Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1997). Large deviations analysis of the generalized processor sharing policy. Preprint Boston University.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Fair Queueing, have emerged as an important mechanism for accommodating heterogeneous quality of service requirements in integrated services networks. The queueing analysis of GPS is extremely difficult. Interesting partial results for exponential traffic models were obtained in Bertsimas et al. [4], Dupuis Ramanan [9] Massouli e [15] and Zhang et al. 18] Here, we focus on non exponential traffic models. We show that, under mild assumptions, the tail behavior of the buffer content of an individual source with long tailed traffic characteristics is equivalent to the tail behavior when ....

Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1997). Large deviations analysis of the generalized processor sharing policy. Preprint Boston University.

....the utilization of the network. By e#cient, we mean that the method is simple and fast in order to be used as a part of on line call admission control. Generally, there are several methods to analyze the cell loss in a network: service curve based method [15] large deviation approximations [3, 6, 9 13], importance sampling [5, 7, 8] and bu#erless fluid flow approximation [4] The service curve based method is based on Markovian processes, which possess an interesting property that after traversing through a router or a switch, the output is also Markovian. This property simplifies the ....

....in a two queue GPS server. That indicates the bounds are still not tight. 4.2 Long range dependent tra#c For LRD tra#c sources, since their decay is less than exponential, we cannot continue using the single exponent form as shown in Eqn. 4.17) to bound their decay rate. Bertsimas and Borst [3, 9, 13] have addressed this problem and presented their solutions in di#erent ways. 9 In [13] the problem of estimating the queue length tail distribution is converted to an optimal control problem, and thus some standard approaches of optimal control can be used to derive the bounds on cell loss ....

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Dimitris Bertsimas, Ioannis Ch. Tsitsiklis, "Large Deviations Analysis of the Generalized Processor Sharing Policy", Queueing Systems: Theory and Applications, Vol. 32, pp 319--349.

....flows and provides service differentiation. Both deterministic and stochastic bounds (see e.g. 4] 5] 6] have been derived for GPS systems. Moreover, large deviation based approaches have been used to found asymptotical decay rates of different performance measures (see e.g. 7] 8] 9] [10], 11] The requirement that incoming traffic is infinitely divisible makes the GPS scheduling policy unrealistic. However, there are many practical implementations for packetized traffic (e.g. packet by packet GPS and virtual clock scheduling) Usually, Petteri Mannersalo and Ilkka Norros are ....

D. Bertsimas, I.C. Paschalidis, and J.N. Tsitsiklis, "Large deviations analysis of generalized processor sharing policy," Queueing Systems, vol. 32, pp. 319--349, 1999.

....policy among classes. By using sample path Large Deviations techniques we derive logarithmic equivalents of the quantities P(W i x) as x 1, where W i is the stationary amount of class i traffic backlogged. This generalizes previous results of Weber [13] O Connell [9] and Bertsimas et al. [1] who derived these equivalents in the case of only two customer classes. The results are illustrated in the case of Poisson traffic. It is shown, in particular, that for such traffic and in a symmetric case huge backlogs for a given class are more likely with Fair Queueing rather than with FIFO ....

....however, and give no information about performance in the case of unconstrained input. Estimates of the tail of the queue length distribution P(W i x) for large x have recently been derived for GPS service in the case of two traffic classes, by Weber [13] O Connell [9] and Bertsimas et al. [1]. In these papers, expressions have been obtained for the quantity ffi i : Gamma lim x 1 1 x log P(W i x) 1) by using sample path Large Deviations (sp LD) techniques. This in turn leads to approximate P(W i x) by exp Gammaffi i x. The main contribution of this paper is to derive ....

D. Bertsimas, I. Paschalidis and Tsitsiklis (1997), Large Deviations Analysis of the Generalized Processor Sharing Policy, Technical Report: MNS-97-108, Systems Group, College of Engineering, Boston University.

