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Queuing Analysis

Queuing Analysis. Lecture 4 Bus 480. 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

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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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