Performance modeling of state dependent system with mixed standbys and two modes of failure

The present investigation deals with a multicomponent repairable system with state dependent rates. For smooth functioning of the system, mixed standbys (warm and cold) are provided so that the failed units are immediately replaced by standbys if available. To prevent congestion in the system due to failure of units, permanent along with additional repairmen are provided to restore the failed units. It is assumed that the units may fail in two modes. The units have exponential life time and repair time distributions. The failed unit may balk in case of heavy load of failed units. The failed units may also wait in the queue and renege on finding the repairmen busy according to a pre-specified rule. The Chapman–Kolmogorov equations, governing the model in the form of matrix are constructed using transition flow rates of different states. The steady state solution of queue size distribution is derived using product formula. A cost function is suggested to determine the optimal number of warm and cold standbys units required for the desired level of quality of service. The numerical illustrations are carried out to explore the effect of different parameters on performance measures.