- The average number of people waiting in the bank (i.e. the queue length)
- How much of the cashiers' time is spent idle (as a percentage of their time)

- The distribution of the length of time it takes the cashiers to carry out a task is exponential, with mean 2 minutes and standard deviation of 5/4 minutes
- There is virtually no limit on the queue length, as the bank has a large floor area
- Customers arrive in a Poisson distribution, with mean of 25 per hour
- Service is done on a first come first serve basis
- This is an M/M/c queue, where 1 =< c =< 5

Symbol |
Meaning, and value if in assumptions |

c | Number of cashiers (servers) |

Lq | Average length of the queue |

W | Percentage of cashiers idle (waiting) time |

Mean arrival rate (25 per hour) | |

1/(mean time to server customers) = 30 per hour | |

p | Average amount of work for each server, per hour |

p0 and Lq, W and the idle time can be calculated using the following equations:

The results of these equations for 1 to 5 cashiers are summarised in the following table:

Cashiers | p | r^n / n! | p0 | Lq | W | Idle |

1 | 0.833 | 0.833 | 0.167 | 4.167 | 0.2 | 0.167 |

2 | 0.417 | 0.347 | 0.412 | 0.175 | 0.04 | 0.583 |

3 | 0.278 | 0.096 | 0.432 | 0.022 | 0.034 | 0.722 |

4 | 0.208 | 0.02 | 0.434 | 0.003 | 0.033 | 0.792 |

5 | 0.167 | 0.003 | 0.435 | 0.000 | 0.033 | 0.833 |

Henry
Morgan,
14 June 1999