On the Reliability of a Renewable Multiple Cold Standby System

Apart from important but particular cases, such as state-dependent exponential distributions, for example, [1, 4] or an arbitrary repair time distribution [12], the reliability analysis of multiple cold standby systems is far from complete. We present a general reliability analysis of a basic multiple cold standby system attended by a single repairman, henceforth called a K-system. The K-system consists of a single operative unit (the on-line unit) endowed with K units in cold standby. The system acts as a closed queueing system evolving in time, that is, any failed unit is immediately replaced by a standby spare (if available upon failure) and taken into repair provided that the repairman is idle. Otherwise, the failed unit has to queue for repair. On the other hand, a repaired unit lines up in standby or becomes instantaneously operative if the K remaining units are down. Our K-system is the statistical dual of the K-system described in [12]. An example of the so-called duplex system (K = 1) is the (internal) cooling device of the International Space Station (ISS) sustained by an (external) cooling device in cold standby. Note that astronauts can provide local repair. OurK-system satisfies the usual conditions, that is, independent identically distributed random variables, perfect repair and queuing. Each operative unit has an arbitrary lifetime (failure-free time) distribution, a zero failure rate in standby (the so-called cold standby [2]), and a constant repair rate [8]. Note that our particular assumptions are less general with regard to the previous assumptions proposed in [1, 4]. But, on the other hand, our present assumptions are less restrictive since we are dealing with an arbitrary lifetime distribution. In order to derive the survival function of the K-system, we first introduce a stochastic process defined on some filtered probability space {Ω,B,P,F}. Moreover, we define