Restaurant wait time question?

[Nomad84]

My roommate's stats prof asked his class a question and offered $10 to anyone who can come up with the correct answer by next class. I'm not sure if there is a correct answer, so I thought I'd post it here to get your opinion.

When you get to a restaurant, the hostess tells you that the median wait time is 15 minutes. After 5 minutes, what will be your average wait time?

That's all the information we were given. To me, it seems as though there cannot be a single correct answer because you cannot directly relate the median and the mean. It seems to me that you would need to make additional assumptions. Am I missing something here?

[pzhon]

[ QUOTE ]
To me, it seems as though there cannot be a single correct answer because you cannot directly relate the median and the mean. It seems to me that you would need to make additional assumptions.

[/ QUOTE ]
Come up with a reasonable model of waiting times with one parameter. Analyze your model. Specifically, compute the conditional expectation of the waiting time given that you are still waiting after 5 minutes.

One model might be that the waiting times follow an exponential distribution. That's simple because it is memoryless, so you can ignore the initial wait. With a half-life of 15 minutes, the expected wait is not 15 minutes, but is easily computed. I think that is the answer the professor wants. However, if this were a more serious problem, you might want to use a better model (perhaps based on a queue), and you would get a different answer.

[PairTheBoard]

I don't think the memoryless exponential is a good model here. As you wait, the other patrons are finishing their meals. If you've waited an hour the situation has changed considerably from when you walked in.

PairTheBoard

[fnord_too]

Not enough information. Median wait time is worthless here if you are looking for mean wait time.

[Cleveland Guy]

[ QUOTE ]
My roommate's stats prof asked his class a question and offered $10 to anyone who can come up with the correct answer by next class. I'm not sure if there is a correct answer, so I thought I'd post it here to get your opinion.

When you get to a restaurant, the hostess tells you that the median wait time is 15 minutes. After 5 minutes, what will be your average wait time?

That's all the information we were given. To me, it seems as though there cannot be a single correct answer because you cannot directly relate the median and the mean. It seems to me that you would need to make additional assumptions. Am I missing something here?

[/ QUOTE ]

It's probably about 11 Minutes left. Assume there is an even distribution bell curve centered on 15. So you will get called the same percent of the time in the first 5 minutes as you will in the 25th minute.

So if we give it a small percent chace of happening - say 3%, that means the other 97% of the time your wait will be longer than 5 minutes.

Figuring out an even distrbution - and then dividing it by the 97% of the time this distribution comes into play - the 5 Minutes you already have waited puts you between 10 and 11 minutes left of waiting.

I got 10.4 minutes, but it's based on actual numbers you plug in.

[fnord_too]

[ QUOTE ]
Not enough information. Median wait time is worthless here if you are looking for mean wait time.

[/ QUOTE ]

Actually, upon reflection, there may be enough information IF you make a bunch of assumptions. This is basically a queuing theory problem, and if you assume the arival and service distributions, and those distributions are easy to work with (like the standard distributions you use in queue theory), and maybe make some more assumptions, then you may be able to calculate this, but that is a lot of assumptions to make and I am too lazy to refamiliarize myself with queue theory right now.

[Paul2432]

I think you misread the question. The median wait time refers to those currently waiting. The question then asks for your wait time five minutes later.

My first instinct was that if the medium wait time is 15 minutes then there will be a person waiting 15 minutes, and an equal number waiting more than fifteen minutes and less than fifteen minutes.

I can't prove it, but it seems intuitive that the next person will have to wait 30 minutes. So after five minutes, you will have to wait 25 minutes.

Paul

[Nomad84]

For the record, this is a pretty basic intro class, I think. That's why I'm tempted to just tell him that there isn't enough info, unless (as several of you mentioned) you make a lot of assumptions. Would it be reasonable to assume that the wait times for individual customers can be approximated by a normal distribution centered at 15 minutes? If so, then I think you'd still have to know the standard deviation, right? I mean, if it's a normal distribution centered at 15 minutes and a standard deviation of 10 seconds, then it's pretty clear that the average wait after 5 minutes will be almost exactly 10 minutes (very very slightly longer). It seems that this might be what he is looking for, but I'm not sure. Perhaps I could assume that the standard deviation is just high enough that the probability of a negative wait time is below X and work with that after choosing X? This just doesn't make much sense for an intro class, though.

