Transient Solution of Markov Chains Transient solution is more meaningful than steady-state solution when the system under investigation needs to be evaluated with respect to its shortterm behavior, Using steady-state measures instead of transient measures could lead to substantial errors in this case. Furthermore, applying transient analysis is the onl y choice if non-ergodic models are investigated, Transient analysis of Markov chains has been attracting increasing attention and is of particular importance in dependability modeling. . | Queueing Networks and Markov Chains Gunter Botch Stefan Greiner Hermann de Meer Kishor S. Trivedi Copyright 1998 John Wiley Sons Inc. Print ISBN 0-471-19366-6 Online ISBN 0-471-20058-1 Transient Solution of Markov Chains Transient solution is more meaningful than steady-state solution when the system under investigation needs to be evaluated with respect to its shortterm behavior. Using steady-state measures instead of transient measures could lead to substantial errors in this case. Furthermore applying transient analysis is the only choice if non-ergodic models are investigated. Transient analysis of Markov chains has been attracting increasing attention and is of particular importance in dependability modeling. Unlike steady-state analysis CTMCs and DTMCs have to be treated differently while performing transient analysis. Surprisingly not many algorithms exist for the transient analysis of DTMCs. Therefore we primarily focus on methods for computing the transient state probability vector 7r t for CTMCs as defined in Eq. . Furthermore additional attention is given to the computation of quantities related to transient probabilities such as cumulative measures. Recall from Eq. that for the computation of transient state probability vector 7r t the following linear differential equation has to be solved given infinitesimal generator matrix Q and initial probability vector 7r 0 7r i Q tt 0 7T0 0 7ri 0 . . Measures that can be immediately derived from transient state probabilities are often referred to as instantaneous measures. However sometimes measures based on cumulative accomplishments during a given period of time 177 178 TRANSIENT SOLUTION OF MARKOV CHAINS XXX Fig. A pure birth process. 0 t could be more relevant. Let L i Tt u du denote the vector of the total expected times spent in the states of the CTMC during the indicated period of time. By integrating Eq. on both sides we obtain a new differential equation for L t A L t Q - 0 L 0 0.
