American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Delay characteristics in place-reservation queues with class-dependent service times
  • JIMO Home
  • This Issue
  • Next Article
    Performance evaluation and optimization of cognitive radio networks with adjustable access control for multiple secondary users
January  2019, 15(1): 15-35. doi: 10.3934/jimo.2018030

Multiserver retrial queue with setup time and its application to data centers

Tuan Phung-Duc 1,, and  Ken'ichi Kawanishi 2,

1. 

Division of Policy and Planning Sciences, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
2. 

Division of Electronics and Informatics, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma 376-8515, Japan

* Corresponding author

Received  February 2017 Revised  July 2017 Published  February 2018

Fund Project: The reviewing process of this paper was handled by Yutaka Takahashi and Wuyi Yue

This paper considers a multiserver retrial queue with setup time which is motivated from application in data centers with the ON-OFF policy, where an idle server is immediately turned off. The ON-OFF policy is designed to save energy consumption of idle servers because an idle server still consumes about 60% of its peak consumption processing jobs. Upon arrival, a job is allocated to one of available off-servers and that server is started up. Otherwise, if all the servers are not available upon arrival, the job is blocked and retries in a random time. A server needs some setup time during which the server cannot process a job but consumes energy. We formulate this model using a three-dimensional continuous-time Markov chain obtaining the stability condition via Foster-Lyapunov criteria. Interestingly, the stability condition is different from that of the corresponding non-retrial queue. Furthermore, exploiting the special structure of the Markov chain together with a heuristic technique, we develop an efficient algorithm for computing the stationary distribution. Numerical results reveal that under the ON-OFF policy, allowing retrials is more power-saving than buffering jobs. Furthermore, we obtain a new insight that if the setup time is relatively long, setting an appropriate retrial time could reduce both power consumption and the mean response time of jobs.

Keywords: Data center, power consumption, delay, trade-off, retrials.
Mathematics Subject Classification: 60K25, 68M20, 90B22.
Citation: Tuan Phung-Duc, Ken'ichi Kawanishi. Multiserver retrial queue with setup time and its application to data centers. Journal of Industrial & Management Optimization, 2019, 15 (1) : 15-35. doi: 10.3934/jimo.2018030
References:
[1]

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. Google Scholar

[2]

L. A. Barroso and U. Hölzle, The case for energy-proportional computing, Computer, 40 (2007), 33-37. doi: 10.1109/MC.2007.443. Google Scholar

[3]

L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic Models, 11 (1995), 497-525. doi: 10.1080/15326349508807357. Google Scholar

[4]

J. Chang and J. Wang, Unreliable M/M/1/1 retrial queues with set-up time, Quality Technology & Quantitative Management, (2017), 1-13. doi: 10.1080/16843703.2017.1320459. Google Scholar

[5]

J. E. Diamond and A. S. Alfa, The MAP/PH/1 retrial queue, Stochastic Models, 14 (1998), 1151-1177. doi: 10.1080/15326349808807518. Google Scholar

[6]

J. E. Diamond and A. S. Alfa, Matrix analytic methods for a multiserver retrial queue with buffer, Top, 7 (1999), 249-266. doi: 10.1007/BF02564725. Google Scholar

[7]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997.Google Scholar

[8]

A. Gandhi, M. Harchol-Balter and I. Adan, Server farms with setup costs, Performance Evaluation, 67 (2010), 1123-1138. Google Scholar

[9]

A. Gandhi, S. Doroudi and M. Harchol-Balter, 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. Google Scholar

[10]

Q.-M. He, H. Li and Y. Q. Zhao, Ergodicity of the BMAP/PH/s/s + K retrial queue with PH-retrial times, Queueing Systems, 35 (2000), 323-347. doi: 10.1023/A:1019110631467. Google Scholar

[11]

G. Latouche and V. Ramaswami, A logarithmic reduction algorithm for quasi-birth-death process, Journal of Applied Probability, 30 (1993), 650-674. doi: 10.1017/S0021900200044387. Google Scholar

[12]

G. Latouche and V. Ramaswami, Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia PA, 1999. Google Scholar

[13]

E. Morozov, A multiserver retrial queue: Regenerative stability analysis, Queueing Systems, 56 (2007), 157-168. doi: 10.1007/s11134-007-9024-y. Google Scholar

[14]

E. Morozov and T. Phung-Duc, Stability analysis of a multiclass retrial system with classical retrial policy, Performance Evaluation, 112 (2017), 15-26. doi: 10.1016/j.peva.2017.03.003. Google Scholar

[15]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover Publications, Inc., New York, 1994. Google Scholar

[16]

T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A simple algorithm for the rate matrices of level-dependent QBD processes, Proceedings of the 5th International Conference on Queueing Theory and Network Applications, (2010), 46-52. doi: 10.1145/1837856.1837864. Google Scholar

