American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Effect of energy-saving server scheduling on power consumption for large-scale data centers
  • JIMO Home
  • This Issue
  • Next Article
    Tail asymptotics of fluid queues in a distributed server system fed by a heavy-tailed ON-OFF flow
April  2016, 12(2): 653-666. doi: 10.3934/jimo.2016.12.653

Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states

Dequan Yue 1, , Wuyi Yue 2, and  Guoxi Zhao 3,

1. 

Department of Statistics, College of Sciences, Yanshan University, Qinhuangdao 066004
2. 

Department of Intelligence and Informatics, Konan University, Kobe 658-8501
3. 

School of Economics and Management, Yanshan University, Qinhuangdao 066004, China

Received  September 2014 Revised  March 2015 Published  June 2015

We consider an M/M/1 queueing system with vacations and impatient customers. Whenever a customer arrives at the system, it activates an random ``impatience timer". If the customer's service has not been completed before the customer's impatience timer expires, the customer abandons the queue, and never returns. It is assumed that the impatience timer depends on the server's states. We analyze both multiple and single vacation scenarios and derive the probability generating functions of the number of customers in the system when the server is in vacation period and busy period. Then, we obtain explicit expressions for various performance measures such as the mean system sizes when the server is either on vacation or busy, the proportion of customers served, and the average rate of abandonments due to impatience. We present some numerical results for multiple vacation scenario to show the effects of the parameters of impatience timers on some performance measures. Finally, we show some inequalities on some performances under the single vacation policy and under multiple vacation policy.
Keywords: vacation, probability generating function, performance measures., Queue, impatience.
Mathematics Subject Classification: Primary: 60K25; Secondary: 90B2.
Citation: Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial & Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653
References:
[1]

E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations,, Queueing Systems, 52 (2006), 261.  doi: 10.1007/s11134-006-6134-x.  Google Scholar

[2]

E. Altman and U. Yechiali, Infinite-server queues with systems' additional task and impatient customers,, Probability in the Engineering and Informational Sciences, 22 (2008), 477.  doi: 10.1007/978-1-4020-8741-7_57.  Google Scholar

[3]

C. J. Ancker and A. V. Gafarian, Some queueing problems with balking and reneging,, Operations Research, 11 (1963), 88.  doi: 10.1287/opre.11.1.88.  Google Scholar

[4]

F. Baccelli, P. Boyer and G. Hebuterne, Single-server queues with impatient customers,, Advances in Applied Probability, 16 (1984), 887.  doi: 10.2307/1427345.  Google Scholar

[5]

F. Baccelli and G. Hebuterne, On queues with impatient customers,, in Perforamnce' 81 (F. Kylstra, (1981), 159.   Google Scholar

[6]

S. Benjaafar, J. Gayon and S. Tepe, Optimal control of a production-inventory system with customer impatience,, Operations Research Letters, 38 (2010), 267.  doi: 10.1016/j.orl.2010.03.008.  Google Scholar

[7]

N. K. Boots and H. Tijms, A multiserver queueing system with impatient customers,, Management Science, 45 (1999), 444.  doi: 10.1287/mnsc.45.3.444.  Google Scholar

[8]

O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers,, ITC, 14 (1994), 743.  doi: 10.1016/B978-0-444-82031-0.50079-2.  Google Scholar

[9]

S. R. Chakravarthy, A disater queue with Markovian arrivals and impatient customers,, Applied Mathematics and Computation, 214 (2009), 48.  doi: 10.1016/j.amc.2009.03.081.  Google Scholar

[10]

D. J. Daley, General customer impatience in the queue GI/G/1,, Journal of Applied Probability, 2 (1965), 186.  doi: 10.2307/3211884.  Google Scholar

[11]

S. Dimou, A. Economou and D. Fakinos, The single server vacation queueing model with geometric abandonments,, Journal of Statistical Planning and Inference, 141 (2011), 2863.  doi: 10.1016/j.jspi.2011.03.010.  Google Scholar

[12]

B. Doshi, Single server queues with vacation: A survey,, Queueing Systems, 1 (1986), 29.   Google Scholar

[13]

S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue,, European Journal of Operational Research, 203 (2010), 143.  doi: 10.1016/j.ejor.2009.07.014.  Google Scholar

[14]

N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutotial, review, and research prospects,, Manufacturing and Service Operations Management, 5 (2003), 79.  doi: 10.1287/msom.5.2.79.16071.  Google Scholar

[15]

O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers,, Manufacturing and Service Operations Management, 4 (2002), 208.  doi: 10.1287/msom.4.3.208.7753.  Google Scholar

[16]

S. Graves, The application of queueing theory to continous perishable inventory systems,, Management Science, 28 (1984), 401.   Google Scholar

