Multi-Server Queue, with Heterogeneous Service Valuations Induced by Travel Costs

This work presents a variation of Naor’s strategic observable model (Naor, 1969) for a loss system M/G/2/2, with a heterogeneous service valuations induced by the location of customers in relation to two servers, A, located at the origin, and B, located at M. Customers incur a “travel cost” which depends linearly on the distance of the customer from the server. Arrival of customers is assumed to be Poisson with a rate that is the integral of a nonnegative intensity function. We find the Nash equilibrium threshold strategy of the customers, and formulate the conditions that determine the optimal social welfare strategy. For the symmetric case (i.e., both servers have the same parameters and the intensity function is symmetric), we find the socially optimal strategies; Interestingly, we find that when only one server is idle, then under social optimality, the server also serves far away consumers, consumers whom he would not serve if he was a single server (i.e., in M/M/1/1).