Expected tardiness computations in multiclass priority M=M=c queues

We discuss the evaluation of expected tardiness of an order at the time of arrival in an M=M=c queuing system with N priority classes, considering both nonpreemptive and preemptive service disciplines. Upon arrival, a customer order is quoted a lead time of d, and placed in the queue according to the priority class of the customer. Orders within the same priority class are processed on a first-come, first-served basis. We derive the Laplace transforms of the expected tardiness of the order given the quoted lead time, priority class of the order, and system status. For the special case of single priority class, the Laplace transform can be inverted into a closed-form expression. For the case with multiple priority classes, a closed-form expression cannot be obtained, hence, we develop three customized numerical inverse Laplace transformation algorithms. Two of these algorithms provide upper and lower bounds for the expected tardiness under a simple condition on system parameters. Using this property, we obtain error bounds for our customized algorithms; such bounds are not available for general purpose numerical inversion algorithms in the literature. Next, we develop a novel methodology to compare the precision of general purpose numerical inversion algorithms and analyze the performances of three algorithms from the literature. Finally, we provide a recommendation scheme given computational time and error tolerances of the decision maker. The methods developed in this paper for the accurate estimation of expected tardiness establish an important step toward developing due date quotation policies in a multiclass queue, contributing to the due date quotation literature that has been largely focused on single-class queues.