On the price of anarchy in a single-server queue with heterogeneous service valuations induced by travel costs

This work presents a variation of Naor’s strategic observable model (1969), by adding a component of customer heterogeneity induced by the location of customers in relation to the server. Accordingly, customers incur a “travel cost” which depends linearly on the distance of the customer from the server. The arrival of customers with distances less than x is assumed to be a Poisson process with rate λ(x) = ∫ x 0 h(y)dy <∞, where h(y) is a nonnegative “intensity” function of the distance y. In a loss system M/G/1/1 we define the threshold Nash equilibrium strategy xe, and the optimal social threshold strategy x ∗. We consider the price of anarchy (PoA) and prove that it converges to 1 when xe → 0. The behaviour of PoA when xe →∞ is more complex and interesting. We show that if the rate of arriving customers is bounded then PoA converges to 1, i.e., in the limit there is no difference between the social and equilibrium optimal benefits (even though the corresponding optimal strategies x∗ and xe do not coincide). The rest of the paper is dedicated for the case in which the rate of arriving customers is unbounded. We develop an explicit formula to calculate lim xe→∞ PoA(h, xe) when it exists. We present sufficient conditions for the limit to exist and for the existence of a simple formula for ∗[email protected], [email protected] y [email protected]