# An eigenmatrices method to obtain transient solutions for the M/M/k:(N/FIFO) queueing system (k=1,2)

Abstract (Summary)

This thesis deals with the birth and death process of arrivals in the M/M/k:(N/FIFO) queueing system (k=1,2) and attempts to obtain the transient solutions for these types of queueing systems. The M/M/k: (N/FIFO) queueing system is characterized by a negative exponential inter-arrival time distribution, a negative exponential service time distribution, k parallel identical indepen- dently operating service channels, one queue only, a maximum allowable number in the system N and first-in first-out (FIFO) service discipline. In this thesis, a novel step by step method called the eigen- matrices method is developed to obtain explicit closed form formulas for the transient solutions for the M/M/k:(N/FIFO) queueing system (k=1,2). The eigenmatrices method consists of three major steps: 1) A general expression of the transient solution is derived by solving the fundamental differential-difference equations of the M/M/k: (N/FIFO) queueing system. The transient solution is represented as a function of eigenvalues and eigenvectors. 2) In order to obtain the eigenvalues, the characteristic polynomial is transferred into a fairly simple function by solving a difference equation, avoiding the problems involved when trying to find the roots of a polynomial with higher degrees ( >4 ). For the M/M/1:(N/FIFO) queueing system, the eigenvalues are obtained fairly easily after applying the simplified function. For the M/M/2:(N/FIFO) queueing system, a sequence {ƒÖ m} can be used to converge ƒÖ (ƒÖ is an eigenvalue) when m tends to ‡. 3) the eigenvectors are obtained by solving difference equations. This proposed method is fairly simple, for it merely requires an understanding of the basic concepts of linear algebra, matrices, calculus, and difference equations. The results obtained by using this method have been validated by using both general mathematical solution techniques and numerical integration techniques (Continuous System Modeling Program (CSMP)).
Bibliographical Information:

Advisor:

School:Ohio University

School Location:USA - Ohio

Source Type:Master's Thesis

Keywords:eigenmatrices method obtain transient solutions queueing system

ISBN:

Date of Publication:01/01/1991