Efficient Solutions for an Approximation Technique for the Transient Analysis of Markovian Models

One of the most widely used technique to obtain transient measures is the uni-formization method. However, although uniformization have many advantages, the computational cost required to calculate transient probabilities are very large for stii models. We study eecient solutions that can be applied to an approximate method developed for calculating transient state probabilities of Markov models, and cumulative expected reward measures over a nite interval. The method our work is based on, approximates the state probabilities at time t by the state probabilities calculated at a random time with Erlangian distribution. The original method requires an inversion of a matrix obtained from the state transition rate matrix, which destroys special structures such as sparseness and banded matrices. This precludes the use of the technique for large models. In our work we propose eecient solutions that can take advantage of special structures. Finally, we present examples which show that the proposed technique is computationally very eecient for stii models when compared with uniformization. Solutions eecaces relatives a une m ethode approch ee pour l'analyse transitoire de mod eles markoviens R esum e : L'une des techniques les plus utilis ees pour obtenir des r esultats sur le comporte-ment transitoire des mod eles markoviens est celle de l'uniformisation. Cependant, malgr e les nombreux avantages de cette derni ere, sa complexit e temporelle la rend tr es co^ uteuse pour les syst emes raides. Nous etudions ici des solutions eecaces relatives a une m ethode approch ee d evelopp ee pour l'analyse en r egime transitoire de mod eles markoviens, tant des probabilit es d' etat que de mesures d'esp erance de r ecompenses sur un intervalle ni. La m ethode sur la-quelle repose notre travail approche les probabilit es d' etat a l'instant t par les probabilit es d' etat calcul ees a un instant al eatoire distribu e selon une loi d'Erlang. La m ethode d'origine n eces-site l'inversion d'une matrice (contruite a partir du g en erateur innnit esimal) qui fait perdre les structures sp eciales telles que celles des matrices creuses ou des matrices bandes. Ce qui interdit l'usage de l'usage de la m ethode d'origine pour les grands espaces d' etat. Notre approche four-nit des solutions eecaces qui conservent les avantages des structures particuli eres. Ennn nous d eveloppons des exemples qui montrent que, par rapport a la technique de l'uniformisation, notre approche est tr es eecace pour …