A General Model of a Single-Channel Queue: Discrete and Continuous Time Cases

In Part I, a single-channel service system consisting of a service facility and queue, in which the possible times of arrival of units and the possible service times are discrete, is analyzed. A method of calculating the moments of the total service time of units in the system is developed. This total service time is related to the delay caused by the system. When arrivals at different times are assumed to be independent, the “values” of the resulting Markov process can be calculated. These values lead to information about the transient behavior, autocorrelation function, expected first passage time, and expected extra delay that arises if another unit is inserted into the system. An expression for the geometric transform or moment generating function of the probability distribution of the total service time of units in the system is determined. The results are derived for arbitrary arrival and service time distributions. In Part II, the continuous time analog of the system of Part I is considered and analogous results are determined.