LA CONSERVACION DE LA RED VIAL NACIONAL EN EL PERU

Another model that deals with equilibrium in a queueing network is mentioned by Larson [100]. In this model, customers (vessels on inland U.S. waterways) move between queues (at locks) and select their travel speeds. High speed is associated with higher travel cost due to high fuel consumption. A customer who observes another customer just behind him increases his speed to avoid being overtaken and having to wait longer at the next queue. Larson claims that eventually customers move at their maximum speeds, and this is socially inefficient.12 The model, however, is not formally described and analyzed, and it is not clear why travelling at maximum speed is indeed an equilibrium when there is some positive distance between customers (so that the one behind has no chance of overtaking the one in front). It may be that such an equilibrium is possible if customers’ travel speeds differ or are subject to uncertainty. This is an interesting open problem.

9.

Related literature

Chen and Frank [33] discussed the robustness of the main result of Edelson and Hildebrand [47], that a profit maximizer chooses a so- cially optimal admission fee, when the assumption of a linear utility function is removed. A main issue here is how to model the social utility of the collected feesλp. Since these fees and the aggregate in- dividual utilities are treated differently, there is no reason to expect that the socially optimal policy leaves customers with zero surplus. Therefore, the profit and welfare-maximizing admission fees differ. This is illustrated in [33] by an example that uses a specific utility function and the social value of the collected fees is assumed to be linear. Under certain conditions it is shown that a socially optimal fee may induce negative expected customer welfare. Such an outcome is natural when the customer’s value of a monetary unit is smaller than its social value.

Balachandran [16] considered an unobservableM/G/1 model with a fixed costcof running the service facility. This cost does not depend on the number of customers served or their service times. Given an arrival rateλ,cis absorbed by the customers so that each individual

Unobservable queues 71 pays _λc. Balachandran investigated the impact of this cost allocation on the equilibrium arrival rateλe. An increase inλaffects the welfare

of a customer in two ways: it increases his expected waiting costs, but it decreases his share in covering the operating costc. Therefore, it is not clear a priori whether a joining customer gains or looses from an increase in λ.

Balachandran proved that the equilibrium arrival rate is unique, and then investigated the related question of how the equilibrium arrival rate behaves as a function λ(c) of the operating cost. First, for any

c > 0, λe(c) < λe(0). This result is expected since with a positive

operating cost each customer is worse-off in comparison to the zero operating cost case. Second,λe(c) is monotone decreasing if and only

if the total expected cost per unit of time due to queueing is greater than the expected cost of maintaining the service center while it is idle. In terms of λe this condition is

λe(c)CWq(λe(c))> c 1₋ λe(c) µ .

Last, the class dominance property (see Section 4.1) does not hold in this model.

Stidham [165] raised an issue that is commonly discussed in economic models. Consider a discrete time version of the model of Section 3. Fix a value for λt and suppose that the server sets a price pt that

maximizes his profits given λt. Suppose now that the arrival rate

at period t+ 1 is set to the value λt+1 that equates the marginal

value of service with the full price, under the assumption that the arrival rate is λt, that is, V′(λt+1) = pt+W(λt). Suppose that

this process repeats itself. Will pt and λt converge to the optimal

values? In particular, is this the case when the process initializes near the optimal values? In other words, is the optimal solution stable? Stidham showed that this may or may not be the case, and gave necessary and sufficient conditions for the M/M/1 model (see also Rump and Stidham [150]).

Friedman and Landsberg [54] considered a discrete time model in which customers in periodn+ 1 decide whether or not to enter the queue based on the expected delay, which they assume to be the same as the delay in the previous periodn. They proved that if the capacity of the queue is sufficiently large, the equilibrium arrival rate is stable. For small capacities, the arrival rate typically oscillates near the equilibrium.

72 TO QUEUE OR NOT TO QUEUE The issue of stability of the equilibrium was further investigated by Masuda and Whang [120] under the assumption that the system man- ager does not have full knowledge of the demand. They considered alternative dynamic pricing rules and models of adaptive learning with bounded rationality, and characterized the equilibrium and its stability conditions.

Chapter 4