## Trams and Runners

At this time of year trams run every 15 minutes in both directions along the tram line going the 6km south from Little Bispham (Bisp-ham) to Central Pier. Trams take 20 minutes to make the journey. The line is a good deal longer than just between these two stops.

1. Can you make sense of this situation by drawing a diagram?

If it helps you to have more detail, trams at Little Bispham set off south at two minutes past the hour (and every 15 minutes thereafter) and arrive at Little Bispham at two minutes before the hour and every 15 minutes thereafter.

2. If you were on any of these trams, how many would you meet going in the opposite direction?

3. If all trams are running to time, what are the times in which any two trams on this stretch of line meet in the first twenty minutes of any hour ?

4. There are two places just outside this strecth of line, that is just north of Little Bispham and just south of Central Pier, where trains are expected to meet. At what distances are these two places, measured from the named stations?

This stretch of line runs along the Promenade, right beside the sea and with the Blackpool Illuminations right beside the track for most of this length. As training for the running races along the paths that run along the same stretch of coast, I run beside the tram line several days a week.

5. I have noticed that on some days I am not overtaken by a tram at all. If we assume that the trams are running on time, what interval between 10:00 and 10:30 gives me the best opportunity to not be passed by a tram? Assuming I just succeed in not being overtaken, how long does it take me to run the 6km stretch? What then do you think my 10km time might be?

6. That result gives you a slowest running speed that achieves 'not being passed by a tram’. I estimate that about one day in three I am overtaken by a tram. Working to the nearest whole minute, work out how long does that suggests I take to run the 6km - and therefore the 10km.

7. If my counting is wrong and I am overtaken two days out of three, what are the new numbers?

Of course, trams stop at the tram stops to let people on and off, so it really doesn’t take very long before the trams are not exactly running to the timetable. So all these results cannot be taken precisely and we might expect trams to be early or late by up to a minute. So each tram lies inside a time-window of two minutes.

8 You have a time diagram from Q1. If you were going to modify this to show trams stopping, what would you do? Draw one such adjusted line. Given the information about lateness, can you draw an ‘envelope’, the time-space on the graph into which the tram should be expected to fit?

Can you see that the line manager might want to have a measure of the probability of a tram being on time? Can you see that a customer might be aiming to catch the 10:02 and be very annoyed if the tram has already left? If everyone uses their timepieces correctly, and if the customer recognises that “10:02" might be very close to 10:01 on his/her watch, then them missing the tram means it was more than a minute early. The tram being late is not the same problem at all. But if the tram is early and loses customers (those who were trying to be exact, or simply pushing the possibilities, pushing at their own time envelope) or if it merely annoys customers, then the business loses popularity. And therefore business. The trouble here is that the tram will be early only if there are only a few customers, so it becomes actually more important to be on the schedule, because the potential for annoyed customers has gone up, so the fraction of annoyed customers rises dramatically. The annoyance does not compare with the few customers who arrive a small amount of time quicker than expected. Oddly enough, what everyone wants is reliability. In the summer months at Blackpool, the tram system runs more slowly, as many more customers are holidaymakers (and in no hurry), so there is an awful lot more passenger traffic. In compensation, trams run 5 or 6 per hour, not the 4 on this page, which is the off-season frequency. In the high season, the proportion of people relying on the timetable is relatively low and, at 6 per hour, the maximum wait for a tram is ten minutes (unless they’re so full you can’t get on the ‘next’ one). How do you think that affects the local who uses the tram as a commuter?

DJS 20180128

1. One way to draw this is to draw a time line down the page. One the left side of the page mark Little Bispham and mark Central Pier on the other side and draw diagonals for the trams going in that direction, so that any tram's position can be seen at any time. We are drawing average speed. In the other direction (diagonals the other way) put the trams going from Central Pier towards Little Bispham. I find that every tram meets two other trams in this stretch of line [Q2].

3. If every tram is exactly on time, then trams meet every eighth of an hour. They must meet just north of Little Bispham on the hour, so they must meet every 15 minutes somewhere between the two stations. The 10:02:00 tram from Little Bispham will also meet  a tram just south of central Pier at 10:22:30 and so we have a second set at 15 minutes offset from this. Trams meet at 10:00:00,  10:07:30. 10:15:00.

4. The northern crossing point is two minutes north of Little Bispham and the sourthern crossing is half a minute south of Central Pier. [Really!] The trams cover 6km in twenty minutes, so two minutes is 600m and half a minute is 150m. Answers: 600m north of Little Bispham and 150m south of Central Pier. Assumption that everything is incredibly precise.

5. We assume I run slower than a tram, so the runner’s line on the diagram is steeper. It needs to leave Little Bispham just behind the 10:02 tram and arrive just in front of the 10:37 at Central Pier. That’s 35 minutes for 6km and a 10km run would scale up to 58:20, if we assume no change in pace (which we might, since 6km along the coast suggests that the total daily run is at least 12km, so the 10km time might be quicker, say 58 minutes.

6. This is starting to get messy, but it means I take ⅔ of the 35 minutes, say 24 minutes. If I arrive at any time between 10:02 and 10:12, I wil not be caught by the 10:02. The cycle repeats every 15 minutes, so between 10:17 and 10:27 I have the same experience. If I arrive at the start (Little Bispham) sufficiently randomly then I have 10 minutes in every 15 that give me a no-overtake run. So at 24 minutes for 6km I’m looking at 40 minutes for a 10km race; very much better and perhaps rather better than actually happens when racing.

7  I must be running between 24 and 35 minutes for the stretch, probably close to 30 minutes. To double the frequency of being overtaken I need 5 minutes in every 15 minute cycle available, so 10:02 to10:07 at Little Bispham means arriving between10:32 and 10:37 at Central Pier. This is, indeed, 30 minutes for 6km, or 50 minutes for 10km. This makes sense, since my current expectation is 45 minutes for the next 10km race I have (but not longer than that, that is a time for running only 10km, not further).

8. The straight line now becomes stepped (short horizontal lines added), to represent stopping. There are 14 stops between these two stations and not all will be used on every trip. So the envelope is the same line only drawn with a very much wider writing nib, since the line has an offset of a minute each side; the tram is expected to be in this window most of the time.

9. Discussion, hopefully leading you towards seeing how probability might be a tool for management information. Questions on this sort of topic are only difficult because we don’t teach the tools for handling them. This page may help.

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