Exact Distributions of Waiting Time Problems of Mixed Frequencies and Runs in Markov Dependent Trials

We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve frequency cells and run cells with prescribed quotas and an additional 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 , in our Model II we determine the waiting time until 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