A random shock model for a continuously deteriorating system
Purpose - A random shock model for a system whose state deteriorates continuously is introduced and stochastically analyzed. It is assumed in the model that the state of the system follows a Brownian motion with negative drift and is also subject to random shocks. A repairman arrives at the system according to a Poisson process and repairs the system if the state has been below a threshold since the last repair. Design/methodology/approach - Kolmogorov's forward differential equation is adopted together with a renewal argument to analyze the model stochastically. The renewal reward theorem is used to obtain the long-run average cost per unit time. Findings - An explicit expression is deduced for the stationary distribution of the state of the system. After assigning several costs to the system, an optimization is also studied as an example. Practical implications - The present model can be used to manage a complex system whose state deteriorates both continuously and jumpwise due to the continuous wear and random shocks, such as a machine and a production line in a factory. The model can also be applied to an inventory which supplies the stock both continuously and jumpwise, such as a gas station and the distribution center for a franchise, if the continuous wear and random shocks are considered as demands for the stock. Originality/value - The present model is quite complicate, however, more realistic than the previous models where the state of the system is subject to either one of continuous wear and random shocks.