4.11 "User-optimal" versus "system-optimal" equilibrium Two different facilities, A and B, provide the same type of service. Each facility contains a single server with service times distributed as negative exponential random variables. The mean service times are 1/₁=1 minute and 1/₂ = 4 minutes at facilities A and B, respectively. Other than the different expected length of service times, the service that A and B provide is identical (in terms of quality, cost to the user, etc.). (Think, for instance, of two truck-weighing and inspection facilities that differ only by the rate at which they inspect trucks.)
        A combined total of = 60 customers per hour wish to avail themselves of the service provided by facilities A and B. The situation is pictured in Figure P4.11. Arrivals of customers at the critical point are Poisson. At that point each customer makes a choice, independently of all others, on which facility he or she is going to use. This choice is made without any knowledge of the status of the queue in front of either facility. Let p denote the probability that a random customer chooses facility A (and 1 - p that he or she chooses B).

a. Consider the case when the same pool of customers use facilities A and B on a repeated basis (say, once every day). (Think, for instance, of suburban commuters in a town with two major access roads to the central business district. Each of these commuters decides every day, independently of all others, which of the two roads he or she is going to use that day, without knowledge of traffic conditions on either road.) Let us assume that each user is only concerned with minimizing his or her expected time spent in a service facility, (= average waiting time + average service time). It can be reasonably expected that in the long run (system in steady state and in equilibrium-as far as the distribution of customers between the two facilities is concerned) the customers will "distribute" themselves among the two facilities, in the sense that p will stabilize around a specific value. What is that value of p?

b. Suppose now that your objective is to minimize the total amount of time (waiting and being serviced) that all customers spend in either of the two facilities per unit of time (with the system in the steady state). This is equivalent to minimizing, say, the "cost" suffered by the community each day, where cost is measured in terms of total commuter hours spent in traffic. You can thus set the value of p yourself (and thus force each arriving customer, independently, to choose between the two facilities on the basis of your p). What would you choose as the value of p?

Note: Feel free to use trial and error (rather than an "elegant" approach) in determining p.

c. Can you explain intuitively why the answers to parts (a) and (b) are different? Can you suggest a situation parallel to the one above in highway traffic and how it might be possible to achieve the value of p that you found in part (b) in this case?