A NEW METHOD FOR CALCULATING THE MEAN FAILURE NUMBERS OF A REPAIRABLE SYSTEM DURING(0,t]
Shi Dinghua (Shanghai Institute of Railway Technology)
Let S(t) represent the system state at the time t.P_i(t)=P_t{S(t)=i}is called the probability that the system appears in the state i at the time t.A(t)is called availability of the system at the time t.Let N_f(t) represent the total number of the system failures which have occurred up to time t.M_f(t)=E[N_f(t)] is called the mean failure numbers of the system during (0,t]. W_f(t)=M'_f(t) is called failure frequency of the system.A new method for calculating the mean failure numbers of a repairable system during (0, t] is presented in this paper.Its major idea is to get A(t) and M_f(t) simultaneously by using P_i(t).Theorem 1 Suppose that S(t) is a timehomogeneous Markov chain with continuous time.We obtain where G is the state set of the system in the good condition.F is the state set of the system in the failure case, the matrix [a_(ij)] is an infinitesimal transition matrix of S(t).Theorem 2 Suppose that {S(t), X_1(i),Y_1(t),…} amounts to defining a special high dimensional time homogeneous Markov process where X_1(t),Y_1(t),…denote supplementarv variables.We obtain if two multiple integrals c_(ij)(t)=integral from n=o to t integral from n=o to t…[a_(ij)(x_1,y_1,…)dx_1dy_1…]and v_(ij)(t)=integral from n=o to ∞ integral from n=o to ∞ [P_i(t;x_1,y_1,…)a_(ij)(x_1,y_1,…)dx_1dy_1…]are finite for fixed time t.Also,the matrix [a_(ij)(x_1,y_1,…)] is an infinitesimal transition matrix of {S(t),X_1(t),Y_1(t),…},and P_i(t;x_1,y_1,…) is called the probability density that the system appears in the state(i,x_1, y_1,…) at the time t.


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