ON THE OPTIMALITY OF LEPT AND μc RULES FOR PARALLEL PROCESSORS AND DEPENDENT ARRIVAL PROCESSES Arie Hordijk & Ger Koole* Dept. of Mathematics and Computer Science, University of Leiden P.O. Box 9512, 2300 RA Leiden, the Netherlands ABSTRACT In this paper we study scheduling problems of multiclass customers on identical parallel processors. A new type of arrival process, called a Markov Decision Arrival Process, is introduced. This arrival process can be controlled and allows for an indirect dependence on the numbers of customers in the queues. As a special case we show the optimality of LEPT and the μc-rule in the last node of a controlled tandem network for various cost structures. A unifying proof using dynamic programming is given. STOCHASTIC SCHEDULING; DYNAMIC PROGRAMMING; DEPENDENT ARRIVALS Classification: 90B22 primary, 90C40 secondary. Published in: Advances in Applied Probability 25:979–996, 1993. 1. INTRODUCTION In this paper we consider stochastic scheduling problems on s identical parallel processors. There are m classes of customers, each class has its own queue and the service time of a customer of class k, k =1,...,m is exponentially distributed with rate μ k . Arrivals occur according to a Markov Decision Arrival Process (MDAP). This arrival process generates customers with intensities that do not depend on the state of the queues. However, actions can be taken in the MDAP. The control, consisting of an action in the MDAP and the assignment of the servers, may depend on the state of the MDAP and the numbers of customers in the queues. This induces a dependence between the arrival process and the state of the queues. The MDAP is especially designed to model the arrivals in the last center of a controlled tandem network of service centers. With the MDAP we can also control the availability of the servers. Besides dependent arrival processes and machine breakdowns and repairs it is possible to model the usual independent arrivals and breakdowns and repairs. It is shown that LEPT (the policy that processes customers in the system in decreasing order of expected processing time) is optimal in the class of preemptive policies if the cost function satisfies two conditions; one is the monotonicity in the numbers of customers in the queues and the other requires that the service rate multiplied by the marginal cost of a customer is decreasing in the same order as the expected processing times. We verify these conditions for various objective functions such as the indicator functions of the makespan or the first time to an empty system. Hence LEPT stochastically minimizes the makespan or busy period. We show that the μc-rule minimizes the expected weighted sum of customer completion times under the agreeability condition that LEPT and the μc-rule have the same priority list. This result gives the optimality of the μc-rule in the last node of a tandem network. We also analyze the optimality of the * Present address: C.W.I., Kruislaan 413, 1098 SJ Amsterdam 1