American Institute of Mathematical Sciences

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October  2013, 9(4): 901-917. doi: 10.3934/jimo.2013.9.901

Equilibrium joining probabilities in observable queues with general service and setup times

Feng Zhang 1, , Jinting Wang 1, and  Bin Liu 2,

1. 

Department of Mathematics, Beijing Jiaotong University, 100044 Beijing
2. 

Department of Mathematics, University of Northern Iowa, Cedar Falls, IA 50614-0506

Received  October 2012 Revised  March 2013 Published  August 2013

This paper analyzes an M/G/1 queue with general setup times from an economical point of view. In such a queue whenever the system becomes empty, the server is turned off. A new customer's arrival will turn the server on after a setup period. Upon arrival, the customers decide whether to join or balk the queue based on observation of the queue length and the status of the server, along with the reward-cost structure of the system. For the observable and almost observable cases, the equilibrium joining strategies of customers who wish to maximize their expected net benefit are obtained. Two numerical examples are presented to illustrate the equilibrium joining probabilities for these cases under some specific distribution functions of service times and setup times.
Keywords: M/G/1 queue, joining probabilities, setup time., economical analysis, equilibrium solution.
Mathematics Subject Classification: Primary: 60K25; Secondary: 90B2.
Citation: Feng Zhang, Jinting Wang, Bin Liu. Equilibrium joining probabilities in observable queues with general service and setup times. Journal of Industrial & Management Optimization, 2013, 9 (4) : 901-917. doi: 10.3934/jimo.2013.9.901
References:
[1]

E. Altman and R. Hassin, Non-threshold equilibrium for customers joining an M/G/1 queue, in "ISDG2002, Vol. I, II" (St. Petersburg), St. Petersburg State Univ. Inst. Chem., St. Petersburg, (2002), 56-64.  Google Scholar

[2]

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.  Google Scholar

[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.  Google Scholar

[4]

A. Borthakur and G. Choudhury, A multiserver Poisson queue with a general startup time under $N$-policy, Calcutta Statistical Association Bulletin, 49 (1999), 199-213.  Google Scholar

[5]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes, European Journal of Operational Research, 218 (2012), 708-715. doi: 10.1016/j.ejor.2011.11.043.  Google Scholar

[6]

A. Burnetas, Customer equilibrium and optimal strategies in Markovian queues in series, Annals of Operations Research, 208 (2013), 515-529. doi: 10.1007/s10479-011-1010-4.  Google Scholar

[7]

A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times, Queueing Systems, 56 (2007), 213-228. doi: 10.1007/s11134-007-9036-7.  Google Scholar

[8]

G. Choudhury, On a batch arrival Poisson queue with a random setup and vacation period, Computers $&$ Operations Research, 25 (1998), 1013-1026. doi: 10.1016/S0305-0548(98)00038-0.  Google Scholar

[9]

G. Choudhury, An $M^X$/G/1 queueing system with a setup period and a vacation period, Queueing Systems, 36 (2000), 23-38. doi: 10.1023/A:1011089403694.  Google Scholar

[10]

A. Economou and S. Kanta, On balking strategies and pricing for the single server Markovian queue with compartmented waiting space, Queueing Systems, 59 (2008), 237-269. doi: 10.1007/s11134-008-9083-8.  Google Scholar

[11]

A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs, Operations Research Letters, 36 (2008), 696-699. doi: 10.1016/j.orl.2008.06.006.  Google Scholar

[12]

A. Economou and S. Kanta, Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue, Naval Research Logistics, 58 (2011), 107-122. doi: 10.1002/nav.20444.  Google Scholar

[13]

A. Economou, A. Gomez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times, Performance Evaluation, 68 (2011), 967-982. doi: 10.1016/j.peva.2011.07.001.  Google Scholar

[14]

A. Economou and A. Manou, Equilibrium balking strategies for a clearing queueing system in alternating environment, Annals of Operations Research, 208 (2013), 489-514. doi: 10.1007/s10479-011-1025-x.  Google Scholar

[15]

N. M. Edelson and K. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica, 43 (1975), 81-92. doi: 10.2307/1913415.  Google Scholar

[16]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues, Operations Research, 59 (2011), 986-997. doi: 10.1287/opre.1100.0907.  Google Scholar

[17]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers, European Journal of Operational Research, 222 (2012), 278-286. doi: 10.1016/j.ejor.2012.05.026.  Google Scholar

[18]

R. Hassin and M. Haviv, Equilibrium threshold strategies: the case of queues with priorities, Operations Research, 45 (1997), 966-973. doi: 10.1287/opre.45.6.966.  Google Scholar

[19]

R. Hassin and M. Haviv, "To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems," International Series in Operations Research & Management Science, 59, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4615-0359-0.  Google Scholar

[20]

M. Haviv and Y. Kerner, On balking from an empty queue, Queueing Systems, 55 (2007), 239-249. doi: 10.1007/s11134-007-9020-2.  Google Scholar

[21]

Q. M. He and E. Jewkes, Flow time in the $M AP$/G/1 queue with customer batching and setup times, Stochastic Models, 11 (1995), 691-711. doi: 10.1080/15326349508807367.  Google Scholar

[22]

Y. Kerner, The conditional distribution of the residual service time in the $M_n$/G/1 queue, Stochastic Models, 24 (2008), 364-375. doi: 10.1080/15326340802232210.  Google Scholar

[23]

Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue, Game and Economic Behavior, 71 (2011), 521-526. doi: 10.1016/j.geb.2010.06.002.  Google Scholar

[24]

W. Liu, Y. Ma and J. Li, Equilibrium threshold strategies in observable queueing systems under single vacation policy, Applied Mathematical Modelling, 36 (2012), 6186-6202. doi: 10.1016/j.apm.2012.02.003.  Google Scholar

[25]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200.  Google Scholar

[26]

S. Stidham, Jr., "Optimal Design of Queueing Systems," CRC Press, Boca Raton, FL, 2009. doi: 10.1201/9781420010008.  Google Scholar

[27]

W. Sun, P. Guo and N. Tian, Equilibrium threshold strategies in observable queueing systems with setup/closedown times, Central European Journal of Operational Research, 18 (2010), 241-268. doi: 10.1007/s10100-009-0104-4.  Google Scholar

[28]

H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation. Vol. 1. Vacation and Priority Systems. Part I," North-Holland, Amsterdam, 1991.  Google Scholar

[29]

N. Tian and Z.G. Zhang, "Vacation Queueing Models. Theory and Applications," International Series in Operations Research & Management Science, 93, Springer, New York, 2006.  Google Scholar

[30]

J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729. doi: 10.1016/j.amc.2011.08.012.  Google Scholar

[31]

F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations, Journal of Industrial and Management Optimization, 8 (2012), 861-875. doi: 10.3934/jimo.2012.8.861.  Google Scholar

show all references

References:
[1]

E. Altman and R. Hassin, Non-threshold equilibrium for customers joining an M/G/1 queue, in "ISDG2002, Vol. I, II" (St. Petersburg), St. Petersburg State Univ. Inst. Chem., St. Petersburg, (2002), 56-64.  Google Scholar

[2]

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.  Google Scholar

[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.  Google Scholar

[4]

A. Borthakur and G. Choudhury, A multiserver Poisson queue with a general startup time under $N$-policy, Calcutta Statistical Association Bulletin, 49 (1999), 199-213.  Google Scholar

[5]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes, European Journal of Operational Research, 218 (2012), 708-715. doi: 10.1016/j.ejor.2011.11.043.  Google Scholar

[6]

A. Burnetas, Customer equilibrium and optimal strategies in Markovian queues in series, Annals of Operations Research, 208 (2013), 515-529. doi: 10.1007/s10479-011-1010-4.  Google Scholar

[7]

A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times, Queueing Systems, 56 (2007), 213-228. doi: 10.1007/s11134-007-9036-7.  Google Scholar

[8]

G. Choudhury, On a batch arrival Poisson queue with a random setup and vacation period, Computers $&$ Operations Research, 25 (1998), 1013-1026. doi: 10.1016/S0305-0548(98)00038-0.  Google Scholar

[9]

