Application of Computational Linear Algebraic methods to a Reliability Simulation

Open Access
Author:
Henry, Seth Michael
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Christopher H Griffin, Thesis Supervisor
  • Victoria V Sadovskaya, Honors Advisor
Keywords:
  • Markov chain
  • reliability theory
  • computational methods
  • Gauss-Seidel
  • Conjugate Gradient
  • Successive over-relaxation
Abstract:
In this paper, numerical methods for the solution of a reliability modeling problem are presented by finding the steady state solution of a Markov chain. The reliability modeling problem analyzed is that of a large system made up of two smaller systems each with a varying number of subsystems. The focus of this study is on the optimal choice and formulation of algorithms for the steady-state solution of the generator matrix for the Markov chain associated with the given reliability modeling problem. In this context, iterative linear equation solution algorithms were compared. The Conjugate-Gradient method was determined to have the quickest convergence with the Gauss-Seidel method following close behind for the relevant model structures. The successive over-relaxation method was studied as well, and some optimal relaxation parameters are presented.