Characterization of Markov-Bernoulli Geometric Distribution Related to Random Sums

Abstract

The Markov-Bernoulli geometric distribution is obtained when a generalization, as a Markov process, of the independent Bernoulli sequence of random variables is introduced by considering the success probability changes with respect to the Markov chain. The resulting model is called the MarkovBernoulli model and it has a wide variety of application fields. In this study, some characterizations are given concerning the Markov-Bernoulli geometric distribution as the distribution of the summation index of independent randomly truncated non-negative integer valued random variables. The achieved results generalize the corresponding characterizations concerning the usual geometric distribution.

Some New Characterizations of Markov-Bernoulli Geometric Distribution Related to Random Sums

“…In the following, we list some results obtained by Gharib et al (2014) which are of direct relevance to the development of the results of the present paper. Consider the random sum Z defined by (1.3).…”

“…The rv Z represents the truncated sum until the moment where for the first time the process {Y n ; n ≥ 1} has greater jump than the corresponding jump of the process {X n ; n ≥ 1}. The sequence {Y n } is called the truncating process (Gharib et al, 2014). In queuing systems with unreliable server, Z can be interpreted as the total time duration of the unreliable server until the successful finish of the service if the corresponding duration without breakdowns is previously known (Dimitrov et al, 1991).…”

“…Using the results of Lemma 1.1 and Lemma 1.2, Gharib et al (2014) gave the following two characterizations of the MBG distribution.…”