Stochastic comparison bounds for an M1, M2/G1, G2/1 retrial queue with two way communication

Abstract

The main goal in the present paper is to provide a technique that considers the stochastic comparison approach for investigating monotonicity and comparability of an $M_{1},M_{2}/G_{1},G_{2}/1$ retrial queues with two way communication. This approach is developed for comparing a non Markov process to Markov process with many possible stochastic orderings. Particularly, we show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and convex ordering, as well as the comparability of two transition operators. Bounds are also obtained for the stationary distribution of the number of customers at departure epochs. Additionally,  the performance measures of the system considered can be estimated by those of an $M_1,M_2/M_1,M_2/1$ retrial queue with two way communication when the service time distribution is NBUE (respectively NWUE). Finally, we validate stochastic comparison results by presenting a numerical example illustrating the interest of the approach.

Keywords

• Retrial queues
• Two way communication
• Markov chain
• Monotonicity
• Stochastic comparison
• Simulation

References

1. [1] Z. Aksin, M. Armony and V. Mehrotra. The modern call center: A multi-disciplinary perspective on operations management research, Production and Operations Management 16 (6), 665–688, 2007.
2. [2] J. R. Artalejo and A. Gómez-Corral. Retrial queueing system: A computational approach, Springer-Verlag, Berlin, Heidelberg, 2008.
3. [3] J. R. Artalejo and T. Phung-Duc. Single server retrial queues with two way communication, Applied Mathematical Modelling 37 (4), 1811–1822, 2013.
4. [4] J. R. Artalejo and P. D. Tuan. Markovian retrial queues with two way communication, Journal of industrial and management optimization 8 (4), 781–206, 2012.
5. [5] M. Boualem. Insensitive bounds for the stationary distribution of a single server retrial queue with server subject to active breakdowns, Advances in Operations Research 2014, Article ID 985453, 12 pages, 2014.
6. [6] M. Boualem, A. Bareche and M. Cherfaoui. Approximate controllability of stochastic bounds of stationary distribution of an M/G/1 queue with repeated attempts and two-phase service, International Journal of Management Science and Engineering Management, DOI: 10.1080/17509653.2018.1488634, 2018.
7. [7] M. Boualem, M. Cherfaoui and D. Aissani. Monotonicity properties for a single server queue with classical retrial policy and service interruptions, Proceedings of the Jangjeon Mathematical Society 19 (2), 225–236, 2016.
8. [8] M. Boualem, M. Cherfaoui, N. Djellab and D. Aissani. Stochastic Analysis of an M/G/1 Retrial Queue with FCFS. In: Ould Saïd et al. (eds.) Functional Statistics and Applications. Contributions to Statistics. Springer, Cham. 127–139, 2015.

9. [9] M. Boualem, M. Cherfaoui, N. Djellab and D. Aissani. Inégalités stochastiques pour le modèle d’attente M/G/1/1 avec rappels, Afrika Matematika 28 (5-6), 851–868, 2017.
10. [10] M. Boualem, M. Cherfaoui, N. Djellab and D. Aissani. A stochastic version analysis of an M/G/1 retrial queue with Bernoulli schedule, Bulletin of the Iranian Mathematical Society 43 (5), 1377–1397, 2017.
11. [11] M. Boualem, N. Djellab and D. Aissani. Stochastic inequalities for M/G/1 retrial queues with vacations and constant retrial policy, Mathematical and Computer Modelling 50 (1-2), 207–212, 2009.
12. [12] M. Boualem, N. Djellab and D. Aissani. Stochastic approximations and monotonicity of a single server feedback retrial queue, Mathematical Problems in Engineering 2012, Article ID 536982, 13 pages, 2012.
13. [13] M. Boualem, N. Djellab and D. Aissani. Stochastic bounds for a single server queue with general retrial times, Bulletin of the Iranian Mathematical Society 40, 183–198, 2014.
14. [14] Z. Khalil and G. Falin. Stochastic inequalities for M/G/1 retrial queues, Operations Research Letters 16, 285–290, 1994.
15. [15] H. M. Liang. Service station factors in monotonicity of retrial queues, Mathematical and Computer Modelling 30, 189–196, 1999.
16. [16] H. M. Liang and V. G. Kulkarni. Monotonicity properties of single server retrial queues, Stochastic Models 9, 373–400, 1993.
17. [17] S. Ouazine and K. Abbas. A functional approximation for retrial queues with two way communication, Annals of Operations Research 247 (1), 211–227, 2016.
18. [18] H. Sakurai and T. Phung-Duc. Two-way communication retrial queues with multiple types of outgoing calls, TOP 23, 466–492, 2015.
19. [19] M. Shaked and J. G. Shanthikumar. Stochastic Orders, Springer-Verlag, New York, 2007.
20. [20] Y. W. Shin. Monotonicity properties in various retrial queues and their applications, Queueing Systems 53, 147–157, 2006.
21. [21] D. Stoyan. Comparison methods for queues and other stochastic models, Wiley, New York, 1983.
22. [22] S. Taleb and A. Aissani. Unreliable M/G/1 avec rappels: monotonicity and comparability, Queueing Systems 64 (3), 227–252, 2010.
23. [23] D. Zirem, M. Boualem, K. Adel-Aissanou and D. Aissani. Analysis of a single server batch arrival unreliable queue with balking and general retrial time, Quality Technology & Quantitative Management, DOI: 10.1080/16843703.2018.1510359, 2018.
24. [24] D. Zirem, M. Boualem and D. Aissani. $M^{X}/G/1$ Queueing system with breakdowns and repairs, International Journal of Advances in Computer Science & Its Applications 6 (3), 18–21, 2016.