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

This work presents a strategic observable model where customer heterogeneity is induced by the customers' locations and travel costs. The arrival of customers with distances less than x is assumed to be Poisson with rate equal to the integral from 0 to x, of a nonnegative intensity function h. In a loss system M/G/1/1 we define the threshold Nash equilibrium strategy xe and the socially-optimal threshold strategy x*. We investigate the dependence of the price of anarchy (PoA) on the parameter xe and the intensity function. For example, if the potential arrival rate is bounded then PoA is bounded and converges to 1 when xe goes to infinity. On the other hand, if the potential arrival rate is unbounded, we prove that x*/xe always goes to 0, when xe goes to infinity and yet, in some cases PoA is bounded and even converges to 1; if h converges to a positive constant then PoA converges to 2; if h increases then the limit of PoA is at least 2, whereas if h decreases then PoA is bounded and the limit of PoA is at most 2. In a system with a queue we prove that PoA may be unbounded already in the simplest case of uniform arrival.