[17]

T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A matrix continued fraction approach to multiserver retrial queues, Annals of Operations Research, 202 (2013), 161-183. doi: 10.1007/s10479-011-0840-4. Google Scholar

[18]

T. Phung-Duc, Server farms with batch arrival and staggered setup, Proceedings of the Fifth symposium on Information and Communication Technology (SoICT 2014), (2014), 240-247. doi: 10.1145/2676585.2676613. Google Scholar

[19]

T. Phung-Duc and K. Kawanishi, An efficient method for performance analysis of blended call centers with redial, Asia-Pacific Journal of Operational Research, 31 (2014), 1440008 (33 pages). Google Scholar

[20]

T. Phung-Duc, M/M/1/1 retrial queues with setup time, Proceedings of the 10th International Conference on Queueing Theory and Network Applications, (2015), 93-104. Google Scholar

[21]

T. Phung-Duc and K. Kawanishi, Impacts of retrials on power-saving policy in data centers in Proceedings of the 11th International Conference on Queueing Theory and Network Applications, ACM, (2016), Article No. 22. doi: 10.1145/3016032.3016047. Google Scholar

[22]

T. Phung-Duc, Exact solutions for M/M/c/Setup queues, Telecommunication Systems, 64 (2017), 309-324. doi: 10.1007/s11235-016-0177-z. Google Scholar

[23]

T. Phung-Duc, Single server retrial queues with setup time, Journal of Industrial and Management Optimization, 13 (2017), 1329-1345. Google Scholar

[24]

Y. W. Shin, Stability of $MAP/PH/c/K$ queue with customer retrials and server vacations, Bulletin of the Korean Mathematical Society, 53 (2016), 985-1004. doi: 10.4134/BKMS.b150337. Google Scholar

[25]

R. L. Tweedie, Sufficient conditions for regularity, recurrence and ergodicity and Markov processes, Mathematical Proceedings of the Cambridge Philosophical Society, 78 (1975), 125-136. doi: 10.1017/S0305004100051562. Google Scholar

show all references

References:
[1]

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. Google Scholar

[2]

L. A. Barroso and U. Hölzle, The case for energy-proportional computing, Computer, 40 (2007), 33-37. doi: 10.1109/MC.2007.443. Google Scholar

[3]

L. Bright and P. G. Taylor, Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes, Stochastic Models, 11 (1995), 497-525. doi: 10.1080/15326349508807357. Google Scholar

[4]

J. Chang and J. Wang, Unreliable M/M/1/1 retrial queues with set-up time, Quality Technology & Quantitative Management, (2017), 1-13. doi: 10.1080/16843703.2017.1320459. Google Scholar

[5]

J. E. Diamond and A. S. Alfa, The MAP/PH/1 retrial queue, Stochastic Models, 14 (1998), 1151-1177. doi: 10.1080/15326349808807518. Google Scholar

[6]

J. E. Diamond and A. S. Alfa, Matrix analytic methods for a multiserver retrial queue with buffer, Top, 7 (1999), 249-266. doi: 10.1007/BF02564725. Google Scholar

[7]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997.Google Scholar

[8]

A. Gandhi, M. Harchol-Balter and I. Adan, Server farms with setup costs, Performance Evaluation, 67 (2010), 1123-1138. Google Scholar

[9]

A. Gandhi, S. Doroudi and M. Harchol-Balter, 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. Google Scholar

[10]

Q.-M. He, H. Li and Y. Q. Zhao, Ergodicity of the BMAP/PH/s/s + K retrial queue with PH-retrial times, Queueing Systems, 35 (2000), 323-347. doi: 10.1023/A:1019110631467. Google Scholar

[11]

G. Latouche and V. Ramaswami, A logarithmic reduction algorithm for quasi-birth-death process, Journal of Applied Probability, 30 (1993), 650-674. doi: 10.1017/S0021900200044387. Google Scholar

[12]

G. Latouche and V. Ramaswami, Matrix Analytic Methods in Stochastic Modelling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia PA, 1999. Google Scholar

[13]

E. Morozov, A multiserver retrial queue: Regenerative stability analysis, Queueing Systems, 56 (2007), 157-168. doi: 10.1007/s11134-007-9024-y. Google Scholar

[14]

E. Morozov and T. Phung-Duc, Stability analysis of a multiclass retrial system with classical retrial policy, Performance Evaluation, 112 (2017), 15-26. doi: 10.1016/j.peva.2017.03.003. Google Scholar

[15]

M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover Publications, Inc., New York, 1994. Google Scholar

[16]

T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A simple algorithm for the rate matrices of level-dependent QBD processes, Proceedings of the 5th International Conference on Queueing Theory and Network Applications, (2010), 46-52. doi: 10.1145/1837856.1837864. Google Scholar

