July  2017, (3): 1329-1345. doi: 10.3934/jimo.2016075

Single server retrial queues with setup time

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

Received  September 2015 Revised  May 2016 Published  October 2016

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

This paper considers single server retrial queues with setup time where the server is aware of the existence of retrying customers. In the basic model, the server is switched off immediately when the system becomes empty in order to save energy consumption. Arriving customers that see the server occupied join the orbit and repeat their attempt after some random time. The new feature of our models is that an arriving customer that sees the server off waits at the server and the server is turned on. The server needs some setup time to be active so as to serve the waiting customer. If the server completes a service and the orbit is not empty, it stays idle waiting for either a new or retrial customer. Under the assumption that the service time and the setup time are arbitrarily distributed, we obtain explicit expressions for the generating functions of the joint queue length. We also obtain recursive formulas for computing the moments of the queue length. We then consider an extended model where the server has a closedown time before being turned off for which an explicit solution is also obtained.

Citation: Tuan Phung-Duc. Single server retrial queues with setup time. Journal of Industrial and Management Optimization, 2017, (3) : 1329-1345. doi: 10.3934/jimo.2016075
References:
[1]

J. R. Artalejo, Analysis of an M/G/1 queue with constant repeated attempts and server vacations, Computers and Operations Research, 24 (1997), 493-504.  doi: 10.1016/S0305-0548(96)00076-7.

[2]

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

[3]

W. Bischof, Analysis of M/G/1-queues with setup times and vacations under six different service disciplines, Queueing Systems, 39 (2001), 265-301.  doi: 10.1023/A:1013992708103.

[4]

S. Derkic and J. E. Stafford, Symbolic computation of moments in priority queues, INFORMS Journal on Computing, 14 (2002), 261-277.  doi: 10.1287/ijoc.14.3.261.115.

[5]

S. DrekicJ. E. Stafford and G. E. Willmot, Symbolic calculation of the moments of the time of ruin, Insurance: Mathematics and Economics, 34 (2004), 109-120.  doi: 10.1016/j.insmatheco.2003.11.004.

[6]

T. V. Do, M/M/1 retrial queue with working vacations, Acta Informatica, 47 (2010), 67-75.  doi: 10.1007/s00236-009-0110-y.

[7]

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

[8]

A. GandhiM. Harchol-Balter and M. A. Kozuch, Are sleep states effective in data centers?, Proceedings of IEEE 2012 International Green Computing Conference (IGCC), (2012), 1-10.  doi: 10.1109/IGCC.2012.6322260.

[9]

A. GandhiS. DoroudiM. 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. 

[10]

A. GandhiS. DoroudiM. 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.

[11]

S. KapodistriaT. Phung-Duc and J. Resing, Linear birth/immigration-death process with binomial catastrophes, Probability in the Engineering and Informational Sciences, 30 (2016), 79-111.  doi: 10.1017/S0269964815000297.

[12]

V. J. Maccio and D. G. Down, On optimal policies for energy-aware servers, Performance Evaluation, 90 (2013), 36-52.  doi: 10.1109/MASCOTS.2013.11.

[13]

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.

[14]

T. Phung-DucH. MasuyamaS. Kasahara and Y. Takahashi, M/M/3/3 and M/M/4/4 retrial queues, Journal of Industrial and Management Optimization, 5 (2009), 431-451.  doi: 10.3934/jimo.2009.5.431.

[15]

T. Phung-DucH. MasuyamaS. Kasahara and Y. Takahashi, State-dependent M/M/c/c+ r retrial queues with Bernoulli abandonment, Journal of Industrial and Management Optimization, 6 (2010), 517-540.  doi: 10.3934/jimo.2010.6.517.

[16]

T. Phung-Duc, An explicit solution for a tandem queue with retrials and losses, Operational Research, 12 (2012), 189-207.  doi: 10.1007/s12351-011-0113-7.

[17]

T. Phung-Duc, Impatient customers in power-saving data centers, Lecture Notes in Computer Science, 8499 (2014), 185-199.  doi: 10.1007/978-3-319-08219-6_13.

[18]

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

[19]

T. Phung-Duc, Exact solutions for M/M/$c$/Setup queues Telecommunication Systems 2016. doi: 10.1007/s11235-016-0177-z.

[20]

T. Phung-Duc, Multiserver queues with finite capacity and setup time, Lecture Notes in Computer Science, 9081 (2015), 173-187.  doi: 10.1007/978-3-319-18579-8_13.

[21]

T. Phung-Duc, M/M/1/1 retrial queues with setup time, Advances in Intelligent Systems and Computing, 383 (2015), 93-104.  doi: 10.1007/978-3-319-22267-7_9.

[22]

C. SchwartzR. Pries and P. Tran-Gia, A queuing analysis of an energy-saving mechanism in data centers, Proceedings of IEEE 2012 International Conference on Information Networking (ICOIN), (2012), 70-75.  doi: 10.1109/ICOIN.2012.6164352.

[23]

H. Takagi, Priority queues with setup times, Operations Research, 38 (1990), 667-677.  doi: 10.1287/opre.38.4.667.

[24]

H. Takagi and K. Sakamaki, Symbolic moment calculation for the sojourn time in M/G/1 queues with Bernoulli feedback, Journal of the Operations Research Society of Japan, 42 (1999), 78-87.  doi: 10.1016/S0453-4514(99)80006-4.

