Abstract
A new general solution method is derived for the general GI/G/1 type processes --- for the steady-state distribution of infinite block-structured Markov chains with repetitive structure. While matrix inversion is needed in each iterational step of other general (and of more special) matrix analytical procedures, the method presented here uses matrix addition and matrix multiplication only. In exchange, the computational complexity and the memory requirement is increasing in each iterational step of the proposed method. This paper, however, lays priority on the theoretical aspect of the general solution.
- A. S. Alfa, S. R. Chakravarthy, editors. Advances in Matrix-Analytic Methods in Stochastic Models. Notable Publications, Inc., New Jersey, 1998.Google Scholar
- S. R. Chakravarthy, A. S. Alfa, editors. Matrix-Analytic Methods in Stochastic Models, Vol. 183 of Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York, 1997.Google Scholar
- H. R. Gail, S. L. Hantler, B. A. Taylor. Use of Characteristic Roots for Solving Infinite State Markov Chains. Ch. 7 of Computational Probability (W. K. Grassmann, Ed.), International Series in Operations Research and Management Science, Vol. 24, pp. 205-255, Kluwer Academic Publishers, Oct. 1999.Google Scholar
- W. K. Grassmann, D. P. Heyman. Equilibrium Distribution of Block-Structured Markov Chains with Repeating Rows. J. Appl. Prob. 27. pp. 557-576, 1990.Google ScholarCross Ref
- W. K. Grassmann, D. A. Stanford. Matrix Analytic Methods. Ch. 6 of Computational Probability (W. K. Grassmann, Ed.), International Series in Operations Research and Management Science, Vol. 24, pp. 153-202., Kluwer Academic Publishers, Oct. 1999.Google Scholar
- G. Latouche, P. Taylor, editors. Advances in Algorithmic Methods for Stochastic Models, Proceedings of the 3rd International Conference on Matrix Analytic Methods. Notable Publications, Inc., New Jersey, 2000.Google Scholar
- I. Mitrani, R. Chakka. Spectral Expansion Solution for a Class of Markov Models: Application and Comparison with the Matrix-Geometric Method. Performance Evaluation 23, pp. 241-260, 1995. Google ScholarDigital Library
- M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models, An Algorithmic Approach. John Hopkins University Press, Baltimore, 1981.Google Scholar
Index Terms
- GI/G/1 type processes: a non-inversive matrix analytical solution
Recommendations
Solutions of M/G/1//N-type Loops with Extensions to M/G/1 and GI/M/1 Queues
<P>Closed form solutions of the joint equilibrium distribution of queue sizes are derived for a large class of M/G/1//N queues, i.e., any closed loop of two servers in which one is exponential but possibly load dependent, and the other has a probability ...
Bounds for Some Generalizations of the GI/G/1 Queue
<P>Expressions are derived for the expected number and expected wait in queue for the following generalizations of the GI/G/1 queue: 1 arrivals in batches of random size, 2 service in batches of fixed size, 3 queues with added delay for the first ...
An Interpolation Approximation for the Mean Workload in a GI/G/1 Queue
<P>This paper develops a closed form approximation for the mean steady-state workload or virtual waiting time in a GI/G/1 queue, using the first two moments of the service-time distribution and the first three moments plus the density at the origin of ...
Comments