# POM Prof. Tempelmeier GmbH

## Production and Operations Management Consulting.

## Asynchronous flow line: Two stations with a finite buffer

An asynchronous flow line comprising two processing stations and a buffer. Processing times follow exponential distributions 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.

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 |

0 | idle |

1 | busy |

2 | busy and one workpiece in buffer |

3 | busy and twoworkpieces in buffer |

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 |

- Papadopoulos/Heavey/Browne
(1993), p. 55-56

- Viswanadham/Narahari (1992), p. 423-425