Analysis of Probabilistic Systems via Generating Functions and Padé Approximation

We investigate the use of generating functions in the analysis of discrete Markov chains. Generating functions are introduced as power series whose coefficients are certain hitting probabilities. Being able to compute such functions implies that the calculation of a number of quantities of interest, including absorption probabilities, expected hitting time and number of visits, and variances thereof, becomes straightforward. We show that it is often possible to recover this information, either exactly or within excellent approximation, via the construction of Pade approximations of the involved generating function. The presented algorithms are based on projective methods from linear algebra, which can be made to work with limited computational resources. In particular, only a black-box, on-the-fly access to the transition function is presupposed, and the necessity of storing the whole model is eliminated. A few numerical experiments conducted with this technique give encouraging results.