On the joint distribution of runs in a sequence of multi-state trials

Reliability of Circular Systems With Markov Dependencies

“…The total number of runs is one of the most important run statistics. Beside this statistic, various run statistics have been defined for both a binary sequence and a sequence which consists of multi-state trials (Han and Aki [17], Doi and Yamamato [9], Eryilmaz [11]). Distributions of run statistics have been obtained under different assumptions, such as i.i.d, exchangeability, and Markov dependence, using different techniques such as combinatorial methods, the Markov Chain Imbedding Technique, or probability generating functions (Demir and Eryilmaz [7], Fu and Koutras [13], Han and Aki [16]).…”

“…Therefore, the reliability of this system can be represented as This probability is equal to the second equation at the bottom of the page. The probability distribution of the longest failure run is given in (9). Using (10) and (8), it is easy to obtain the probability distribution of the total number of failed components , and the joint distribution of , by the third equation at the bottom of the page.…”

Bivariate Markov chain embeddable variables of polynomial type

“…Fu (1996) extended the original method to cover the case of arbitrary patterns (instead of runs) whereas Koutras (1997) treated several waiting time problems within this framework. Finally Doi and Yamamoto (1998) and Han and Aki (1999) considered the case of multivariate run related distributions and offered simple solutions by exploiting proper extensions of the Markov chain embedding technique (for an illustrative presentation of this method see Koutras, 2003).…”

On the joint distribution of runs in the sequence of Markov-dependent multi-state trials

“…Based on this approach Fu (1996) introduced a 'forward and backward principle' for method of Markov chain imbedding to study exact joint distributions of runs and patterns in a sequence of multi-state trials. Doi and Yamamoto (1998) obtained joint distribution of c kinds of runs in the sequence of ðc þ 1Þ-state trials by using a finite Markov chain method proposed by Fu and Koutras (1994) under all the above counting schemes except for '-overlapping counting scheme. Koutras and Alexandrou (1995) refined a finite Markov chain imbedding approach by introducing Markov chain imbeddable variable of binomial type (MVB) to remove the dependence of order of transition probability matrices of imbedded Markov chain on number of trials ðnÞ.…”