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

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)
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