A. Weiss (1986). A new technique for analyzing large traffic systems. Adv. Appl. Prob., 506--532.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of (12) there are interesting indications that in many cases a stronger condition holds, namely R B : 13) This is equivalent with the concavity of the scaled asymptotic rate function R w.r.t. B. Indeed, for the superposition of ON OFF Markov fluids when B is small it was shown in [26] that R C 1 C 2 B, where C 2 0. Also, for large B, it is shown in [16] that R converges to a linear function in B; similar indications can be found in [24] On the other hand, concavity is not always the case, as for the sub bursty Markovian sources considered in [17] Since we could ....

A. Weiss, "A new technique for analyzing large traffic systems," Advances in Applied Probability, vol. 18, pp. 506--532, 1986.

....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. Weiss. A new technique of analyzing large traffic systems. Advances of Applied Probability, 18: 506 -- 532, 1986. 32

....of (17) there are interesting indications that in many cases a stronger condition holds, namely FR B : 22) This is equivalent with the concavity of the scaled asymptotic rate function FR w.r.t. B. Indeed, for the superposition of ON OFF Markov fluids and small B it was shown in [13] that FR C 1 C 2 p B. Also, for large B, it is shown in [4] that FR converges to a linear function in B; similar indications can be found in [5] On the other hand, concavity is not always the case, as for the sub bursty Markovian sources considered in [2] Since we could not hope for a ....

A. Weiss. A new technique for analyzing large traffic systems. Adv. Appl. Prob., 18:506--532, 1986.

.... the asymptotic estimate of the loss probability when there are many sources and the buffering per source large is given by (10) Note that (10) is simply the Chernoff s estimate of the probability that the streams are in a combination of subchains whose total mean rate exceeds the channel capacity [46], 23] Note that this estimate does not depend on the fast time scale statistics of the streams nor on the specific value of buffer size , provided that it is large enough to absorb the fast time scale variations of the streams. This result can be interpreted as a decomposition of the gain from ....

A. Weiss, "A new technique for analyzing large traffic systems," Adv. Appl. Prob., vol. 18, pp. 506--532, 1986.

....An intrinsic drawback of the analytical approach described so far is that it is computationally intensive. This provides motivation for simpler, asymptotic approaches to deal with the access models. In both access models, we study largedeviations asymptotics for the scaling introduced by Weiss [31], i.e. the regime in which the number of users grows large and resources (buffer and bandwidth) scale proportionally. We derive exponential approximations , comparable to those in [3] 4] 9] for the ordinary FIFO discipline. Exponential approximations of the first feedback model, i.e. without ....

....is its computational complexity. When the size of the system (i.e. the number of sources) grows, a large eigensystem needs to be solved. This explains the interest in simpler asymptotic approaches. In this section we will focus on the socalled many sources scaling , which was introduced by Weiss [31]. In this regime, we derive explicit results on the overflow probability. In the many sources scaling, buffer and bandwidth resources are scaled by the number of users N . In other words, if we scaleC Nc, the exponential decay rate ofIP Nx) can be determined explicitly in terms of x and ....

A. Weiss. A new technique of analyzing large traffic systems. Advances in Applied Probability, 18: 506 -- 532, 1986.

....of (14) there are interesting indications that in many cases a stronger condition holds, namely FR B FR B : 18) This is equivalent with the concavity of the scaled asymptotic rate function FR w.r.t. B. Indeed, for the superposition of ON OFF Markov fluids and small B it was shown in [13] that FR C 1 C 2 p B . Also, for large B, it is shown in [6] that FR converges to a linear function in B; similar indications can be found in [11] On the other hand, concavity is not always the case, as for the sub bursty Markovian sources considered in [7] Since we could not hope for a ....

Alan Weiss, "A new technique for analyzing large traffic systems," Adv. Appl. Prob., vol. 18, pp. 506--532, 1986.

....X R [f1g has compact level sets. If X is a process, this is called a sample path LDP. The left and right hand sides of this inequality are referred to as the large deviations lower and upper bounds. 2. 2 Related work The many sources limiting regime was described in an early paper of Weiss [57]. It has more recently been studied by Botvich and Duffield [4] and Courcoubetis and Weber [11] and others, whose work will be described in Chapter 3. Another limiting regime which has been much more widely studied is the large buffer asymptotic, in which X L is a speeded up version of a base ....

....by Likhanov and Mazumdar [34] Their techniques involve a lot more technical details and give only a little extra insight, so we stick with large deviations. CHAPTER 3. QUEUES 19 Another benefit of the sample path LDP approach is that it tells us the most likely sample path to overflow. Weiss [57], who introduced the many sources asymptotic, obtained similar results for the special case of an on off Markov source using direct methods; and Mandjes and Ridder [40] have too for Markovmodulated sources and periodic sources. Our results hold much more generally. The contraction principle ....

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A. Weiss. A new technique for analyzing large traffic systems. Advances in Applied Probability, 18:506--532, 1986.

....the matching and then discuss our specific choice of t m . We choose a time period of one second as it is the sum of the average talkspurt and silence period durations. Hence, it is easy to calculate given the source parameters. This choice has been suggested in a number of earlier papers. Weiss [23] refers to it as the relaxation time. He takes it to be the memory in the source for the initial condition. Ramaswamy [18] recommends one second as an appropriate time scale to model the superposition, based on the time period over which the peaks in the serial correlation of counts for the ....

