How long should you wait for an elevator?

Jason, a loyal MR reader, asks:

Google wasn’t able to help me here.

I figure that the longer you wait, the shorter the expected remaining waiting time.

However, in the worse case, if the lift has broken down, the waiting time could be infinite.

For an individual lift, one could, I suppose, collect some stats on average wait times, but I’m interested in the best strategy for an arbitrary lift.

The technical approach is to model the arrival of the elevator as a mathematical process, set up the problem, and solve it.  The seat of the pants approach is to ask about your psychological biases.  Are you, in the first place, more likely to spend too much or too little time waiting for elevators?  In my view standing and waiting isn’t so bad, provided you have something to do or think about.  So my advice is this: once you start waiting for an elevator, begin to think through some interesting problem you face.  The ideal is that when the elevator arrives, you will be disappointed and of course that means you have hedged your risk in the first place.  The question that people screw up is not how long they should wait but what they should do in the meantime.

If you’ve finished thinking about your problem and the elevator still isn’t there, take the stairs.

Readers, what do you advise?  Is there a second best case to be made for "elevator waiting indecisiveness," or should you just have a simple time rule and stick with it?  Is there a formula based upon the number of shafts and number of floors in the building?  The frustrated look of the person standing next to you?

Comments

Doesn't it depend on the arrival time distribution? If interarrival times are exponential, then the conditional expectation of how long you have to wait given that you have already waited N minutes is exactly the same for all values of N. So the fact that you have waited doesn't tell you anything. However, that's not the case with most other distributions.

Wait time is dependent on a variety of factors. Elevators have external indicators of their status. Elevators make noise and elevators frequently tell you where they are. So what does the external indicator say? What is the importance of reason one must go up an elevator? What is the age of the building? How many floors up is it?

It seems that Prof Cowen's approach would not be optimal in the case of an expected wait time of less than thirty seconds or so.

An interesting problem I face that requires thinking about incolves a siginificant amount of context. Before I begin thinking about the problem, I must "load" that context into my mind, which requires some mental effort. If the elavator arrives while I am doing this, or very shortly after it is complete, I will have exerted that effort with no payooff, which (for me, anyway) is a worse outcome than wasting thirty seconds.

In essence, we are hedging a bet that we almost always win, grasping a defeat from the jaws of victory.

Similar problems exist for taking out a book, listening to a song on the iPod, making a phone call, etc. A;; these tasks require non-negligible effort to start, which will almost always be thwarted.

The solution in the case of an individual waiting alone is different than if one is joined by another. I'm not sure how it's different, but when waiting is a social experience, other factors get involved. (i.e., a short-tempered, very fit individual in a hurry might lead the way to the stairs.)

It's also fascinating that we all, so far, are assuming someone waiting for the "up" elevator. Going down would also differ.

If you're fat or out of shape, but otherwise healthy, take the stairs.

Reminds me of a lecture by a logician whare he explained that the Gödel theorem is analog to waiting the bus, you have to take steps before having a coherent theory. I suppose the point "lift can be broken" is key. You have to make a faith commitment and either suppose the lift is working: then you can use a Poisson model for example to decide your strategy. Of you accept that the lift can be broken, but you have to wait an infinite time to be sure that this is the case. So the additional twist is to ask: how long should you wait for a lift while knowing you did not loose your time?

There is, of course, the social solution.

Put up a clipboard with an attached stop watch and have everybody measure their wait times!

That is effectively what we do on the internet (except apache writes the logs for us).

[1] the externalites of pressing the button are walking away are beyond the scope of this analysis.

As far as I know, once an elevator is going up (down) it keeps going in the same direction until it reaches the top (bottom) floor. Assuming that you know the number of floors the building has, and that you can come up with an educated guess of the travel time between floors, you can estimate the maximum travel time to reach your floor. Let´s say there´s a single lift, N is the number of floors, T is travel time between floors, and W is total waiting time. For N=2, W=T; for N=3T, W=3; for N>3, W=(N+2)*T. If F is the number of floors I need to travel, and S is average time between floors taking the escalators, then for (F*T) + W < (F*S) wait for the elevator, otherwise take the stairs (this is assuming you are only concerned with time). Of course, if W>(N+2)*T but still less than F*S, it´s safe to assume that the elevator is broken and so take the stairs anyway.

The critical variable is the attractiveness of the person waiting with you.

Google had no answer? But I just Googled and came up with a great discussion at:
http://www.marginalrevolution.com/marginalrevolution/2008/06/how-long-should.html

oh.

Oh, Brad, you spoiled it :) I wanted to see how long everyone was going to come up with whacky creative solutions before consulting a CS freshman.

I believe our exam question involved whether you should pay a hero to slay a dragon or continue to pay the dragon protection money and wait for him to die.

Brad's answer is great:
So in the elevator case... If we're trying to minimize only time (not calories burnt), and you know it would take you 5 minutes to walk up the stairs, just wait at the elevator for 5 minutes.

But, it does depend on the distribution of wait times. If (fancifully speaking) the elevator is guaranteed to take EXACTLY 5 minutes 1 second, you are always screwing yourself by following that strategy.

So ideally you should find out the Gaussian distribution for wait times of this elevator (perhaps conditioned on time of day), and set the threshold T that minimizes expected waiting time:
(Prob(waitT)

But there are many confounding factors (risk of being late, interesting problems to think about, presence of pretty girls and/or good conversation, tiredness) and I don't think it's a good use of time to really solve the problem.

I think the gut is likely to be pretty good indicator, but as Tyler says you should leverage conscious thought to circumvent predictable biases. So if I know I will get "stuck" continuing to wait due to the sunk cost fallacy, I should pre-commit to a threshold time after which I will take the stairs.

This certainly is a NP-Complete problem!

The next problem is, most of the time there are other people waiting in front of you.
So when the elevator finally arrives you have to wait even longer.
The Solution: Just walk up one or two flights and wait there.

In the building where I work, the only possible answer is "as long as it takes." The stairways are alarmed and for use only in emergencies, so the elevators are the only option.

How much do you value the health benefits from taking the stairs?

Wikipedia seems to have some interesting links:

http://en.wikipedia.org/wiki/Wait/walk_dilemma

For buildings with multiple elevators, one cool development is destination-based elevators.

I am reminded (though not adequately) of an old story about the consultant called in to examine such a problem of complaints about waiting for elevators.

After several varying periods of observations and data gathering, he prepared a report of what was transpiring and on the reactions of the "traffic" to the conditions.

At the end he gave his recommendation:
Place mirrors in all wait locations.

That was done. Complaints ceased.

:-) Reminds me of the one of the lifts in our University, which has the weirdest of logics. I guess the basement gets the least of preferences!

Wow! With the obesity epidemic on the rise....lets face it....we should all take the stairs. Think, sing, listen, and invite the "pretty girls" to join you as you walk up the stairs. Let’s face it; she’s probably taking the stairs anyway. How else would she maintain that figure? The more you take the stairs, the more fit and able you will be. Then, issues like being winded after 5 floors or it taking too long become less of an issue. Plus, as a perk, you'll increase your daily activity and possible improve your health and maybe trigger the urge to initiate other healthy lifestyle changes. It could be a productive and healthy endeavor to take the stairs that can lead to a chain reaction of healthy choices. Obviously my analytical and scientific mind isn't quite wired like all of yours, but just a little food for thought.

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