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L12_WaitingTime_B4 - Lecture 12 Waiting Time Problems Slide...

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Slide 1 Lecture 12 Waiting Time Problems
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Slide 2 Le arning  O bje c tive s Analyze stationary arrival queues Predict waiting times and quantify performance metrics Redesign service systems to reduce variability
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Slide 3 Waiting time in a Stationary Queue
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Slide 4 Arrival On average, a flow unit arrives every  a  time units, or  1/a  flow units arrives per  unit of time .    The standard deviation of the inter-arrival time is  σ a . The coefficient of variation is  CV a σ a /a . Process There are   servers (resource) in the system It takes an average of  p  units of time to serve a flow unit, or  1/p  flow units can  be processed per unit of time. The standard deviation of the processing time is   σ p . The coefficient of variation is  CV σ p /p . Assumptions   Stationary Queuing System
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Slide 5 Performance Measures  Stationary Queuing System Stable   average demand rate = 1/a < m/p  = average capacity I q   = flow units (inventory) in the queue or queue length I p  = flow units (inventory) in service I  = flow units (inventory) in the system T q  = waiting time while in the queue T  = flow time Arrival (demand) a  (time/unit) 1/a   (unit/time) CV a   σ a /a Process (resource) m  servers/resources p   (time / unit)   1/p   (unit / time) CV σ p /p ……
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Slide 6 Example: RentAPhone The company currently has 80 phones. There are , on average, 25 customers per day  (24 hours) requesting a phone.  The corresponding coefficient of variation is 1.  Customers keep their phones on average 72 hours. The standard deviation of this time  is 100 hours.  a  = 24/25=0.96 hr/customer 1/a  = 25/24 customer/hr CV a  =1 m  = 80   p  = 72 hr/customer 1/p  = 1/72 customer/hr CV p  = 100/72=1.3889 Utilization : …… Flow rate: 
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Slide 7 Example: RentAPhone cont = × = q q T r I = × = p r I p Waiting time : = + - - + 2 1 2 2 1 ) 1 ( 2 p a m q CV CV u u m p T Flow time : = + = p T T q Inventory in queue : Total inventory : Inventory in service : = × = × = T a T r I 1 a  = 24/25=0.96 hr/customer 1/a  = 25/24 customer/hr CV a  =1 m  = 80   p  = 72 hr/customer 1/p  = 1/72 customer/hr CV p  = 100/72=1.3889 ……
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Slide 8 Example: RentAPhone cont  Waiting inventory incurs a carrying cost of $1 per unit per hour a  = 24/25=0.96 hr/customer 1/a  = 25/24 customer/hr CV a  =1 m  = 80   p  = 72 hr/customer 1/p  = 1/72 customer/hr CV p  = 100/72=1.3889 …… Company pays $1 per hour to provide phone cards to each waiting customer. What is  the total monthly expense for phone cards?
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Slide 9 Example: RentAPhone cont  Redo the computation with m = 81, we obtain a  = 24/25=0.96 hr/customer 1/a  = 25/24 customer/hr CV a  =1 m  = 80
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