Limit theorems for generalized single server queues

Abstract

For the generalized single server queueing system described herein weak convergence results are obtained for the processes {Wa, n ≧ 0}, {W(t), t ≧ 0}, and {Q (t), t ≧ 0}, where Wn is the waiting time of customer n, W(t) is the workload of the server at time t, and Q(t) is the number of customers present in the system at time t. We also provide a functional strong law, a functional central limit theorem, and a functional law of the iterated logarithm for various cumulative processes in the system.

Simulation of Multidimensional Diffusions with Sticky Boundaries via Markov Chain Approximation

“…Our first application considers a multi-server queuing system where customers receive exceptional service whenever a server is idle. Studies on this type of system can be found in for example Welch (1964), Lemoine (1974), Lemoine (1975) and Harrison and Lemoine (1981). Diffusions with sticky boundaries arise as the heavy traffic limit of such system (see Rácz and Shkolnikov (2015)).…”

Limit theorems for generalized single server queues

“…But, that g( • ) is d.R.i. follows by a standard argument incorporating ideas of Takacs (1963) and Smith (1955); see [8] for complete details. Therefore, [15].…”

The GI/G/1 queue with uniformly limited virtual waiting times; the finite dam

“…Regenerative processes in this sense were introduced by Smith [17], [18] and have been used by many authors to study the stationary behaviour of many queueing systems. (See, for example, [3], [5], [6], [12], [13], [16] and [19].) By using the regenerative properties along with analytical methods in this paper, we shall show that not only the stationary behaviour of the system can be studied but its transient characteristics can also be obtained.…”