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Unformatted text preview: Probability and Statistics with Reliability, Queuing and Computer Science Applications Second edition by K.S. Trivedi Publisher-John Wiley & Sons Chapter 8 (Part 6) :Continuous Time Markov Chains Solution Techniques Dept. of Electrical & Computer engineering Duke University Email:kst@ee.duke.edu URL: www.ee.duke.edu/~kst Copyright © 2003 by K.S. Trivedi 1 CTMC Solutions • Transient Solutions – Solving Kolmogorov differential equation • Steady-State – Solving the linear system of equations πQ = 0 • Closed-form solutions – Possible for very small CTMCs or highly structured CTMCs • Numerical Solutions – Most other cases Copyright © 2003 by K.S. Trivedi 2 Outline of This Part of Chapter 8 • Methods for Steady-State Analysis – Power method – SOR • Methods for Transient Analysis – Fully symbolic method – Numerical Methods – Uniformization Copyright © 2003 by K.S. Trivedi 3 Methods for Steady-State Analysis Copyright © 2003 by K.S. Trivedi 4 Methods for Steady-State Analysis • Direct methods (e.g., Gaussian elimination) – Full-matrix – Memory-intensive – Upper bound on the time • Iterative methods (e.g., power method, SOR, Gauss-Seidel) – Sparse storage methods and sparsity preserving – Less storage and less computing resources needed – Long time to converge • Q matrix for practical CTMC problems is highly sparse. Copyright © 2003 by K.S. Trivedi 5 Power Method • Rewrite Eq. (8.23) as where • Note is a transition probability matrix of a DTMC. • Iteration: • Initial value: Copyright © 2003 by K.S. Trivedi 6 Example • Consider the following two-state availability model • Assume repair rate µ > failure rate λ • Choose q=max{µ, λ} = µ Copyright © 2003 by K.S. Trivedi 7 Example – cont’d • The matrix is • After n iterations, • As n approaches infinity, Copyright © 2003 by K.S. Trivedi 8 Successive Overrelaxation (SOR) • Start with an initial guess • Iterate until some criteria for convergence are satisfied: • Where is the solution vector at the kth iteration, L is a lower triangular matrix, U is an upper triangular matrix, and D is a diagonal matrix. • Gauss-Seidel is a special case of SOR with Copyright © 2003 by K.S. Trivedi 9 Example • Consider the CTMC in (a) • w/ matrix Copyright © 2003 by K.S. Trivedi 10 Example – cont’d • This CTMC will not converge using the Guass-Seidel or SOR. • The iteration will converge using underrelaxation, i.e., • Renumbering the states can also cause the iteration to converge for Gauss-Seidel and SOR. See Fig. (b). Copyright © 2003 by K.S. Trivedi 11 Methods for Transient Analysis Copyright © 2003 by K.S. Trivedi 12 Fully Symbolic Method • Taking the Laplace transform on both sides of the Kolmogorov differential equation and rearranging the terms: • The transient state probability is obtained by computing the inverse Laplace transform of • Solutions obtained will be closed-form and full symbolic in both the system parameters and time t. • However, in general, computing the inverse Laplace transform is extremely difficult. • Semi-symbolic solutions – Assuming entries in the Q matrix are all numerical, but final solution being a symbolic function in time parameter t. – Simpler than the full symbolic method. – Implemented in the SHARPE software package. Copyright © 2003 by K.S. Trivedi 13 Numerical Methods • Discretization methods are standard to solve ODEs such as Kolmogorov differential equations. – Discretize the time interval into a finite number of subintervals and compute the solution step by step. • Implicit discretization methods – More stable and high solution accuracy – For stiff Markov chains – For example: TR-BDF2 and the implicit Runge-Kutta • Explicit discretization methods – For nonstiff Markov chains – For example: Runge-Kutta Copyright © 2003 by K.S. Trivedi 14 Uniformization • ODE solvers are for the evolution profile of the state probabilities. – Require a very small step to obtain accurate results. • Uniformization method – Also known as randomization method or Jensen’s method – Starts with the formal solution where the matrix exponential is defined by the infinite series: Copyright © 2003 by K.S. Trivedi 15 Practical problems using this approach 1. Q has both negative and positive entries and hence the computation has both additions and subtractions – poor numerical behavior 2. Raising the matrix Q to its powers is costly and fills in zeros in the matrix – In practice, Q is very large yet sparse. 3. The infinite series needs to be truncated. Copyright © 2003 by K.S. Trivedi 16 Solution to Problem 1 • Use an integrating factor • Let • Kolmogorov differential equations becomes • We have • Hence • DTMC matrix has no negative entries. Copyright © 2003 by K.S. Trivedi 17 Solution to Problem 2 • Rewrite last equation as • Problem 2 (raising matrix Q to its powers) is avoided. Copyright © 2003 by K.S. Trivedi 18 Solution to Problem 3 • To solve problem 3, either left or right truncate the infinite series: • The value of l and r can be determined from the specified truncation error tolerance by: Copyright © 2003 by K.S. Trivedi 19 Example • Consider the 2-state homogenous CTMC • The Kolmogorov differential equations are: Copyright © 2003 by K.S. Trivedi 20 Example – cont’d • Note we obtain • (1) • Assume µ > λ, let q= µ. Now define • Differentiate both sides Copyright © 2003 by K.S. Trivedi 21 Example – cont’d • From Eq. 1, we obtain • Solving the equation, • We have • Given initial condition Copyright © 2003 by K.S. Trivedi 22 Example – cont’d • So we have • Similarly, Copyright © 2003 by K.S. Trivedi 23 ...
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