BU8401-Lect09-WaitingLineII.pdf - Nanyang Business School BU8401 Management Decision Tools Seminar 9 Waiting Line Models II Outline ▪ M/M/k

BU8401-Lect09-WaitingLineII.pdf - Nanyang Business School...

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Nanyang Business School BU8401: Management Decision Tools Seminar 9: Waiting Line Models II
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Outline M/M/k Multi-server Waiting Line Model M/G/k Multi-server Waiting Line Model No Waiting Line Economics Issues of Waiting Lines Comparing M/M/k with k x M/M/1 2
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Multiple Server M/M/k Queue Poisson arrival rate/Exponential service times k multiple identical servers with single queue Each server services at μ customers per unit time Every server is identical in performance Customer does not choose which server for service 3 S 1 S 2 S 3 Customer leaves Customer arrives Waiting line
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Multiple Server M/M/k Queue 𝜆 μ is called the “implied utilization”, and is often greater than 1. 𝜌 = 𝜆 is our usual “utilization” on a per -server basis. In other words, each server is ρ percent of the time busy. 4 S 1 S 2 S 3 Customer leaves Customer arrives Waiting line ? μ
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5 Multiple Server Model - M/M/k Formulae k = number of servers k = mean effective service rate for the system which must exceed the arrival rate Probability of no customers in the system (all servers idle) is ? < ?? 𝑃 0 = 1 σ 𝑛=0 ?−1 1 𝑛! ? ? 𝑛 + 1 ?! ? ? ? ?? ?? − ? 𝑃 ? = ? ? ? 1 ?! ∙ 𝑃 0 Probability of j customers in the system 1 ≤ ? ≤ ? : 𝑃 ? = 𝑃 ? ? ?? ?−? = ? ?? ? ? ? ?! ∙ 𝑃 0 Probability of j customers in the system ? ≥ ? :
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6 Multiple Server Model - M/M/k Average number of customers in the system: Average waiting time in the system (waiting and being served): Average number of customers in queue: Average time a customer spends in waiting to be served: 𝑊 = 𝐿 ? 𝐿 = 𝐿 𝑞 + ? ? 𝑊 𝑞 = 𝐿 𝑞 ? 𝐿 𝑞 = ?? ? ? ? ? − 1 ! ?? − ? 2 𝑃 0
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Road Example (Part II) 7 Assume now that the arrival rate is 624 per hour and three (instead of one) toll booths are open. What are the performance measures for the new system ? = 10.4/minute, = 4/minute, k = 3 / = 2.60 P 0 = 0.0345 (from table) L q = 4.9328 W = 7.5328/10.4 = 0.7243 L = 4.9328 + 2.6 = 7.5328 W q = 4.9328/10.4 = 0.4743m A toll road has one attendant at an exit gate. Cars arrive in a Poisson fashion at the rate of 120 cars per hour. It takes the attendant, on the average, 15 seconds to service a car. Service times are exponentially distributed. Obtain the basic queue performance measures for the system. Assume that there is an infinite car calling population and an infinite queue length is possible.
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  • Fall '17
  • jane
  • Automobile, Queueing theory, Automobile repair shop

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