The k-out-of-n System Model with Degradation Facility

In this paper, we study the reliability analysis of k-out-of-n system with degradation facility. Let failure rate, degradable rate and repair rate of components are assumed to be exponentially distributed. There are two types of repair. The first is due to failed state. The second is due to degraded state. The expressions of reliability and mean time to system failure are derived with repair and without repair. We used several cases to analyze graphically the effect of various system parameters on the reliability system. Citation: El-Damcese MA, Shama MS (2016) The k-out-of-n System Model with Degradation Facility. J Appl Computat Math 5: 326. doi: 10.4172/21689679.1000326


Introduction
The general structure of series and parallel systems: the so-called k-out-of n system. In this type of system, if any combinations of k units out of n independent units work, it guarantees the success of the system. For simplicity, assume that all units are identical. Furthermore, all the units in the system are active. The parallel and series systems are special cases of this system for k=1 and k=n, respectively.
In the past decades; many articles concerning the reliability and availability of standby systems have been published. Among them Galikowsky et al. [1] analyzed the series systems with cold standby components. Wang and Sivazlian [2] examined the reliability characteristics of a multiple-server (M+W) unit system with exponential failure and exponential repair time distributions. Ke et al. [3] studied the reliability measures of a repairable system with warm standby switching failures and reboot delay. Yuge et al. [4] introduced the reliability of a k-out-of-n system with common-cause failures using multivariate exponential distribution. Zhang et al. [5] analyzed the availability and reliability of k-out-of-(M+N): G warm standby systems with two types of failure.
In this paper, we provide a detailed coverage on reliability evaluation of the k-out-of-n systems with degradation. We study the simple model "one unit with one degradable state (simplex)" then we investigate triple modular redundancy (TMR) without repair components. Finally we study the k-out-of-n in details with repair and without repair of components. In addition, we perform numerical results to analyze the effects of the various system parameters on the system reliability.

Simplex system
We get Simplex system when n=k=1 Based on the state transition diagram in Figure 1, we can derive the following differential equations: From Figure 2, taking LaPlace transforms of the state equations yields: Solving equations (14-21and taking inverse Laplace transforms of these equations, we get the reliability function of the system As we know the reliability of the triple modular redundancy of a component with one failure rate can be obtained from this equation 2 3 ( ) 3 2 Where R=e -λt reliability of a component with failure rate λ, from (22) we get 2 3 ( ) 3( The MTTF tmr can be obtained from this equation

General system
We will examine a general model for analysis of such systems when they are nonrepairable. When a k-out-of-n system is put into operation, all n components are in good condition. The system is failed when the number of working components goes down below k or the number of Initial conditions: Taking the Laplace transform of equations (1-3) and applying initial conditions, we have  Inverse Laplace transforms of these equations yield The reliability function of the system can be written as The MTTF can be obtained from this equation

