M7- queue length

Arizona State University - W. P. Carey School of Business

  • Module 7: Business Process Management (BPM)

Business Process Management (BPM): Queue Length (Module 7)

Key Concepts

  • The Queue Length formula relies on two main effects:

    • Utilization Effect

    • Variability Effect

Variables and Definitions

  • u = Ri / Rp

    • Ri: Arrival Rate

    • Rp: Processing Rate

  • Rp = c / Tp

    • c: Number of servers

    • Tp: Average processing time

  • Coefficient of Variation (Ci/Cp):

    • A measure of variability, defined as the ratio of standard deviation to the mean:

      • Ci: Coefficient of Variation for interarrival times.

      • Cp: Coefficient of Variation for processing times.

Queue Length Example

Example with One Security Check Person

  • Scenario: 10 passengers, 1 security check person (c = 1)

  • Interarrival Times: 10, 2, 10, 1, 10, 3, 7, 2, 9

    • Sample Standard Deviation: 3.937

    • Mean Interarrival Time (Ti): 54/9 = 6 seconds

    • Arrival Rate (Ri): 1/6 passengers per second

  • Processing Times: 7, 1, 7, 2, 8, 7, 4, 8, 1, 5

    • Sample Standard Deviation: 2.828

    • Mean Processing Time (Tp): 50/10 = 5 seconds

    • Processing Rate (Rp): 1/5 = 0.2 passengers per second

Calculating Queue Length

  • Calculations:

    • Coefficient of Variation (Ci): Std. Dev. / Mean = 3.937 / 6 = 0.656

    • Coefficient of Variation (Cp): Std. Dev. / Mean = 2.828 / 5 = 0.566

    • Utilization (u): Ri / Rp = (1/6) / (1/5) = 0.833

    • Average Number of Passengers Waiting (Ii):

      • Formula: ( Ii = \frac{u^2(c + 1)}{1 - u} \left( C_i^2 + C_p^2 \right) )

      • Result: ( Ii = 0.833^2(2) \cdot 0.375 = 1.559 \text{ passengers waiting in line} )

Queue Length Example (With Added Server)

Scenario with Two Security Check Persons

  • Scenario: 10 passengers, 2 security check persons (c = 2)

  • Interarrival Times: 10, 2, 10, 1, 10, 3, 7, 2, 9

    • Sample Standard Deviation: 3.937

    • Average Interarrival Time (Ti): 6 seconds

    • Arrival Rate (Ri): 1/6 passengers per second

  • Processing Times: 7, 1, 7, 2, 8, 7, 4, 8, 1, 5

    • Sample Standard Deviation: 2.828

    • Average Processing Time (Tp): 5 seconds

    • Processing Rate (Rp): 2/5 = 0.40 passengers per second

Calculating Queue Length for Two Servers

  • Calculations:

    • Coefficient of Variation (Ci): Std. Dev. / Mean = 3.937 / 6 = 0.656

    • Coefficient of Variation (Cp): Std. Dev. / Mean = 2.828 / 5 = 0.566

    • Utilization (u): Ri / Rp = (1/6) / 0.40 = 0.417

    • Average Number of Passengers Waiting (Ii):

      • Formula: ( Ii = \frac{u^2(c + 1)}{1 - u} \left( C_i^2 + C_p^2 \right) )

      • Result: ( Ii = 0.417^2(3) \cdot 0.375 = 0.075 \text{ passengers waiting in line} )

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