In this section we introduce an efficient method for the steady-state analysis of Markov chains. Whereas direct and iterative techniques can be used for the exact analysis of Markov chains as previously discussed, the method computations of Courtois [Cour75, Cour77] is mainly applied to approximate u NN the desired state probability vector u. Courtois’s approach is based of on decomposability properties of the models under consideration. | 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 jeet o r __S_ Steady-State Aggregation Disaggregation Methods Tn this chapter we consider two main approximation methods Courtois s decomposition method and Takahashi s iterative aggregation disaggregation method. COURTOIS S APPROXIMATE METHOD In this section we introduce an efficient method for the steady-state analysis of Markov chains. Whereas direct and iterative techniques can be used for the exact analysis of Markov chains as previously discussed the method of Courtois Cour75 Cour77 is mainly applied to approximate computations of the desired state probability vector v. Courtois s approach is based on decomposability properties of the models under consideration. Initially substructures are identified that can separately be analyzed. Then an aggregation procedure is performed that uses independently computed subresults as constituent parts for composing the final results. The applicability of the method needs to be verified in each case. If the Markov chain has tightly coupled subsets of states where the states within each subset are tightly coupled to each other and weakly coupled to states outside the subset it provides a strong intuitive indication of the applicability of the approach. Such a subset of states might then be aggregated to form a macro state as a basis for further analysis. The macro state probabilities together with the conditional micro state probabilities from within the subsets can be composed to yield the micro state probabilities of the initial model. Details of the approach are clarified through the following example. 153 154 STEADY-STATE AGGREGATION DISAGGREGATION METHODS Decomposition Since the method of Courtois is usually expressed in terms of a DTMC whereas we are emphasizing the use of a CTMC in our discussion of
