Single server retrial queues with speed scaling: Analysis and performance evaluation

Tuan Phung-Duc; Wouter Rogiest; Sabine Wittevrongel

doi:10.3934/jimo.2017025

Article Contents

2017, Volume 13, Issue 4: 1927-1943. Doi: 10.3934/jimo.2017025

This issue Previous Article Analysis of a discrete-time queue with general service demands and phase-type service capacities Next Article Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue

Single server retrial queues with speed scaling: Analysis and performance evaluation

1.
Division of Policy and Planning Sciences, Faculty of Engineering, Information and Systems, University of Tsukuba, Ibaraki 305-8573, Japan
2.
Department of Telecommunications and Information Processing, Ghent University, St.-Pietersnieuwstraat 41, B-9000 Gent, Belgium

* Corresponding author
* Corresponding author

Received: October 2015

Published: November 2017

The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors.

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Abstract

Recently, queues with speed scaling have received considerable attention due to their applicability to data centers, enabling a better balance between performance and energy consumption. This paper proposes a new model where blocked customers must leave the service area and retry after a random time, with retrial rate either varying proportionally to the number of retrying customers (linear retrial rate) or non-varying (constant retrial rate). For both, we first study a basic case and then subsequently incorporate the concepts of a setup time and a deactivation time in extended versions of the model. In all cases, we obtain a full characterization of the stationary queue length distribution. This allows us to evaluate the performance in terms of the mentioned balance between performance and energy, using an existing cost function as well as a newly proposed variant thereof. This paper presents the derivation of the stationary distribution as well as several numerical examples of the cost-based performance evaluation.

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Mathematics Subject Classification: Primary: 68M20, 60K25; Secondary: 90B22.

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Figure 1. Transitions among states.

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Figure 2. Transitions among states.

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Figure 3. Transitions among states.

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Figure 4. Transitions among states.

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Figure 5. Cost $z$ as function of the service rate $\mu$

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Figure 6. Cost $z$ as function of the service rate $\mu$

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Figure 7. Cost $z$ as function of the service rate $\mu$

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Figure 8. Cost $y$ (second cost function) as function of the service rate $\mu$

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