Unreliable M/G/1 retrial queue: Monotonicity and comparability. (English) Zbl 1186.60014

Taleb, Samira; Aissani, Amar

Queueing Syst. 64, No. 3, 227-252 (2010).

Summary: We investigate the monotonicity properties of an unreliable M/G/1 retrial queue using the general theory of stochastic ordering. We show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and increasing convex ordering. We obtain conditions of comparability of two transition operators and we obtain comparability conditions of the number of customers in the system. Inequalities are derived for the mean characteristics of the busy period, number of customers served during a busy period, number of orbit busy periods and waiting times. Inequalities are also obtained for some probabilities of the steady-state distribution of the server state. An illustrative numerical example is presented.

MSC:

60E15	Inequalities; stochastic orderings
60K10	Applications of renewal theory (reliability, demand theory, etc.)
60K25	Queueing theory (aspects of probability theory)

Keywords:

stochastic ordering; monotonicity; comparability; ageing distributions; unreliable retrial queue; stationary distribution

References:

[1]	Artalejo, J.R., Gomez-Corral, A.: Retrial Queueing Systems: A Computational Approach. Springer, Heidelberg (2008) · Zbl 1161.60033
[2]	Bon, J.L.: Fiabilité des Systèmes: Méthodes Mathématiques. Masson, Paris (1995)
[3]	Choi, B.D., Chang, Y.: Single server retrial queues with priority calls. Math. Comput. Model. 30(3–4), 7–32 (1999) · Zbl 1042.60533
[4]	Denuit, M., Muller, A.: Smooth generators of integral stochastic orders. Ann. Appl. Probab. 13(4), 1174–1184 (2002) · Zbl 1065.60015
[5]	El-Taha, M., Stidham, S.: Sample-Path Analysis of Queueing Systems. Kluwer Academic, Boston (1999) · Zbl 0938.60095
[6]	Falin, G.I., Templeton, J.G.C.: Retrial Queues. Chapman & Hall, New York (1997) · Zbl 0944.60005
[7]	Gomez-Corral, A.: Stochastic analysis of single server retrial queue with the general retrial times. Nav. Res. Logist. 46, 561–581 (1999) · Zbl 0934.90017
[8]	Jean-Marie, A., Liu, Z.: Stochastic comparisons for queueing models via random sums and intervals. Adv. Appl. Probab. 24, 960–985 (1992) · Zbl 0763.60006
[9]	Khalil, Z., Falin, G.: Stochastic inequalities for M/G/1 retrial queues. Oper. Res. Lett. 16, 285–290 (1994) · Zbl 0819.60090
[10]	Kijima, M., Ohnishi, M.: Stochastic orders and their applications in financial optimization. Math. Methods Oper. Res. 50(2), 351–372 (1999) · Zbl 0958.91020
[11]	Liang, H.M.: Service station factors in monotonicity of retrial queues. Math. Comput. Model. 30, 189–196 (1999) · Zbl 1042.60544
[12]	Liang, H.M., Kulkarni, V.G.: Monotonicity properties of single server retrial queues. Commun. Stat. Stoch. Models 9(3), 373–400 (1993) · Zbl 0777.60091
[13]	Lindvall, T.: Lectures on the Coupling Method. Dover, Mineola (2002) · Zbl 1013.60001
[14]	Muller, A., Stoyan, D.: Comparison Methods for Stochastic Models and Risks. Wiley, New York (2002) · Zbl 0999.60002
[15]	Ross, S.M.: Stochastic Processes. Wiley, New York (1996)
[16]	Shaked, M., Shanthikumar, J.G.: Stochastic Orders and their Applications. Academic Press, San Diego (1994) · Zbl 0806.62009
[17]	Shin, Y.W.: Monotonicity properties in various retrial queues and their applications. Queueing Syst. 53, 147–157 (2006) · Zbl 1104.60058
[18]	Shin, Y.W., Kim, Y.C.: Stochastic comparisons of Markovian retrial queues. J. Korean Stat. Soc. 29(4), 473–488 (2004)
[19]	Stoyan, D.: Comparison Methods for Queues and other Stochastic Models. Wiley, New York (1983) · Zbl 0536.60085
[20]	Szekli, R.: Stochastic Ordering and Dependence in Applied Probability. Springer, New York (1995) · Zbl 0815.60017
[21]	Taleb, S., Saggou, H., Aissani, A.: Unreliable M/G/1 retrial queue with geometric loss and random reserved time. Int. J. Oper. Res. 7(2), 171–191 (2010) · Zbl 1183.90127
[22]	Thorisson, H.: Coupling, Stationarity and Regeneration. Springer, New York (2000) · Zbl 0949.60007