Mr Beishuizen is head of operations at an air-traffic control centre near the Dutch town of Maastricht, which looks after aircraft movements in the Netherlands, Belgium, Luxembourg and western Germany. We were in the windowless and dimly lit operations room from which hundreds of flights are guided every day.
The controllers, mostly middle-aged men, talked in nonchalant English to the pilots whose aircraft were visible on their screens, breaking off occasionally for a chat with a neighbour or a stroll across the room.
But outside was chaos. It was a Friday afternoon at the end of July, the peak of the holiday season, and all over the Continent aircraft full of passengers were waiting on the Tarmac to take off or swooping in gentle circles waiting to land.
Not only is there a shortage of airport capacity in Europe, with too many aircraft trying to take off and land, but the air-traffic control system is creaking under the strain. It is likely to get worse.
After years of muttering, European airlines have finally begun to complain. Last month, they issued a stinging warning that the future growth of air travel in Europe will be under threat unless the problems are sorted out. They claimed that billions of pounds are lost every year because of the delays.
The airlines' figures are suspect - they valued passengers' lost time at pounds 32 an hour, a price they are strangely reluctant to pay when they are responsible for delays. But they rightly pointed out that keeping aircraft waiting on the taxiway or in the air costs money - which must in the end be paid by the travelling public.
Air-traffic control in Europe is a mess. The 23 countries that belong to the European Civil Aviation Conference have 51 control centres with equipment from 18 different manufacturers, 22 different computer systems and 33 programming languages. This is because governments often buy from domestic firms - Marconi, Siemens and Thomson are among the big manufacturers - rather than scour the world for the best equipment at the best price.
Attempts to give Europe an integrated system have largely failed. The European Organisation for the Safety of Air Navigation, known as Eurocontrol, was set up in 1961 with the aim of building a single system. In 1981, its charter was changed to reflect more modest aims.
The Maastricht centre is Eurocontrol's main claim to fame - four countries out of 14 members have put good sense before national pride and pooled their systems. Not even they have done it completely: Maastricht's jurisdiction starts 19,000ft (5,800m) above the ground in Luxembourg and Belgium, but 30,000ft (9,150m) above the Netherlands.
To see why the European system fails, follow the progress of a single flight. The airline fills in a form detailing its route and timing, which is sent by telex to the relevant air-traffic control centres. The flight is guided on to the Tarmac by airport ground controllers. Once in the air, it is handled by a succession of specialist controllers to get it into upper airspace, across the territory of perhaps three or four different countries, back down again for the approach to its destination, and then on to the Tarmac at the airport there.
But things are improving. On 1 January, Maastricht will take control of Dutch airspace between 24,500ft and 30,000ft (7,500-9,150m). Spain and Italy will join Eurocontrol next year, and Denmark is putting out feelers. With those three, the organisation would cover all 12 EC countries, plus Malta, Cyprus, Switzerland, Turkey and Hungary.
The biggest surprise about Maastricht's control room is how old-fashioned everything is. The big radar screens are pin-sharp but not very bright. Aircraft are indicated by dots marked with flight numbers, a solid line showing where each one is likely to be a minute from now, and a dotted line showing where it was a minute ago.
You peer at the screen in horror, fearing that two crossed lines indicate a pair of airliners are about to crash, and then sit back in relief as you realise from the numbers next to the dots that one plane is 10,000ft (3,000m) above the other.
None of this would pass muster in an arcade game, let alone on a personal computer. In a market where manufacturers and software designers had to make their products easy to use, colour or flashing lights or tones would be used to indicate which planes are flying at which heights. And although some rudimentary early warning systems are in place at Maastricht, it is largely up to the controllers to keep an eye out for impending collisions.
So little of the flight data comes into the computer systems automatically that busy sectors are handled by three people: two to handle the input of information, one to talk to pilots. And when a plane passes from one centre to the next, the controller has to pick up a phone, dial the other centre and warn a counterpart that a flight is coming.
'It may only take five seconds,' says Mr Beishuizen, 'but that means five simultaneous seconds for both controllers. Since one may be busy, and the aircraft is flying at 400 knots (460mph/740kph), it can be very stressful waiting on the line.'
Maastricht is only now installing a prototype system to pass flight data directly from the computer of one control centre to the computer of the next.
Calculating from the technical specifications and present traffic levels, the system would be expected to produce one accident every three years. In fact, nobody at Maastricht can remember the last time a plane crashed in Europe because of air-traffic control. 'There was one in the late Sixties, I think, above Zagreb,' said the public relations man.
Yet the Continent's admirable safety record has its price. When a sector becomes too crowded, the controllers simply tell pilots to stay outside it until the traffic has thinned. When there are too many aircraft in the sky, others have to be told not to take off. No wonder 21 per cent of European flights are delayed.
The old philosophy at Eurocontrol was that the Continent needs a single giant control centre. But that idea is out of date. Technology allows controllers to talk just as easily to colleagues 1,000 miles away as to those across the room.
More recently, Eurocontrol has been pressing for governments to make their systems compatible, with information transferred electronically between one centre and another. The project is called Eatchip: the European Air Traffic Control Harmonisation and Integration Programme.
Last month, Karel Van Miert, the Belgian member of the European Commission, who is responsible for transport policy, put forward draft rules that will force governments to buy control equipment compatible with a single European standard, but for the rules to come into force, all 12 member states have to approve them.
It will be at least a decade before a harmonised system begins to take shape. By then, the bureaucrats will face a different question: has the whole idea behind today's air-traffic control become out of date?
(Photograph and maps omitted)