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1. Introduction. There is a queue in front of a single server, and the waiting customers are served in order of arrival, with no defections from the queue. We are interested in the waiting-times of customers. As a mathematical idealization of the delays to be suffered in the system, we use the virtual waiting-time Wit), which can be defined as the time a customer would have to wait for service if he arrived at time t. Wit) is continuous from the left; at epochs of arrival of customers, Wit) jumps upward discontinuously an amount equal to the service-time of the arriving customer; otherwise Wit) has slope —1 while it is positive. If it reaches zero, it stays equal to zero until the next jump. It is usual to define the stochastic process Wit) in terms of the arrival epoch tk and the service-time Sk of the fcth arriving customer, for & = 1, 2, • • However, the following procedure is a little more elegant; we describe the service-times and the arrival epochs simultaneously by a single function Kit), which is defined for t^O, left-continuous, nondecreasing, and constant between successive jumps. The locations of the jumps are the epochs of arrivals, and the magnitudes are the service-times. Kit) can be interpreted as the "work load" submitted to the server during the interval [0, t), and Wit) can be thought of as the amount of work remaining to be done at time t. In terms of this interpretation, one can see that the work remaining at t equals the work offered up to t minus the elapsed time t plus the total time during which the server was idle in (0, /). Formally, then, Wit) is defined in terms of Kit) by the nonlinear integral equation

Combinatory methods and stochastic Kolmogorov equations in the theory ofqueues with one server