1Ardavan 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 = 2.718282, (the base of the natural logarithm)

t ≤ t in Exponential Distribution (M/M/1)

2Ardavan 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 = 0.451188

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 = 0.1888

Example 8.5 (M/M/1)

3Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3

M/M/1 an additional Measure

1

system in the customers 0exactly are y thereProbabilit0p

)1()(

system in the customers exactly are y thereProbabilit

n

n np

4Ardavan 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

5.012

6

Rp

R

5.05.0110 P

5Ardavan 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?

6Ardavan 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

7Ardavan Asef-Vaziri Sep-09Operations Management: Waiting Lines3

M/M/1 Performance Evaluation

32 )15

10)(

15

101()

15

10)(

15

101(

15

10)

15

101()

15

101(}3Pr{X

78.009.014.022.033.0

The probability of a consumer getting $10 is 1-0.78 =0.22

On average how much money will the bank pay every hour?

0.22*average number of customers*$10 =$22