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09 QM Lecture Note A

# 09 QM Lecture Note A - GG Hegde 348 Mervis Hall University...

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1 GG Hegde 348 Mervis Hall University of Pittsburgh Quantitative Methods Lecture NoteA: Notes, Sample Questions and Ans Key

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2 Lecture Note A: Introduction to Queuing ( Waiting Lines) GG Hegde It is estimated that Americans spend a total of 37 billion hours a year waiting in lines/queues. Places we wait in line... - stores - hotels - post offices - banks - traffic lights - restaurants - airports - theme parks - on the phone Waiting lines do not always contain people... - returned videos - subassemblies in a manufacturing plant - electronic message on the Internet Queuing theory deals with the analysis and management of waiting lines. Ex.1: Suppose 2 customers/hr arrive and each customer needs 20 minutes of service. Do customers wait ? Ex.2: Suppose an average of 5 customers/hr arrive at a bank, and average service time needed per customer is 6 min. The bank opens its door at 9 am. Customer arrivals and service times are as follows: 10min 3min 5min 5min 7min C1 C2 C3 C4 C5 9:05 9:06am 9:25am 9:27am 9:40am Average time a customer spends waiting in the queue = ………… Average number of customers waiting in the queue=………….
3 Arrivals rate: Average # of customers arriving / time unit, λ Interarrival time 1/ λ . Example Assume calls arrive with λ =2 per hour. Interarrival time , 1/2 hr = 30 min. On average, calls arrive every 30 minutes. Service rate: Average # of customers that can be served per time unit per server, μ Average service time= 1/ μ Example Assume average service rate, μ =3 customer per hour. Average service time, 1/3 hr = 20 minutes

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4 Common Queuing System Configurations, S=1 and S=3 Customer Leaves ... Waiting Line Server 1 Server 2 Server 3 Customer Leaves Customer Leaves Customer Arrives Customer Arrives ... Waiting Line Server Customer Leaves, Figure 1: S= 1, and S=3