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Discrete-time $\mathrm{Geo}^X/G/1$ queue with unreliable server and multiple adaptive delayed vacations. (English) Zbl 1147.60056
Summary: We consider a discrete-time Geo$^X/G/1$ queue with unreliable server and multiple adaptive delayed vacations policy in which the vacation time, service time, repair time and the delayed time all follow arbitrary discrete distribution. By using a concise decomposition method, the transient and steady-state distributions of the queue length are studied, and the stochastic decomposition property of steady-state queue length has been proved. Several common vacation policies are special cases of the vacation policy presented in this study. The relationship between the generating functions of steady-state queue length at departure epoch and arbitrary epoch is obtained. Finally, we give some numerical examples to illustrate the effect of the parameters on several performance characteristics.

##### MSC:
 60K25 Queueing theory 90B22 Queues and service (optimization)
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##### References:
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