On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

[1]  J. Fu,et al.  DISTRIBUTION THEORY OF RUNS AND PATTERNS ASSOCIATED WITH A SEQUENCE OF MULTI-STATE TRIALS , 1996 .

[2]  James C. Fu,et al.  Reliability of Consecutive-k-out-of-n:F Systems with (k-1)-step Markov Dependence , 1986, IEEE Transactions on Reliability.

[3]  W. Y. Wendy Lou,et al.  Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Imbedding Approach , 2003 .

[4]  Markos V. Koutras,et al.  Sooner waiting time problems in a sequence of trinary trials , 1997, Journal of Applied Probability.

[5]  Narayanaswamy Balakrishnan,et al.  Sooner and later waiting time problems for Markovian Bernoulli trials , 1993 .

[6]  M. Chao,et al.  A limit theorem of certain repairable systems , 1989 .

[7]  W. Lou,et al.  An application of the method of finite Markov chain imbedding to runs tests , 1997 .

[8]  Milton Sobel,et al.  Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas , 1990 .

[9]  W. Lou.  On Runs and Longest Run Tests: A Method of Finite Markov Chain Imbedding , 1996 .

[10]  M. Chao,et al.  The reliability of a large series system under Markov structure , 1991, Advances in Applied Probability.

[11]  Narayanaswamy Balakrishnan,et al.  Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials , 1996 .

[12]  S. Aki.  On Sooner and Later Problems Between Success and Failure Runs , 1997 .

[13]  S. Aki,et al.  Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials , 1999 .

[14]  James C. Fu,et al.  On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials , 2002, Journal of Applied Probability.

[15]  E. Yamamoto,et al.  On the joint distribution of runs in a sequence of multi-state trials , 1998 .

[16]  Michael V. Boutsikas,et al.  Reliability Approximation for Markov Chain Imbeddable Systems , 2000 .

[17]  Charles M. Goldie,et al.  Distribution Theory of Runs and Patterns and Its Applications: A Finite Markov Chain Imbedding Approach , 2005 .

[18]  Markos V. Koutras,et al.  Waiting Time Distributions Associated with Runs of Fixed Length in Two-State Markov Chains , 1997 .

[19]  Masayuki Uchida.  On Generating Functions of Waiting Time Problems for Sequence Patterns of Discrete Random Variables , 1998 .

[20]  Markos V. Koutras,et al.  Distribution Theory of Runs: A Markov Chain Approach , 1994 .