General information 
Description 
The most comprehensive prediction model of recurrent event process is the g–renewal process, which was proposed by Kijima [1]. It allows, for example, modelling both perfect and imperfect repairs through the use of the so–called restoration factor q, which defines virtual age A_{i} corresponding to ith event. Two models are suggested. According to Kijima model 1 
where t_{i} is time to ith failure, A_{0}=0. For model 2 we have 
If q=0, the repair is perfect. If q>0, we have imperfect repair including case q=1
when the system is restored to the “sameasold” condition.
Our calculations are based on the advanced Monte Carlo method. It significantly improves raw simulation suggested in [2], especially in case of Frequency of Failure and Unavailability calculation. We also introduced repair time in the calculation and implemented it for the following underlying distribution functions: Weibull, Normal, LogNormal. The detailed description of the method with examples demonstrating accuracy and efficiency of the method you can find in [3]. Here we provide a brief description of the main idea. Consider the g–renewal stochastic process represented in the Figure 1. Here t_{i} and τ_{i} are time to failure and time to repair, respectively, corresponding to the i–th event of the g–renewal process. We consider the conditional probability of failure at the time S_{i1}+t_{i} assuming the component is not failed at current time S_{i1} , but is aged to virtual age A_{i1} : 
For each simulation trial, time to i–th failure is obtained as: 
while current time S_{i1}≤t (see Figure 1). For each time T_{k} from time interval [0, t] we calculate number of failures n_{k} that have occurred before T_{k} . Then simulation trials are repeated N times. The expected number of failures at time T_{k} is then 
Figure 1. Stochastic simulation of the g–renewal process. 
This is the standard Monte Carlo simulation approach for the g–renewal process suggested in [2]. In general, the Monte Carlo method is quite flexible allowing for various underlying failure time distributions and repair restoration models of the g–renewal process. However, the method can be time consuming, if many iterations with many trials are needed. We proposed a significant improvement of the method by considering the g–renewal process as a state space representation (see Figure 2). It is defined as a set of states of a component between the i–th and the (i+1)–th failures. For the sake of simplicity, we do not show restoration states of the component, but they can be easily taken into account as well. 
Figure 2. Renewal process as a state–space representation. 
If P_{i}(t) is denoted as the probability of being at i–th state, the expected number of failures of the component is then 
On the other hand 
where P_{i,i+1}(t) is the probability of transition from state i to state i+1 at given time t under the condition that the unit is in the i–th state. Together with (3) it yields 
But for the first transition P_{0,1}(t)=F(t) , therefore we have 
where according to (1) 
Failure intensity w(t) is also easy to calculate taking derivative of (6) and (7) with respect to time t. Combining the result we obtain 
wheref(t) is the corresponding probability density function.
The implementation of this improved Monte Carlo method is based on equations (6) and (7). For each trial and given time point T_{k} (Figure 1), we consider all failures happened before time T_{k} such that S_{i}≤T_{k}. Then, we calculate the sum of corresponding probabilities of the transition to the next failure at given T_{k} using (6) and (7). First term in (6) is the exact one (its variance equals zero). It is an asymptotic approximation for the small values of W(t) . The standard simulation yields the largest error exactly in this range. If expected time to repair T_{R} is much less than time to failure, the Unavailability of a component is Q(t)=w(t)T_{R} . In general case: 
where G(t) is the cumulative distribution function of repair times.
As a result of our calculations you will see 2 cases (curves) on the diagram. The first one corresponds to the equation Q(t)=w(t)T_{R} (Unavailability is proportional to Failure Frequency), which can be interpreted as repair with given time or with small deviation from mean time to repair (blue curve). The second case corresponds to exponential distribution of repair times (red curve).
1. M. Kijima and N. Sumita, "A useful generalization of renewal theory: counting process governed by nonnegative Markovian increments,"
Journal of Applied Probability, No. 23, pp. 7188, 1986.M.P.
