empirical mean and measure to partial sums. Stochastic Processes and their Applications, 57:191--224, 1995.

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A. Dembo and T. Zajic, Large deviations: from empirical mean and measure to partial sums, Stochastic Processes and their Applications 57 (1995), 191--224.

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A. Dembo and T. Zajic. Large deviations: from empirical mean and measure to partial sums. Stochastic Processes and their Applications, 57:191--224, 1995.

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A. Dembo and T. Zajic, "Large deviations from empirical mean and measure to partial sums processes," preprint, 1993.

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Amir Dembo and Tim Zajic. Large deviations: from empirical mean and measure to partial sums process. Stoch. Proc. Appl. 57:191-224, 1995.

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Amir Dembo and Tim Zajic. Large deviations: from empirical mean and measure to partial sums process. Stoch. Proc. Appl. 57:191--224, 1995.

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