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Published by: bingbing88 on 28 June 2010  |   0       | Report post

Call routing
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Nowadays we take it for granted that someone in England can make a phone call to Australia, or that someone in India can read web pages that are on a computer in Canada. We live in a society in which almost every home has its own telephone line which is connected to a local exchange in the nearest village or town, from there to a main exchange in the nearest city, and from there to any other city in any country in the world. In this way, a person is able to dial a friend in another country just as easily as if they were in the same street.
BT’s Worldwide Network Management Centre

Inside BT’s Worldwide Network Management Centre at Oswestry.
(© British Telecommunications plc)

In order that these large and complicated networks can work properly, mathematics and computer simulation have to be used to understand the networks. The aim is to find out how large networks can be designed and controlled to provide reliable communications systems and to use resources efficiently. The reliable and efficient operation of networks is of vital commercial importance to both users and telephone companies, and even modest percentage improvements can correspond to large revenue gains.
BT’s Worldwide Network Management Centre

BT cable laying in the City of London.
(© British Telecommunications plc)

A large network is affected by many factors which are often hard to predict. There can be busy and quiet periods through the day. If a television program has a phone-in vote, there can be a sudden overload at one point in the network. If a digger cuts through a major telephone wire while repairing the road, there can be a sudden and unexpected failure. Mathematicians have developed ingenious ways of routing calls which can cope with these unpredictable events. These routing schemes work by searching out the spare capacity in the network so as to route calls away from parts of the network that are broken or full and into parts that are underloaded.

A good routing strategy doesn’t only need to be able to find the spare capacity in the network. In order to work well in real networks, it needs to be simple, so that thousands of calls a second can be routed instantly. It also needs to be decentralised; a central controller would be much too slow and could go disastrously wrong if it had a power failure, or got cut off from the rest of the network. Mathematicians have recently been able to show the surprising fact that these different goals can all be achieved simultaneously. Even simple, decentralised schemes can find the spare capacity as efficiently as more complicated schemes.
Erlang’s formula
Village cables.

The chances of getting an engaged tone depends
on the number of lines out of the village.

The Danish mathematician A. K. Erlang was the first to study the problem of telephone networks. In 1917, he looked at a village telephone exchange. He supposed that the village has a certain number of telephone lines going from it to the outside world.

We’ll call the number of lines C. People in the village want to make calls to the outside world. We don’t know when they will want to call or how long their calls will last, but let’s suppose that there are on average v calls starting per minute, and that the average length of a call is one minute. Erlang wanted to know what fraction of callers would find that all the C lines leading out of the village were already full, and so would not be able to make their call until later. He worked out this formula, which gives the answer:

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