Optimal Server Allocation in a System of Multi-Server Stations

Consider a closed queueing network (Gordon and Newell [Gordon, W. J., G. F. Newell. 1967. Closed queueing networks with exponential servers. Oper. Res. 15 252--267.]) with a set of stations. The service rate at each station is an increasing concave function of the number of jobs at that station. Suppose there also exists a station that has c (\ge 1) parallel servers, each of which has a fixed service rate. We show that the throughput of this network is an increasing concave function with respect to c. This result is then applied to solve the optimal server allocation problem in a system of multi-server stations with a fixed buffer capacity (for the total number of jobs) at each station. For a single-station system, the simultaneous optimal allocation of both servers and buffer capacity is also studied.