The Infinite-Server Queue with Poisson Arrivals and Semi-Markovian Services

This paper considers the queue with an infinite number of servers with a Poisson arrival process and with semi-Markovian service times. It studies jointly the queue-length process and the type of the first customer to join the queue after t and obtains transient and asymptotic results that are matrix extensions of the corresponding results of the M/G/∞ queue. In particular, it proves that the limiting distribution of the queue-length process is Poisson.