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Waiting Time Problems for Patterns in a Sequence of Multi-State Trials

Author

Listed:
  • Bara Kim

    (Department of Mathematics, Korea University, Seoul 02841, Korea)

  • Jeongsim Kim

    (Department of Mathematics Education, Chungbuk National University, Chungbuk 28644, Korea)

  • Jerim Kim

    (Department of Mathematics, University of Seoul, Seoul 02504, Korea)

Abstract

In this paper, we investigate waiting time problems for a finite collection of patterns in a sequence of independent multi-state trials. By constructing a finite GI/M/1-type Markov chain with a disaster and then using the matrix analytic method, we can obtain the probability generating function of the waiting time. From this, we can obtain the stopping probabilities and the mean waiting time, but it also enables us to compute the waiting time distribution by a numerical inversion.

Suggested Citation

  • Bara Kim & Jeongsim Kim & Jerim Kim, 2020. "Waiting Time Problems for Patterns in a Sequence of Multi-State Trials," Mathematics, MDPI, vol. 8(11), pages 1-16, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:11:p:1893-:d:438040
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    References listed on IDEAS

    as
    1. Chang, Yung-Ming, 2005. "Distribution of waiting time until the rth occurrence of a compound pattern," Statistics & Probability Letters, Elsevier, vol. 75(1), pages 29-38, November.
    2. Gerber, Hans U. & Li, Shuo-Yen Robert, 1981. "The occurrence of sequence patterns in repeated experiments and hitting times in a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 11(1), pages 101-108, March.
    3. Pozdnyakov, Vladimir, 2008. "On occurrence of patterns in Markov chains: Method of gambling teams," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2762-2767, November.
    4. Kim, Bara & Kim, Jeongsim, 2019. "Sooner waiting time problems in a sequence of multi-state trials with random rewards," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 171-179.
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