Transient behavior of queuing systems

Abstract

The purpose of this dissertation is to study in detail the transient behavior of certain selected queuing systems. This is a detailed work on transient probabilities of a variety of the most known queuing systems, where explicit expressions for these transient probabilities are derived completely from first principles without any references to other recourses or gaps. The objective is the solution to the Kolmogorov differential difference equations to obtain the probabilities of different queuing system states as time functions. The study proceeds in two stages. First we consider the queuing problem with Poisson inputs, single channel and exponential service times, and then we concentrate on the queuing problem with multiple channels. We start with the M/M/1/1 system (no queue) whose solution is rather straightforward, and then relax the no queue assumption to obtain a solution to the M/M/1/∞ system in terms of modified Bessel functions. After this we obtain the Laplace transform of the transient probabilities of the M/M/c queuing problem. Explicit expressions in terms of modified Bessel functions are derived for the two-channel (M/M/2) case. A definite expression for the transient probability of the infinite server (self-service) M/M/∞ system is also derived. For each particular system, known equilibrium and probability conditions are shown to hold. Moreover, a particular case study is studied and numerical (computational) examination is carried out for the M/M/1 ∞ and M/M/∞ queuing systems, in order to get a feeling for the speed of convergence to steady-state of these particular transient probabilities. This will serve also as a tool to highlight the differences and similarities between single and multiple queues. Theoretical results are also accompanied and verified by numerical analysis performed by Matlab® and Qts® software.