Computing the MFPT Matrix

In the following subsections, we will present some Markov chain properties required to hold in order to compute the MFPT values, provide formulas to compute the MFPT matrix of a Markov chain and discuss how detectability can be modeled for improving supply chain risk management. 

Consider a stochastic process X, n = 0,1, 2,. .. that takes a finite or countable number of possible values. Let X = i indicate that the process is at state i at time (or stage) n. This stochastic process is called a Markov chain if the probability of being in a cerfain state at a stage depends only on the state at the immediately previous stage. That is, Pr[X +i = 4+il = iw i = 4 i/ / 

Two states accessible from each other in a Markov chain are said to communicate. A class of stafes is a group of sfafes fhaf communicafe wifh each other. If a Markov chain has only one class, fhen if is said to be irreducible. The period in a Markov chain is fhe minimum number of fransifions required to return to a state upon leaving it. A Markov chain of period one is called aperiodic. 

Let P = (pil) be the transition probability matrix of a Markov chain, where Pij is the probability of moving to a state j in the next stage while the process is in state i at the current stage that is, p = Pr[X,. i = X, = /]. A Markov chain is said to have steady state probabilities if the transition probability matrix converges to a constant matrix. Note that the term steady state probability is used here in a rather loose sense since only aperiodic recurrent Markov chains admit this property. 

Every Markov chain with a finite state set has a unique stationary distribution. In addition, if the Markov chain is aperiodic, then it admits steady state probabilities. Given the transition probability matrix, steady state probabilities of a Markov chain can be computed using the methods detailed in Kulkami (1995). 

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Once n is calculated, we can compute the MFPT matrix M as described in Kemeny and Snell (1976). Several auxiliary matrices have to be computed to reach M. Let Z = (1 - P + ell ) be the fundamental matrix, where I is the identity matrix, P the transition probability matrix, e a vector of all ones and n fhe vector of steady state probabilities. Let also be the matrix that has the same elements as Z in the diagonal and zeros elsewhere and... 

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