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Compound geometric approximation under a failure rate constraint. (English) Zbl 1351.62060

Summary: We consider compound geometric approximation for a nonnegative, integer-valued random variable \(W\). The bound we give is straightforward but relies on having a lower bound on the failure rate of \(W\). Applications are presented to M/G/1 queuing systems, for which we state explicit bounds in approximations for the number of customers in the system and the number of customers served during a busy period. Other applications are given to birth-death processes and Poisson processes.

MSC:

62E17 Approximations to statistical distributions (nonasymptotic)
60E15 Inequalities; stochastic orderings
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
62E10 Characterization and structure theory of statistical distributions