On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations

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October 2012, 8(4): 861-875. doi: 10.3934/jimo.2012.8.861

On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations

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, United States

Received September 2011 Revised July 2012 Published September 2012

Abstract
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A single-server retrial queue with two types of customers in which the server is subject to vacations along with breakdowns and repairs is studied. Two types of customers arrive to the system in accordance with two different independent Poisson flows. The service times of the two types of customers have two different independent general distributions. We assume that when a service is completed, the server will take vacations after an exponentially distributed reserved time. It is assumed that the server has an exponentially distributed lifetime, a generally distributed vacation time and a generally distributed repair time. There is no waiting space in front of the server, therefore, if the server is found busy, or on vacation, or down, the blocked two types of customers form two sources of repeated customers. Explicit expressions are derived for the expected number of retrial customers of each type. Additionally, by assuming both types of customers face linear costs for waiting and retrial, we discuss and compare the optimal and equilibrium retrial rates regarding the situations in which the customers are cooperative or noncooperative, respectively.

Keywords: optimal retrial rate, equilibrium retrial rate, vacation., breakdowns and repairs, Retrial queue.

Mathematics Subject Classification: Primary: 60K25; Secondary: 90B2.

Citation: Feng Zhang, Jinting Wang, Bin Liu. On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations. Journal of Industrial and Management Optimization, 2012, 8 (4) : 861-875. doi: 10.3934/jimo.2012.8.861

References:

[1]	J. R. Artalejo and A. Gómez-Corral, "Retrial Queueing Systems: A Computational Approach," Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9.
[2]	A. Economou and S. Kanta, Equilibrium balking strategiesin the observable single-server queue with breakdowns and repairs, Operations Research Letters, 36 (2008), 696-699. doi: 10.1016/j.orl.2008.06.006.
[3]	A. Economou, A. Gomez-Corral and S. Kanta, Optimal balking strategies insingle-server queues with general service and vacation times, Performance Evaluation, 68 (2011), 967-982. doi: 10.1016/j.peva.2011.07.001.
[4]	N. M. Edelson and K. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica, 43 (1975), 81-92. doi: 10.2307/1913415.
[5]	A. Elcan, Optimal customer return rate for an M/M/1 queueing system with retrials, Probability in the Engineering and Informational Sciences, 8 (1994), 521-539. doi: 10.1017/S0269964800003600.
[6]	G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman & Hall, London, 1997.
[7]	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.
[8]	R. Hassin and M. Haviv, On optimal and equilibrium retrial rates in a queueing system, Probability in the Engineering and Informational Sciences, 10 (1996), 223-227. doi: 10.1017/S0269964800004290.
[9]	R. Hassin and M. Haviv, "To Queue or Not to Queue: Equilibrium Behaviorin Queueing Systems," Kluwer Academic Publishers, Boston, 2003. doi: 10.1007/978-1-4615-0359-0.
[10]	V. G. Kulkarni, A game theoretic model for two types of customers competing for service, Operation Research Letters, 2 (1983), 119-122.
[11]	V. G. Kulkarni, On queueing systems with retrials, Journal of Applied Probability, 20 (1983), 380-389. doi: 10.2307/3213810.
[12]	P. Naor, The regulation of queue size by levying toll, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200.
[13]	S. Stidham, Jr., "Optimal Design of Queueing Systems," CRC Press, Taylor and Francis Group, Boca Raton, 2009.
[14]	N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications," Springer, New York, 2006.
[15]	J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380. doi: 10.1023/A:1010918926884.
[16]	J. Wang and F. Zhang, Equilibrium analysis of the observable queueswith balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729. doi: 10.1016/j.amc.2011.08.012.

show all references

References:

[1]	J. R. Artalejo and A. Gómez-Corral, "Retrial Queueing Systems: A Computational Approach," Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9.
[2]	A. Economou and S. Kanta, Equilibrium balking strategiesin the observable single-server queue with breakdowns and repairs, Operations Research Letters, 36 (2008), 696-699. doi: 10.1016/j.orl.2008.06.006.
[3]	A. Economou, A. Gomez-Corral and S. Kanta, Optimal balking strategies insingle-server queues with general service and vacation times, Performance Evaluation, 68 (2011), 967-982. doi: 10.1016/j.peva.2011.07.001.
[4]	N. M. Edelson and K. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica, 43 (1975), 81-92. doi: 10.2307/1913415.
[5]	A. Elcan, Optimal customer return rate for an M/M/1 queueing system with retrials, Probability in the Engineering and Informational Sciences, 8 (1994), 521-539. doi: 10.1017/S0269964800003600.
[6]	G. I. Falin and J. G. C. Templeton, "Retrial Queues," Chapman & Hall, London, 1997.
[7]	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.
[8]	R. Hassin and M. Haviv, On optimal and equilibrium retrial rates in a queueing system, Probability in the Engineering and Informational Sciences, 10 (1996), 223-227. doi: 10.1017/S0269964800004290.
[9]	R. Hassin and M. Haviv, "To Queue or Not to Queue: Equilibrium Behaviorin Queueing Systems," Kluwer Academic Publishers, Boston, 2003. doi: 10.1007/978-1-4615-0359-0.
[10]	V. G. Kulkarni, A game theoretic model for two types of customers competing for service, Operation Research Letters, 2 (1983), 119-122.
[11]	V. G. Kulkarni, On queueing systems with retrials, Journal of Applied Probability, 20 (1983), 380-389. doi: 10.2307/3213810.
[12]	P. Naor, The regulation of queue size by levying toll, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200.
[13]	S. Stidham, Jr., "Optimal Design of Queueing Systems," CRC Press, Taylor and Francis Group, Boca Raton, 2009.
[14]	N. Tian and Z. G. Zhang, "Vacation Queueing Models: Theory and Applications," Springer, New York, 2006.
[15]	J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380. doi: 10.1023/A:1010918926884.
[16]	J. Wang and F. Zhang, Equilibrium analysis of the observable queueswith balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729. doi: 10.1016/j.amc.2011.08.012.

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