The Transient Behavior of the M/M/s Queue, with Implications for Steady-State Simulation

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Abstract
Although the transient behavior of a queueing system is often of interest, available analytical results are usually quite restricted or are very complicated. We consider the M/M/s queue with an arbitrary number of customers present at time zero. We obtain probabilities in a relatively simple closed form that can be used to evaluate exactly several measures of system performance, including 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. We also discuss the implications of these results for the initialization of steady-state simulations.

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