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Understanding Queuing Theory: Mathematical Analysis of Waiting Lines

Queuing Theory, as discussed by Brian Murphy, offers insights into waiting lines. Key aspects include arrival rates, service rates, performance measures, and characterizing queues using notation like A/B/C/N/K. Common queues such as M/M/1 are explored, along with formulas for system metrics like number of people and response time.

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Understanding Queuing Theory: Mathematical Analysis of Waiting Lines

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  1. Queuing Theory By: Brian Murphy

  2. Overview • Queuing Theory: the mathematical study and analysis of waiting lines (queues). • Performance Measures Involved: • Arrival Rate = λ (measured in persons per unit of time) • Service Rate = μ (measured in persons per unit of time) • Throughput = 1/ λ (rate of successful services) • Utilization = ρ = λ/ μ (portion of time a server is servicing a customer) • Number of people in the system = L • System Response Time = W (average time customer spends in system)

  3. Characterizing Queues • A/B/C/N/K • A denotes type of arrival process • B denotes type of service process • C denotes number of servers • N denotes system capacity • K denotes customer population • Types of arrival and service processes • M: exponential distribution • G: general distribution • D: deterministic distribution • E: Erlang distribution • H: hyperexponential distribution Generally used because of its “memoryless” property

  4. Common Queue: M/M/1 • M/M/1 Queue: exponential arrival and service process, one server • Formulas Involved: • L = ρ/(1- ρ) • W = (1/μ) /(1- ρ)

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