Tang, Yinghui; Yun, Xi; Huang, Shujuan Discrete-time \(\mathrm{Geo}^X/G/1\) queue with unreliable server and multiple adaptive delayed vacations. (English) Zbl 1147.60056 J. Comput. Appl. Math. 220, No. 1-2, 439-455 (2008). 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. 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