Once you learn about certain models used in operations research and industrial engineering, you start seeing them everywhere. I see the umbrella problem and its variants everywhere (read about the umbrella problem in this post).

I try to keep a few pens in my laptop bag at all times. These pens will drift in and out of my bag when I work. Every now and then, the pens entirely move out of my bag and I am left without a much needed pen. This happened earlier in March when I was on a trip and had planned to review a paper on my flight.

It has been a very cold winter in Wisconsin. I learned the hard way that it’s useful to keep a spare sweater or two in my office. I often forget to wear them home, and so sweaters and scarves have been accumulating in my office.

The city of Madison tries to be pedestrian friendly. They have these pedestrian flags (see picture below) that are supposed to help you cross the street safely. Each intersection has a bunch of flags in two bins with on either side of the street. I drive by a few of these intersections on the way home, and sometimes I see an intersection where all the flags are in one of the two bins. It always makes me smile.

This isn’t exactly the umbrella problem since the same pedestrian doesn’t go from one side of the street to the other indefinitely as in the umbrella problem, but the umbrella problem only needs to be slightly modified to capture the real problem here and to provide insight into how many flags should be stocked to maintain a certain reliability level.

Look for a few posts about March Madness next week!

I am teaching stochastic processes again this semester. After enjoying a humorous exchange on twitter about the exponential distribution, I wondered about the practicality of the exponential distribution. Most of the examples from my notes seem a little idealized.  For example, I’m almost positive that light bulb times are not exponentially distributed. Is anything really exponentially distributed?

Things that are almost certainly exponentially distributed:

• The amount of ink and graffiti on dollar bills that have been in circulation more than one year (a student who works for the Federal Reserve Bank provided this useful tifbit!).
• The useful life of things made from polyurethane foam, such as Nerf balls, car seats, mattress pads, and carpet pads (the useful life occurs after the break-in period and prior to the break-down period).
• The time between 911 calls (see below).
• The time between celebrity deaths.

I looked at some emergency medical 911 data.  The call volume changes over the course of the day, but the call volume is constant for a large part of the day.  Looking just at that time period, I examined the interarrival times of the 911 calls.  The exponential distribution is more or less a perfect fit!  I don’t have any data for my celebrity deaths hypothesis, but since they are independent, like most 911 calls, then I would expect celebrity deaths to follow an exponential distribution.

The time between emergency medical 911 calls is exponentially distributed

What else is exponentially distributed?  Do you know how light bulb lifetimes are distributed?

I must have stochastic processes on the mind after wrapping up another semester.  I will admit that this is a rather silly post. I was sauteing a couple of plantains last night.  I cut the plantains into thin slices and then saute them in a little olive oil in a cast iron skillet until they are browned.  It’s hard to flip plantain slices over, and only a few slices can come into contact with the skillet.  As a result, it’s hard to evenly cook the slices.

As I was cooking, I starting to mentally model how I would estimate how long it would take to brown all of the plantain slices.  Since I flip plantains at semi-regular intervals (about once every minute or two), it provides a discrete-time structure to the problem.  The more flips have occurred, the lower the probability that any unbrowned plantain comes into contact with the skillet (since it becomes more likely that already-browned slices come in contact with the skillet).  This is not unlike the coupon collecting problem, where there is a set of n coupons from which coupons are collected one at a time with replacement.  The set of n coupons is my set of plantain slices, plantain slices are collected when they come in contact with the skillet, and plantain slices can come into contact with the skillet more than once.  I do not collect coupons one by one, but rather I collect a subset of coupons at each flip–the number of plantain slices that can come in contact with the skillet.  The bottom line is that it takes an increasingly long time to brown each additional plantain slice.

Of course, I could always game the system and manually exchange browned plantain slices with unbrowned plantain slices to sleep along the cooking process and encourage uniformity.  But (a) I don’t have the time to devote to manually flipping with two kids to feed and (b) that’s a little obsessive.

So that leaves me with the solution to the coupon collecting problem: the plantains would burn before they would all be browned. I suppose I’ll have to work on my plantain-turning technique and accept a few undercooked plantains.

What algorithm do you use for skillet cooking?  How do you cook plantains? (I love ’em, but I’m a plantain novice. I’d love some new recipes).

Links:

• A informative podcast interview with a banana expert for lovers of all banana varieties.  Banana fact:  the banana tree is the world’s largest herb.

Related posts:

• A combinatorial coupon challenge