Asymptotic Behavior of General Queues with One Server

There is a queue in front of one server; waiting customers are served in order of arrival, and no defections from the queue occur. The delays incurred in such a system can be described by a stochastic process XXX the virtual waiting time, defined as the time a customer would have to wait for service if he arrived at time t. Stochastic processes of this kind have been studied. 1,2 In the present work we examine the asymptotic behavior of the probabilities Pr{ IT( I) as t becomes large, for a class of processes satisfying a weak condition of stationarity. II is customary to define the process W( ·) in terms of the arrival epoch ti and the service time Sk of the kth arriving customer, for k = 1, 2, · · · . We can, however, describe the service times and the arrival epochs simultaneously by a single function K(-), defined for / 0, 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. With Ki0) = TF(0) for convenience, Kit) can be interpreted as the "work load" submitted to the server during [0,0, and Wit) as the work remaining to be done at t.