Last weekend was the busiest of the year for Eurotunnel, as holidaymakers from France and across Europe head back to Britain for the start of the school year.

As a result, some delays are to be expected.

The staff at Eurotunnel did a very good job at communicating with their passengers. They sent text messages both 2 days beforehand and throughout the day, and responded rapidly on Twitter.

However my tweet went unanswered, and that prompted me to dig a little deeper. Maybe they were just busy, or maybe the question was close to the mark.

It turns out we have a problem similar to airlines overbooking flights, only to get their fingers burnt if everyone actually turns up. This is also an interesting illustration of some statistical sampling.

### Delays in Calais

Cars were arriving at their scheduled time, which is no earlier than 2 hours before departure and no later than 45 minutes before. People were finding that it was taking some time to get to check-in, prompting several tweets by worried customers asking if they'd have to pay to change trains because they didn't arrive in time. They were reassured that, with delays to check-in advertised on the website, this would be taken into allowance, and there'd be no surcharge.

Oh no you will not be charged. As we display a wait pre check-in it is taken into consideration once you actually take your ticket hanger. ^VH

— Eurotunnel Le Shuttle (@LeShuttle) August 31, 2019

They were using a waiting area at Coquelles to sort and filter traffic as it arrived, and the website said that cars were being segregated by booked departure time. That was all good in theory, except that the waiting area was actually just one large queue where you can turn your engine off and stretch your legs. They had about 30 rows of about 30 cars each row. Once each row was full of parked cars, they'd start to fill the next one. Once 30 rows had been parked up, it was your row's turn to leave the waiting area ready for fresh cars to fill your row. So good for the environment: This queue doesn't involve idling engines and cars nudging forwards. But there was no sign of segregation; it was simply first-in, first-out, as they later admitted to someone on Twitter.

HI Sue, I am really sorry this has been changed to the fact that first one arriving on that area will be the first ones leaving to go to check in. Sorry for the confusion. ^VH

— Eurotunnel Le Shuttle (@LeShuttle) August 31, 2019

The company's status page reassured people that, in spite of the delays, trains were still departing on time. One user on Twitter asked how this could be so. People were asked to arrive no more than 2 hours before booked departure. The website advertised delays of 90 minutes to check-in, and delays of 30 minutes for immigration checks. So how, the question came, could someone still leave on time if they were being delayed by the 2 hours they allowed?

@LeShuttle according to your advice no one should arrive more than 2 hrs before their train, but you have 2hrs of delays so nobody will catch their booked train! pic.twitter.com/gsBuTHczYc

— Andrew Robertson (@rockybruce11) August 31, 2019

It turns out that trains were leaving on time, but passengers were not. You simply got boarded onto a train other than the one you booked on, typically about 2 hours later than scheduled.

Will no-one rid me of these troublesome queues?

### Enter Flexiplus+

It turns out there's an answer.

Eurotunnel have, for many years, had a category of ticket they call Flexiplus. The advertised benefit is flexibility. You can book on a crossing, and travel on any train within 12 months of booking. So, if you want to stay an extra weekend in France, you can. All you do is turn up at any time, and you'll board the next available train, with your own queue and your own lounge to wait in. If you've got a 12 hour drive from the French south coast, you don't have to allow several extra hours in case the peripherique misbehaves.

This gives you a way to jump those queues. You can book with Flexiplus, turn up at your advertised time, and you'll go straight through to the next train departing. Mind you, you pay handsomely for the privilege. The tickets start at £219 one-way, and you pay up to an extra £50 for peak times. That's probably why most of the cars in the Flexiplus queue were Range Rovers and Jaguars. Those with money can skip the peak queues.

### Enter more queues

But there is a problem. When they designed this system, they expected a few people at a time to be using Flexiplus. There are 16 check-in lanes at Calais, of which only one is for Flexiplus users, who'll drive straight up to that barrier, check-in, and proceed to the next stage. Except that, on a busy day, they had commandeered two extra lanes for Flexiplus and there was still a long queue taking a good 30 minutes. They didn't get the car park queue, but it was far from instant. One Twitter user was worried that they might miss the train they'd been allocated, making their Flexiplus useless:

@LeShuttle hi we are booked for 706 with Flexiplus, we are in the queue for border control, should we change our booking for a later train? It doesn’t look like we will be through border control within the next 15 minutes

— Guy Flintoft (@Guy_Flintoft) August 31, 2019

They were fine. But it means Flexiplus is being used to beat the queues, not just to have that bit of extra flexibility. Which makes you wonder: Is the delay being made worse because of Flexiplus? In other words, when you only have one or two Flexiplus users arriving each hour, they get slotted into the spaces left free because one or two people will have missed their booked departure. But if you start to get Flexiplus queues, how do they fit everyone on the train?

