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DADSS: Queuing Models

DADSS: Queuing Models . April 2, 2013. Administrative. Midterm 2 is coming up Monday; old exams posted. Queuing: Chapter 13 in the textbook (4 th ed ). Basic Elements of Queues. Arrival times Interarrival times: the times between successive arrivals. Estimate the distribution

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DADSS: Queuing Models

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  1. DADSS: Queuing Models April 2, 2013

  2. Administrative • Midterm 2 is coming up • Monday; old exams posted. • Queuing: Chapter 13 in the textbook (4thed)

  3. Basic Elements of Queues • Arrival times • Interarrival times: the times between successive arrivals. • Estimate the distribution • If you can’t do (1), then make a reasonable assumption. • Not all customers come in one at a time: groups of friends, carloads, elevators, etc. • We’ll assume one at a time rather than clustered.

  4. Basic Elements of Queues • Interarrival times • Assume interarrival times are distributed exponentially (with parameter ): • CDF: • Inverse CDF: • We call the arrival rate • http://www.wolframalpha.com/input/?i=exponential+distribution

  5. Basic Elements of Queues • Service: once customers come in they have to wait until served. • Will use a first come, first served FCFS (FIFO) procedure • Other procedures are sometimes used (Shortest Processing Time, priority-based)

  6. M/M/1 Model • M/M/1 • 1st M = distribution of interarrival times is exponential • 2nd M = distribution of service times is exponential • 1 = 1 server. • Let 𝜆 be the arrival rateand let µ be the service rate.

  7. M/M/1 Queue • Average interarrival time: 1/𝜆 • Average service time: 1/µ • Utilization: ρ = 𝜆/µ • # in system: L = 𝜆/(µ- 𝜆) • # in Queue: Lq = ρL • Expected time in system: W = 1/(µ-𝜆) • Expected time in Queue: ρW • Prob(time in queue> t) = ρ e^(-µ(1- ρ )t)

  8. M/M/1 Queue • See documents on blackboard for more analytic facts about the MM1 model. • Excel example/template

  9. Queuing Problem • Suppose a new full-body-scan security gate system is proposed for the airport, and airport administrators are trying to figure out how long people could expect to wait in line. • Build a very simple model of the set-up. Assume that passengers arrive on average at a rate of 120 per hour and that the security check (assume only one) can screen on average 150 passengers per hour. • Assume that both the arrival process and the service process are Poisson. (See page 783 in the 4nd edition of the book) You will have to use the exponential distribution to simulate the interarrival and service times. • Using simulation, answer the following questions. • What is the average amount of time that a person will wait in line? • What percentage of the people will have to wait? • How many feet should the airport administrators plan for the queue length? • What impact would a 10% increase in the arrival rate (132/hr) have on your measures?

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