Production systems operated in practice consist of repairable components and such systems may mostly be modeled as a direct-parallel system. Thus, analysis of the reliability characteristics of a direct-cold standby system is important in the reliability analysis of the system.
Most of the components for a repairable system are not the same after repair as new ones and component’s lifetime decreases with the increase in the time of use. What is more, the time for repair increases more and more once they stop working, which finally leads to complete failure.
On the assumption that the lifetime and repair time of components follow a geometric process, Kim Man Su, a lecturer at the Faculty of Applied Mathematics, has studied the reliability of several dual series direct-parallel systems in consideration of such decrepitude.
The figure above shows a state transition graph of a system, where a circle stands for in operation and a square means break down.
It is assumed that the system consists of components 1, 2 and 3, and components 2 and 3 make up a dual cold standby system, which is then connected with component 1 directly. Here, component 1 gets priority in repair. There is a repairman. Components 2 and 3 are repaired in order of failure.
The system is a unidirectional closed dual series cold standby system. It means if component 1 fails, the system breaks down.
At the beginning, three components are all new, and components 1 and 2 are in operation and component 3 is under cold standby. When the three components in the system are in good condition, two are in operation and one is under cold standby. The repairman sets to work once one of them fails. At the same time, the standby one begins to work. When the failed one has been repaired, it is put on cold standby until the next failure. If one fails while the other is still under repair, it must wait for repair and the system breaks down. It is assumed that each component after repair is not ‘as good as a new one’.
A deteriorative repairable system will not support constant repair of its components. For a deteriorative repairable system, it seems more reasonable to assume that the successive working time of the system after repair will become shorter and shorter while the consecutive repair time of the system after failure will get longer and longer. Ultimately, it cannot work any longer, nor can it be repaired.
In a dual series cold standby system, after N cycle of component 2, when it finishes working, a plan to replace it with a new one (the same as the first dual series cold standby system) is called N replacement policy.
If the lifetime and repair time of components follow a geometric process in a unidirectional closed dual series cold standby system, the solution of state probability (density) is difficult to present analytically, so replacement cycle N needs to be defined and the characteristics of the system be analyzed.
In order to determine an effective replacement cycle, he studied the stationary conditions of the unidirectional closed system using the transitive property of the Markov chain assuming every state in all possible cases in the system.
The results showed that the proposed mathematical method could be effectively used in different pieces of research of other kinds of queuing models.
© 2021 KumChaek University of Technology