[17]

Y. W. Shin and T. S. Choo, M/M/s queue with impatient customers and retrials,, Applied Mathematical Modelling, 33 (2009), 2596.  doi: 10.1016/j.apm.2008.07.018.  Google Scholar

[18]

L. Takacs, A single-server queue with limited virtual waiting time,, Journal of Applied Probability, 11 (1974), 612.  doi: 10.2307/3212710.  Google Scholar

[19]

H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems,, Part 1. North-Holland, (1991).   Google Scholar

[20]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, Springer, (2006).   Google Scholar

[21]

U. Yechiali, Queues with system disasters and impatient customers when system is down,, Queueing Systems, 56 (2007), 195.  doi: 10.1007/s11134-007-9031-z.  Google Scholar

[22]

D. Yue and W. Yue, Analysis of M/M/c/N queueing system with balking, reneging and synchronous vacations,, in Advanced in Queueing Theory and Network Applications (ed. W. Yue etal.), (2009), 165.  doi: 10.1007/978-0-387-09703-9_9.  Google Scholar

[23]

D. Yue, W. Yue, Z. Saffer and X. Chen, Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy,, Journal of Industrial and Management Optimization, 10 (2014), 89.  doi: 10.3934/jimo.2014.10.89.  Google Scholar

show all references

References:
[1]

E. Altman and U. Yechiali, Analysis of customers' impatience in queues with server vacations,, Queueing Systems, 52 (2006), 261.  doi: 10.1007/s11134-006-6134-x.  Google Scholar

[2]

E. Altman and U. Yechiali, Infinite-server queues with systems' additional task and impatient customers,, Probability in the Engineering and Informational Sciences, 22 (2008), 477.  doi: 10.1007/978-1-4020-8741-7_57.  Google Scholar

[3]

C. J. Ancker and A. V. Gafarian, Some queueing problems with balking and reneging,, Operations Research, 11 (1963), 88.  doi: 10.1287/opre.11.1.88.  Google Scholar

[4]

F. Baccelli, P. Boyer and G. Hebuterne, Single-server queues with impatient customers,, Advances in Applied Probability, 16 (1984), 887.  doi: 10.2307/1427345.  Google Scholar

[5]

F. Baccelli and G. Hebuterne, On queues with impatient customers,, in Perforamnce' 81 (F. Kylstra, (1981), 159.   Google Scholar

[6]

S. Benjaafar, J. Gayon and S. Tepe, Optimal control of a production-inventory system with customer impatience,, Operations Research Letters, 38 (2010), 267.  doi: 10.1016/j.orl.2010.03.008.  Google Scholar

[7]

N. K. Boots and H. Tijms, A multiserver queueing system with impatient customers,, Management Science, 45 (1999), 444.  doi: 10.1287/mnsc.45.3.444.  Google Scholar

[8]

O. J. Boxma and P. R. de Waal, Multiserver queues with impatient customers,, ITC, 14 (1994), 743.  doi: 10.1016/B978-0-444-82031-0.50079-2.  Google Scholar

[9]

S. R. Chakravarthy, A disater queue with Markovian arrivals and impatient customers,, Applied Mathematics and Computation, 214 (2009), 48.  doi: 10.1016/j.amc.2009.03.081.  Google Scholar

[10]

D. J. Daley, General customer impatience in the queue GI/G/1,, Journal of Applied Probability, 2 (1965), 186.  doi: 10.2307/3211884.  Google Scholar

[11]

S. Dimou, A. Economou and D. Fakinos, The single server vacation queueing model with geometric abandonments,, Journal of Statistical Planning and Inference, 141 (2011), 2863.  doi: 10.1016/j.jspi.2011.03.010.  Google Scholar

[12]

B. Doshi, Single server queues with vacation: A survey,, Queueing Systems, 1 (1986), 29.   Google Scholar

[13]

S. Economou and S. Kapodistria, Synchronized abandonments in a single server unreliable queue,, European Journal of Operational Research, 203 (2010), 143.  doi: 10.1016/j.ejor.2009.07.014.  Google Scholar

[14]

N. Gans, G. Koole and A. Mandelbaum, Telephone call centers: Tutotial, review, and research prospects,, Manufacturing and Service Operations Management, 5 (2003), 79.  doi: 10.1287/msom.5.2.79.16071.  Google Scholar

[15]

O. Garnett, A. Mandelbaum and M. Reiman, Designing a call center with impatient customers,, Manufacturing and Service Operations Management, 4 (2002), 208.  doi: 10.1287/msom.4.3.208.7753.  Google Scholar

[16]

