THE PHYSICS OF THE M(T)/G/IOTA QUEUE

We establish some general structural results and derive some simple formulas describing the time-dependent performance of the M(t)/G/infinity 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 lambda and the service-time distribution. For example, when lambda is quadratic, the mean m(t) coincides with the pointwise stationary approximation lambda(t)E[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/infinity model does not hold for the nonstationary M(t)/G/infinity 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 lambda is decreasing before time t, m(t) is increasing in the service-time variability, but when lambda is increasing before time t, m(t) 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.