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Poisson distributions, such as to model client demands and responsiveness

Queuing theory and its models often assume that the rates of arrival of work and delivery of service can be described by a Poisson distribution (See my post of Jan. 20, 2006: one of many kinds of distributions of numbers; and Aug. 16, 2006: predicts likelihood of event during a given time period.).

Let me describe what a Poisson distribution looks like graphically. Visualize a column chart that shows on the horizontal, bottom axis the number of client requests for legal services that arrive in a legal department each week. The number of requests per week increase as you move to the right.

The vertical axis shows the relative frequency of each of those weeks, expressed as percentages increasing from quite low to perhaps 20 percent. Thus, the tallest column in the middle might be 15 requests a week, which happens during 18 percent of all weeks; the lowest column on the left corresponds to one request during a week, which happens five percent of the time; the lowest column on the right represents 25 requests for legal assistance during a week, which happens one percent of the time. The overall shape of the columns is somewhat like a bell curve, but with a longer tail to the right.

Equivalently, according to William J. Stevenson, Operations Management (McGraw-Hill, 2005, 8th Ed.) at 782, the time between arrival of a request and completion of the service time can be described by a negative exponential distribution. Imagine a line on a chart sloping from high on the left (a high relative frequency percent of numbers or service requests during a week) down to the right where there are unusual demand levels at low percentages of occurrences.

In Stevenson’s example, many times service is provided quickly, but some services take a long time. “That is, if service time is exponential, then the service rate is Poisson. Similarly, if the customer arrival rate is Poisson, then the interarrival time (i.e., the time between arrivals) is exponential.”

How well we understand and can model turnaround time in a law department makes a significant difference. These two statistical tools provide some insight.

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One response to “Poisson distributions, such as to model client demands and responsiveness”

  1. Rees – add to this Pareto’s 80/20 rule (80% of work is generated by 20% of clients whilst 20% of your work generates 80% of your fees) and you have the full set! Unfortunately, of course, how many lawyers are willing to stop word-smithing to apply such numerical analysis or am I being unfair?
    Apparently similar distributions apply at airport customs/immigration halls as well. So, for those undertaking international travel, head for the booths at either end!