Solving Markovian systems of ODE for availability and ... - Science Direct

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Reliability Engineeringand System Safety48 (1995)47-55 (~) 1995ElsevierScienceLimited Printed in Northern Ireland. All rights reserved 0951-8320/95/$9.50

Solving Markovian systems of O.D.E. for availability and reliability calculations B. T o m b u y s e s * & J. D e v o o g h t UniversitO Libre de Bruxelles, Avenue F. D. Roosevelt 50, B-1050, Brussels, Belgium (Received 17 February 1994; revised 23 May 1994; accepted 26 June 1994) The computation of availability or reliability in a Markovian approach involves the solution of O.D.E.: dp/dt = L(t)p, where the transition matrix is time dependent, at least when some aggregation of the states is introduced to reduce the size of the problem. Methods of solution like uniformization are in this case unapplicable and we investigate here four explicit and six implicit R.K. methods from the point of view of stability, amount of numerical work and accuracy. The test problem chosen allows an analytical solution, a uniformization method (when L(t) is constant) and afortiori all R.K. methods. The implicit trapezoidal rule used with a variable step scheme appears to be the best compromise between accuracy and computational work. 1 INTRODUCTION

Therefore a good method of integration should preserve sparsity to minimize arithmetic operations, be suitable for adaptative stepsize and should also be implicit to cope with stiffness] m Although Monte Carlo methods are well adapted to large size Markovian problems (>25,000) since they avoid storage problems, we shall not make comparisons with them here. Uniformization methods (valid for L ( t ) = L problems) are widely used 11"12 for Markovian problems and we shall compare our results to those obtained by uniformization. It should be also mentioned that although implicit methods are in general m o r e costly stepwise, it is not necessarily so in an adaptative setting, inasmuch as implicit methods are easily parallelized. 13 The plan of this p a p e r is the following. After a short description of the properties of the L(t) matrix we describe the various integration methods used in the numerical comparison. The problem used has been chosen to allow an exact solution to which our approximate solutions are compared: the availability of 6 independent parallel components. We compare finally the computation time with accuracy for all methods including uniformization.

The computation of reliability or availability by Markovian models ~'2 means numerical integration of linear systems of O.D.E. d p / d t = L ( t ) p . Usually Markovian problems are homogeneous, i.e., transition rates are time independent which means L ( t ) = L. In this case p(t) - erCpo and the direct evaluation of the exponential is a possible m e t h o d of solution. Moler & Van Loan 3 have reviewed nineteen different ways to compute e a-, most of which are suitable for less than a few hundred equations. H o w e v e r Markovian reliability problems give rise to very large size O.D.E. systems since the n u m b e r of states grows like 2 N if we have N components, each one with two states only. Many aggregation methods have been p r o p o s e d to reduce the size of the system, most of them for steady state ('asymptotic') solutions 4 (and thus concern only availability problems) and we have developed, for use in the reliability code C A M E R A s a c o m p o n e n t influence graph m e t h o d which involves a reduced transition matrix L(t) which is time dependent. 6 The choice of the numerical method must be therefore adapted to the following characteristics:

2 CHARACTERISTICS OF THE TRANSITION MATRIX 14-16

(1) L(t) can involve m o r e than thousands of states and is time dependent (2) L(t) is very sparse (3) L(t) is stiff, i.e. has a wide dispersion of its eigenvalues.

2.1 Availability of a system of independent components A. Transition matrix o f one component Let L = [lij] be the transition matrix, i.e., l,j is the transition rate from the Markovian state j to

* Research Assistant FNRS. 47

B. Tombuyses, J. Devooght

48 Markovian state i with

(1) l i i > - O f o r i ~ j (2) ljj = - X,,j l,j L is the infinitesimal generator of the Markov continuous time process. (3) Matrix U = e 'c is stochastic for any t-> 0, i.e. U~-> 0 and ~ U~/= I. For short we write that the generator L is a g-stochastic matrix.

B. Transition matrix of N independent components Let l(i~. • • iN) be a state of the system, where ik is the state of the component k. The transition matrix L is given by [lu]:

--l,j =k=,~ (qIJ~ fii,,i,,)lik,~k I # J L k = [ l ~ j is the transition matrix of component k.

