ABSTRACT
A control policy is then a mapping from the state space Z x {a , l }z x A � { Fo , FI , Fz , FI 2 } . For a threshold policy the decision only depends on the total number of customers q + e l + e2 i n the system . The discounted sojourn t ime cost J of the continuous t ime problem will be minimized if and only if the cost E L:�=o -t XneT for the discrete time problem is minimized. Here , is a number between ° and 1 , depending on o. This has been proven in [4] using a proof i ndependent of the arrival process, hence valid in our model as well . As in Lin and [(umar!4), a " backward induction" argument shows that the fast server should be used whenever a customer is wait ing and that the slow server should never be started i f the fast server is idle. Then a policy improvement argument shows that if one starts the i teration with a threshold policy then the improved policy i s also of threshold type. S ince the set of candidate optimal policies i s finite ( because using the slow server is certainly socially optimal if it is optimal for the individual customer tak ing the decision, and there is a finite threshold for individual optimisation ) , this i mplies convergence of the policy i teration algorithm to a threshold policy. For details of the proof see Talat ( 1 994) .
