09 QM Lecture Note A

09 QM Lecture Note A - 1 GG Hegde 348 Mervis Hall...

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Unformatted text preview: 1 GG Hegde 348 Mervis Hall University of Pittsburgh hegde@pitt.edu Quantitative Methods Lecture NoteA: Notes, Sample Questions and Ans Key 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, Interarrival time 1/ . Example Assume calls arrive with =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, Average service time= 1/ Example Assume average service rate, =3 customer per hour. Average service time, 1/3 hr = 20 minutes 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 5 Performance Measures- Formulae with Markovian Assumptions, S=1 FORMULAE SET I: One line, one channel, s=1 = average number of arrivals per time period = 2/hr = average number of customers served per time period, per channel = 3/hr 1. % of time that all servers are busy (=Prob. that an arriving customer has to wait for service) = / 0.66 2. Average number of customers waiting in queue Lq = 2 / [ ( - ) ] 1.33 cust. 3. Average number of customers in system(queue+server) L = Lq + ( / ) 2 cust 4. Average time a customer spends in queue Wq= = Lq / 0.66 hr=40 min 5. Average time a customer spends in the system(queue + server) W= Wq + ( 1 / ) 1 hr 6. Probability zero customers in system P = 1 - / 0.33 Customer Arrives ... Waiting Line Server Customer Leaves 6 FORMULAE SET II: S>1 Performance Measures (formulae) for one line, multiple channels, s>1 are given below....
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This note was uploaded on 08/10/2011 for the course BUSQOM 0050 taught by Professor Glowackia during the Spring '08 term at Pittsburgh.

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09 QM Lecture Note A - 1 GG Hegde 348 Mervis Hall...

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