Let's ask:

@LeShuttle Out of interest, what proportion of #Calais to #Folkestone passenger cars today are #flexiplus? As we're sat at Coquelles watching the queues we're wondering if the flexiplus holders are bumping passengers from the trains they're booked on, materially adding to delays.

— James Oakley (@Oakley_J) August 31, 2019

Just in case the tweet embedding isn't working in your browser, here's what I asked:

@LeShuttle Out of interest, what proportion of #Calais to #Folkestone passenger cars today are #flexiplus? As we're sat at Coquelles watching the queues we're wondering if the flexiplus holders are bumping passengers from the trains they're booked on, materially adding to delays.

No answer.

### So how many Flexiplus bookings?

So let's look a little more closely.

Wandering through the double-height section of the crossing we finally boarded, there were 35 vehicles still displaying their hanger. Flexiplus bookings get a different windscreen hanger, so it was possible to see that 12 out of 35 vehicles were booked on Flexiplus, and 23 out of 35 were booked on a named crossing.

That's about 1/3 of travellers using Flexiplus. At this point, you'd point out that this may not be typical; 35 cars on one train is not a representative sample. Your objection would be correct. So let's calculate a more accurate number.

Let's say the proportion of cars arriving at Coquelles with Flexiplus bookings is p. The Flexiplus vehicles were dotted through the train, not all at one end, so let's make the reasonable assumption that the probability any given car is Flexiplus is p, and that one car is independent of the next.

How many cars out of 35 would you expect to be Flexiplus bookings? That depends on p. If p is 0, you'd expect 0 cars to be Flexiplus; if p is 1, you'd expect all 35 to be Flexiplus. We had 12 out of 35. So what is the value of p?

We cannot tell. But we can make a very informed estimate. Let me try to explain the statistics behind this for those who aren't familiar.

Suppose p was 0. That means 0 cars were Flexiplus. So the chance of getting 12 out of 35 is 0. It won't happen. We can rule out 0.

What if p was 0.1, one car in 10 on average? Technically, this is a binomial distribution. If p = 0.1, the probability of 12 or more cars, out of 35, is 0.00009149. That's slightly under 0.01%, or one in 10,000. That's so unlikely, we can rule that out, too.

For an "asymmetrical" test like this, statisticians typically look for 5%. Any value of p that makes gives a less than 5% chance that you get 12 or more Flexiplus cars out of 35 can be treated as not possible. It turns out you hit this at p = 0.211. If p is any smaller than 0.211, then you're just not going to get 12 or more cars that are Flexiplus

So we don't know exactly how big p is. But we don't just say that it's 12/35. Instead we say, with 95% confidence, that it's at least 0.211. So the proportion of cars arriving at Calais with Flexiplus bookings was 21.1%, or higher. That's more than 1 in 5.

### Is it materially adding to delays?

What we don't know is how much space they allow for these Flexiplus cars. Do they stop taking bookings on each train when it's 79% full? Assuming not, here's what's happening: There are 5 departures an hour at peak times. If each train holds 200 cars, that means 1000 cars are booked on specific trains each hour. So 1000 regular-booked cars turn up. So do 175 Flexiplus cars.

5 trains leave. On time. On those 5 trains are 175 Flexiplus cars, and 825 regular cars. The remaining 175 regular cars will get a train the following hour.

Run that throughout a busy day. By the time you've had 6 hours of this, you've bumped off 1050 cars onto a later crossing. That's a full hour's delay. By the time you've had 12 hours, you've had to delay 2 hours' worth of cars.

And so you ask: Is it any coincidence that, by 7pm, the delays had built up to 2 hours? Looking at their Twitter feed, the same thing happened the following day as well.

So many people are using Flexiplus that, on peak travel days, by the end of the day the queue has built up to 2 hours for regular paying passengers, consistently.

The conclusion is that Eurotunnel need to collect some actual statistics (which they have, even if they didn't choose to share them with me), and work out exactly how popular Flexiplus is. They then need to stop taking booking onto each train at whatever point leaves the correct amount of slots for Flexiplus passengers. They can then take their regular passengers, and their Flexiplus passengers, without huge queues building.

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