Queue length distribution of an unreliable machine

In this paper, we consider a state-dependent queueing system in which the machine is subject to random breakdowns. Jobs arrive at the system randomly following a Poisson process with state-dependent rates. Service times and repair times are exponentially distributed. The machine may fail to operate with probability depending on the number of jobs completed since the last repair. The main result of this paper is the matrix-geometric solution of the steady-state queue length probability from which many performance measurements such as mean queue length, machine utilization are obtained.