[17]

T. Phung-Duc, H. Masuyama, S. Kasahara and Y. Takahashi, A matrix continued fraction approach to multiserver retrial queues, Annals of Operations Research, 202 (2013), 161-183. doi: 10.1007/s10479-011-0840-4. Google Scholar

[18]

T. Phung-Duc, Server farms with batch arrival and staggered setup, Proceedings of the Fifth symposium on Information and Communication Technology (SoICT 2014), (2014), 240-247. doi: 10.1145/2676585.2676613. Google Scholar

[19]

T. Phung-Duc and K. Kawanishi, An efficient method for performance analysis of blended call centers with redial, Asia-Pacific Journal of Operational Research, 31 (2014), 1440008 (33 pages). Google Scholar

[20]

T. Phung-Duc, M/M/1/1 retrial queues with setup time, Proceedings of the 10th International Conference on Queueing Theory and Network Applications, (2015), 93-104. Google Scholar

[21]

T. Phung-Duc and K. Kawanishi, Impacts of retrials on power-saving policy in data centers in Proceedings of the 11th International Conference on Queueing Theory and Network Applications, ACM, (2016), Article No. 22. doi: 10.1145/3016032.3016047. Google Scholar

[22]

T. Phung-Duc, Exact solutions for M/M/c/Setup queues, Telecommunication Systems, 64 (2017), 309-324. doi: 10.1007/s11235-016-0177-z. Google Scholar

[23]

T. Phung-Duc, Single server retrial queues with setup time, Journal of Industrial and Management Optimization, 13 (2017), 1329-1345. Google Scholar

[24]

Y. W. Shin, Stability of $MAP/PH/c/K$ queue with customer retrials and server vacations, Bulletin of the Korean Mathematical Society, 53 (2016), 985-1004. doi: 10.4134/BKMS.b150337. Google Scholar

[25]

R. L. Tweedie, Sufficient conditions for regularity, recurrence and ergodicity and Markov processes, Mathematical Proceedings of the Cambridge Philosophical Society, 78 (1975), 125-136. doi: 10.1017/S0305004100051562. Google Scholar

Figure 1.  The power consumption versus retrial rate for $c = 50$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The power consumption versus setup rate for $c = 30, 50$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The ratio $\mathrm{E}[P]/c$ versus retrial rate for $\alpha = 1/100$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Mean response time versus retrial rate for $c = 30, 50$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Mean response time versus retrial rate for $c = 30, 50$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The power consumption versus traffic intensity for $\mu = 1$ and $\alpha = 1/10$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  The power consumption versus retrial rate for $c = 50$ and $\alpha = 1/10$
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Truncation point $N$ and $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ for $c = 30$ and $\mu = 1/10$
$c=30$
$\alpha = 1/100$ $\alpha = 1/10$ $\alpha = 1$
$N$120315059
$\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $1.0060\times 10^{-11}$ $1.6243\times 10^{-10}$ $2.2090\times 10^{-15}$
Table Options

Download as excel

$c=30$
$\alpha = 1/100$ $\alpha = 1/10$ $\alpha = 1$
$N$120315059
$\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $1.0060\times 10^{-11}$ $1.6243\times 10^{-10}$ $2.2090\times 10^{-15}$
Table 2.  Truncation point $N$ and $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ for $c = 50$ and $\mu = 1/10$
$c=50$
$\alpha = 1/100$ $\alpha = 1/10$ $\alpha = 1$
$N$120314958
$\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $4.6265\times 10^{-11}$ $2.4851\times 10^{-10}$ $1.1490\times 10^{-16}$
Table Options

Download as excel

$c=50$
$\alpha = 1/100$ $\alpha = 1/10$ $\alpha = 1$
$N$120314958
$\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $4.6265\times 10^{-11}$ $2.4851\times 10^{-10}$ $1.1490\times 10^{-16}$
Table 3.  Truncation point $N$ and $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$
$c=50$$c=100$
$\rho$ $N$ $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $N$ $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$
0.139 $2.3871 \times 10^{-19}$39 $1.3689 \times 10^{-25}$
0.266 $3.3924 \times 10^{-15}$66 $1.0369 \times 10^{-17}$
0.392 $3.7031 \times 10^{-13}$92 $3.1474 \times 10^{-14}$
0.4118 $9.5226 \times 10^{-12}$118 $4.7215 \times 10^{-12}$
0.5144 $1.4487 \times 10^{-10}$144 $2.1293 \times 10^{-10}$
0.6170 $1.7715 \times 10^{-09}$170 $5.4344 \times 10^{-09}$
0.7197 $1.7430 \times 10^{-08}$196 $1.0141 \times 10^{-07}$
0.8228 $1.0765 \times 10^{-07}$227 $1.0942 \times 10^{-06}$
0.9349 $1.5416 \times 10^{-06}$321 $7.4458 \times 10^{-07}$
Table Options

