The authors 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). They know that, for appropriate initial conditions, the number of busy servers at time t has a Poisson distribution for each t. The present 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 m(t) coincides with the pointwise stationary approximation λ(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/• 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,m(t) is increasing in the service-time variability, but when λ is increasing before time t,m(t) is decreasing in service-time variability. The authors suggest using these infinite-server results to approximately describe the time-dependent behavior of multiserver systems in which some arrivals are lost or delayed.
