ONE-NODE CLOSED MULTICHANNEL SERVICE SYSTEM: SEVERAL TYPES OF CUSTOMERS AND SERVICE RATES, AND RANDOM PICK-UP FROM

A closed queueing system has Q service channels and a waiting line. There are Ni customers of type i in the system, i = 1, 1*, m, E = Ni= N> Q. Q customers are served and Qo = N - Q stay in the waiting line. Q channels are partitioned into n groups with Q, channels in the jth group, j = 1, - * * , n. The service time of the ith type customer by a channel of the jth group is rij - Exp (Ai). When a customer leaves the channel, it is immediately replaced by another one picked up randomly from the waiting time. The customer which has cleared service joins the waiting line without delay. Let X1, be the number of ith type customers served by jth group channels in equilibrium. An explicit formula for P(X1J = ki, i = 1, . -, m; j = 1, - *, n) is found. It is shown in a form of a local limit theorem that the asymptotic distribution of {Xij} is a multidimensional normal, if Ni/N and Q,/N have positive limits as N-- 0o. Formulas for mean values and covariances are given. It turns out that the means of X1, and covariances between Xii and Xrk can be found, using an efficient iterative algorithm, from the deterministic version of the system. A numerical example demonstrates that the normal approximation is rather accurate.