Transient analysis of M/G/1 queueing models: lattice path approach

Abstract

In this thesis, we develop the explicit expression for pure incomplete busy period (PIBP) density function for M/G/1 queueing systems and for incomplete busy period (IBP) density function for M/G/1 queueing systems operating under (0,k) and (k ′, k) control policies. Under (0,k) control policy, the server goes on the vacation when the system becomes empty and re-opens for service immediately at the arrival of the kth customer. Under (k ′, k) control policy, the server starts serving only when the number of customers in the queue becomes k and remains busy as long as there are at least k ′ customers waiting for service. The explicit form of the incomplete busy period density and other measures of the system performance are not known.Our approach is to approximate general service time with Coxian 2-phase distribution and represent the queuing process as a lattice path by recording the state of the system at the point of transitions. Herein an arrival into the system is represented by a horizontal step and departure by a vertical step and shift from phase 1 to phase 2 by a diagonal step. Incomplete busy period can then be represented as lattice path starting from (k0, 0) to (m,n), m > n remaining below the barrier Y = X. Control policies imposes additional restrictions on the barrier. Next we use the lattice path combinatorics to count the feasible number of paths and corresponding probabilities.The above leads to the required density function that has simple probabilistic structure and can be computed using R. In this thesis, we also present the challenges in computing the density using R and illustrate the code and the results.