Journal of Applied Mathematics and Simulation
This paper examines a doubly controlled process of servicing machines. The classical system treated by Takács is equipped with m+1 unreliable machines served by one repairman. In the present modification of this model, the failure rates and the repair time may be controlled with respect to the state of the system. The process describing the number of intact machines is considered. To derive its steady state distribution in the form of a simple explicit formula, the author introduces an auxiliary model with m unreliable machines and a single repairman who keeps working even when all machines are intact. This result is based on a duality principle applied to the process above.
Dshalalow, J. (1988). On a duality principle in processes of servicing machines with double control. Journal of Applied Mathematics and Simulation, 1(3), 245.