Individual versus Social Optimization in the Allocation of Customers to Alternative Servers

Customers arrive at a service area according to a Poisson process. An arriving customer must choose one of K servers without observing present congestion levels. The only available information about the kth server is the service time distribution (with expected duration \mu k -1 ) and the cost per unit time of waiting at the kth server (h k ). Although service distributions may differ from server to server and need not be exponential, it is assumed that they share the same coefficient of variation. Individuals acting in self-interest induce an arrival rate pattern (\lambda \^ 1 , \lambda \^ 2 , ..., \lambda \^ k ). In contrast, the social optimum is the arrival rate pattern (\lambda 1 *, \lambda 2 *, ..., \lambda k *) which minimizes long-run average cost per unit time for the entire system. The main result is that \lambda \^ k 's and \lambda \^ k *'s differ systematically. Individuals overload the servers with the smallest h k /\mu k values. For an exponential service case with pre-emptive LIFO service an alternative charging scheme is presented which confirms that differences between individual and social optima occur precisely because individuals fail to consider the inconvenience that they cause to others.