The transient behavior of certain multiserver queues, with simulation applications

Although the transient behavior of a queueing system is often of interest, available analytical results are often quite restricted or are very complicated. We consider a queueing model with Poisson arrivals, FIFO queueing discipline, s(s ≥ i) parallel servers with independent and identically distributed exponential service times, and an arbitrary number of customers present at time zero. Probabilities in a relatively simple closed form are obtained which allow the exact evaluation of several measures of system performance, among them the expected delay in queue of each arriving customer. A numerical examination is carried out to see how the choice of initial condition affects the nature of convergence of the expected delays to their steady-state values. These results have implications for the initialization of steady-state simulations. Extensions to other queueing models (single server, Poisson input, Erlang service, arbitrary initialization) are also discussed.