5.9 An assembly line consists of two stations in tandem. Each station can hold only one item at a time. When an item is completed in station 1, it moves into station 2 if the latter is empty; otherwise it remains in station 1 until station 2 is free. Items arrive at station 1 according to a Poisson process with rate λ. However, an arriving item is accepted only if there is no other item in the station; otherwise it is lost from the system. The time that an item spends at station 1 is exponentially distributed with mean 1=μ1; and the time that it spends at station 2 is exponentially distributed with mean 1=μ2. Let the state of the system be defined by ðm; nÞ, where m is the number of items in station 1 and n is the number of items in station 2.

a. Give the state-transition-rate diagram of the process.

b. Calculate the limiting-state probabilities pmn.