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Analysis of a discrete-time queue with general service demands and phase-type service capacities
October  2017, 13(4): 1927-1943. doi: 10.3934/jimo.2017025

## 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

Received  October 2015 Published  April 2017

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

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.

Citation: Tuan Phung-Duc, Wouter Rogiest, Sabine Wittevrongel. Single server retrial queues with speed scaling: Analysis and performance evaluation. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1927-1943. doi: 10.3934/jimo.2017025
##### References:
 [1] J. R. Artalejo, A. Economou and M. J. Lopez-Herrero, Analysis of a multiserver queue with setup times, Queueing Systems, 51 (2005), 53-76.  doi: 10.1007/s11134-005-1740-6. [2] J. R. Artalejo and T. Phung-Duc, Markovian retrial queues with two way communication, Journal of Industrial and Management Optimization, 8 (2012), 781-806.  doi: 10.3934/jimo.2012.8.781. [3] L. A. Barroso and U. Holzle, The case for energy-proportional computing, Computer, 40 (2007), 33-37.  doi: 10.1109/MC.2007.443. [4] R. Conway and W. L. Maxwell, A queueing model with state dependent service rate, Journal of Industrial Engineering, 12 (1961), 132-136. [5] A. Gandhi, M. Harchol-Balter and I. Adan, Server farms with setup costs, Performance Evaluation, 67 (2010), 1123-1138. [6] A. Gandhi, S. Doroudi, M. Harchol-Balter and A. Scheller-Wolf, Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward, Proceedings of the ACM SIGMETRICS, (2013), 153-166. [7] A. Gandhi, S. Doroudi, M. Harchol-Balter and A. Scheller-Wolf, Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward, Queueing Systems, 77 (2014), 177-209.  doi: 10.1007/s11134-014-9409-7. [8] X. Lu, S. Aalto and P. Lassila, Performance-energy trade-off in data centers: Impact of switching delay, Proceedings of 22nd IEEE ITC Specialist Seminar on Energy Efficient and Green Networking (SSEEGN), (2013), 50-55.  doi: 10.1109/SSEEGN.2013.6705402. [9] V. J. Maccio and D. G. Down, On optimal policies for energy-aware servers, Performance Evaluation, 90 (2014), 36-52.  doi: 10.1109/MASCOTS.2013.11. [10] I. Mitrani, Managing performance and power consumption in a server farm, Annals of Operations Research, 202 (2013), 121-134.  doi: 10.1007/s10479-011-0932-1. [11] P. R. Parthasarathy and R. Sudhesh, Time-dependent analysis of a single-server retrial queue with state-dependent rates, Operations Research Letters, 35 (2007), 601-611.  doi: 10.1016/j.orl.2006.12.005. [12] T. Phung-Duc, W. Rogiest, Y. Takahashi and H. Bruneel, Retrial queues with balanced call blending: Analysis of single-server and multiserver model, Annals of Operations Research, 239 (2016), 429-449.  doi: 10.1007/s10479-014-1598-2. [13] T. Phung-Duc, Impatient customers in power-saving data centers, Analytical and Stochastic Modeling Techniques and Applications, Lecture Notes in Computer Science, LNCS, 8499 (2014), 185-199.  doi: 10.1007/978-3-319-08219-6_13. [14] T. Phung-Duc, Server farms with batch arrival and staggered setup, Proceedings of the Fifth Symposium on Information and Communication Technology -ACM, (2014), 240-247.  doi: 10.1145/2676585.2676613. [15] T. Phung-Duc, Exact solutions for M/M/c/Setup queues, Telecommunication Systems, 64 (2017), 309-324.  doi: 10.1007/s11235-016-0177-z. [16] T. Phung-Duc, Multiserver queues with finite capacity and setup time, Analytical and Stochastic Modeling Techniques and Applications, Lecture Notes in Computer Science, LNCS, 9081 (2015), 173-187.  doi: 10.1007/978-3-319-18579-8_13. [17] T. Phung-Duc and W. Rogiest, Analysis of an M/M/1 retrial queue with speed scaling, Proceedings of QTNA 2015, Advances in Intelligent Systems and Computing, 383 (2015), 113-124.  doi: 10.1007/978-3-319-22267-7_11. [18] C. Schwartz, R. Pries and P. Tran-Gia, A queuing analysis of an energy-saving mechanism in data centers, Proceedings of International Conference on Information Networking (ICOIN), (2012), 70-75.  doi: 10.1109/ICOIN.2012.6164352. [19] W. Van Heddeghem, S. Lambert, B. Lannoo, D. Colle, M. Pickavet and P. Demeester, Trends in worldwide ICT electricity consumption from 2007 to 2012, Computer Communications, 50 (2014), 64-76.  doi: 10.1016/j.comcom.2014.02.008. [20] A. Wierman, L. Andrew and A. Tang, Power-aware speed scaling in processor sharing systems, Proceedings of IEEE INFOCOM 2009, (2009), 2007-2015.  doi: 10.1109/INFCOM.2009.5062123. [21] F. Yao, A. Demers and S. Shenker, A scheduling model for reduced CPU energy, Proceedings 36th Annual Symposium on Foundations of Computer Science, (1995), 374-382.  doi: 10.1109/SFCS.1995.492493.

