The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
The art of computer programming, volume 1 (3rd ed.): fundamental algorithms
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
ACM '75 Proceedings of the 1975 annual conference
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This paper explores the possible generation of parametric analytic solutions to finite state Markov processes by formal algebraic methods. The algebraic manipulation system employed, MATHLAB, is particularly well-tuned to this problem area because of its spectrum of automatic symbolic facilities including rational simplification, matrix inversion, inverse Laplace transformation, polynomial factorization, differentiation, and the solution of simultaneous linear equations. For the case of Poisson processes, we demonstrate that the irreversibility of the process implies that the above facilities are sufficient to generate closed-form analytic expressions for the probability vector. An example of this technique, including two-dimensional computer display of the solution, is presented. A contrasting example is included to illustrate how MATHLAB can be employed to compute symbolically certain significant process parameters even when the methods described in this paper fail to yield the probability vector. Typical areas of application are the study of queues, probabilistic games, and reliability.