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

Affiliation(s)

Department of Mathematics and Statistics, California State University, Long Beach, USA.

Department of Mathematics and Statistics, California State University, Long Beach, USA.

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.

We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve

Cite this paper

B. Chaderjian, M. Ebneshahrashoob and T. Gao, "Exact Distributions of Waiting Time Problems of Mixed Frequencies and Runs in Markov Dependent Trials,"*Applied Mathematics*, Vol. 3 No. 11, 2012, pp. 1689-1696. doi: 10.4236/am.2012.311234.

B. Chaderjian, M. Ebneshahrashoob and T. Gao, "Exact Distributions of Waiting Time Problems of Mixed Frequencies and Runs in Markov Dependent Trials,"

References

[1] N. Balakrishnan and M. V. Koutras, “Runs and Scans with Applications,” John Wiley & Sons, New York, 2002.

[2] J. C. Fu and W. Y. Lou, “Distribution Theory of Runs and Patterns and its Applications,” World Scientific Publisher, Singapore City, 2003.

[3] A. P. Godbole and S. G. Papastavridis, “Runs and Patterns in Probability: Selected Papers,” Kluwer, Dordrecht, 1994. doi:10.1007/978-1-4613-3635-8

[4] S. Aki and K. Hirano, “Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials,” Annals of the Institute of Statistical Mathematics, Vol. 51, No. 1, 1999, pp. 17-29. doi:10.1023/A:1003874900507

[5] K. Balasubramanian, R. Viveros and N. Balakrishnan, “Sooner and Later Waiting Time Problems for Markovian Bernoulli Trials,” Statistics & Probability Letters, Vol. 18, No. 2, 1993, pp. 153-161. doi:10.1016/0167-7152(93)90184-K

[6] Q. Han and S. Aki, “Waiting Time Problems in a TwoState Markov Chain,” Annals of the Institute of Statistical Mathematics, Vol. 52, No. 4, 2000, pp. 778-789. doi:10.1023/A:1017537629251

[7] N. Kolev and L. Minkova, “Run and Frequency Quotas in a Multi-State Markov Chain,” Communications in Statistics—Theory and Methods, Vol. 28, No. 9, 1999, pp. 2223-2233. doi:10.1080/03610929908832417

[8] K. D. Ling and T. Y. Low, “On the Soonest and Latest Waiting Time Distributions: Succession Quotas,” Communications in Statistics—Theory and Methods, Vol. 22, No. 8, 1993, pp. 2207-2221. doi:10.1080/03610929308831143

[9] M. Sobel and M. Ebneshahrashoob, “Quota Sampling for Multinomial via Dirichlet,” Journal of Statistical Planning and Inference, Vol. 33, No. 2, 1992, pp. 157-164. doi:10.1016/0378-3758(92)90063-X

[10] M. Ebneshahrashoob, T. Gao and M. Sobel, “Double Window Acceptance Sampling,” Naval Research Logistics (NRL), Vol. 51, No. 2, 2004, pp. 297-306. doi:10.1002/nav.10119

[11] M. Ebneshahrashoob, T. Gao and M. Sobel, “Sequential Window Problems,” Sequential Analysis: Design Methods and Applications, Vol. 24, No. 2, 2005, pp. 159-175. doi:10.1081/SQA-200056194

[12] M. Ebneshahrashoob, T. Gao and M. Wu, “An Efficient Algorithm for Exact Distribution of Scan Statistics,” Methodology and Computing in Applied Probability, Vol. 7, No. 4, 2005, pp. 459-471. doi:10.1007/s11009-005-5003-0

[13] M. J. Evans and J. S. Rosenthal, “Probability and Statistics, the Science of Uncertainty,” W. H. Freeman and Company, New York, 2004.

[14] Y. Saad, “Iterative Methods for Sparse Linear Systems,” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi:10.1137/1.9780898718003

[1] N. Balakrishnan and M. V. Koutras, “Runs and Scans with Applications,” John Wiley & Sons, New York, 2002.

[2] J. C. Fu and W. Y. Lou, “Distribution Theory of Runs and Patterns and its Applications,” World Scientific Publisher, Singapore City, 2003.

[3] A. P. Godbole and S. G. Papastavridis, “Runs and Patterns in Probability: Selected Papers,” Kluwer, Dordrecht, 1994. doi:10.1007/978-1-4613-3635-8

[4] S. Aki and K. Hirano, “Sooner and Later Waiting Time Problems for Runs in Markov Dependent Bivariate Trials,” Annals of the Institute of Statistical Mathematics, Vol. 51, No. 1, 1999, pp. 17-29. doi:10.1023/A:1003874900507

[5] K. Balasubramanian, R. Viveros and N. Balakrishnan, “Sooner and Later Waiting Time Problems for Markovian Bernoulli Trials,” Statistics & Probability Letters, Vol. 18, No. 2, 1993, pp. 153-161. doi:10.1016/0167-7152(93)90184-K

[6] Q. Han and S. Aki, “Waiting Time Problems in a TwoState Markov Chain,” Annals of the Institute of Statistical Mathematics, Vol. 52, No. 4, 2000, pp. 778-789. doi:10.1023/A:1017537629251

[7] N. Kolev and L. Minkova, “Run and Frequency Quotas in a Multi-State Markov Chain,” Communications in Statistics—Theory and Methods, Vol. 28, No. 9, 1999, pp. 2223-2233. doi:10.1080/03610929908832417

[8] K. D. Ling and T. Y. Low, “On the Soonest and Latest Waiting Time Distributions: Succession Quotas,” Communications in Statistics—Theory and Methods, Vol. 22, No. 8, 1993, pp. 2207-2221. doi:10.1080/03610929308831143

[9] M. Sobel and M. Ebneshahrashoob, “Quota Sampling for Multinomial via Dirichlet,” Journal of Statistical Planning and Inference, Vol. 33, No. 2, 1992, pp. 157-164. doi:10.1016/0378-3758(92)90063-X

[10] M. Ebneshahrashoob, T. Gao and M. Sobel, “Double Window Acceptance Sampling,” Naval Research Logistics (NRL), Vol. 51, No. 2, 2004, pp. 297-306. doi:10.1002/nav.10119

[11] M. Ebneshahrashoob, T. Gao and M. Sobel, “Sequential Window Problems,” Sequential Analysis: Design Methods and Applications, Vol. 24, No. 2, 2005, pp. 159-175. doi:10.1081/SQA-200056194

[12] M. Ebneshahrashoob, T. Gao and M. Wu, “An Efficient Algorithm for Exact Distribution of Scan Statistics,” Methodology and Computing in Applied Probability, Vol. 7, No. 4, 2005, pp. 459-471. doi:10.1007/s11009-005-5003-0

[13] M. J. Evans and J. S. Rosenthal, “Probability and Statistics, the Science of Uncertainty,” W. H. Freeman and Company, New York, 2004.

[14] Y. Saad, “Iterative Methods for Sparse Linear Systems,” SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 2003. doi:10.1137/1.9780898718003