International Symposium on Stochastic Models in Reliability Engineering, Life Sciences and Operations Management (SMRLO'10)
SEMI-MARKOV RELIABILITY MODEL OF THE COLD STANDBY SYSTEM Franciszek GRABSKI Prof, Department of Mathematics and Physics, Polish Naval University, Gdynia, Poland
E-mail:
[email protected]
Abstract: The semi-Markov reliability model of the cold standby system with renewal is presented in the paper. The model is some modification of the model that was considered by Barlow & Proshan (1965), Brodi & Pogosian (1978). To describe the reliability evolution of the system, we construct a semi-Markov process by defining the states and the renewal kernel of that one. In our model the time to failure of the system is represented by a random variable that denotes the first passage time from the given state to the subset of states. Appropriate theorems from the semi-Markov processes theory allow us to calculate the reliability function and mean time to failure. As calculating an exact reliability function of the system by using Laplace transform is often complicated we apply a theorem which deals with perturbed semiMarkov processes to obtain an approximate reliability function of the system. Key words: semi-Markov process; perturbed process; reliability model; renewal standby system
1. Description and Assumptions We assume that the system consists of one operating series subsystem (unit), an identical stand-by subsystem and a switch (see Figure 1): 1
2
N
1
2
N
Figure 1. Diagram of the system
399
International Symposium on Stochastic Models in Reliability Engineering, Life Sciences and Operations Management (SMRLO'10) Each subsystem consists of N components. We assume that time to failure of those
ζk,
elements are represented by non-negative mutually independent random variables
k = 1,..., N , with distributions given by probability density functions
f k (x ) , x ≥ 0 ,
k = 1,..., N . When the operating subsystem fails, the spare is put in motion by the switch immediately. The failed subsystem is renewed. There is a single repair facility. A renewal time is a random variable having distribution depending on a failed component. We suppose that the lengths of repair periods of units are represented by identical copies of non-negative random
variables
γ k , k = 1,..., N ,
which
have
cumulative
distribution
functions
H k ( x ) = P(γ k ≤ x ) , x ≥ 0 . The failure of the system occurs when the operating subsystem fails and the subsystem that has sooner failed in not still renewed or when the operating subsystem fails and the