Priority Queues

Abstract

In a priority queue different types of items (individuals or elements) arrive at a service mechanism and each item has a relative priority for order of service. Let there be $K$ classes of items, $1, 2, \cdots, K$. If the service mechanism is to select an item for service, a type $i$ item will be selected in preference to a type $j$ item for $i < j$ even if the type $j$ item arrived before the type $i$ item, and within each class the "first come, first served" policy determines the order of service. When a type $j$ item is in service and a type $i$ item arrives $(i < j)$, there are two primary disciplines for handling the priority demand. The "head-of-the-line" discipline allows the type $j$ item to complete service but places the type $i$ item ahead of any other lower priority items. The "preemptive" discipline withdraws the type $j$ item from service and replaces it by the type $i$ item. Under the preemptive scheme the only time at which a type $j$ item $(1 < j)$ can be in service is when there are no items of types $1, \cdots, j - 1$ in the queue. When a lower priority item which has been preempted returns to service, the preemptive discipline must distinguish two cases. The "preemptive resume" policy allows the preempted item to resume service at the point at which it was preempted so that its service time upon reentry has been reduced by the amount of time the item has already spent in service. The "preemptive repeat" policy requires the preempted item to commence service again at the beginning. A priority queue with an indifferent server is of course a special case of the preemptive resume discipline. In the special case $K = 2$ the type 1 items will be referred to as priority items and the type 2 items as non-priority items. It will be assumed throughout this paper that the input process for type $i$ items, $i = 1, \cdots, K$, is Poisson with arrival rate $\lambda_i$ and the input processes operate independently. The service time distribution for a type $i$ item (in isolation) will be denoted by $F_{s_i}$ and unless explicitly stated to the contrary will be assumed to be general subject only to the restrictions $F_{s_i}(0+) = 0$ and $E(S_i) < \infty$. Let $\rho_i = \lambda_iE(S_i)$, and let $\tilde S_i$ be the Laplace-Stieltjes transform of $F_{s_i}$. The service mechanism consists of a single channel or server. A. Cobham [1], [2] introduced the head-of-the-line priority queue and derived equilibrium expected waiting times. Subsequent contributions have been made by Holley [3], Kesten and Runnenburg [4], [5], and Morse [6]. The first published results for the preemptive discipline were by H. White and L. S. Christie [7], and additional results have been presented by Stephan [8]. Koenigsberg [9] has generalized the priority model to a continuous number of priority types with application to machine breakdown problems. Under various assumptions in this paper the following quantities have either been obtained explicitly or characterized as the unique (subject to regularity conditions) solution to a functional equation: the generating function for the stationary probabilities on the number of priority and non-priority items $(K = 2)$ in the queue, the Laplace-Stieltjes transforms of the waiting time distributions, the Laplace-Stieltjes transform of the distribution of a busy period, and the generating function for the probabilities on the number of items serviced during a busy period. For most of the distributions mentioned the first two moments are computed.