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Queuing Analysis

This lecture dives into the fundamentals of queuing analysis, covering essential factors like queue discipline, server types (single vs. multiple), and calling population (finite vs. infinite). We explore the concepts of arrival and service rates, focusing on Poisson and exponential distributions. Key formulas for single-server models are presented, including the probability of customers in the system, average wait times, and utilization factors. Practical examples illustrate the application of these formulas, enabling better understanding and management of queuing systems.

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Queuing Analysis

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  1. Queuing Analysis Lecture 4 Bus 480

  2. Factors of Queuing Models • Queue Discipline • Order that customers is served • Single server, multiple server • Calling Population • Infinite or finite • Arrival Rate • Frequency in which customers arrive to a waiting line according to a probability distribution • Typically a Poisson distrubution • Service Rate • Average number of customers who can be served during a time period • Defined by a exponential distribution

  3. Single Server Model • Assumptions • Infinite calling population • First-come, first served queue discipline • Poisson arrival rate • Exponential service rate • Arrival rate is less than service rate

  4. Single Server Model Customers enter At rate = λ Customers exit At rate = μ Process customers

  5. Single Server Queue Formulas • P 0 = probability of no customers in the system = 1 – (λ / μ)  • P n = Probability of n-customers in the system = (λ / μ)n * P0 • L = Average number of customers in the system = λ/(μ - λ) • Lq = Average number of customers in the queue or waiting line = λ2 /(μ(μ - λ))

  6. Single Server Queue Formulas • W = Average time customer spends in the whole system = 1/(μ - λ) = L / λ • Wq = Average time spend in the queue = λ/ (μ (μ - λ))  • U = Utilization factor = Probability that the server is busy = λ / μ   • I = Idle factor = probability that the server is idle = 1- U = 1 – (λ / μ) 

  7. Example page 266 #8 • Ticket booth • λ = 10 customers/hour • μ = 12 customers/hour • Find the average time a ticket a ticket buyer must wait, Wq • Wq = λ/ (μ (μ - λ)) = 10/(12*(12-10)) = 10/24 = .41 hours • Find proportion of time the ticket seller is busy, U • U = λ/ μ = 10/12 = .833 = 83.3% busy

  8. Example Page 267 #14 • Adviser approves every 2 minutes • 30 students/hour , μ • Students arrive at a rate of 28/hour , λ • Find L, Lq, W, Wq, and U • L = 28/(30-28) = 14 students on average in the system • Lq = (28)2/(30*(30-28)) = 13.1 students on average in queue • W = 1/(30-28) = .5 hours of waiting (30 minutes) • Wq = 28/(30*(30-28)) = .47 hours waiting in the queue on average = 28.2 minutes • U = 28/30 = .93 probability the student will wait

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