The Physics of the Mt/G/∞ Queue

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Abstract
We establish some general structural results and derive some simple formulas describing the time-dependent performance of the Mt/G/∞ queue with a nonhomogeneous Poisson arrival process. We know that, for appropriate initial conditions, the number of busy servers at time t has a Poisson distribution for each t. Our results show how the time-dependent mean function m depends on the time-dependent arrival-rate function λ and the service-time distribution. For example, when λ is quadratic, the mean mt coincides with the pointwise stationary approximation λtE[S], where S is a service time, except for a time lag and a space shift. It is significant that the well known insensitivity property of the stationary M/G/∞ model does not hold for the nonstationary Mt/G/∞ model; the time-dependent mean function m depends on the service-time distribution beyond its mean. The service-time stationary-excess distribution plays an important role. When λ is decreasing before time t, mt is increasing in the service-time variability, but when λ is increasing before time t, mt is decreasing in service-time variability. We suggest using these infinite-server results to approximately describe the time-dependent behavior of multiserver systems in which some arrivals are lost or delayed.

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