ELEMENTARY METHODS FOR FAILURE DUE TO A SEQUENCE OF MARKOVIAN EVENTS

This paper is concerned with elementary methods for evaluating the distribution of the time to system failure, following a particular sequence of 
events from a Markov chain. After discussing a simple example in which 
a specific sequence from a two-state Markov chain leads to failure, the 
method is generalized to a sequence from a 2} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"> ( k > 2 ) -state chain. The expectation and variance of the time T to failure can be obtained from the 
probability generating function (p.g.f.) of T . The method can be extended 
to the case of continuous time.


Introduction
In many processes arising in reliability theory, a system may at any time t- 0,1,2,..., be in one of k_> 2 states A1,A2,...,Ak, the sequence of which forms a Markov chain.While this chain can be simple or of order r >_ 2, we restrict ourselves here to simple Markov chains; any chain of higher order can be reduced to a simple chain by redefining the states.
A system usually fails after it has passed through a particular sequence All, Ai2 ,.. ., Aim of m states (m _> k, or < k) at the times T-m + 1, T-m + 2, ...,T, with possible repetitions of states, before failure at T >_ m.We wish to study the distribution of the failure time T.
In order to formulate the problem clearly and solve it, we rely on the approaches of Blom and Thorburn [1], Fu and Koutras [3], and Gani [4].In a sense, the methods used are an extension of the theory of runs first studied by Mood [6] and Feller [2, Chapter 2].These methods have been considered by Gmbas and Odlyzko [5], and are applied here to Markov chains rather than independent trials.While our results can- not claim great originality, they have the virtue of being elementary, thus making them readily accessible to engineers and operations researchers.

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J. GANI 2. The Case of the 2-State Chain When k-2, we may for simplicity label the states A1,A 2 as 0,1, forming the simple Markov chain with transition probability matrix Let us assume that failure occurs when the sequence 010 arises for the first time.
We now form an augmented transition probability matrix for the states O, 1, 01,010,   this last being an absorbing state: (2.2) If we denote the initial probability vector by p'-[P0Pl 0 ] , we can readily see that the failure time T010 will have the probability distribution P{Tol o n} p,pn-2Q, n >_ 2, (2.3) where in fact this probability is 0 for n < 3.

n--2
This can be derived explicitly as Elementary Methods for Failure Due to a Sequence of Markovian Events 313 One can easily find the expectation and variance of TOlo from (2.5) as E(Tolo) f)lO(1), V(Tolo) flo(1) + f)lO(1) (f)lO(1)) 2.
Such calculations can be tedious if the full notation is retained, but become simpler if specific values are used for the probabilities, as in the following example.
3. An m-Sequence of States for k > 2 Let us now suppose that the sequence of states leading to failure is AilAi2...Aim where these m states (mk k, or < k) are selected from among the k > 2 states J. GANI A2,...,A k.These form a Markov chain with transition probability matrix and initial probabilities Pi-P{Xo-i}, i-1,...,k.