Further transient analysis of theBMAP/G/1 Queue

Previously, we derived the two-dimensional transforms of the emptiness function, the transient workload and queue-length distributions in the single-server queue with general service times and a batch Markovian arrival process (BMAP). This arrival process includes the familiar phase-type renewal process and the Markov modulated Poisson process as special cases, as well as superpositions of these processes, and allows correlated interarrival times and batch sizes. We continue the transient analysis of this model in this paper by deriving explicit expressions for the transforms of the queue length at the -th departure (assuming a departure at time ), and the delay of the -th arrival (keeping track of the appropriate phase changes). Also, the departure process is characterized by the double transform of the probability that the -th departure occurs at time less than or equal to time .