Solving Markovian systems of O.D.E. for availability and reliability calculations

Abstract

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 a fortiori 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.