Download as excel

$c=50$$c=100$
$\rho$ $N$ $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $N$ $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$
0.139 $2.3871 \times 10^{-19}$39 $1.3689 \times 10^{-25}$
0.266 $3.3924 \times 10^{-15}$66 $1.0369 \times 10^{-17}$
0.392 $3.7031 \times 10^{-13}$92 $3.1474 \times 10^{-14}$
0.4118 $9.5226 \times 10^{-12}$118 $4.7215 \times 10^{-12}$
0.5144 $1.4487 \times 10^{-10}$144 $2.1293 \times 10^{-10}$
0.6170 $1.7715 \times 10^{-09}$170 $5.4344 \times 10^{-09}$
0.7197 $1.7430 \times 10^{-08}$196 $1.0141 \times 10^{-07}$
0.8228 $1.0765 \times 10^{-07}$227 $1.0942 \times 10^{-06}$
0.9349 $1.5416 \times 10^{-06}$321 $7.4458 \times 10^{-07}$
[1]

Yanqin Bai, Yudan Wei, Qian Li. An optimal trade-off model for portfolio selection with sensitivity of parameters. Journal of Industrial & Management Optimization, 2017, 13 (2) : 947-965. doi: 10.3934/jimo.2016055
[2]

Lihui Zhang, Xin Zou, Jianxun Qi. A trade-off between time and cost in scheduling repetitive construction projects. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1423-1434. doi: 10.3934/jimo.2015.11.1423
[3]

Masataka Kato, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Effect of energy-saving server scheduling on power consumption for large-scale data centers. Journal of Industrial & Management Optimization, 2016, 12 (2) : 667-685. doi: 10.3934/jimo.2016.12.667
[4]

Daniel Mckenzie, Steven Damelin. Power weighted shortest paths for clustering Euclidean data. Foundations of Data Science, 2019, 1 (3) : 307-327. doi: 10.3934/fods.2019014
[5]

Zuo Quan Xu, Fahuai Yi. An optimal consumption-investment model with constraint on consumption. Mathematical Control & Related Fields, 2016, 6 (3) : 517-534. doi: 10.3934/mcrf.2016014
[6]

Rafael Diaz, Laura Gomez. Indirect influences in international trade. Networks & Heterogeneous Media, 2015, 10 (1) : 149-165. doi: 10.3934/nhm.2015.10.149
[7]

Michael Helmers, Barbara Niethammer, Xiaofeng Ren. Evolution in off-critical diblock copolymer melts. Networks & Heterogeneous Media, 2008, 3 (3) : 615-632. doi: 10.3934/nhm.2008.3.615
[8]

Vitali Kapovitch, Anton Petrunin, Wilderich Tuschmann. On the torsion in the center conjecture. Electronic Research Announcements, 2018, 25: 27-35. doi: 10.3934/era.2018.25.004
[9]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89
[10]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[11]

Daniele Bartoli, Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Further results on multiple coverings of the farthest-off points. Advances in Mathematics of Communications, 2016, 10 (3) : 613-632. doi: 10.3934/amc.2016030
[12]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907
[13]

Masakatsu Suzuki, Hideaki Matsunaga. Stability criteria for a class of linear differential equations with off-diagonal delays. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1381-1391. doi: 10.3934/dcds.2009.24.1381
[14]

Jingzhen Liu, Ka-Fai Cedric Yiu, Kok Lay Teo. Optimal investment-consumption problem with constraint. Journal of Industrial & Management Optimization, 2013, 9 (4) : 743-768. doi: 10.3934/jimo.2013.9.743
[15]

Lei Sun, Lihong Zhang. Optimal consumption and investment under irrational beliefs. Journal of Industrial & Management Optimization, 2011, 7 (1) : 139-156. doi: 10.3934/jimo.2011.7.139
[16]

Filipe Martins, Alberto A. Pinto, Jorge Passamani Zubelli. Nash and social welfare impact in an international trade model. Journal of Dynamics & Games, 2017, 4 (2) : 149-173. doi: 10.3934/jdg.2017009
[17]

Pau Erola, Albert Díaz-Guilera, Sergio Gómez, Alex Arenas. Modeling international crisis synchronization in the world trade web. Networks & Heterogeneous Media, 2012, 7 (3) : 385-397. doi: 10.3934/nhm.2012.7.385
[18]

B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609
[19]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[20]

Claudio Buzzi, Claudio Pessoa, Joan Torregrosa. Piecewise linear perturbations of a linear center. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3915-3936. doi: 10.3934/dcds.2013.33.3915

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (172)
  • HTML views (1073)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences