A POLLING MODEL WITH RETRIAL CUSTOMERS

Abstract

A polling model with n stations and switchover times is considered. The customers are of n different types, arrive to the system according to the Poisson distribution in batches of random size, and if they find the server unavailable, they start to make retrials until succeed to find a position for service. Each batch may contain customers of different types while the numbers of customers belonging to each type in a batch are distributed according to a multivariate general distribution. The server, upon polling a station, stays there for an exponential period of time and if a customer asks for service before this time expires, the customer is served and a new "stay period" begins. Finally the service times and the switchover times are both arbitrarily distributed with different distributions for the different stations. For such a model we obtain formulas for the expected number of retrial customers in each station in a steady state. Our results can be easily adapted to hold for zero switchover times and also in the case of the ordinary exhaustive service polling model with (without) switchover times and correlated batch arrivals. In all cases mentioned above (retrial model, exhaustive model, switchover times, zero switchover times) to find the expected queue lengths we need finally to solve a set of only n linear equations (O(n^3) arithmetic operations to compute the coefficients). Tables of numerical values are finally obtained and used to observe the system performance when we vary the values of the parameters.