Stochastic approximations and monotonicity of a single server feedback retrial queue. (English) Zbl 1264.90051

Summary: We focus on a stochastic comparison of the Markov chains to derive some qualitative approximations for an \(M/G/1\) retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters.


90B22 Queues and service in operations research
60J27 Continuous-time Markov processes on discrete state spaces
60E15 Inequalities; stochastic orderings
60K25 Queueing theory (aspects of probability theory)
Full Text: DOI


[1] J. R. Artalejo, “Accessible bibliography on retrial queues: progress in 2000-2009,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1071-1081, 2010. · Zbl 1198.90011
[2] J. R. Artalejo and A. Gómez-Corral, Retrial queueing system: A Computation Approach, Springer, Berlin, Germany, 2008. · Zbl 1161.60033
[3] G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman and Hall, London, UK, 1997. · Zbl 0944.60005
[4] G. Ayyappan, A. Muthu Ganapathi Subramanian, and G. Sekar, “M/M/1 retrial queueing system with loss and feedback under non-pre-emptive priority service by matrix geometric method,” Applied Mathematical Sciences, vol. 4, no. 45-48, pp. 2379-2389, 2010. · Zbl 1220.60051
[5] T. Yang and J. G. C. Templeton, “A survey of retrial queues,” Queueing Systems. Theory and Applications, vol. 2, no. 3, pp. 201-233, 1987. · Zbl 0658.60124
[6] D. Stoyan, Comparison Methods for Queues and Other Stochastic Models, John Wiley & Sons, Chichester, UK, 1983. · Zbl 0536.60085
[7] M. Shaked and J. G. Shanthikumar, Stochastic Orders, Springer, New York, NY, USA, 2007. · Zbl 1111.62016
[8] M. Shaked and J. G. Shanthikumar, Stochastic Orders and Their Applications, Academic Press, Boston, Mass, USA, 1994. · Zbl 0806.62009
[9] N. Oukid and A. Aissani, “Bounds on busy period for queues with breakdowns,” Advances and Applications in Statistics, vol. 11, no. 2, pp. 137-156, 2009. · Zbl 1168.60365
[10] M. Boualem, N. Djellab, and D. Aïssani, “Stochastic inequalities for M/G/1 retrial queues with vacations and constant retrial policy,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 207-212, 2009. · Zbl 1185.90041
[11] S. Taleb and A. Aissani, “Unreliable M/G/1 retrial queue: monotonicity and comparability,” Queueing Systems. Theory and Applications, vol. 64, no. 3, pp. 227-252, 2010. · Zbl 1186.60014
[12] N. Djellab, “On the M/G/1 retrial queue with feedback,” in Proceedings of the International Conference on Mathematical Methods of Optimization of Telecommunication Networks, pp. 32-35, Minsk, Byelorussia, 2005.
[13] L. Takács, “A single-server queue with feedback,” The Bell System Technical Journal, vol. 42, pp. 505-519, 1963.
[14] B. Heidergott and F. J. Vázquez-Abad, “Measure-valued differentiation for Markov chains,” Journal of Optimization Theory and Applications, vol. 136, no. 2, pp. 187-209, 2008. · Zbl 1149.90173
[15] B. Heidergott and F. J. Vázquez-Abad, “Measure-valued differentiation for random horizon problems,” Markov Processes and Related Fields, vol. 12, no. 3, pp. 509-536, 2006. · Zbl 1117.60073
[16] B. Heidergott, A. Hordijk, and H. Weisshaupt, “Measure-valued differentiation for stationary Markov chains,” Mathematics of Operations Research, vol. 31, no. 1, pp. 154-172, 2006. · Zbl 1278.90428
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.