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The distribution of delays of dispersed messages in an M/M/1 queue

We analyze the distribution of the delay of messages in an infinite capacity M/M/1 queue. A message is composed of n packets, and the arrival of the packets to the queue is Poisson. Our calculations are based on recursive schemes. We obtain explicit expressions for the Laplace-Stieltjes transform (LST) of the delays, which enables to obtain exact expressions for the moments of the delay in a complexity smaller than the one obtained by using recursive schemes. We repeat the above calculations for the case that messages are dispersed, i.e. packets from several sources arrive to an M/M/1 queue and are served according to the FIFO discipline. Hence, several packets of other messages may arrive between consecutive packets of a given message.