Midterm Formula Sheet: Process Analysis And Queueing Theory

Midterm Formula Sheet: Process Analysis and Queueing Theory

This document provides a comprehensive summary of key formulas and concepts related to process analysis, Little’s Law, queueing theory, and general queueing formulas, as typically used in operations management and systems analysis courses.

Process Analysis and Little’s Law

Little’s Law is foundational in analyzing flow processes and states that the average inventory in a system is equal to the product of the average flow rate and the average flow time:

• Average Inventory = Average Flow Rate × Average Flow Time

Utilization of resources is calculated as the ratio of actual output rate to maximum capacity:

• Utilization = Flow Rate / Capacity = Actual Output Rate / Maximum Output Rate

The load factor, indicating the demand relative to capacity, is:

• Load Factor = Demand / Capacity = Demand Rate / Maximum Output Rate

General Concepts in Process Variability

The coefficient of variation (CV) measures the relative variability of a process or distribution, calculated as:

• Coefficient of Variation = Standard Deviation / Mean

Queueing Theory: Notation & Terminology

Key parameters include:

• Number of Servers: "k"
• Coefficient of Variation for Inter-Arrival Times: φ_ω
• Arrival Rate: λ (= 1 / Inter-Arrival Time)
• Coefficient of Variation for Service Times: ψ_ω
• Service Rate per server: μ (= 1 / Service Time)
• Utilization (ρ): ρ = λ / (k × μ)

Queueing Models and Formulas

M/M/1 Queue (Single Server, Poisson Arrivals, Exponential Service)

Utilization:

• ρ = λ / μ

Average time in queue:

• τ_queue = ρ / (μ - λ)

Average number in system:

• L = λ / (μ - λ)

Probability that system is empty:

• P₀ = 1 - ρ

Average number in queue:

• L_q = ρ² / (1 - ρ)

Average time in system:

• τ = 1 / (μ - λ)

G/G/1 Queue (Single Server, General Arrivals, General Service Times)

Expected waiting time in queue:

• τ_queue = Activity Time ( utilization / (1 - utilization) ) ( 2 + CV_service² + CV_{inter-arrival}² )

G/G/k Queue (Multiple Servers, General Arrivals and Service Times)

Expected waiting time in queue:

• τ_queue = Activity Time ( utilization / (1 - utilization) ) sqrt(2(CV_service²) + CV_{inter-arrival}²)

Queueing System Performance Measures

• Utilization: ρ = λ / (k * μ)
• Inventory in Service: I_s = Activity Time * ρ
• Average Flow Time: τ = τ_queue + 1 / μ
• Inventory in Queue: I_q = λ * τ_queue
• Inventory in System: I_sys = I_s + I_q

Throughput Loss and Erlang Loss Model

In systems with multiple servers and possible capacity loss, the Erlang loss formula estimates the probability of call blocking or loss:

• P_loss = Erlang B formula: ErlangB(traffic, servers)

This is typically looked up in a provided table or calculated via the Erlang B function where:

• traffic = λ / μ
• μ = Service rate
• λ = Arrival rate
• Number of servers: k

Conclusion

This compilation of formulas and concepts provides a framework for analyzing process flow efficiency, variability, queue performance, and system capacity. Understanding these principles aids in designing optimal operations, reducing wait times, and improving productivity across various industries.

References

• Gross, D., Shortle, J. F., Thompson, J. M., & Harris, C. M. (2008). Fundamentals of Queueing Theory (4th ed.). Wiley.
• Hopp, W. J., & Spearman, M. L. (2011). Factory Physics (3rd ed.). Waveland Press.
• Stevenson, H. (2009). Operations Management. McGraw-Hill Education.
• Mitra, S. (2009). Fundamentals of Queueing Theory. Prentice Hall.
• Kleinrock, L. (1975). Queueing Systems Volume 1: Theory. Wiley-Interscience.
• Johnson, R., & Montgomery, D. (2014). Operations Research in Manufacturing and Service Systems. Wiley.
• Kleijnen, J.P.C. (2014). Design and Analysis of Simulation Experiments. Springer.
• Silver, E. A., Pyke, D. F., & Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. Wiley.
• Pinedo, M. (2016). Scheduling: Theory, Algorithms, and Systems. Springer.
• Buzacott, J. A., & Shanthikumar, J. G. (1993). Stochastic Models of Manufacturing Systems. Prentice Hall.