Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least and at Most Involving Markov Dependent Trials: GERT Approach
We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving homogenous Markov dependent trials. GERT besides providing visual picture of the system helps to analyze the system in a less inductive manner. Mean and variance of the waiting times of the occurrence of the patterns have also been obtained. Some earlier results existing in literature have been shown to be particular cases of these results. 1. Introduction Probability generating functions of waiting time distributions of runs and patterns have been studied and utilized in various areas of statistics and applied probability, with applications to statistical quality control, ecology, epidemiology, quality management in health care sector, and biological science to name a few. A considerable amount of literature treating waiting time distributions have been generated, see Fu and Koutras [1], Aki et al. [2], Koutras [3], Antzoulakos [4], Aki and Hirano [5], Han and Hirano [6], Fu and Lou [7], and so forth. The books by Godbole and Papastavridis [8], Balakrishnan and Koutras [9], Fu and Lou [10] provide excellent information on past and current developments in this area. The probability generating function is very important for studying the properties of waiting time distributions of runs and patterns. Once a potentially problem-specific statistic involving runs and patterns has been defined, the task of deriving its distribution can be very complex and nontrivial. Traditionally, combinatorial methods were used to find the exact distributions for the numbers of runs and patterns. By using the theory of recurrent events, Feller [11] obtained the probability generating function for waiting time of a success run of size in a sequence of Bernoulli trials. Fu and Chang [12] developed general method based on the finite Markov chain imbedding technique for finding the mean and probability generating functions of waiting time distributions of compound patterns in a sequence of i.i.d. or Markov dependent multistate trials. Ge and Wang [13] studied the consecutive- -out-of- : system involving Markov Dependence. Graphical Evaluation and Review Technique (GERT) has been a well-established technique applied in several areas. However, application of GERT in reliability studies has not been reported much. It is only recently that Cheng [14] analyzed reliability of fuzzy consecutive- -out-of- : system using GERT. Agarwal et al. [15, 16], Agarwal & Mohan
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