A system consists of two units, both of which may operate simultaneously. The pr

Question

A system consists of two units, both of which may operate simultaneously. The probability that an operating unit will fail in a short time interval of length h is h + o(h). The failures of the units are of two kinds – regular and severe. The fraction of regular failures is p and the fraction of severe failures is 1 p. The system has a single repair facility that processes units on a first-come-first-served basis. Repair times are exponentially distributed with parameter for regular failures and for severe failures, with < .
Model the system as a continuous time Markov process by defining the states as pairs (x, y), where x is the number of units operating and y is the number of failed units. Specify the infinitesimal generator Q.

Explanation / Answer

Ans: The state space will be S={ 0,(1,R),(1,S),(2,R),(2,S)

Rate Matrix Q=

0 1R 1S 2R 2S 0 -µ pµ (1-p)µ 0 0 1R p -(µ+p) 0 pµ (1-p)µ 1S (1-p) 0 (1-p) pµ (1-p)µ 2R 0 p p -2p 0 2S 0 (1-p) (1-p) 0 -2(1-p)

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