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Variability in demand and processing

Variability in demand

If there is more demand than capacity, the implied utilization rate rises above 100%, which makes waiting time unavoidable. The more interesting cases are those, in which there is waiting time even though the implied utilization rate is below 100%. Such waiting time stems from demand variability generated by the somewhat random nature of most demand processes, e.g. many customers showing up at once at some point in time and no customers showing up at all at some other point in time.

In order to calculate with demand variability, we need to define arrival time, inter-arrival time and average inter-arrival time. The self-explanatory arrival time is defined as the time, when customers arrive at a business. The inter-arrival time is thus defined as the time between subsequent customer arrivals. If demand is random, both the arrival times and the inter-arrival times will be drawn from an underlying statistical distribution (often a Poisson distribution). The average inter-arrival time is usually denoted with a.

Another important parameter is the coefficient of variation of the arrival time, which is calculated as the standard deviation over the mean and is denoted as Cv_a. The coefficient of variation is a way to measure the standard deviation against the mean of the distribution. This is useful, because the standard deviation itself is not really a good measure for variability, since it does not express whether a 10 minute deviation is a lot or not.

If the inter-arrival times are drawn from an Poisson distribution, the coefficient of variation is always 1. This knowledge can be used to calculate other parameters considering this formula:

Cv_a = standard deviation (of inter-arrival times) / mean (of inter-arrival times) = 1

Variability in processing

Variability is not limited to the demand process, but also occurs in processing. The calculations are basically the same, with the average processing time being denoted with p and the coefficient of variation being denoted with Cv_p. The coefficient of variation can be seen as a measure of the degree to which a work process has been standardized.

These lecture notes were taken during 2013 installment of the MOOC “An Introduction to Operations Management” taught by Prof. Dr. Christian Terwiesch of the Wharton Business School of the University of Pennsylvania at

1 Comment

  1. […] root cause for quality problems is process variability. Were it not for process variability, every run through a process would result in the optimal output or in the very same error, which […]

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