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

Bara Kim,Jeongsim Kim,Jerim Kim

doi:10.3390/math8111893

Bara Kim, Jeongsim Kim + Show 1 more

Open Access

https://doi.org/10.3390/math8111893

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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.

Highlights

Waiting time problems for runs and patterns in a random sequence of trials are considered important, as they are of theoretical interest and have practical applications in various areas of statistics and applied probability such as reliability, sampling inspection, quality control, DNA/RNA sequence analysis, and hypothesis testing ([1])
Antzoulakos [11] and Fu and Chang [13] obtained the probability generating function of the waiting time for a finite collection of patterns in a sequence of i.i.d. and Markov dependent multi-state trials, respectively. They used a Markov chain with absorbing states corresponding to the patterns and considered the waiting time as the first entrance time into the absorbing state
The analysis is based on the matrix analytic method

Summary

Introduction

Waiting time problems for runs and patterns in a random sequence of trials are considered important, as they are of theoretical interest and have practical applications in various areas of statistics and applied probability such as reliability, sampling inspection, quality control, DNA/RNA sequence analysis, and hypothesis testing ([1]). Antzoulakos [11] and Fu and Chang [13] obtained the probability generating function of the waiting time for a finite collection of patterns in a sequence of i.i.d. and Markov dependent multi-state trials, respectively. They used a Markov chain with absorbing states corresponding to the patterns and considered the waiting time as the first entrance time into the absorbing state. This enables us to construct a finite GI/M/1-type Markov chain with a disaster and consider the waiting time as the time until the occurrence of the disaster Based on this and the matrix analytic method, we obtain the probability generating function of the waiting time W on.

Problem Formulation

Probability Generating Function of the Waiting Time

Numerical Examples

Conclusions