The Physics of the Mt/G/∞ Queue

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 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. 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.

Dynamic Optimization with Applications to Dynamic Rate Queues

“…However, this offered load process is not a Poisson process. As discussed in Eick et al [6], the resulting deterministic, mean offered load process…”

What you should know about queueing models to set staffing requirements in service systems

“…see Theorem 1 of Eick et al [3]. Equation (6.1) applies to the situation in which the system began operation in the distant past.…”

“…As emphasized in Eick et al [3,4] and subsequent works, we can learn a lot about how to cope with time-varying demand in many-server queues by considering the associated infinite-server M t /GI/∞ model, having the same arrival process and the same service times, but infinitely many servers. When the required number of servers is large in the M t /GI/s t + GI model and the quality of service provided is consistently good, it is evident that the M t /GI/∞ model should behave similarly.…”

“…Of course, the arrival-rate function λ(t) will usually not be a polynomial, but a smooth function can be approximately by polynomials in the neighborhood of individual arguments, by virtue of Taylor-series approximations. We proceed on this basis, following Eick et al [3]; also see Massey and Whitt [15].…”

“…where λ (k) (t) is the kth derivative of λ(t) evaluated at time t, from which we obtain from (6.1) the approximation We can also consider a second-order Taylor-series approximation for the arrival-rate function λ(t): (2) (t) u 2 2 for u ≥ 0, (6.11) from which Eick et al [3,Theorem 9] obtain the approximation…”