Asymptotic regimes and approximations for discriminatory processor sharing

Kessel, G. van; Núñez-Queija, Rudesindo; Borst, Sem

doi:10.1145/1035334.1035352

Cited by 19 publications

(16 citation statements)

References 7 publications

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“…After developing a procedure to determine all moments of the queue length distributions from systems of linear equations, [26] show that the variability of the queue length vector is of a lower order than the mean queue lengths, which directly leads to state space collapse of the multi-dimensional queue length process. In [22] it was indicated that a similar approach as in [26] could be followed for the heavy-traffic analysis of the DPS queue with phase-type distributions. Here we follow a different and more direct approach, by investigating the joint probability generating function of the queue lengths.…”

Section: Introductionmentioning

confidence: 99%

Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing

Verloop

Ayesta

Núòez-Queija

2011

Operations Research

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We analyze a generalization of the discriminatory processor-sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume that customers have phase-type distributed service requirements and allow that customers have different weights in various phases of their service. In our main result we establish a state-space collapse for the queue-length vector in heavy traffic. The result shows that in the limit, the queue-length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by Rege and Sengupta [Rege, K. M., B. Sengupta. 1996. Queue length distribution for the discriminatory processor-sharing queue. Oper. Res. 44(4) 653–657], who considered a DPS queue with exponentially distributed service requirements. Their analysis was based on obtaining all moments of the queue-length distributions by solving systems of linear equations. We undertake a more direct approach by showing that the probability-generating function satisfies a partial differential equation that allows a closed-form solution after passing to the heavy-traffic limit. Making use of the state-space collapse result, we derive interesting properties in heavy traffic: (i) For the DPS queue, we obtain that, conditioned on the number of customers in the system, the residual service requirements are asymptotically independent and distributed according to the forward recurrence times. (ii) We then investigate how the choice for the weights influences the asymptotic performance of the system. In particular, for the DPS queue we show that the scaled holding cost reduces as classes with a higher value for dk/E(Bkfwd) obtain a larger share of the capacity, where dk is the cost associated to class k, and E(Bkfwd) is the forward recurrence time of the class-k service requirement. The applicability of this result for a moderately loaded system is investigated by numerical experiments.

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Section: Introductionmentioning

confidence: 99%

Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing

Verloop

Ayesta

Núòez-Queija

2011

Operations Research

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“…Rege and Sengupta [19] obtained the moments of the queue length distributions as the solutions to linear equations for the case of exponential service requirements, and they also proved a heavytraffic limit theorem for the joint queue length distribution. These results were extended to phase type distributions by van Kessel et al [13]. Kim and Kim [15] found the moments of the sojourn time in the M/M/1 DPS queue as a solution of linear simultaneous equations.…”

Section: Introductionmentioning

confidence: 67%

Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

Cheung

Berg

Boucherie

2005

Performance Evaluation

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We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for multi-class egalitarian PS queues, we show that the marginal queue length distribution for each class equals the queue length distribution of an equivalent single class PS model with a random number of permanent customers. Similarly, the mean sojourn time (conditioned on the initial service requirement) for each class can be obtained by conditioning on the number of permanent customers. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian PS models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights.

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“…The sensitivity increases when ρ = 0.5, but still remains limited. For ρ = 0.75, however, the sensitivity is considerably amplified, and becomes even stronger for ρ = 0.9 and ρ = 0.95 [19]. Evidently, for w ≈ 1, the strict insensitivity to the service requirement distribution of the ordinary multi-class PS system manifests itself, regardless of the load.…”

Section: Sensitivity To the Service Requirement Distributionmentioning

confidence: 94%

“…First we consider a relatively lightly-loaded system (ρ = 0.25; Tables I-V), and then repeat the experiment for a more heavily-loaded system (ρ = 0.75; Tables VI-X). For ρ = 0.5, ρ = 0.9 and ρ = 0.95, we refer to [19].…”

Section: Sensitivity To the Service Requirement Distributionmentioning

confidence: 99%

Differentiated bandwidth sharing with disparate flow sizes

Kessel

Núñez-Queija

Borst

Proceedings IEEE 24th Annual Joint Conference of the IEEE Computer and Communications Societies.

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Abstract-We consider a multi-class queueing system operating under the Discriminatory Processor-Sharing (DPS) discipline. The DPS discipline provides a natural approach for modeling the flow-level performance of differentiated bandwidth-sharing mechanisms. Motivated by the extreme diversity in flow sizes observed in the Internet, we examine the system performance in an asymptotic regime where the flow dynamics of the various classes occur on separate time scales. Specifically, from the perspective of a given class, the arrival and service completions of some of the competing classes (called mice) evolve on an extremely fast time scale. In contrast, the flow dynamics of the remaining classes (referred to as elephants) occur on a comparatively slow time scale. Assuming a strict separation of time scales, we obtain simple explicit expressions for various performance measures of interest, such as the distribution of the numbers of flows, mean delays, and flow throughputs. In particular, the latter performance measures are insensitive, in the sense that they only depend on the service requirement distributions through their first moments. Numerical experiments show that the limiting results provide remarkably accurate approximations in certain cases 1 .

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Asymptotic regimes and approximations for discriminatory processor sharing

Cited by 19 publications

References 7 publications

Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing

Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing

Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

Differentiated bandwidth sharing with disparate flow sizes

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