Passage time distributions in large Markov chains

Abstract

Probability distributions of response times are important in the design and analysis of transaction processing systems and computercommunication systems. We present a general technique for deriving such distributions from high-level modelling formalisms whose state spaces can be mapped onto finite Markov chains. We use a load-balanced, distributed implementation to find the Laplace transform of the first passage time density and its derivatives at arbitrary values of the transform parameter ¢. Setting ¢ ¤ £ ¦ ¥ yields moments while the full passage time distribution is obtained using a novel distributed Laplace transform inverter based on the Laguerre method. We validate our method against a variety of simple densities, cycle time densities in certain overtake-free (tree-like) queueing networks and a simulated Petri net model. Our implementation is thereby rigorously validated and has already been applied to substantial Markov chains with over 1 million states. Corresponding theoretical results for semi-Markov chains are also presented.

Iterative convergence of passage-time densities in semi-Markov performance models

“…Let the kernel of this process be: R(n, i, j, t) = P(X n+1 = j, T n+1 − T n ≤ t|X n = i) (1) for i, j ∈ S. The continuous time semi-Markov process (SMP), {Z(t), t ≥ 0}, defined by the kernel R, is related to the Markov renewal process by: (2) where N(t) = max{n : T n ≤ t}, i.e. the number of state transitions that have taken place by time t. Thus Z(t) represents the state of the system at time t. We consider time-homogeneous SMPs, in which R(n, i, j, t) is independent of any previous state except the last.…”

“…Our tool supports two Laplace transform inversion algorithms, which are briefly outlined below: the Euler technique [12] and the Laguerre method [13] with modifications summarised in [2].…”

“…(16). As described in [2], the Laguerre method can be modified by noting that the Laguerre coefficients q n are independent of t. This means that if the number of trapezoids used in the evaluation of q n is fixed to be the same for every q n (rather than depending on the value of n), values of Q(z) (and hence L(s)) can be reused after they have been computed. Typically, we set n = 200.…”

“…Techniques for evaluating passage-times in Markovian systems have existed for some time [2][3][4], however the practical calculation of passage-times in a semi-Markov environment, where arbitrary distributions of atomic events are permitted, is a recent development. In previous work [1,5] we described the iterative algorithm presented here, but had no direct proof that the iterative scheme converged to the correct passage-time.…”

Uniformization and hypergraph partitioning for the distributed computation of response time densities in very large Markov models

“…There are two main methods for computing first passage time (and hence response time) densities in Markov chains: those based on Laplace transforms and their inversion [1,21] and those based on uniformization [42][43][44]. The former has wider application to semi-Markov processes but is less efficient than uniformization when restricted to Markov chains.…”