....service capacity is redistributed among the sources with non empty buffers in proportion to their respective weights. See [10] for a formal description of the evolution of the buffer content process. The queueing analysis of GPS is extremely difficult. Interesting partial results were obtained in [2], 10] 14] 17] If N = 2, then the above coupledprocessors model with r 1 = r 2 = 2 coincides with the GPS model with equal weights; hence the exact queue length analysis in [11] 13] for the case of exponentially distributed service requests, applies to this special GPS case. ....

D. Bertsimas, I.Ch. Paschalidis and J.N. Tsitsiklis (1997). Large deviations analysis of the generalized processor sharing policy. Report Boston University.

....such as Weighted Fair Queueing, have emerged as an important mechanism for achieving di#erentiated quality of service in integrated services networks. The queueing analysis of GPS is extremely di#cult. Interesting partial results for exponential tra#c models were obtained in Bertsimas et al. [5], Dupuis Ramanan [19] Massoulie [30] Zhang [36] and Zhang et al. 37] Here, we focus on non exponential tra#c models. Extending the results from [7] we show that, under certain conditions, an individual source with long tailed tra#c characteristics is e#ectively served at a constant rate, ....

Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1997). Large deviations analysis of the generalized processor sharing policy. Report Boston University.

....scheduling algorithms, such as Weighted Fair Queueing, play a major role in achieving differentiated quality of service in integrated services networks. The queueing analysis of GPS is extremely difficult. Interesting partial results for exponential traffic models were obtained in Bertsimas et al. [3], Dupuis Ramanan [14] Massouli e [22] Zhang [27] and Zhang et al. 28] Here, we focus on non exponential traffic models. We show that, in certain scenarios, a flow may be strongly affected by the activity of heavier tailed flows, and may inherit their traf2 fic characteristics, causing ....

Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1999). Large deviations analysis of the generalized processor sharing policy. Queueing Systems 32, 319-349.

.... to obtain comprehensive equilibrium large deviations results for a multiclass FIFO queue [21] and can also be applied to systems with dedicated bu ers [20, 22] the latter corresponds to the random walk in a quadrant, and is the subject of many recent papers: see, for example, Bertsimas et al. [2, 3]) Proof of Theorem 1 We have from the assumptions of the theorem and the Dawson G artner theorem for projective limits [11, Theorem 4.6.1] that the sequence S n satis es the LDP on C d (lR ) equipped with the topology of uniform convergence on 8 compacts, with the good rate function ....

D. Bertsimas, I. Paschalidis and J. N. Tsitsiklis. Large deviations analysis of the generalized processor sharing policy. Queueing Systems, to appear.

....scheduling algorithms, such as Weighted Fair Queueing, play a major role in achieving differentiated quality of service in integrated services networks. The queueing analysis of GPS is extremely difficult. Interesting partial results for exponential traffic models were obtained in Bertsimas et al. [6], Dupuis Ramanan [24] Massouli e [36] Zhang [46] and Zhang et al. 47] Here, we focus on non exponential traffic models, extending the results of [8] 11] We show that, for certain weight combinations, an individual flow with long tailed traffic characteristics is effectively served at a ....

Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1999). Large deviations analysis of the generalized processor sharing policy. Queueing Systems 32, 319--349.

....policy among classes. By using sample path Large Deviations techniques we derive logarithmic equivalents of the quantities P(W i x) as x # #, where W i is the stationary amount of class i tra#c backlogged. This generalizes previous results of Weber [18] O Connell [11] and Bertsimas et al. [1] who derived these equivalents in the case of only two customer classes. The results are illustrated in the case of Poisson tra#c. It is shown, in particular, that for such tra#c and in a symmetric case huge backlogs for a given class are more likely with Fair Queueing rather than with FIFO ....

....however, and give no information about performance in the case of unconstrained input. Estimates of the tail of the queue length distribution P(W i x) for large x have recently been derived for GPS service in the case of two tra#c classes, by Weber [18] O Connell [11] and Bertsimas et al. [1]. In these papers, expressions have been obtained for the quantity # i : lim x## 1 x log P(W i x) 1) by using sample path Large Deviations (sp LD) techniques. This in turn leads to approximate P(W i x) by exp # i x. The main contribution of this paper is to derive similar Large ....

