## Single-server queue with Markov-dependent inter-arrival and service times.(English)Zbl 1036.90029

Summary: We study a single-server queue where the inter-arrival times and the service times depend on a common discrete time Markov chain. This model generalizes the well-known $$MAP$$/$$G$$/1 queue by allowing dependencies between inter-arrival and service times. The waiting time process is directly analyzed by solving Lindley’s equation by transform methods. The Laplace-Stieltjes transforms (LST) of the steady-state waiting time and queue length distribution are both derived, and used to obtain recursive equations for the calculation of the moments. Numerical examples are included to demonstrate the effect of the autocorrelation of and the cross-correlation between the inter-arrival and service times.

### MSC:

 90B22 Queues and service in operations research 60K25 Queueing theory (aspects of probability theory)
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