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Transient analysis of the state dependent M/M/1/K queue

In this paper, we study the transient behavior of a state dependent M/M/1/K queue during the busy period. We derive in closed-form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. For two special cases called the threshold policy and the static policy we determine simple expressions for their joint transform. The performance metrics of the three random variables such as their expectations, variances, and covariances follow directly from the joint transform. Finally, we give additional results that contain the distribution of the maximum queue level reached during the busy period, the transform of the c-congestion period and the total number of such periods during the busy period.