On a Class of Queuing Problems and Discrete Transforms

Abstract

The main object of the paper is to demonstrate a method by which a number of problems arising in the study of impatient customers in queues can be solved. The main difficulty in analyzing such queuing systems is the presence of variable coefficients in the difference-differential equations describing the processes; the usual generating function method fails in such cases. When there is a finite number of states, it has been shown that at least formal solutions can be obtained through matrix algebra, which yields what are called “discrete transforms.” The results are explicitly given when the balking and reneging coefficients are linear in n. An alternative method in which the spectral resolution of matrices is used is also given for solution of the set of equations. In the special case of a M/G/1 queue with finite waiting room, both the methods fail (as the underlying matrix is in the Jordan form), and the solution is obtained by a simple manipulation of the underlying matrix. The methods presented here lend themselves readily to computer solutions. This paper demonstrates the generality of the basic M/G/1 model by relating it to problems arising in other branches of operations research, such as reliability theory, inventory control, etc. Finally, the methods used for the M/G/1 case are shown to hold for a similar class of problems in the GI/M/1 case. As very little work has been reported on impatient customer phenomena in GI/M/1, systems, possibly because of the difficulty in solving equations with variable coefficients, this method may prove to be useful.