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This work presents a variation of Naor's strategic observable model (1969),by adding a component of customer heterogeneity induced by the location ofcustomers in relation to the server. Accordingly, customers incur a travel costwhich depends linearly on the distance of the customer from the server. Thearrival of customers with distances less than x is assumed to be a Poissonprocess with rate lambda(x)=int_0^x h(y)dy<\infty, where h(y) is a nonnegativeintensity function of the distance y. In a loss system M/G/1/1 we define thethreshold Nash equilibrium strategy x_e and the optimal social thresholdstrategy x^*. We show that if the rate of arriving customers is bounded thenPoA converges to 1 when x_e \to\infty, i.e., in the limit there is nodifference between the social and equilibrium optimal benefits. The rest of thepaper is dedicated for the case in which the rate of arriving customers isunbounded. We develop an explicit formula to calculate lim_{x_e\to \infty}PoAwhen it exists. We present sufficient conditions for the limit to exist and forthe existence of a simple formula for calculating it. We prove that if therelation between two intensity functions converges to a positive constant, thenthe corresponding limits of PoA coincide. If, on the other hand, one intensityfunction is larger than the other from some point on, then under certainconditions the limit of PoA (if exists) will be larger for the larger intensityfunction. For all intensity functions h, we prove that if h converges to aconstant 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 beunbounded already in the simple case of uniform arrival, namely h~c, where c>0.

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