A system with two subsystems working in alternating periods

Abstract

We come across in daily life systems consisting of two subsystems which work alternatively. One of the subsystems works for a period which has a probability distribution. Then the other subsystem begins to work for a further period, having a different probability distribution. The system after this period swings back to the previous subsystem and the process goes on. We have the striking example of a power system which has a hydroelectric power subsystem and a thermal electric power subsystem. The hydro-subsystem works for a period when water is let out for irrigation. Then, when the irrigation is stopped, the thermal subsystem works. Again, when irrigation shutters are open, the hydro-subsystem resumes working. In the problem under consideration the system consists of two independent subsystems desigfiated as subsystem 1 and subsystem 2. The subsystem 1 has two i.i.d units, one of which is a cold standby. This subsystem works for a period which has a general probability distribution. In the subsequent period of working the subsystem 2 (consisting of units in series) works for a period which has another general probability distribution. The system again alternates to the 2-unit standby subsystem. In this way the process continues. Thus, we have two alternating periods: working period 1 and working period 2. During the working period 1, the subsystem 1 works and during the working period 2, the subsystem 2 works. Working period I and working period 2 alternate. In model I we have assumed that if in a period a subsystem fails, the other is not switched on. In model II, if the subsystem 1 (2-unit standby system) fails the subsystem 2 (units in series) is immediately switched on. On the other hand, if the subsystem 2 fails during its operation period, the subsystem 1 is not switched on, and the whole system is considered as failed. In both the models, the i.i.d units and the subsystem 2 have exponential failure time distributions. There is a repair facility, the repair (hazard) rate being also a constant. The reliability of the system and the mean time to the first system failure in each of the models are derived. For model I, Markov renewal technique is applied and for model II, supplementary variable technique. In model I, the two subsystems can be considered to be similar to a single system which has a working period as well as a rest period, which has been discussed by the authors Eli. The working period is the period when the subsystem 1 of the original system works and the rest period coincides with the working period of subsystem 2.