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Exact Distributions of Waiting Time Problems of Mixed Frequencies and Runs in Markov Dependent Trials
Department of Mathematics and Statistics, California State University, Long Beach, USA
Department of Mathematics and Statistics, California State University, Long Beach, USA
Department of Mathematics and Statistics, California State University, Long Beach, USA
- 1 Department of Mathematics and Statistics, California State University, Long Beach, USA
- 2 Department of Mathematics and Statistics, California State University, Long Beach, USA
- 3 Department of Mathematics and Statistics, California State University, Long Beach, USA
Applied Mathematics·Volume 03 (2012)·Pages 1689–1696·Published 7 November 2012·DOI10.4236/am.2012.311234
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Abstract
We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve α frequency cells and β run cells with prescribed quotas and an additional γ slack cells without quotas. For any given and , in our Model I we determine the waiting time until at least frequency cells and at least run cells reach their quotas. For any given τ ≤ α + β , in our Model II we determine the waiting time until τ cells reach their quotas. Computer algorithms are developed to calculate the distributions, expectations and standard deviations of the waiting time random variables of the two models. Numerical results demonstrate the efficiency of the algorithms.
KeywordsInverse SamplingMultinomial Stopping ProblemSoonest through Latest Waiting Time VariableProbability Generating FunctionFirst-Order Markov Dependent Trial
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