# Mysterious Trams

I ride a tram six stops to work. I don't worry about the time or check a clock: I just go to work when I'm ready to go, and go home when I'm done for the day. Never at set times.

Each tram line starts every morning at some time I've never bothered to check, then run more or less every 10 minutes all day long in both directions. The mean and standard deviation of the tram is identical for all tram lines in both directions at all times of day.

Once I leave for work or leave for home, I'm pursuing an optimal strategy to get where I'm going.

AND YET! I have to wait substantially longer (say 25-75% longer) at the tram stop for a tram home than I do for a tram to work.

Why so?

(This isn't a "think outside the box" problem: it's not due to crowding, me seeing the tram coming, stopping for a newspaper, being in a wheelchair, trams running late in the evening rush hour, boundary conditions such as leaving with the first tram or working until the last tram, anything like this. It's not related to me personally: if anyone lived where I do, worked where I do, and did any job where their start and stop times were random, and so on, they'd have the same problem.)

• Why do you say "Even though I'm trying to minimize my wait times"? It sounds to me like you are leaving both sides at a pretty much arbitrary time, and as a result, waiting at the tram stop. – EightAndAHalfTails Sep 6 '18 at 9:27
• I'm trying to say that the delay isn't simply because I'm doing something stupid and/or non-productive. I don't care enough about getting home that I have learned the schedule and set my watch to the second. But once I'm going home I'm doing the optimal job of going home. – Swiss Frank Sep 6 '18 at 13:58
• @SwissFrank, did you know that this is the 14,000 question on PSE!! :D – QuantumTwinkie Sep 6 '18 at 15:09
• You don't transfer between lines, right? – Acccumulation Sep 6 '18 at 15:59
• No, no transfers! I'll clarify that, thanks! – Swiss Frank Sep 6 '18 at 19:28

Maybe your work location is equidistant from two different stations, from which two different tram lines pass, and they both take you home (to the same station). When you leave home in the morning, you can take either train, so the frequency of the cars that suit you is higher than 10 mins on average. However, on the way back you must make a choice, and walk to one of the two stations that are near your workplace, where the average train frequency is 10 minutes.

• Exactly the issue underlying a problem, and a better explanation than I could probably manage! – Swiss Frank Feb 21 at 6:52

I think it might have to do with

the timing of the trams and you leaving work.

When leaving from home:

On average, you don't know when it will arrive, and so you leave at a random time. Overall, then, you should expect to wait anywhere from 0 to 9 minutes for the tram. This means that over time, you will wait for an average of 5 minutes for the tram in the mornings.

When leaving from work:

Most people must leave at the same time every day. Let's say, for the sake of argument, that your shift ends at 5pm. It takes you 3 minutes to get to the tram, so you arrive at 5:03 every day. The tram leaves work every 10 minutes, 5pm, 5:10pm, 5:20pm, etc. This means that every day, you must wait exactly 7 minutes for the tram in the evenings. Over time, you will wait for an average of 7 minutes for the tram in the evenings. This gives a feasible explanation for the delay, since 7 is 40% more than 5.

• It's not because I happen to finish at the same time every day. I'll clarify that. – Swiss Frank Sep 6 '18 at 3:18

My guess:

The tram line makes a figure 8. You live at the crossing of the rounds, where 4 trams pass every 10 minutes. Your work is somewhere along the line, where only 2 trams pass every ten minutes. You say "I'm pursuing an optimal strategy to get where I'm going", but you don't say you care about the travel time. All trams will bring you eventually to your destination (even if they first take the "other" round) so the chances you get a ride quickly are greater at home then at work.

Do you live in a situation like this?

The trams arrive with the same frequency at each station, but the way home has half its traffic follow another path

next to the "to work" tram stop.

across the street from the "to home" tram stop, at a major intersection in the central business district.

What's more,

you are a law-abiding pedestrian and won't cross against the light. Occasionally you are caught at the crosswalk as the tram approaches (with traffic), stops, and leaves. Leaving in the morning, this situation would mean catching the tram immediately, but coming home this means waiting another ten minutes.

Although the specific parameters will vary depending on the distribution of trams over time,

if trams arrive uniformly your average wait time outside your apartment is 5 minutes. Your average wait time outside your office would be 5 minutes, but the light cycle is 2 minutes long, 1 minute per direction. Making some naive assumptions about randomness, roughly 5% of the time you'll watch the tram pass you by at the intersection and add 10 minutes to your wait time, and your average wait time increases to 5.5 minutes. You can rejigger the parameters and correlations to hit that 25%-75% increase.

Source: lots of time waiting for buses. (and trams in Switzerland)

• Interesting idea! But for me, if you're waiting at a light to get to the tram stop, you're not at the tram stop. Also if you're across the street from where a tram would stop, I wouldn't call that a tram stop. I'm not saying you're wrong, that's just how I view it as an English/German speaker in Zurich (which has lots of trams). Maybe other languages or cities view it differently. – Swiss Frank Sep 7 '18 at 19:17
• @SwissFrank I think your language is perfectly fine, I was mistakenly interpreting it through a lens of "waiting for the tram," independent of "at the stop." FWIW the picture in my head was that little green fence in the middle of the street on the south side of Zurich HB :) – kyle Sep 7 '18 at 19:51

I only answer as it is the absolute same for me (in real life):

When I go to work in the morning, my station is near the "Depot" of the trams where they are during the night. In the morning they ALL have to pass this station independent of their "real" line / route: So there is one tram almost every minute.

