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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?
Posted by Tyler Cowen on June 9, 2008 at 01:53 PM in Education | Permalink
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.
Posted by: Jim at Jun 9, 2008 1:52:52 PM
Also is a function of how many floors you need to travel, and thus how much time it would take to use the stairs.
Posted by: Tim at Jun 9, 2008 1:54:41 PM
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?
Posted by: tim at Jun 9, 2008 1:55:16 PM
A related question is how long to wait in checkout lines at retail stores when there is some delay. For instance, if the customer at the cash register disagrees about the price that came up and the store personnel have to send two trips to the aisle to see what the listed price was. This happened to me in Wal-mart last week. Do you stay and wait or go to another line?
Posted by: Dirk at Jun 9, 2008 2:01:39 PM
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.
Posted by: JohnMcG at Jun 9, 2008 2:03:04 PM
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.
How about something in between? How about a Bayesian inference that the elevator is successively less likely to show up after it's past a reasonable estimate for mean time for functioning elevators?
Posted by: Michael F. Martin at Jun 9, 2008 2:10:15 PM
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.
Posted by: Bill Harshaw at Jun 9, 2008 2:12:16 PM
My rules are simple:
distance more than 4 flights = elevator
have heavy load = elevator
else, stairs.
My cutoff for assuming that the elevator is broken is about 5 minutes absent other information.
Posted by: Bob Montgomery at Jun 9, 2008 2:22:20 PM
If you're fat or out of shape, but otherwise healthy, take the stairs.
Posted by: 8 at Jun 9, 2008 2:24:08 PM
Turing showed that there is no way to determine whether an arbitrary computer program will successfully finish it's computation and halt. I would think that it would be similarly impossible to prove that an elevator will ever arrive.
The modern computer solution to this is to simply wait an arbitrary amount of time, say, ninety seconds, and then ask the user if they want to give up. Not particularly elegant.
Posted by: Tom at Jun 9, 2008 2:27:20 PM
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?
Posted by: Gödel at Jun 9, 2008 2:34:03 PM
This is only a statistical problem if:
1. No lights to indicate elevator movement.
2. No people nearby to ask: "What do you think?"
3. Stairs are non-trivial.
In such a case, the worst possible way to spend your time:
Wondering about the statistical distribution of evevator wait times.
Posted by: Dave at Jun 9, 2008 2:41:36 PM
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).
Posted by: Ted Dunning at Jun 9, 2008 2:51:12 PM
You should take the stairs, it's good for you.
As a expected time minimization problem, I don't think we can solve it without knowing the distribution of the elevator arrival times.
But even without knowing the distribution of elevator arrival time, we can minimize both the maximum time wasted and the maximum ratio of actual time spent to optimal time spent.
Assume the stairs would take 2m and the elevator 30s and the elevator can be heard 10s before arrival.
The plan is simple: wait walking time minus elevator time plus detection time minus walking time (120-30-10=80s). In the worst time case, where the elevator is just out of earshot when you walk, your trip takes 200s and the optimal trip (walking immediately) would have been 120s. The ratio of actual to optimal was 5:3.
If we always walked right away, we might waste 90s with a ratio of 4:1. If we press the button and listen, going immediately if we hear nothing[1] we might waste 80s, with a ratio of 120:40 = 3:1. If you wait indefinately, both measures might be infinitely bad.
Posted by: Henry at Jun 9, 2008 2:53:02 PM
[1] the externalites of pressing the button are walking away are beyond the scope of this analysis.
Posted by: Henry at Jun 9, 2008 2:54:14 PM
Is there a way of applying the Monty Hall problem to this? This would only work with multiple banks of elevators. Pick three elevators, choose one. Once the first elevator arrives, and opens, change your choice. Thereafter, would it not be true that the probability of your second choice arriving sooner is then higher? Thus reducing your estimated wait time (for the purposes of this thought exercise, we'll ignore the fact that the first elevator has already arrived..) It's challenges like this, that allow me to enjoy waiting for elevators. Anyone remember the elevators in the HItchhikers guide to the Galaxy? - the ones that took to sulking in the basement.
Posted by: Nick L at Jun 9, 2008 2:56:42 PM
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.
Posted by: at Jun 9, 2008 3:11:30 PM
Oh you economists... Computer scientists would recognize this as the ski rental problem, solved a decade ago and now being applied to such things as optimal power-up and sleep policies for power hungry chips. This preso has an overview of the problem:
http://209.85.173.104/search?q=cache:qlQhqGcLshEJ:www.cs.ucla.edu/~darwiche/MURI/y02/irani.ppt+ski+rental+problem+irani&hl=en&ct=clnk&cd=1&gl=us&client=safari
The ski rental problem, IIRC, is that you're starting skiing and don't know how you'll like it and want to be most efficient in spending your money. Do you rent or buy? How long do you rent? Turns out, if you rent until you've paid enough in rent to buy skis then buy the skis (if you plan to ski more), you minimize your exposure over all outcomes.
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.
Next question...
Posted by: Brad Hutchings at Jun 9, 2008 3:30:29 PM
The critical variable is the attractiveness of the person waiting with you.
Posted by: Brad at Jun 9, 2008 3:31:06 PM
In my Industial Management course in antedeluvian times, we were taught that the solution to impatience while waiting for a "lift" was the installation of a mirror. Then the girls could pass the time looking at themselves, and the men could pass the time looking at the girls too. So much simpler than Queuing Theory.
Posted by: dearieme at Jun 9, 2008 3:34:15 PM
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.
Posted by: Kevin Postlewaite at Jun 9, 2008 3:41:01 PM
It depends on what's upstairs and whether I'm on time. If it's a meeting or an interview, or something of that nature, I'd wait up to four solid minutes for the elevator. Even if I'm running late, I'd rather not bust up five flights and arrive slightly less late but breathless and flustered. If I'm a few minutes early, I'd take the stairs at a normal pace and arrive in normal condition, no elevator necessary.
Posted by: Miranda at Jun 9, 2008 4:20:54 PM
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.
Posted by: Toby at Jun 9, 2008 4:43:49 PM
Well, truth be told, the ski rental problem was post-graduate stuff a decade and a half ago. It is pretty cool how CS absorbs that into the freshman weeder class so quickly! My degrees just get more valuable with each passing minute...
Posted by: Brad Hutchings at Jun 9, 2008 4:53:08 PM
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.
Posted by: mk at Jun 9, 2008 5:14:40 PM
