Key Formula

Key Equations

(9-1)  | |

  | Where: | | |

  |   | X | = number of arrivals per unit of time (e.g., hour) |

  |   | P(X) | = probability of exactly X arrivals |

  |   | λ | = average arrival rate (i.e., average number of arrivals per unit of time) |

  |   | e | = 2.7183 (known as the exponential constant) |

  |   | | |

  | • Poisson probability distribution used in describing arrivals. |

(9-2)  | P (t) = e-μt |   | for t ≥ 0 |

  | Where: |   |   |

  |   | t | = service time |

  |   | P(t) | = probability that service time will be greater than t |

  |   | μ | = average service rate (i.e., average number of customers served per unit of time) |

  |   | e | = 2.7183 (exponential constant) |

  |   | | |

  | • Exponential probability distribution used in describing service times. |

Equations 9-3 through 9-9 describe the operating characteristics in a single-server queuing system that has Poisson arrivals and exponential service times.

λ = average number of arrivals per time period (e.g., per hour)

μ = average number of people or items served per time period

(9-3)  | ρ = λ / μ |

  |   |

  | • Average server utilization in the system. |

(9-4)  | |

  |   |

  | • Average number of customers or units waiting in line for service. |

(9-5)  | L = Lq + λ / μ |

  |   |

  | • Average number of customers or units in the system. |

(9-6)  | |

  |   |

  | • Average time a customer or unit spends waiting in line for service. |

(9-7)  | W = Wq + 1/ μ |

  |   |

  | • Average time a customer or unit spends in the system. |

(9-8)  | P0 = 1 – λ / μ |

  |   |

  | • Probability that there are zero customers or units in the system. |

(9-9)  | Pn = (λ / μ)n P0 |

  |   |

  | • Probability that there are n customers or units in the system. |

(9-10)  | Total cost = Cw × L + Cs × s |

  | Where: |   |   |

  |   | Cw | = customer waiting cost per unit time period |

  |   | L | = average number of...