PEAK CONGESTION IN MULTISERVER SERVICE SYSTEMS WITH SLOWLY VARYING ARRIVAL RATES

In this paper we consider the M-t/G/infinity queueing model with infin itely many servers and a nonhomogeneous Poisson arrival process. Our g oal is to obtain useful insights and formulas for nonstationary finite -server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maxim um arrival rate occurs. We describe the asymptotic behavior of these q uantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved app roximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribu tion for the number of customers in service in associated finite-serve r models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-serv er mean serving as the offered load. We compare the value and lag in p eak congestion predicted by the MOL approximation with exact values fo r M-t/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman-Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, w e suggest focusing on the form of the arrival-rate function near its p eak, in particular, on its second and third derivatives at the peak. W e suggest estimating these derivatives from data by fitting a quadrati c or cubic polynomial in a suitable interval about the peak.