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# note_variability_queue - 311 OPERATION MANAGEMENT...

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311 O PERATION M ANAGEMENT I NFORMATION AND O PERATIONS M ANAGEMENT M ARSHALL S CHOOL OF B USINESS P AGE 1 T EACHING N OTE ON V ARIABILITY AND Q UEUES A. Uncertainty: 1. Sources of Uncertainty: Operations are subject to three major sources of uncertainty: (1) Uncertainty in Processing Times, (2) Uncertainty in Demand and (3) Uncertainty in Supply Uncertainty in Processing Times: Time required to complete a task (e.g. time for a bank teller to serve a customer) is likely to differ each time the task is performed. A number of factors contribute to this uncertainty -- human inconsistency, differences among customers, variation in raw materials, machine failures, quality problems, absenteeism, etc. In facilities that produce many different products or serve different types of customers, the uncertainty in processing times is also due to variety. Uncertainty in Demand: The rate at which customers show up at a service facility / purchase a product also exhibits uncertainty. While some of this uncertainty is natural, marketing actions can also contribute to demand variations. Coupons, price changes, advertising, etc are likely to result in greater demand in one period and possibly lower in the next. 2. Measures of Variability: Measures of Processing Time Uncertainty : The standard deviation of processing times is a measure of variability. Example 1: The times to clean cars at a car wash (in minutes): 8, 7, 8, 9, 10 and 12. Since all processing times are equally likely (probability of 1/6) we get that the mean is: 9 6 12 10 9 8 7 8 We calculate the Standard Deviation in the following way: S.D. = 6329 . 1 6 ) 9 12 ( ) 9 10 ( ) 9 9 ( ) 9 8 ( ) 9 7 ( ) 9 8 ( 2 2 2 2 2 2 Average = 9 minutes. Standard Deviation = 1.6329 minutes.

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311 OPERATIONS MANAGEMENT P AGE 2 Example 2: The time to process a loan application (in minutes): 138, 137, 138, 139, 140 and 142. Average = 139 minutes Standard Deviation = 1.6329 minutes. The above examples illustrate the limitation of standard deviation. Although the standard deviation in both examples is exactly the same, intuitively the variation at the car-wash example appears to be significantly higher than that at the loan-processing center. A better measure of variability is the ratio of the standard deviation to the average. This ratio is called the coefficient of variation. Coefficient of Variation = Standard Deviation Average The coefficient of variation in processing times at the car wash is: 1.79/9 = 0.1987, while the coefficient of variation in processing times at the loan center = 1.79/139 = 0.0129. Coefficient of variation is a good measure because: It takes in to account the relationship between the average and the standard deviation, and It is not affected by the units in which we measure processing times. The standard deviation and the average will be different (numbers) if we measure processing times in seconds instead of minutes, but the coefficient of variation would remain exactly the same.
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