The Physics of the Mt/G/∞ Queue

We establish some general structural results and derive some simple formulas describing the time-dependent performance of the M t / 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 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 M t / 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. 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.