Aggregation/Disaggregation Methods for Computing the Stationary Distribution of Markov Chains with Application to Multiprogramming System

ABSTRACf. This paper studies the aggregation/disaggregation of nearly completely decomposable Markov chains that have many applications in queueing networks and packet switched networks. A general class of simi­ larity transformation that transforms the stochastic transition probability matrix into a reduced order aggregated matrix is presented. This transfor­ mation is used to develop an aggregation algorithm to compute the exact stationary probability distribution, as weB as O( e k ) approximation of it. The proposed aggregation method is applied to a multiprogramming computer system with six active terminals and the capacity of the CPU and the secon­ dary memory is 3. This example is used to compare our algorithm with three well-known algorithms. The simulation studies showed that our algorithm usually converges in less number of iterations and CPU time. Moreover, it is shown that the other algorithms do not converge in some cases while our algorithm usually converges.

[1]  Udo R. Krieger,et al.  Modeling and Analysis of Communication Systems Based on Computational Methods for Markov Chains , 1990, IEEE J. Sel. Areas Commun..

[2]  Pierre-Jacques Courtois,et al.  Decomposability, instabilities, and saturation in multiprogramming systems , 1975, CACM.

[3]  S. Sastry,et al.  Hierarchical aggregation of singularly perturbed finite state Markov processes , 1983 .

[4]  G. Stewart,et al.  On a Rayleigh-Ritz refinement technique for nearly uncoupled stochastic matrices , 1984 .

[5]  Pierre Semal,et al.  Computable Bounds for Conditional Steady-State Probabilities in Large Markov Chains and Queueing Models , 1986, IEEE J. Sel. Areas Commun..

[6]  Herbert A. Simon,et al.  Aggregation of Variables in Dynamic Systems , 1961 .

[7]  H. Khalil,et al.  Aggregation of the policy iteration method for nearly completely decomposable Markov chains , 1991 .

[8]  William J. Stewart,et al.  Iterative aggregation/disaggregation techniques for nearly uncoupled markov chains , 1985, JACM.

[9]  J. R. Rohlicek,et al.  Time Scale Decomposition: The Role of Scaling in Linear Systems and Transient States in Finite-State Markov Processes , 1985, 1985 American Control Conference.

[10]  Linda Kaufman,et al.  Analysis of Packet Network Congestion Control Using Sparse Matrix Algorithms , 1981, IEEE Trans. Commun..

[11]  Alan S. Willsky,et al.  The reduction of perturbed Markov generators: an algorithm exposing the role of transient states , 1988, JACM.

[12]  Alan G. Konheim,et al.  Finite Capacity Queuing Systems with Applications in Computer Modeling , 1978, SIAM J. Comput..