The M/M/\(\infty\) service system with ranked servers in heavy traffic. With a preface by Franz Ferschl (Q796435)

These Lecture Notes are concerned with a service facility consisting of a large (or infinite) number of servers in parallel, where there is a preferential ordering of the servers. Each newly arriving customer enters the lowest ranked available server, and remains there until his service is completed. It is assumed that customers arrive according to a Poisson process of rate \(\lambda\), that service times are exponentially distributed with parameter \(\mu\), and that \(a=\lambda /\mu\) is large. The object of study is the random function N(s,t) describing the number of busy servers among the first s ordered servers at time t. The analysis is motivated by applications to telephone traffic: note that the equilibrium distribution of N(s,t) is the Erlang distribution, and the function \(N(\infty,t)\)- N(s,t) describes overflow traffic from a group of s primary servers. The bulk of the analysis deals with asymptotic properties, particularly in the limit a, \(s\to \infty\) with \(\kappa =(s- a)/s^{{1\over2}}\) fixed. Some time-dependent solutions are described, and the accuracy of the ''equivalent random method'' is discussed.