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Asynchronous flow line: two stations with a finite buffer

An asynchronous flow line comprising two processing stations and a buffer. Processing times follow an exponential distribution with means 1/my1 and 1/my2. Interarrival times at station 1 are exponential distributed with mean 1/lam. Alternatively, it may be assumed, that station 1 is never starved (infinite arrival rate). The buffer size is limited (sizes between 0 and 2 are possible). Station 2 is never blocked. Production blocking (blocking-after-service) is assumed. Workpieces finding station 1 busy (or blocked) are rejected.

The flow line is modelled as a continuous time markov chain.

First the steady-state balance equations are constructed.

The state of the system is described with the tupel (x,y), where x=state of station 1 and y=state of station 2.

Possible states of station 1:

 0 idle 1 busy

Possible states of station 2:
 0 idle 1 busy 2 busy and one workpiece in buffer 3 busy and twoworkpieces in buffer

The column RS is the right-hand side of the system of balance equations.

After the balance equations have been built up, the system is solved. The probabilities of the states are analysed to find the throughput of the system and the mean work-in-process.

Symbols:

 lam arrival rate at station 1 my1 processing rate (1/mean processing time) at station 1 my2 processing rate (1/mean processing time) at station 2 L work-in-process at a station u(i,j) transition rate between states i and j X throughput of the system RS right-hand side of the system of balance equations