show all references

##### References:
 [1] J. R. Artalejo, A. Economou and M. J. Lopez-Herrero, Analysis of a multiserver queue with setup times, Queueing Systems, 51 (2005), 53-76.  doi: 10.1007/s11134-005-1740-6. [2] J. R. Artalejo and T. Phung-Duc, Markovian retrial queues with two way communication, Journal of Industrial and Management Optimization, 8 (2012), 781-806.  doi: 10.3934/jimo.2012.8.781. [3] L. A. Barroso and U. Holzle, The case for energy-proportional computing, Computer, 40 (2007), 33-37.  doi: 10.1109/MC.2007.443. [4] R. Conway and W. L. Maxwell, A queueing model with state dependent service rate, Journal of Industrial Engineering, 12 (1961), 132-136. [5] A. Gandhi, M. Harchol-Balter and I. Adan, Server farms with setup costs, Performance Evaluation, 67 (2010), 1123-1138. [6] A. Gandhi, S. Doroudi, M. Harchol-Balter and A. Scheller-Wolf, Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward, Proceedings of the ACM SIGMETRICS, (2013), 153-166. [7] A. Gandhi, S. Doroudi, M. Harchol-Balter and A. Scheller-Wolf, Exact analysis of the M/M/k/setup class of Markov chains via recursive renewal reward, Queueing Systems, 77 (2014), 177-209.  doi: 10.1007/s11134-014-9409-7. [8] X. Lu, S. Aalto and P. Lassila, Performance-energy trade-off in data centers: Impact of switching delay, Proceedings of 22nd IEEE ITC Specialist Seminar on Energy Efficient and Green Networking (SSEEGN), (2013), 50-55.  doi: 10.1109/SSEEGN.2013.6705402. [9] V. J. Maccio and D. G. Down, On optimal policies for energy-aware servers, Performance Evaluation, 90 (2014), 36-52.  doi: 10.1109/MASCOTS.2013.11. [10] I. Mitrani, Managing performance and power consumption in a server farm, Annals of Operations Research, 202 (2013), 121-134.  doi: 10.1007/s10479-011-0932-1. [11] P. R. Parthasarathy and R. Sudhesh, Time-dependent analysis of a single-server retrial queue with state-dependent rates, Operations Research Letters, 35 (2007), 601-611.  doi: 10.1016/j.orl.2006.12.005. [12] T. Phung-Duc, W. Rogiest, Y. Takahashi and H. Bruneel, Retrial queues with balanced call blending: Analysis of single-server and multiserver model, Annals of Operations Research, 239 (2016), 429-449.  doi: 10.1007/s10479-014-1598-2. [13] T. Phung-Duc, Impatient customers in power-saving data centers, Analytical and Stochastic Modeling Techniques and Applications, Lecture Notes in Computer Science, LNCS, 8499 (2014), 185-199.  doi: 10.1007/978-3-319-08219-6_13. [14] T. Phung-Duc, Server farms with batch arrival and staggered setup, Proceedings of the Fifth Symposium on Information and Communication Technology -ACM, (2014), 240-247.  doi: 10.1145/2676585.2676613. [15] T. Phung-Duc, Exact solutions for M/M/c/Setup queues, Telecommunication Systems, 64 (2017), 309-324.  doi: 10.1007/s11235-016-0177-z. [16] T. Phung-Duc, Multiserver queues with finite capacity and setup time, Analytical and Stochastic Modeling Techniques and Applications, Lecture Notes in Computer Science, LNCS, 9081 (2015), 173-187.  doi: 10.1007/978-3-319-18579-8_13. [17] T. Phung-Duc and W. Rogiest, Analysis of an M/M/1 retrial queue with speed scaling, Proceedings of QTNA 2015, Advances in Intelligent Systems and Computing, 383 (2015), 113-124.  doi: 10.1007/978-3-319-22267-7_11. [18] C. Schwartz, R. Pries and P. Tran-Gia, A queuing analysis of an energy-saving mechanism in data centers, Proceedings of International Conference on Information Networking (ICOIN), (2012), 70-75.  doi: 10.1109/ICOIN.2012.6164352. [19] W. Van Heddeghem, S. Lambert, B. Lannoo, D. Colle, M. Pickavet and P. Demeester, Trends in worldwide ICT electricity consumption from 2007 to 2012, Computer Communications, 50 (2014), 64-76.  doi: 10.1016/j.comcom.2014.02.008. [20] A. Wierman, L. Andrew and A. Tang, Power-aware speed scaling in processor sharing systems, Proceedings of IEEE INFOCOM 2009, (2009), 2007-2015.  doi: 10.1109/INFCOM.2009.5062123. [21] F. Yao, A. Demers and S. Shenker, A scheduling model for reduced CPU energy, Proceedings 36th Annual Symposium on Foundations of Computer Science, (1995), 374-382.  doi: 10.1109/SFCS.1995.492493.
Transitions among states.
Transitions among states.
Transitions among states.
Transitions among states.
Cost $z$ as function of the service rate $\mu$
Cost $z$ as function of the service rate $\mu$
Cost $z$ as function of the service rate $\mu$
Cost $y$ (second cost function) as function of the service rate $\mu$
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