A binomial approximation of lot yield under Markov modulated Bernoulli item yield

Abstract

The existing literature that models item-by-item production yield as a Bernoulli process assumes that the intraperiod likelihood of producing an acceptable item is stationary. We investigate the stochastic process that results from relaxing this assumption to account for system deterioration during each production run. More specifically, we consider a Bernoulli yield model with a nonstationary parameter that depends on the deterioration level of the system, which evolves according to a discrete-time Markov chain. For tractability reasons, we construct a simple binomial approximation of the non-stationary process, and compare the two yield distributions both analytically and numerically. Our results suggest that the approximation performs well, even when the deterioration occurs relatively fast, which serves to validate existing (and future) decision models that impose the stationarity assumption.

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

“…The sequence X 1 , X 2 , ... with the transition matrix (1.1) and the given initial distribution is called the Markov-Bernoulli model(MBM) or the Markov modulated Bernoulli process (Özekici, 1997). Numerous researchers have studied the MBM from the various aspects of probability, statistics and their applications, in particular the classical problems related to the usual Bernoulli model (Anis & Gharib, 1982;Arvidsson & Francke, 2007;Cekanavicius & Vellaisamy, 2010;Gharib & Yehia, 1987;Inal, 1987;Maillart et al, 2008;Minkova & Omey, 2011;Omey et al, 2008;Özekici, 1997;Özekici & Soyer, 2003;Pacheco et al, 2009;Pedler, 1980;Pires & Diniz, 2012;Satheesh et al, 2002;Xekalaki & Panaretos, 2004 and others). Further, due to the fact that the MBM operates in a random environment depicted by a Markov chain so that the probability of success at each trial depends on the state of the environment, this model has a wide variety of applications include, but not limited, reliability modeling (where system and components function in a randomly changing environment), non-life insurance, matching DNAsequences, disease clustering, traffic modeling, the occupation and waiting times problems in two state Markov chains, reconstructing patterns from sample data and statistical ecology (Arvidsson & Francke, 2007;Chang & Zeiterman, 2002;John, 1971;Özekici, 1997;Özekici & Soyer, 2003;Pacheco et al, 2009;Pedler, 1980;Pires & Diniz, 2012;Switzer, 1967Switzer, , 1971Wang, 1981;Xekalaki & Panaretos, 2004).…”

Characterization of Markov-Bernoulli Geometric Distribution Related to Random Sums

“…The resulting model is called the Markov-Bernoulli Model (MBM) or the Markov modulated Bernoulli process (Ozekici, 1997). Many researchers have been studied the MBM from the various aspects of probability, statistics and their applications, in particular the classical problems related to the usual Bernoulli model (Anis and Gharib, 1982;Gharib and Yehia, 1987;Inal, 1987;Yehia and Gharib, 1993;Ozekici, 1997;Ozekici and Soyer, 2003;Arvidsson and Francke, 2007;Omey et al, 2008;Maillart et al, 2008;Pacheco et al, 2009;Cekanavicius and Vellaisamy, 2010;Minkova and Omey, 2011). Further, due to the fact that the MBM operates in a random environment depicted by a Markov chain so that the probability of success at each trial depends on the state of the environment, this model represents an interesting application of stochastic processes and thus used by numerous authors in, stochastic modeling (Switzer, 1967;1969;Pedler, 1980;Satheesh et al, 2002;Özekici and Soyer, 2003;Xekalaki and Panaretos, 2004;Arvidsson and Francke, 2007;Nan et al, 2008;Pacheco et al, 2009;Doubleday and Esunge, 2011;Pires and Diniz, 2012).…”