Triple modular redundancy: TMR
In this scheme, three identical redundant units or modules perform the same task simultaneously with degradable rate. The TMR system only experiences a failure when more than one component fails. In other words, this type of redundancy can tolerate failure of a single component. Figure 2 shows a diagram of the TMR scheme.  failed components has reached n−k+1. We consider the components in a k-out-of-n system are i.i.d. Let the failure rate and degradable rate occur independently of the states of other units and follow exponential distributions with λ 1 ,λ 2 , respectively.
At time t=0 the system starts operation with no failed units. The Laplace transforms of P i,j (t) are defined by: Based on the model descriptions, the system state transition diagram is given in Figure 3 and it leads to the following Laplace transform expressions for P i,j * (s): From solving equations (26-34) and taking inverse Laplace, we obtain the reliability function as follows: The mean time to failure MTTF N (k,n) can be obtained from the following relation.
When we perform a sensitivity analysis for changes in the RN(k,n) resulting from changes in system parameters λ 1 and λ 2 and. By differentiating equation (36) with respect to λ 1 we obtain, We use the same procedure to get We use two cases to study the effect of k and n on system reliability Case 1: Fix λ 1 =0.001, λ 2 =0.008, n=3 and choose k=1,2,3.
From Figures 4 and 5 we can be observed that the system reliability increases as k increases or n increases.
Then, we perform a sensitivity analysis with respect to λ 1 and λ 2 . In Figure 6 we can easily observe that the biggest impact almost happened at different time and the order of magnitude of the effect is (λ 1 >λ 1 ). rate that occur independently of the states of other units and follow exponential distributions with λ 1 and λ 2 respectively in addition, repair rates of failure and degradation are assumed to be exponentially distributed with parameters µ 1 and µ 2 respectively.
The system starts at time t=0 and there is no failed or degradable components. When a unit failed it is immediately sent to the first service line where it is repaired with time-to-repair which is exponentially distributed with parameter µ 1 .we have two service lines the first one repair failed units and the second one repair degraded units. When an operating unit degraded it is it is repaired with timeto-repair which is exponentially distributed with parameter µ 2 during it working. We assume that the secession of failure times and repair times are independently distributed random variable. Let us assume that failed units arriving at the repairmen form a single waiting line and are repaired in the order of their breakdowns; i.e. according to the first-come, first-served discipline. Suppose that the repairmen in the two service lines can repair only one failed unit at a time and the repair is independent of the failure of the units. Once a unit is repaired, it is as good as new.
The mean repair rate µ 1,j is given by: The mean repair rate µ 2,j is given by:

Repairable k-out-of-n system
In this section, we will develop a general model for analysis of such systems when they are repairable. Let failure rate and degradable     By solving equations (39-49) and taking inverse Laplace transforms (using maple program). We obtain the reliability function as follows: Where i,j=0,1,2,…,n -k.
The mean time to failure MTTFR(k,n) can be obtained from the following relation.
We perform a sensitivity analysis for changes in the reliability of the system R R (k,n) from changes in system parameters λ 1 , λ 2 , µ 1 and µ 2 . by differentiating equation (50) with respect to λ 1 we obtain We use the same procedure to get

Numerical results
In this section, we use MAPLE computer program to provide the numerical results of the effects of various parameters on system reliability and system availability. We choose λ 1 =0.001, λ 2 =0.008 and fix µ 1= 0.1, µ 2 =0.8. The following cases are analyzed graphically to study the effect of various parameters on system reliability.
From Figure 8 we find that R 1 don't effect on system reliability when number of repairman more than one. Figure 9 shows that the repairable system reliability increases as R 2 increases.
Finally we perform sensitivity analysis for system reliability R R (k,n) with respect to system parameters.

1-With respect to all system parameters
From Figures 10 and 11 we can easily observe that the biggest impact almost happened at the same time for λ 2 , µ 1 and µ 2 but it's happen at shorter time for λ 1 . Moreover, we find λ 1 is the most     prominent parameter whileλ 2 , µ 1 and µ 2 are the second, the third and the fourth respectively in magnitude. Figure 12 shows that when λ 1 decreases its impact on reliability R Y (t) happened at longer interval time, and the biggest impact almost happened at longer time.

3-With respect to λ 1 at various values of λ 2
From Figure 13 we can also observe that as λ 2 decreases the impact of λ 1 on reliability R Y (t) happened at longer interval time, and the biggest impact almost happened at longer time, we can observe that the biggest impact of λ 1 on reliability R Y (t) is not affected by the value of λ 2 but it's happen at longer time as λ 2 decreases.

Conclusions
In this paper, we studied reliability and mean time to system failure for k-out-of-n models with degradation facility. Mathematical model were constructed for these models. Results indicate that the reliability of k-out-of-n nonrepairable system with degradation increases as k or n increases. In repairable system we observe that the order of magnitude of the effect is (λ 1 >λ 1 >µ 1 >µ 2 ).