D. Bertsimas, I. Paschalidis and Tsitsiklis (1997), Large Deviations Analysis of the Generalized Processor Sharing Policy, Technical Report: MNS-97-108, Systems Group, College of Engineering, Boston University.

....service capacity is redistributed among the sources with non empty buffers in proportion to their respective weights. See [10] for a formal description of the evolution of the buffer content process. The queueing analysis of GPS is extremely difficult. Interesting partial results were obtained in [2], 10] 14] 17] If N = 2, then the above coupled processors model with r 1 = r 2 = 2 coincides with the GPS model with equal weights; hence the exact queue length analysis in [11] 13] for the case of exponentially distributed service requests, applies to this special GPS case. ....

D. Bertsimas, I.Ch. Paschalidis and J.N. Tsitsiklis (1997). Large deviations analysis of the generalized processor sharing policy. Report Boston University.

....such as Weighted Fair Queueing, have emerged as an important mechanism for achieving differentiated quality of service in integrated services networks. The queueing analysis of GPS is extremely difficult. Interesting partial results for exponential traffic models were obtained in Bertsimas et al. [5], Dupuis Ramanan [17] Massoulie [27] Zhang [32] and Zhang et al. 33] Here, we focus on non exponential traffic models. Extending the results from [7] we show that, under certain conditions, an individual source with long tailed traffic characteristics is effectively served at a constant ....

Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1997). Large deviations analysis of the generalized processor sharing policy. Report Boston University.

....such as Weighted Fair Queueing, have emerged as an important mechanism for achieving differentiated quality of service in integrated services networks. The queueing analysis of GPS is extremely difficult. Interesting partial results for exponential traffic models were obtained in Bertsimas et al. [5], Dupuis Ramanan [16] Massoulie [26] and Zhang et al. 31] Here, we focus on non exponential traffic models. Extending the results from [7] we show that, under certain conditions, an individual source with long tailed traffic characteristics is effectively served at a constant rate, which ....

Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1997). Large deviations analysis of the generalized processor sharing policy. Report Boston University.

....Fair Queueing, have emerged as an important mechanism for accommodating heterogeneous quality of service requirements in integrated services networks. The queueing analysis of GPS is extremely difficult. Interesting partial results for exponential traffic models were obtained in Bertsimas et al. [4], Dupuis Ramanan [9] Massouli e [15] and Zhang et al. 18] Here, we focus on non exponential traffic models. We show that, under mild assumptions, the tail behavior of the buffer content of an individual source with long tailed traffic characteristics is equivalent to the tail behavior when ....

Bertsimas, D., Paschalidis, I.Ch., Tsitsiklis, J.N. (1997). Large deviations analysis of the generalized processor sharing policy. Preprint Boston University.

....overflow, under both policies, obtained from the solution of the Sec. 2 Preliminaries 4 corresponding control problems. In Section 7 we state the upper bound for the GPS policy (the proof is quite technical and involved and we omit it in the interest of space; we refer the interested reader to [BPT97b] Section 8 contains the proof for the upper bound in the GLQF case. We gather our main performance analysis results in Section 9, where we also treat the special case of strict priority policies. Finally, we compare the two scheduling policies in Section 10. Conclusions are in Section 11. 2 ....

....inf a 0 1 a II GLQF (a) 4.2 GPS lower Bound We next turn our attention to the GPS policy and establish a lower bound on the overflow probability. In the interest of space we provide an outline of the proof. The complete proof Sec. 5 The optimal control problem 14 can be found in [BPT97b] Proposition 4.2 (GPS Lower Bound) Assuming that the arrival and service processes satisfy Assumptions A and B, and under the GPS policy, the steady state queue length L 1 of queue Q 1 satisfies lim U 1 1 U log P[L 1 U ] Gamma GPS ; 20) where GPS is given by GPS = ....

[Article contains additional citation context not shown here]

D. Bertsimas, I. Ch. Paschalidis, and J. N. Tsitsiklis, Large deviations analysis of the generalized processor sharing policy, Tech. Report MNS-97-108, Department of Manufacturing Engineering, Boston University, June 1997, to appear Queueing Systems.

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