All Trams pass the station next to my work on their route (in my case this is the one at the train station).

This fact is different in the evening.

All Trams follow their routes, and the only line I can take is the one that directly goes to my station. None of the others will work.

Why the differences (25% - 75%)?

I start work very early at almost the same time as lines are getting "populated" with trams... The later I am, the more the traffic "normalizes" and after a certain time I can only get MY line again, as the others do not pass here anymore. during the day.

• Interesting ideas but not the idea I was going for. My problem is also real world, by the way! – Swiss Frank Sep 7 '18 at 19:04

There could be two (or more) tram lines to/from work, each running every $10$ minutes. Going to work they arrive evenly spaced every $5$ minutes, which is a $2.5$ minute wait on average. Going home however, they are bunched together with one line always arriving just a minute after the other followed by a $9$ minute gap. This would lead to an average waiting time of $0.1*0.5+0.9*4.5 = 4.1$ minutes.

Bunching of trams/buses tends to happen because the first bus/tram after a long gap gets delayed even more by the larger number of passengers that have gathered to get on. The following bus/tram doesn't need to stop as long and catches up even more. At the start of the day when going to work this process might not have happened as much yet, so they are more likely to run more regularly spaced when going to work than when going home.

• Bunching is a real thing in real life--a sign saying "about 5 busses an hour" in London means your average wait is still 30 minutes, but you will see a LOT of buses when they finally come! :-D However, bunching is not the problem here. And your point of what I'd call pernicious scheduling phase alignment" is a good one and I was trying to think of how to eliminate that cause from my wording but kind of forgot. – Swiss Frank Sep 6 '18 at 14:01
• @SwissFrank I actually think that Acccumulation was trying to describe the same thing in his answer, though it wasn't exactly clear. – Jaap Scherphuis Sep 6 '18 at 14:07
• OK, to be clear, either could WELL be the cause in real life, and I've dealt with both personally. (E.g., the 3 tram and 31 bus from Zurich main station to Kunsthaus are synchronized to run together!) but neither is the answer I'm looking for. I'll up-vote though not give an answer check. – Swiss Frank Sep 6 '18 at 14:10

You are a dispatcher at the end of the tram line. To get to your station, you wait an average of 5 minutes for a tram. After you release your last tram of the day you have to wait about 10 minutes for the next one.

• Wouldn't be great if a dispatcher doesn't worry about the time or check a clock and just goes to work when he's ready to go :-) – Michel Sep 7 '18 at 13:38
• Nothing says he is good at his job. :) . The only thing he is good at is getting places quickly once he decides to do so. – jay613 Sep 7 '18 at 18:53

Your house is near enough to the tram station that you can leave your house when you notice the tram showing up. On the way home, you lack that advantage.

or alternately

there is some other source of delay at the work tram station - buying a coffee, increased crowds, or whatever.

• This isn't a thinking-outside-the-box problem, more just scheduling or topology or something. I'll clarify that, thanks! – Swiss Frank Sep 6 '18 at 3:16

Possibilities:

There are more tram lines to work than back

There are more people taking the trams back home, so you sometimes have to wait for the next tram.

If the timing of the tram lines are unevenly spaced, that could explain it. For instance, if trams going to your work leave every five minutes, you'll wait on average two and a half minutes. If the ones coming back are spaced two minutes apart, you'll wait on average 2.9 minutes.

• How would you wait 2.9 minutes if a tram leaves every 2 minutes? – Chris Sep 5 '18 at 23:18
• The trams run bidirectionally at ten minute intervals all day. I'll clarify that. – Swiss Frank Sep 6 '18 at 3:20

Because:

You leave work and set off for the tram station when the tram station is closing down. Therefore half the trams don't leave, ready for tomorrow, and so only depart every 20 minutes, and hence your increased waiting time.

• sorry, no! You leave work hours before service stops. This isn't a thinking-outside-the-box question in terms of what I'd call boundary conditions. – Swiss Frank Sep 6 '18 at 14:02

Does it have to do with the mean and standard deviation being identical? If so this is a property of exponential distributions which can be memoryless which would mean the current time is important and for someone reason the person doesn't seem to care about the time at which they leave ... so I would assume this happening is pure coincide. (never really got to deep into stats but this is what I've read from some wikipedia pages lol, seemed like a good lead)

• Interesting angles, but this isn't meant to require academic statistics to understand. – Swiss Frank Sep 6 '18 at 19:27

My guess:

The train stays at the station near your house longer than it stays at the station near your work. Therefore, even though they leave every 10 minutes, your more likely to find the train at the station leaving home and not have to wait for the next train

Perhaps:

You live near the terminus of the tram line.

So at the home side:

You can either get on the tram as it is coming 'from work' and wait on the tram as it turns around/switches tracks/however your trams run, or get on when it is going 'to work'.

This means:

Effectively you get 'two trams' every 10 minutes, and depending on how long it takes for the tram to switch direction you get a lower waiting time than any other en-route stop.