[25]

H. Takagi and K. Sakamaki, Moments for M/G/1 queues, Mathematica Journal, 6 (1996), 75-80. 

[26]

H. Takagi and S. Kudoh, Symbolic higher-order moments of the waiting time in an M/G/1 queue with random order of service, Stochastic Models, (1997), 167-179.  doi: 10.1080/15326349708807419.

show all references

References:
[1]

J. R. Artalejo, Analysis of an M/G/1 queue with constant repeated attempts and server vacations, Computers and Operations Research, 24 (1997), 493-504.  doi: 10.1016/S0305-0548(96)00076-7.

[2]

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

[3]

W. Bischof, Analysis of M/G/1-queues with setup times and vacations under six different service disciplines, Queueing Systems, 39 (2001), 265-301.  doi: 10.1023/A:1013992708103.

[4]

S. Derkic and J. E. Stafford, Symbolic computation of moments in priority queues, INFORMS Journal on Computing, 14 (2002), 261-277.  doi: 10.1287/ijoc.14.3.261.115.

[5]

S. DrekicJ. E. Stafford and G. E. Willmot, Symbolic calculation of the moments of the time of ruin, Insurance: Mathematics and Economics, 34 (2004), 109-120.  doi: 10.1016/j.insmatheco.2003.11.004.

[6]

T. V. Do, M/M/1 retrial queue with working vacations, Acta Informatica, 47 (2010), 67-75.  doi: 10.1007/s00236-009-0110-y.

[7]

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

[8]

A. GandhiM. Harchol-Balter and M. A. Kozuch, Are sleep states effective in data centers?, Proceedings of IEEE 2012 International Green Computing Conference (IGCC), (2012), 1-10.  doi: 10.1109/IGCC.2012.6322260.

[9]

A. GandhiS. DoroudiM. 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. 

[10]

A. GandhiS. DoroudiM. 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.

[11]

S. KapodistriaT. Phung-Duc and J. Resing, Linear birth/immigration-death process with binomial catastrophes, Probability in the Engineering and Informational Sciences, 30 (2016), 79-111.  doi: 10.1017/S0269964815000297.

[12]

V. J. Maccio and D. G. Down, On optimal policies for energy-aware servers, Performance Evaluation, 90 (2013), 36-52.  doi: 10.1109/MASCOTS.2013.11.

[13]

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.

[14]

T. Phung-DucH. MasuyamaS. Kasahara and Y. Takahashi, M/M/3/3 and M/M/4/4 retrial queues, Journal of Industrial and Management Optimization, 5 (2009), 431-451.  doi: 10.3934/jimo.2009.5.431.

[15]

T. Phung-DucH. MasuyamaS. Kasahara and Y. Takahashi, State-dependent M/M/c/c+ r retrial queues with Bernoulli abandonment, Journal of Industrial and Management Optimization, 6 (2010), 517-540.  doi: 10.3934/jimo.2010.6.517.

[16]

T. Phung-Duc, An explicit solution for a tandem queue with retrials and losses, Operational Research, 12 (2012), 189-207.  doi: 10.1007/s12351-011-0113-7.

[17]

T. Phung-Duc, Impatient customers in power-saving data centers, Lecture Notes in Computer Science, 8499 (2014), 185-199.  doi: 10.1007/978-3-319-08219-6_13.

[18]

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

[19]

T. Phung-Duc, Exact solutions for M/M/$c$/Setup queues Telecommunication Systems 2016. doi: 10.1007/s11235-016-0177-z.

[20]

T. Phung-Duc, Multiserver queues with finite capacity and setup time, Lecture Notes in Computer Science, 9081 (2015), 173-187.  doi: 10.1007/978-3-319-18579-8_13.

[21]

T. Phung-Duc, M/M/1/1 retrial queues with setup time, Advances in Intelligent Systems and Computing, 383 (2015), 93-104.  doi: 10.1007/978-3-319-22267-7_9.

[22]

C. SchwartzR. Pries and P. Tran-Gia, A queuing analysis of an energy-saving mechanism in data centers, Proceedings of IEEE 2012 International Conference on Information Networking (ICOIN), (2012), 70-75.  doi: 10.1109/ICOIN.2012.6164352.

[23]

H. Takagi, Priority queues with setup times, Operations Research, 38 (1990), 667-677.  doi: 10.1287/opre.38.4.667.

[24]

H. Takagi and K. Sakamaki, Symbolic moment calculation for the sojourn time in M/G/1 queues with Bernoulli feedback, Journal of the Operations Research Society of Japan, 42 (1999), 78-87.  doi: 10.1016/S0453-4514(99)80006-4.

[25]

H. Takagi and K. Sakamaki, Moments for M/G/1 queues, Mathematica Journal, 6 (1996), 75-80. 

[26]

H. Takagi and S. Kudoh, Symbolic higher-order moments of the waiting time in an M/G/1 queue with random order of service, Stochastic Models, (1997), 167-179.  doi: 10.1080/15326349708807419.

Figure 1.  Transitions among states
Figure 2.  Transition among states
Figure 3.  Probability of OFF state against $\alpha$ ($\rho = 0.7$)
Figure 4.  Mean number of jobs in orbit against $\alpha$ ($\rho = 0.7$)
Figure 5.  Probability of OFF state against $\alpha$ ($\rho = 0.1$)
Figure 6.  Mean number of jobs in orbit against $\alpha$ ($\rho = 0.1$)
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