Composite Priority Queue

This paper presents formulas for calculating waiting time for customers in a queue with combined preemptive and head-ofline (nonpreemptive) priority scheduling disciplines and describes the reasoning behind them. This work has been applied in the development of programmable terminal control units. Introduction The development of programmable terminal control units has created the need for a new priority queuing model, one taking into consideration both preemptive and head-of-line priority relationships between devices and programming tasks. As concurrent service demands occur, extended waiting delays and service times are experienced by low priority demands. The effect of higher priority demands on the mean and standard deviation of response time at a particular demand point is presented here. Previous work on priority queues has been done by Cobham [ 1 1 , Gaver [2], Takacs [3], Chang [4], Jaiswal [ 51, and Herzog [ 61. Our model is an adaptation of preemptive and head-of-line priority queues, combining the separate results into a composite. In this paper the model is first characterized, and then the mathematics is developed using heuristic reasoning. Finally, an example is presented applying the model to a typical control unit configuration. Mathematical description Suppose that customers of different priorities are arriving at a counter in accordance with a Poisson process of density A. The customers are served by a single server in order of priority and for each priority in order of arrival. Each priority consists of two parts: 1 . Preemptive-resume If a customer of higher preemptive priority arrives when a customer of lower preemptive priority is being served, the server interrupts the current service and immediately starts serving the customer of higher priority. The service of the cus78 tomer of lower priority is resumed when o more customers of higher preemptive priority are present in the system. The server is busy as long as there are customers in the system. There are II levels of such preemptive priority. 2. Head-of-line (nonpreemptive) Within each preemptive priority level, m classes of customers have headof-line priority with respect to each other. If a customer of higher head-of-line priority arrives when a customer of lower priority is being served, the server does not interrupt the current service, as long as both customers have the same preemptive priority level. At the completion of the current service, the server chooses the customer who arrived first among the customers with highest head-of-line priority present in the system. This composite priority scheme is diagrammed in Fig. 1. It is convenient to assume that customer with a smaller priority number has precedence over a customer with a greater priority number. Accordingly, a customer in the jth head-of-line priority class within the ith preemptive priority level has priority number ij. The following notation is used. Let h i j = arrival rate of customers with priority number ij Ai = ZJE1 Akj, aggregated arrival rate of customers down to and including the ith preemptive priority level (A, = A) wi j = mean service time for a customer of priority number i j wij(') = rth moment of service time with priority number 8. Here, as well as below, the first moment T. W. GAY AND P. H. SEAMAN IBM J. RES. DEVELOP. is denoted by omission of the superscript. Thus, aiir) = A," E:=, ; : l A k j wkj"), rth moment of aggregated service times down to and including the ith preemptive priority level p . . = hij w i j , utilization of the server due to customers of priority number i j U i j = Xt:l pkz + p,,utilization of the server due to all customers down to the ith preemptive priority level, including those in the ith level across to and including thejth head-of-line class. (1) w . . = w . . . I J