Stochastic Behavior of a Two-Unit Parallel System with Dissimilar Units and Op- tional Vacations under Poisson Shocks

This article examines the impact of some system parameters on an industrial system composed of two dissimilar parallel units with one repairman. The active unit may fail due to essential factors like aging or deteriorating, or exterior phenomena such as Poisson shocks that occur at various time periods. Whenever the value of a shock is larger than the specified threshold of the active unit, the active unit will fail. The article assumes that the repairman has the right to take any of two decisions at the beginning of the system operation: either a takes a vacation if the two units work in a normal way, or stay in the system to monitor the system until the first system failure. In case of having a failure in any of the two units during the absence of the repairman, the failing unit will have to wait until the repairman is called back to work. We suppose that the value of every shock is assumed to be i.i.d. with some known distribution. The length of the repairman’s vacation, repair time, and recall time are arbitrary distributions. Various reliability measures have been calculated by the supplementary variable technique and the Markov’s vector process theory. At last, numerical computation and graphical analysis have been given for a particular case to validate the derived indices.


Introduction
The economic progress of any country depends to a great extent on developing the industrial and mechanical fields. This progress leads to the appearance of new technicalities that help solving problems that may occur in these complex industrial systems. But in spite of all that help, consumers hope to deal with low cost and high efficiency industrial systems. Therefore, system designers and researchers in this field face a great challenge to develop many systems and raise their efficiency, reliability, and safety.
Many researches have been made in the past to explain the concept of reliability and to raise the efficiency of many industrial systems, and to analyze the cost of the different redundant systems under the effect of some restrictions such as the periods of which the system is on or off, the different types of failures affecting that system, the different types of repairing these units, whether or not it's better for the repair man to take single or multiple vacations…etc.
In this context, we present some of the previous research work that deserves mention. The concept of vacation was first presented to the model analysis of the queuing system in article [1] as 2-standard vacation policies. These policies were defined as multiple vacations and single vacation. Some thorough and excellent studies from the modern results for a variety of vacation models, inclusive of some applications were presented by [2]. Many researchers, though, only studied different vacation models from a queuing theory viewpoint. Therefore, in article [3] the concept of vacation was introduced by the reliability theory viewpoint, and an n-component series system with multiple vacations of repairman was discussed. Since then, the researchers were interested in

Description of the System and Assumptions
The system which consists of two dissimilar units and a single repairman under Poisson shock is subjected to the following assumptions: A1: At the initial time, both units are working with high efficiency, and a repairman has the choice to either stay in the system or takes a vacation. A2: The system is exposed to shocks continually. The arrival of shocks is considered as a Poisson process { ( ), ≥ 0} with the strength > 0. The value of every shock is , i.i.d random variable with distribution function . A3: When a shock occurs and the value of this shock overrides a threshold, the active unit will break down. The threshold of units (A and B) is a non-negative random variable with a distribution function . A4: When any unit fails with the existence of the repairman in the system, it will be repaired immediately. Once the repairman is done repairing the failed units, the repairman has the choice to either stay at the system or take a vacation, then he returns from vacation if at least one unit is failing. The repair rule is ''first-in-first-out''. If a unit fails while the other is being repaired, the recently failed unit must wait for repair, and the system has to stop working. To expedite the system operation when the repairman in a vacation and the system is a breakdown, the system requires repairing one of two units A or B, with probability p or q respectively. A5: Shocks are the main reason for units to fail, and the system fails only if both the units fail.
Based on the preceding assumptions, we can conclude that the conditional failure probability of unit A and unit B are random variables ( ) ( = 1,2). The probability distributions are: . The possibility of a single shock causes unit (A or B) to fail. According to the above assumptions , , we get

System Analysis
The states of this system Ω( ) at time t as a following: S0: at any time t, unit A is active, unit B is active, and the repairman has the choice to either stay at the system or takes a vacation. S1: at any time t, unit A is active unit B is being repaired, and the repairman still in the system, the system is working.
S2: at any time t unit A is being repaired, unit B is active, and the repairman still in the system, the system is working.
S3: at any time t, unit A is active, unit B is active, and the repairman chooses to take a vacation, the system is working.
S4: at any time t, unit A is still repaired from above S2, unit B is waiting for being repaired, and the system is down.
S5: at any time t, unit A is active, unit B is waiting for repair, the repairman in a vacation, and the system is working.
S6: at any time t, unit A is waiting for repair, unit B is active, and the repairman in a vacation, the system is working. S7: at any time t, unit A is still waiting for repair from S6, unit B is waiting for repair and the repairman in a vacation either go to state 4 with probability p or going to state 8 with probability q and the system is down. ( ) = ( ( ) = ), ( = 0,1,2,4,5,6,7,8). From the above, we can formulation the differential equations that represent this system by using the probability arguments and limiting transitions as following.
When Δ tend to zero, we get The same style, we get the following: The boundary conditions are: The initial conditions are: Taking the limit → ∞ in the equations (1) - (17), the following equations are obtained:

Reliability Characteristics
According to the results derived from the analysis of the system in the previous section, the reliability index of the system is obtained as follows: 4.1. Steady-state availability is

Mean Time to the First Failure (MTTFF)
In this section, we deduce the mean time to the first failure (MTTFF) of the system.
We assumed that t be the time to the first failure of the system, therefore the reliability function of this system is calculated as follows ( ) = ( > ). To obtain the reliability function, we consider the failure states {4, 7, 8} of the system are absorbing states.
Let: In the same manner as previously mentioned in Section 4, we conclude reliability function as following: The boundary conditions are: The initial conditions are: Taking the Laplace transform of the equations (39-49), as well as initial conditions, we have:

Special Case
In this section, we present the following special cases, which confirm the results of the previous sections.
Case 1: = , ( ) = 1, then it means any shock will cause the active units to fail and the repairman is in the system. Corresponding results can easily get for the previous particular cases.

Numerical Illustration
This section shows the usefulness of the proposed system by examining the impact of the repairman and other parameters on the system through the following numerical illustrations taking into consideration that: The steady-state availability and the mean time to system failure are examined, when and change, as shown in Tables (1-6). We vary the values of and and note their cross-impact on the steady-state availability and the mean time to system failure. It shows that increasing can greatly decrease the steady-state availability and the mean time to system failure; however, increasing seldom affects the values of steady-state availability and the mean time to system failure.

Conclusion
In this article, we deduced the reliability measurements of a system consisting of two dissimilar parallel units and a single repairman. The repairman might take a vacation or not at the beginning of the system operation and the active units might be attacked from successive shocks. Such a system can be considered as an evolution of a general repairable Industrial system and is also difficult to theoretically analyze the existence of many random variables with general distributions. The numerical illustration explains the relationship between the derived reliability measurements and system parameters.