A Closed Stochastic Reliability System with Variable Resources

Abstract

We consider a single-server closed queueing system with a finite number of permanent customers. This system can also be described in terms of servicing unreliable machines by a single repairman. In addition we assume that not every machine is vulnerable. Of the total quantity, there are some that are inactive and subject to cold standby. They are not on reserve though, and this situation alters randomly, i.e. at random epochs of time, each machine, which is intact, can change its status from being active (or vulnerable) to inactive (or invulnerable). Active machines, in turn, can fail, and in this case, they form a line of defective machines, which are processed one by one by a single repairman. If all machines are intact, the repairman is idle until a next breakdown. We will use a dual multi-channel open queueing model with a variable number of active channels to investigate the processes of intact and defective machines in the closed model and develop a duality principle that enables us to simplify the original system and arrive at a closed form solution. We also use semi-regenerative techniques and demonstrate applicability on a number of real world situations that include major epidemics (such as TB), human resources, and a calibration system.