Passage time distributions in large Markov chains

Abstract

Accepted versio

State selection in Markov models for panel data with application to psoriatic arthritis

“…Note that the median times for the first progressions are longer than the length of follow-up at the UoT clinic because they are based on all patients, including those who did not reach severe disability during the study, who we found made up approximately 90% of the sample. We also calculated the mean first passage times using the method of Harrison et al [23] and found a similar pattern to that found for the medians. Table III reports the number of observed transitions between the states, defined as observed changes in HAQ category between consecutive visits, of the six-state model.…”

Hypergraph-based parallel computation of passage time densities in large semi-Markov models

“…As all our distribution manipulations take place in Laplace-space, we link our distribution representation to the Laplace inversion technique that we ultimately use. Our tool supports two Laplace transform inversion algorithms, which are briefly outlined below: the Euler technique [3] and the Laguerre method [1] with modifications summarised in [14].…”

“…As described in [14], 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.…”

“…The focus of the present study is on semi-Markov processes [24], a generalisation of Markov processes which support arbitrary state holding times. Techniques already exist for the practical extraction of passage time densities and quantiles from large purely Markovian systems [11,14,17]. Until now, however, only relatively small semi-Markov systems have been analysed for passage time quantities [5,12,15].…”

A formal language towards the unification of model checking and performance evaluation

“…Another work we would like to mention is performance trees [61], as they can describe results of various types (real-valued or otherwise, including distributions). Unlike CTML or asCTML, however, performance trees is a more general framework or interface that utilizes the existing performance evaluation algorithms (such as passage time distributions [34]) as well as some of the existing model checking algorithms for the expression of both logic and real-valued measures.…”