S. Graves, The application of queueing theory to continous perishable inventory systems,, Management Science, 28 (1984), 401.   Google Scholar

[17]

Y. W. Shin and T. S. Choo, M/M/s queue with impatient customers and retrials,, Applied Mathematical Modelling, 33 (2009), 2596.  doi: 10.1016/j.apm.2008.07.018.  Google Scholar

[18]

L. Takacs, A single-server queue with limited virtual waiting time,, Journal of Applied Probability, 11 (1974), 612.  doi: 10.2307/3212710.  Google Scholar

[19]

H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Volume 1: Vacation and Priority Systems,, Part 1. North-Holland, (1991).   Google Scholar

[20]

N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, Springer, (2006).   Google Scholar

[21]

U. Yechiali, Queues with system disasters and impatient customers when system is down,, Queueing Systems, 56 (2007), 195.  doi: 10.1007/s11134-007-9031-z.  Google Scholar

[22]

D. Yue and W. Yue, Analysis of M/M/c/N queueing system with balking, reneging and synchronous vacations,, in Advanced in Queueing Theory and Network Applications (ed. W. Yue etal.), (2009), 165.  doi: 10.1007/978-0-387-09703-9_9.  Google Scholar

[23]

D. Yue, W. Yue, Z. Saffer and X. Chen, Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy,, Journal of Industrial and Management Optimization, 10 (2014), 89.  doi: 10.3934/jimo.2014.10.89.  Google Scholar

[1]

Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102
[2]

Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial & Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895
[3]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[4]

Gopinath Panda, Veena Goswami, Abhijit Datta Banik, Dibyajyoti Guha. Equilibrium balking strategies in renewal input queue with Bernoulli-schedule controlled vacation and vacation interruption. Journal of Industrial & Management Optimization, 2016, 12 (3) : 851-878. doi: 10.3934/jimo.2016.12.851
[5]

Gábor Horváth, Zsolt Saffer, Miklós Telek. Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1365-1381. doi: 10.3934/jimo.2016077
[6]

Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012
[7]

Zhanyou Ma, Wuyi Yue, Xiaoli Su. Performance analysis of a Geom/Geom/1 queueing system with variable input probability. Journal of Industrial & Management Optimization, 2011, 7 (3) : 641-653. doi: 10.3934/jimo.2011.7.641
[8]

Zsolt Saffer, Miklós Telek. Analysis of BMAP vacation queue and its application to IEEE 802.16e sleep mode. Journal of Industrial & Management Optimization, 2010, 6 (3) : 661-690. doi: 10.3934/jimo.2010.6.661
[9]

Pikkala Vijaya Laxmi, Singuluri Indira, Kanithi Jyothsna. Ant colony optimization for optimum service times in a Bernoulli schedule vacation interruption queue with balking and reneging. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1199-1214. doi: 10.3934/jimo.2016.12.1199
[10]

Veena Goswami, Pikkala Vijaya Laxmi. Analysis of renewal input bulk arrival queue with single working vacation and partial batch rejection. Journal of Industrial & Management Optimization, 2010, 6 (4) : 911-927. doi: 10.3934/jimo.2010.6.911
[11]

Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905
[12]

Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control & Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007
[13]

Alexander O. Brown, Christopher S. Tang. The impact of alternative performance measures on single-period inventory policy. Journal of Industrial & Management Optimization, 2006, 2 (3) : 297-318. doi: 10.3934/jimo.2006.2.297
[14]

Yuan Zhao, Wuyi Yue. Performance analysis and optimization for cognitive radio networks with a finite primary user buffer and a probability returning scheme. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-16. doi: 10.3934/jimo.2018195
[15]

Pikkala Vijaya Laxmi, Seleshi Demie. Performance analysis of renewal input $(a,c,b)$ policy queue with multiple working vacations and change over times. Journal of Industrial & Management Optimization, 2014, 10 (3) : 839-857. doi: 10.3934/jimo.2014.10.839
[16]

Zhanyou Ma, Wenbo Wang, Linmin Hu. Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-14. doi: 10.3934/jimo.2018196
[17]

Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435
[18]

Xing Huang, Michael Röckner, Feng-Yu Wang. Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3017-3035. doi: 10.3934/dcds.2019125
[19]

Thomas Jordan, Mark Pollicott. The Hausdorff dimension of measures for iterated function systems which contract on average. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 235-246. doi: 10.3934/dcds.2008.22.235
[20]

Shiva Moslemi, Abolfazl Mirzazadeh. Performance evaluation of four-stage blood supply chain with feedback variables using NDEA cross-efficiency and entropy measures under IER uncertainty. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 379-401. doi: 10.3934/naco.2017024

2018 Impact Factor: 1.025

Tools

Metrics

  • PDF downloads (28)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences