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 and 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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[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.

[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.

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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[25]

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

[26]

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

[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.

[28]

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

[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.

[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.

[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.

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