Consider a discrete time Geo/Geo/1 queue with single working vacation. The server works at a lower rate rather than completely stoping service during the vacation period. Using quasi birth and death chain and matrix-geometric solution method, we obtain distributions of the number of customers in the system and the sojourn time of a customer and their stochastic decomposition structures in the stationary state. Furthermore, we derive the formulae of expected regular busy period and expected busy cycle. Finally, using some numerical examples, we analyze the effect of the parameters on the expected queue length and waiting time, and verify that our model represents some practical problems reasonably well.
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Li et al. (2008) studied this question.