Sojourn time in a queue with clustered periodic arrivals

This paper considers a first-in-first-out single-server queue with independent and homogeneous sources. Each source generates exactly one message every interval of fixed length. Each message is then divided into a constant number of fixed-size cells and these cells arrive to the queue back to back as if they form a train of cells. We call this arrival process clustered periodic arrivals. The queue with clustered periodic arrivals is an obvious generalization of ∑D/D/1 queue which corresponds to the case that each message consists of only one cell. This paper derives the stationary probability distributions of sojourn times of respective cells in a message. An interesting feature of these sojourn time distributions is that they are not continuous functions of time, but they have masses at multiples of the cell transmission time. This paper also derives the joint probability distribution of differences between sojourn times of successive celles in a message and the mean waiting times of respective cells in a message. At last, the overall mean waiting time in the queue with clustered periodic arrivals is compared with those in the corresponding queues with dispersed periodic arrivals and periodic batch arrivals, and the efficiency of dispersing cells is quantitatively shown by simple formulae.