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• W61012307 abstract "Two main approximation methods for steady-state analysis of Markov chains are introduced: Courtois's decomposition method and Takahashi's iterative method. The Courtois's approach is based on decomposability properties of the models under consideration. While Courtois's method is non- iterative and is applied for the approximate computation of the steady-state probability vector for a given ergodic Discrete-Time Markov Chain -DTMC (or Continuous-Time Markov Chain), Takahashi's iterative method allows computation of the state probability vector. Examples and numerical results are shown. ν of the desired state probability vector ν . The efficiency of Courtois's method is due to the fact that instead of solving one linear system of equations with size of the state space S, several much smaller linear systems are solved independently. One system is solved for each subset Sj of the partitioned state space S, and one for the aggregated chain (2,7). Takahashi's method differs substantially from Courtois' approach both with respect to the used methodology and the applicability conditions. While Courtois's method is non- iterative and is applied for the approximate computation of the steady-state probability vector ≈ ν for a given ergodic DTMC (or ≈ π in the case of a CTMC), Takahashi's iterative method allows a computation of the state probability vector (3). Numerical errors are still encountered even in such exact methods. The method is in that there are no modelling approximations. To allow a straightforward comparison of the two methods, any given ergodic CTMC can easily be transformed into an ergodic DTMC (4). COURTOIS'S APPROXIMATION METHOD Courtois's approach is based on decomposability properties of the models under consideration. Initially substructures that can separately be analyzed are identified. Then, an aggregation procedure is performed that uses independently computed sub results 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. Let P be a transition probability matrix. The state space associated with P is partitioned into groups, often referred to as lumps, such that any state within a group contains the same number of customers in a specific portion of the state descriptor (8). The transition probability matrix P, written out according to these groups consists of blocks of transition probabilities along the diagonal and of other blocks along the off-diagonals above and below the diagonal. P is said to be Nearly Completely Decomposable (NCD) if the sum of the non-zero transition probabilities on each row that lie within the diagonal block is close to 1. That means that the sum of the off-diagonal probabilities along a row is extremely small. This occurs when most of the transitions are between states of the same group, with very few transitions between states of different groups (6). The error induced by the method can, in principle, be bounded, but a formal error bounding is often omitted. A modified algorithm for calculation of probability vector, built according to Courtois's approximation method is presented in Figure 1." @default.
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• W61012307 date "2005-01-01" @default.
• W61012307 modified "2023-09-26" @default.
• W61012307 title "Approximation Algorithms for Steady-State Solutions of Markov Chains" @default.
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