On the price of anarchy in a single server queue with heterogenous   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 lambda(x)=int_0^x h(y)dy<\infty, 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 x_e and the optimal social threshold strategy x^*. We show that if the rate of arriving customers is bounded then PoA converges to 1 when x_e \to\infty, i.e., in the limit there is no difference between the social and equilibrium optimal benefits. 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_{x_e\to \infty}PoA when it exists. We present sufficient conditions for the limit to exist and for the existence of a simple formula for calculating it. We prove that if the relation between two intensity functions converges to a positive constant, then the corresponding limits of PoA coincide. If, on the other hand, one intensity function is larger than the other from some point on, then under certain conditions the limit of PoA (if exists) will be larger for the larger intensity function. For all intensity functions h, we prove that if h converges to a constant then PoA converges to 2, and that if from some point on h decreases (increases) monotonically then the limit of PoA, if exists, is smaller (larger) than 2. In a system with a queue we prove that the price of anarchy may be unbounded already in the simple case of uniform arrival, namely h~c, where c>0. 
Refael Hassin, Irit Nowik, Yair Y. Shaki
math.OC