An Empirical Investigation of the Transient Behavior of Stationary Queueing Systems

This paper examines the transient behavior of infinite-capacity, single-server, Markovian queueing systems. It estimates Q(t), the expected number of customers in queue at time t, by numerically solving the sets of simultaneous, first-order differential equations that describe these systems. Empirical results have been drawn from these observations. For small values of t, the behavior of Q(t) is strongly influenced by the initial state of the queueing system. For systems with deterministic initial conditions, one can roughly predict which of a small set of patterns this behavior will follow. After an initial period of time and independently of initial conditions, Q(t) approaches Q(∞) in a manner that can be approximated through a decaying exponential function. On the basis of experimental evidence, we have developed an expression that provides a good approximation to the observed values of the time constant associated with this exponential function. This expression can also be used to determine an upper bound for the amount of time required until Q(t) is close to Q(∞).