In this paper, we discuss the time dependent behaviour of an M/M/1 queueing system with multiple vacations. We assume that the system is subject to sudden breakdowns and that during this period the system is flushed out. After the repair, the server waits for the arrival of the next customer. We have obtained analytical expressions for the transient state probabilities of the system size. We also obtain expressions for the transient expected value and variance of the system length. The corresponding steady state results are derived. We have performed a busy period analysis of this system in the steady state. We have obtained analytical expressions for the expected duration of a busy period and the expected duration of the vacation period. We present numerical examples to illustrate the sensitivity of the performance measures of the system to the changes in the parameters of the system.
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Kalidass et al. (2014) studied this question.