....proceeding through the two stages is slightly greater than the time taken to solve the fluid model for the parameter ranges considered here. On the other hand, the fluid model potentially suffers from numerical problems which may cause it to break down for large systems; the reader may refer to [23] for examples of such systems (the problem is alleviated to a extent in [22] by truncation of the state space) The renewal approximation is the simplest and most versatile of all the models. However, it has limited accuracy. It may be used in heavy traffic situations with reasonable accuracy. ....

Alan Weiss. A new technique for analyzing large traffic systems. Journal of Applied Probability, pages 506--532, 1985.

....by a continuous fluid flow. The tail probabilities for multiple on off sources sharing a large buffer is obtained with a fluidflow model [1] Using large deviation techniques it is shown that the fluid flow approximation corresponds to the most probable path for continuous time Markov processes [19]. The effective bandwidth for multiple Markov modulated fluid flow sources sharing a statistical multiplexer with a large buffer is obtained [10,12] Large deviation techniques have also been used to bound traffic parameters to obtain exponential upper bounds on the queue length and delay in ....

.... at first sight is justifiable a priori for the resulting elegant structures; and aposteriori for simple systems such as the M M 1 B queue and the M M C C B queue (see Appendix A) and for systems with no buffers (B = 0) The scalability property for the fluid flow model is asymptotically shown in [19] for large C, B and N . In each of these cases, the GoS is defined as the negative logarithm of the loss probability. The quantities C, G, N , B bears a certain remarkable parallel respectively with the thermodynamic quantities E, S, N , V as shall be pointed out later. This resemblance is due ....

A. Weiss, "A new technique for analyzing large traffic systems,"Adv. in Appl. Probab., vol. 18, 1986.

....of (17) there are interesting indications that in many cases a stronger condition holds, namely FR B FR B : 22) This is equivalent with the concavity of the scaled asymptotic rate function FR w.r.t. B. Indeed, for the superposition of ON OFF Markov fluids and small B it was shown in [13] that FR C 1 C 2 p B. Also, for large B, it is shown in [4] that FR converges to a linear function in B; similar indications can be found in [5] On the other hand, concavity is not always the case, as for the sub bursty Markovian sources considered in [2] Since we could not hope for a ....

A. Weiss. A new technique for analyzing large traffic systems. Adv. Appl. Prob., 18:506--532, 1986.

....in addition to a large buffer and rare transition probabilities, and derive better estimates of the loss probability which in turn yields a formula for the effective bandwidth less conservative than (5. 16) The asymptotic regime where there are large number of sources has also been analysed in [Wei86] and [Hui88] for single time scale sources. As a base case for comparison, consider first the problem of approximating the loss probability when a large number of statistically identical and independent single (fast) time scale streams are multiplexed together. Each stream fX (j) t g(j = 1; 2; ....

A. Weiss. A new technique for analyzing large traffic systems. Advances in Applied Probability, 18:506--532, 1986.

....we use also allow the study of delays in the network, although we are not concerned with that here. Another extension that can be dealt with using the techniques developed here allows the number of users and the buffer sizes to tend to infinite together. The single switch model is considered in [24]. One of the standard approaches to dealing with the one switch problem (and a number of extensions as well) is to employ a construction due to Loynes [18] This construction heavily uses the fact that IR is well ordered and does not seem to apply to multidimensional problems such as ours which ....

A. Weiss. A new technique for analyzing large traffic systems. Adv. Appl. Probab., 21:506-- 532, 1986.

....of L. Then the asymptotic estimate of the loss probability when there are many sources is p exp( GammaL ( c) Delta n) 3) Note that (3) is simply the Chernoff s estimate of the probability that the streams are in a combination of subchains whose total mean rate exceeds the channel capacity [27, 16], and does not depend on the fast time scale statistics of the streams. The buffering essentially absorbs the fast time scale variations of the streams but has little effect on the slow time scale. There has been some recent work suggesting that compressed video traffic can have a self similar ....

A. Weiss. A New Technique for Analyzing Large Traffic Systems. Advances in Applied Probability, 18:506--532, 1986.

....the mean to the peak value of X jk [0; t] t as s increases from 0 to 1. Let L(C; B; n) be the proportion of workload lost, through overflow of a buffer of size B 0, when it is served at rate C and n = n 1 ; n 2 ; n J ) An important limiting regime, first considered in a key paper of Weiss (1986), is one in which the number of sources and the buffer size increase together. Courcoubetis and Weber (1996) have shown that lim N 1 1 N log L(CN;BN;nN) sup t inf s 2 4 st J X j=1 n j ff j (s; t) Gamma s(Ct B) 3 5 : 2) The corresponding result has been proved in continuous time ....

Weiss, A. (1986) A new technique for analyzing large traffic systems. Adv. Appl. Prob.

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A. Weiss (1986). A new technique for analyzing large traffic systems. Adv. Appl. Prob., 506--532.

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A. Weiss. A new technique of analyzing large traffic systems. Advances in Applied Probability, 18: 506 -- 532, 1986.

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Weiss, A., "A new technique for analyzing large traffic systems," Adv. Appl. Prob. 18 pp. 506-- 532, 1986.

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A. WEISS. A new technique of analyzing large traffic systems. Advances of Applied Probability, 18: 506 - 532, 1986.

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Weiss, A. (1986). A new technique for analyzing large traffic systems. Adv. Appl. Prob., 18, 506-532.

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Weiss, A., "A new technique for analyzing large traffic systems," Adv. Appl. Prob. 18 pp. 506-- 532, 1986.

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