Abstract
The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2.mconditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Referenoes
Aki, S. (1992) Waiting time problems for a sequence of discrete random variables, Ann. Inst. Statist. Math., 44, 363–378.
Aki, S. and Hirano, K. (1993). Discrete distributions related to succession events in a two-state Markov chain, Statistical Science & Data Analysis (eds. K. Matusita, M. L. Puri and T. Hayakawa), 467–474, VSP Publishers, Amsterdam.
Balasubramanian, K., Viveros, R. and Balakrishnan, N. (1993). Sooner and later waiting time problem for Markovian Bernoulli trials. Statist. Probab. Lett., 18, 153–161.
Crysaphinou, O. and Papastavridis, S. (1990). The occurrence of sequence patterns in repeated dependent experiments, Theory Probab. Appl., 35, 145–152.
Crysaphinou, O., Papastavridis, S. and Tsapelas, T. (1994). On the waiting time appearance of given patterns, Runs and Patterns in Probability: Selected Papers (eds. A. P. Godbole and S. G. Papastavridis), 231–241, Kluwer, Dordrecht.
Derman, C., Lieberman, G. J. and Ross, S. M. (1982). On the consecutive-k-of-n:F system, IEEE Transactions on Reliability, R- 31, 57–63.
Ebneshahrashoob, M. and Sobel, M. (1990). Sooner and later problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Lett., 9, 5–11.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n:F systems with (k-1)-step Marcov dependence, IEEE Transactions on Reliability, R- 35, 602–606.
Fu, J. C. and Koutras, M. V. (1994). Distribution theory of runs: A Markov chain approach, J. Amer. Statist. Assoc., 89, 1050–1058.
Godbole, A. P. (1993). Approximate reliabilities of m-consecutive-k-out-of-n:failure systems, Statistica Sinica, 3, 321–327.
Godbole, A. P. and Papastavridis, S. G. (1994). Runs and Patterns in Probability: Selected Papers, Kluwer, Dordrecht.
Hahn, G. J. and Gage, J. B. (1983). Evaluation of a start-up demonstration test, Journal of Quality Technology, 15, 103–106.
Hirano, K. (1986). Some properties of the distributions of order k, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 43–53, Reidel, Dordrecht.
Hirano, K. and Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain, Statistica Sinica, 3, 313–320.
Johnson, N. L., Kotz, S. and Kemp, A. W. (1992). Univariate Discrete Distributions, Wiley, New York.
Lambiris, M. and Papastavridis, S. G. (1987). Reliability of a consecutive-k-out-of-n:F system for Markov dependent components, IEEE Transactions on Reliability, R- 36, 78–79.
Mohanty, S. G. (1994). Success runs of length k in Markov dependent trials, Ann. Inst. Statist. Math., 46, 777–796.
Papastavridis, S. and Lambiris, M. (1987). Reliability of a consecutive-k-out-of-n:F system for Markov-dependent components, IEEE Transactions on Reliability, R- 36, 78–79.
Philippou, A. N. (1986). Distributions and Fibonacci polynomials of order k, longest runs, and reliability of consecutive-k-out-of-n:F system, Fibonacci Numbers and Their Applications (eds. A. N. Philippou, G. E. Bergum and A. F. Horadam), 203–227, Reidel, Dordrecht.
Philippou, A. N. and Makri, F. S. (1986). Successes, runs, and longest runs, Statist. Probab. Lett., 4, 101–105.
Philippou, A. N. and Muwafi, A. A. (1982). Waiting for the k-th consecutive success and the Fibonacci sequence of order k, Fibonacci Quart., 20, 28–32.
Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties, Statist. Probab. Lett., 1, 171–175.
Rajarshi, M. B. (1974). Success runs in a two-state Markov chain, J. Appl. Probab., 11, 190–192.
Schwager, S. J. (1983). Run probability in sequences of Markov-dependent trials, J. Amer. Statist. Assoc., 78, 168–175.
Stanley, R. T. (1986). Enumerative Combinatories, Wadsworth Publishers, Kentucky.
Viveros, R. and Balakrishnan, N. (1993). Statistical inference from start-up demonstration test data, Journal of Quality Technology, 25, 119–130.
Viveros, R., Balasubramanian, K. and Balakrishnan, N. (1994). Binomial and negative binomial analogues under correlated Bernoulli trials, Amer. Statist., 48, 243–247.
Author information
Authors and Affiliations
Additional information
This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.
About this article
Cite this article
Aki, S., Balakrishnan, N. & Mohanty, S.G. Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials. Ann Inst Stat Math 48, 773–787 (1996). https://doi.org/10.1007/BF00052332
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00052332