1 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Mean inter-arrival time = 1/Ri = 1/R  Probability that the time between two arrivals.

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1 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  Mean inter-arrival time = 1/Ri = 1/R  Probability that the time between two arrivals t is less than or equal to a specific value of t  P(t≤ t) = 1 - e - Rt,  e = , (the base of the natural logarithm) t ≤ t in Exponential Distribution (M/M/1)

2 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3  If the processing time is exponentially distributed with a mean of 5 seconds, the probability that it will take no more than 3 seconds is 1- e -3/5 =  If the time between consecutive passenger arrival is exponentially distributed with a mean of 6 seconds ( Ri =R = 1/6 passenger per second)  The probability that the time between two consecutive arrivals will exceed 10 seconds is  1- (1- e - 10/6 ) = e - 10/6 = Example 8.5 (M/M/1)

3 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/M/1 an additional Measure

4 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/M/1 Performance Evaluation Example: The arrival rate to a GAP store is 6 customers per hour (Poisson). The service time is 5 min per customer (Exponential). What is the probability of having exactly zero

5 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/M/1 Performance Evaluation  The arrival rate to the main branch of Key bank is 10 customers per hour.  The service rate is 15 customers per hour.  Management promise to each consumer that enter the bank and sees more that 3 customers in front of him $10. What is the probability that a customer will get $10? On average how much money will the bank pay every hour?

6 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/M/1 Performance Evaluation  Probability that a customer gets $10 = 1 – Probability that the customer does not get $10  Probability that the customer does not get $10 = Probability of 0 customers + Probability of 1 + Probability of 2 + probability of 3

7 Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3 M/M/1 Performance Evaluation The probability of a consumer getting $10 is =0.22 On average how much money will the bank pay every hour? 0.22*average number of customers*$10 =$22