Allocation of Servers for Stochastic Service Stations with One Overflow Station

As one of its services, the Bell System provides microwave and cable transmission equipment for the transmission of news, sports and special-event TV programs from sites that do not have permanent transmission facilities. This paper describes a model for determining quantities of this TV equipment for service stations located throughout the country. This is actually a fundamental model for allocating servers (equipment in our case) for a stochastic service system consisting of several multiserver stations with one multiserver overflow station. The flows of customers to the initial stations are independent Poisson processes, and if a customer arrives at a station when all of its servers are busy, it is either served at the overflow station or denied service, depending on whether an overflow-server is available or not. The service times are exponential random variables. There are holding costs for having the servers in the system and costs for the services, depending on where they are done. We present a myopic nonlinear programming algorithm that finds numbers of servers for the stations that serve a desired percentage of customers at near-minimum cost. As part of our analysis, we show how Hayward's and Wilkinson's approximations for the blocking probability at the overflow station extend to heterogeneous service rates.