Thanks for the comments so far. Keep them coming please! /images/graemlins/smile.gif

[PairTheBoard]

fnord_too: "Not enough information. Median wait time is worthless here if you are looking for mean wait time."

As fnord_too points out upon further reflection and as pzhon points out under an assumption this is an exponential process, Information about the Median can very well be used to calculate the Mean or "average". In this case, assuming the waiting time for a table has an exponential distribution the average, a, can be found by solving the following integral equation for a.

.5 = [0,15]Int( 1/a * exp(-x/a) )dx

solving for a,

a= -15/ln(.5) = about 22 minutes.

In assuming the exponential model, the fact you've waited 5 minutes does not affect your wait time from that point on. It remains as if you had just walked in.

Assuming the Exponential Distribution is probably not too bad. This model is sometimes used for remaining life of a light bulb even though intuitively the light bulb's remaining life seems like it should be less after it has burned for a period of time. Even with Queing Models you commonly assume Exponential/Poison Distributions for Service Times and Number of Arrivals. The Exponential Distribution Is the waiting time distribution for the next occurance in a Poison Process.

Unless this is a Very Advanced Stats course, I think pzhon is right in thinking this is probably what the Teacher is looking for. It makes for a good lesson showing that the Median of 15 minutes is a bit of a sales gimmick when the Average is really 22 minutes. And waiting 5 minutes may very well Not mean your average wait remaining has been lessened. Remember that Seinfield Episode?

PairTheBoard

[Nomad84]

Wow. You lost me /images/graemlins/smile.gif

Like I said, it's a pretty basic class. I haven't had any stats myself except the stuff I learned in my engr courses, so I'm not familiar with the stuff you are talking about. It seems counterintuitive to me, as you mentioned. I don't understand why the wait time would follow an exponential distribution.

Actually, after trying to type out why I didn't understand it, I'm actually starting to understand better. As people come to the restaurant at a rate higher than the rate at which people finish their dinner and leave, the new arrivals have to wait on not only the customers currently eating, but also those in line in front of them. I can certainly see how the wait time could grow quickly in that case. I was thinking people were coming to the restaurant at the same rate they were leaving, roughly, and that the wait list was staying at a roughly constant size. But I'm still not sure what his prof is wanting... I'm pretty sure it doesn't involve Poison Processes though.

[PairTheBoard]

The Poisson would be one of the basic distributions studied in a basic class. In this case, it's probably natural to assume that the number of tables coming open in a time period like 10 or 20 minutes has an aproximate Poisson distribution. Once you make that not unreasonable assumption, the work I show above follows. The math involved is very simple calculus. The dificulty for the students is in actually understanding how the Poisson and Exponential relate and what they are describing - then applying that understanding. Students who are used to being taught algorithms for finding solutions are sometimes shocked to find out they're actually expected to reason their own ways and means to solutions. The problem may be a shock to the students now - thus the $10 incentive - but would make for a good Final Exam Problem. It's a quick and easy solution if you know what you're doing with the basic Stats. Otherwise it's impossible.

PairTheBoard

[Nomad84]

Thanks again! I'll definitely point my roommate to this thread. I wish I'd gotten to take a stats class, but it didn't ever fit into my schedule. I tried to get in Applied Engineering Statistics for next semester for one of my engr electives, but there were scheduling conflicts there too. Oh well. Thanks for your help.

[PairTheBoard]

To tell you the truth I'm really not that happy with assuming the Poisson distribution for the number of tables coming open in a period of time like 10 minutes. This model would apply best when there are a large number of tables and patrons spend a relatively long time at their tables.

A Full Credit solution should probably include reasons why the Poisson is a good model, either by way of the Poisson's characteristic properties or as a limit for the Binomial.

PairTheBoard