Terminating Passage-Time Calculations on Uniformised Markov Chains

Uniformisation[1, 2] is a key technique which allows modellers to extract passage-time quantiles/densities which in turn permits the plotting of probability density and cumulative distribution functions. Uniformisation converts a CTMC (Continuous-Time Markov Chain) into a DTMC (Discrete-Time Markov Chain) with equivalent semantics. This can be used to calculate the probability of completing a passage within a given time t by calculating the probability of completing the passage within a number of iterations, n, of the DTMC and then calculating the probability that the nth iteration is performed within time t. However to calculate the passage-time quantiles we desire we must theoretically perform this calculation for values of n from zero to infinity and sum the probabilities. This can be approximated by calculating for values of n from zero to some finite value if we know that larger values of n will yield negligible probabilities and hence add nothing significant to the summation. This paper discusses two important conditions which ensure that the approximation is appropriate while also reducing the amount of negligible values