Customers of different priorities arrive at a counter in accordance with a Poisson process. The customers are served by a single server in order of priority and for each priority in order of arrival. Preemptive discipline is assumed. Three service policies are considered: (i) preemptive-resume, (ii) preemptive-repeat-identical, and (iii) preemptive-repeat-different. The time-dependent solutions for these priority systems are very complicated. However, the problem can be simplified in case of stationary solutions. In this paper, a step-by-step method is proposed to find the stationary distributions of the queue sizes, the waiting times, and the busy periods of each priority class.

This paper considers a special queue situation, one in which a single facility serves two major priority classes of customers. Within each class, there are several levels of priorities. The first class has the higher priority. On arrival, a customer of the first class immediately replaces any customer of lower priority being served. The second class has the lower priority, as compared to the first class. On its arrival, a customer of the second class cannot interrupt the current service of a lower priority customer in the system; it must wait until the service is completed. The first class is the preemptive priority, and the second class is the nonpreemptive priority. This paper formulates a theoretical solution for this queuing system, which has a wide range of application in the computer industry. The real-time control program under the multiprogramming environment is an analog of this priority queuing model.

A time-sharing computer system brings the man with a problem much closer to the computer power he needs. The priority processing problems associated with time-sharing computer system are described. A mathematical model was developed based on queues with feedback. The queue size and its variability within the computer can be determined with the enclosed formulation. Other important system design parameters such as the system response time, etc., can also be obtained. The mathematical model complements standard simulation techniques for time-sharing computer analysis. Although the problems discussed in this paper arose in the computer systems, the queuing model developed is quite general and may be useful for other industrial applications.