Note: On the Efficiency of Imbalance in Multi-Facility Multi-Server Service Systems

We consider the problem of simultaneously allocating servers and demands in a service system with independent multiple facilities. We assume a fixed number of facilities and total servers which must service a given Poisson arrival stream. We also assume that service times are identically distributed and independent of the server or facility. The allocation decision is one of simultaneously determining the number of servers and the fraction of the total arrival stream for each facility in order to optimize a given performance measure. Several performance measures are considered including minimizing expected system delay and equalizing delays across facilities. Our findings demonstrate that the overall system performance improves as the individual facilities become more unbalanced in the number of allocated servers. More formally, we show that if there is a server allocation that is maximal under the partial order of majorization, then it is optimal.