1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.1 Generating Random Observations from a Probability Distribution The method for generating random observations from a continuous distribution is called the inverse transformation method. Notation –r is the random number –F(x) is the cumulative distribution function (CDF) of the distribution from which we wish to generate a random observation. Thus, for each possible value of x, F(x) is the probability of being less than or equal to x. The Inverse Transformation Method 1.Generate a random number r. 2.Find the value of x such that F(x) = r. This value of x is the desired random observation from the probability distribution.

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.2 Illustration of Inverse Transformation Method

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.3 Inverse Transformation Method for a Discrete Distribution

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.4 Herr Cutter’s Barber Shop Herr Cutter is a German barber who runs a one-man barber shop. He opens his shop at 8:00 AM each weekday morning. His customers arrive randomly at an average rate of two customers per hour. He requires an average of 20 minutes for each haircut. As his business has increased, his customers now often wait awhile (sometimes over half-an-hour). His loyal customers are willing to wait, but new customers are much less likely to return if they have to wait. An article in The Barber’s Journal states –In a well-run barber shop, loyal customers will tolerate an average wait of 20 minutes, while new customers will tolerate only a 10 minute average wait. (With longer waits, they typically take their business elsewhere in the future.) Question: Should Herr Cutter hire a new associate to share the workload?

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.5 Probability Distributions The time required to give a haircut varies between 15 and 25 minutes. His best estimate is that the times between 15 and 25 minutes are equally likely. –Estimated distribution of service times: The Uniform distribution over the interval from 15 to 25 minutes. –The CDF for this distribution is P(service time ≤ x) = F(x) = 0for x ≤ 15 (x – 15)/10 for 15 ≤ x ≤ 25 1for x ≥ 25 The barber shop has random arrivals of customers, averaging two per hour. –Estimated distribution of interarrival times: An exponential distribution with a mean of 30 minutes. –The CDF for this distribution is P(interarrival time ≤ x) = F(x) = 1–e –x/30 for x ≥ 0.

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.6 CDF of Service-Time Distribution (Uniform)

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.7 CDF of Interarrival-Time Distribution (Exponential)

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.8 Applying the Inverse Transformation Method for the Service Time (Uniform Distribution) F(x) = (x–15)/10 = 0.7270 when x = 22.27

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2008 CD12s.9 Applying the Inverse Transformation Method for the Interarrival Time (Exponential Distribution) F(x) = 1–e –x/30 = 0.8130 when e -x/30 = 0.187 or –x/30 = ln 0.187 = –1.6767 or x = 50.30