G. Choudhury, An $M^X$/G/1 queueing system with a setup period and a vacation period, Queueing Systems, 36 (2000), 23-38. doi: 10.1023/A:1011089403694.  Google Scholar

[10]

A. Economou and S. Kanta, On balking strategies and pricing for the single server Markovian queue with compartmented waiting space, Queueing Systems, 59 (2008), 237-269. doi: 10.1007/s11134-008-9083-8.  Google Scholar

[11]

A. Economou and S. Kanta, Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs, Operations Research Letters, 36 (2008), 696-699. doi: 10.1016/j.orl.2008.06.006.  Google Scholar

[12]

A. Economou and S. Kanta, Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue, Naval Research Logistics, 58 (2011), 107-122. doi: 10.1002/nav.20444.  Google Scholar

[13]

A. Economou, A. Gomez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times, Performance Evaluation, 68 (2011), 967-982. doi: 10.1016/j.peva.2011.07.001.  Google Scholar

[14]

A. Economou and A. Manou, Equilibrium balking strategies for a clearing queueing system in alternating environment, Annals of Operations Research, 208 (2013), 489-514. doi: 10.1007/s10479-011-1025-x.  Google Scholar

[15]

N. M. Edelson and K. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica, 43 (1975), 81-92. doi: 10.2307/1913415.  Google Scholar

[16]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues, Operations Research, 59 (2011), 986-997. doi: 10.1287/opre.1100.0907.  Google Scholar

[17]

P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers, European Journal of Operational Research, 222 (2012), 278-286. doi: 10.1016/j.ejor.2012.05.026.  Google Scholar

[18]

R. Hassin and M. Haviv, Equilibrium threshold strategies: the case of queues with priorities, Operations Research, 45 (1997), 966-973. doi: 10.1287/opre.45.6.966.  Google Scholar

[19]

R. Hassin and M. Haviv, "To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems," International Series in Operations Research & Management Science, 59, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4615-0359-0.  Google Scholar

[20]

M. Haviv and Y. Kerner, On balking from an empty queue, Queueing Systems, 55 (2007), 239-249. doi: 10.1007/s11134-007-9020-2.  Google Scholar

[21]

Q. M. He and E. Jewkes, Flow time in the $M AP$/G/1 queue with customer batching and setup times, Stochastic Models, 11 (1995), 691-711. doi: 10.1080/15326349508807367.  Google Scholar

[22]

Y. Kerner, The conditional distribution of the residual service time in the $M_n$/G/1 queue, Stochastic Models, 24 (2008), 364-375. doi: 10.1080/15326340802232210.  Google Scholar

[23]

Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue, Game and Economic Behavior, 71 (2011), 521-526. doi: 10.1016/j.geb.2010.06.002.  Google Scholar

[24]

W. Liu, Y. Ma and J. Li, Equilibrium threshold strategies in observable queueing systems under single vacation policy, Applied Mathematical Modelling, 36 (2012), 6186-6202. doi: 10.1016/j.apm.2012.02.003.  Google Scholar

[25]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200.  Google Scholar

[26]

S. Stidham, Jr., "Optimal Design of Queueing Systems," CRC Press, Boca Raton, FL, 2009. doi: 10.1201/9781420010008.  Google Scholar

[27]

W. Sun, P. Guo and N. Tian, Equilibrium threshold strategies in observable queueing systems with setup/closedown times, Central European Journal of Operational Research, 18 (2010), 241-268. doi: 10.1007/s10100-009-0104-4.  Google Scholar

[28]

H. Takagi, "Queueing Analysis: A Foundation of Performance Evaluation. Vol. 1. Vacation and Priority Systems. Part I," North-Holland, Amsterdam, 1991.  Google Scholar

[29]

N. Tian and Z.G. Zhang, "Vacation Queueing Models. Theory and Applications," International Series in Operations Research & Management Science, 93, Springer, New York, 2006.  Google Scholar

[30]

J. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729. doi: 10.1016/j.amc.2011.08.012.  Google Scholar

[31]

F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations, Journal of Industrial and Management Optimization, 8 (2012), 861-875. doi: 10.3934/jimo.2012.8.861.  Google Scholar

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