- - l ~ -- - ~

l,,

l~J

- - P r o p e r t i e s (1), (2). (3) also hold true for L.

C. Size and structure of L (4) If nk is the number of states of component k, the number of states of the system is n = I]N-~ nk and therefore L is a n x n matrix. For two states components, n = 2 N. If each component has only one transition by state, there exists N transitions by state of the global system (multiple transitions are excluded) and L has N off diagonal elements nonzero by column. For two states components, L has (N + 1)2 N = n(log2 n + 1) nonzero elements. Thanks to this sparsity, the multiplication of a vector by L needs n ( l o g 2 n + l ) , much less multiplications than n 2 (for N = 10:11264 0 for each differential equation such that a change in the starting values by a fixed amount produces a bounded change in the numerical solution for all h such that 0 0 (Icl/a < 3" -O and p(t) = eC'po can be written as

p(t) = e-qteqL'po = ~ e -q' (qt)k L'kpo. k=0 k! All the terms of the series are nonnegative. There are no problems of numerical cancellation to sum the series and we can estimate the error of truncation if we stop at kmo~ < ~. For large qt, k,,,x to reach an absolute accuracy of 10m~m >-3) is approximately given by k,,,x = E[qt + (z,~/at]qt)] where E[x] represents the integer immediately superior to x, and z~ is the normal deviate at the a level.

(1) System A: I1: hm

1 5 : h m / ( 2 + X/2)

12:1/2 hm

16:hm/(3 - X/-3)

(2) System B: I3: 3 V ~

14: lX/]-~, if we choose 3, = X/Icdl/ab.

(3) System C: 13:2/3 hm

14:1/2 hm

For systems A and C we are sure that the spectral radius tends to zero and the convergence is very fast. It is not the case for system B. T o calculate the initial transient, h is chosen such that h m < < l and the convergence is good.

4.3 Practical solution of the algebraic systems The systems from the methods 11, 12, 15 and 16 (system A) are solved by a Gauss-Seidel algorithm. For 13 and 14, we choose the system C, also solved by a intermediate algorithm between Jacobi and G a u s s Seidel (Lx2 is computed with x2 as in Gauss-Seidel and LZx2 with x2 as in Jacobi). G o o d convergence obtained by Gauss-Seidel did not suggest any further study with a more complex iterative method, inasmuch as a good initial trial vector is known from the previous time step. The stepsize will be chosen small enough to insure the convergence. The initial approximation to solve Ax = z will be Xo = z. It is an

6 TEST PROBLEM The choice of the test problem was dictated by the fact that we needed a benchmark solution obtained analytically. Indeed the use of one of the methods (with small time steps) as a reference is at best uncertain and we preferred an analytical solution. The size of the problem (64 equations) was dictated by the necessity of doing a large number of comparisons. Although the example was time independent and concerned with availability only we have used R u n g e - K u t t a methods for time dependent transitions and allowing for the accuracy control of each component of the probability vector. An example of application in the general case is given in Ref. 6. The system is a set of six independent components in parallel. We will calculate the availability from t = 0 to t large enough to reach the asymptotic value. At t = 0, all components are working. For the availability problem, independent components allow us to compute the exact solution as reference. The matrix (-A' ~i / transition of a component is L i = Ai - ~ # (i = 1 • • • 6), with eigenvalues 0 and - ( & + ~i). Table 1 gives the transition rates for each component. The first eigenvalues of the global system are 0, -0-0027, - 0 . 0 1 0 3 . • • and the last -26-228.

B. Tombuyses, J. Devooght

52

Table 1. Transition rates of the components i

l

A,(h ') /z~(h ')

0.001

2

3

0"01 0-004 5 0"1

1

4

5

0-1 20

6

0.0003 0-0007 0"01 0.002

y', ( = Ly,).

7 RESULTS 7.1 Integration with constant stepsize We have integrated the system without and with extrapolation for different stepsizes, from t = 0 to t = lh. Figures 1 and 2 represent the accuracy as a function of stepsize. With logarithmic axes, the curve is a line of slope equal to the order of the method. Stability problems exist with too large stepsize for explicit methods. For implicit methods, accuracy is limited by the solution of the algebraic systems. These solutions need 3 or 4 iterations in average for algebraic systems A. Systems C require 5 or 6 iterations with a stepsize -