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Queuing Theory: Mathematical Analysis of Waiting Lines

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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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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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