Exponential Distributions, Part 4 - Manage to the Mean, Not the Mode

In Part 3 we looked at the forgetfulness of exponential queues, namely the memoryless property.  In this last article, we look at how to manage queues and systems that behave according to an exponential distribution.

Exponential Attribute #4 - The Mean is More Important Than the Mode

Or, Your Gut Feeling Will Fail You

In a normal distribution (bell curve), the peak value coincides with the mean value.  The most common occurrence (the mode) also happens to be the average.

Not so with the exponential distribution.  The asymmetrical tail skews the average away from the peak.  The mean will always be larger than the mode.

So what?  It turns out this subtle mathematical difference can play tricks on your mind if you're not careful.  If you're a manager, that translates into consistently being over-budget and consistently failing to meet your operational performance targets.

Managing to the Mode or to the Mean?

As a manager, you gradually get a feel for how your department operates.  You have a sense of the typical system behaviour, whether it's how long a call on your help desk lasts or how much it will cost to reprogram 100 lines of code.  There is randomness involved, but your experience tells you the randomness averages out over time.  If your system behaves according to a normal distribution, then it's hard to know if you're managing to the mode or to the mean.  In fact, it doesn't matter because the two values are the same.

However, if your system behaves according to an exponential distribution, it suddenly matters which value you are managing to.  When you want to manage to achieve the budget targets, you must use the mean.  The mode is irrelevant.

I have found that people tend to manage to the most common events.  It is easy to get a "gut feel" for the things that happen the most frequently, but it is not as easy to get a "gut feel" for the average.  That means your "gut" manages to the mode.

Therefore if you manage according to your gut, an exponential system will cause you to fail every time.  If you budget to the mode, you will always end the year over budget.  If your target time for callers kept on hold is set by the mode, you will never achieve your target.  You must budget and manage to the mean.

Beware of Management CFIT

In aviation, investigators created a term called CFIT, which means Controlled Flight Into Terrain.  It can be the cause of a crash when the aircraft is working fine, including the instruments, but the pilot ignores the signs and flies by "gut feel" instead.  When visibility is poor, that means the pilot can fly the plane directly into the ground, sea, or mountain without realizing it.

When you are managing a system that behaves in an exponential way, you must fly by instruments and not by your gut.  Your instruments tell the nurse manager that the average stay of her inpatients is 5 days, but her gut feel is that 3 days is more common.  Her gut is right (i.e. the mode length of stay is 3 days) but she must manage to the mean of 5 days to stay on budget.

That is why reliable and frequent reporting is key to managing exponential systems, because without reliable instruments you cannot pilot the system properly.

Manage to the Mean AND Build Flexibility

Managing to the mean is still not enough to be a successful manager however.  As discussed in Part 2, with exponential distributions the tail wags the dog.  You must be prepared for the large but rare outlier events that will have a huge impact on your system, such as the spinal injury patient whose length of stay is greater than 1 year.  Over the long term the average value will become correct, but during a single budget year there may not be enough time to absorb the effect of such a large-valued event.  To handle those events, you need to build in some flexibility into the system.

Flexibility can take different forms.  It can involve cross-training of staff to permit additional capacity when a long-duration event occurs.  For instance, asking a mortgage specialist to cover as a bank teller when one customer has an hour's worth of coins to deposit or having an extra doctor on call when the Emergency waiting time exceeds a few hours.  It can also take the form of budget reserves.  For instance, the cost of that rare spinal injury patient cannot be predicted by simply budgeting to that nursing unit's averages, but across a hospital with a few dozen nursing units, those rare outlier events may happen with much more predictability.  (This is the Central Limit Theorem in action.)  A once-in-a-half-century event for a nursing unit may become a bi-annual event across the entire hospital, which becomes much easier to budget for.  When an unusual event occurs, transfer that reserve to the affected nursing unit and the departmental budgets stay on track.  You have managed to predict an unpredictable event and reduced it's impact on your system.

Summary

What I hope to have shown in these few articles is that exponential distributions are real, they have some counter-intuitive effects, but they are actually quite predictable and manageable if you understand their behaviour.