﻿ Maths of Planet Earth | Limitless Applications

A planet organized by humans

Published on November 19th, 2013 | by Emily Corbett

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# By Patrick Laub, University of Queensland

When you pick up your phone to call a friend, there’s a lot of work going on in the background to make the connection. Once your phone provider finds the person you’re calling, it then needs to find a route through the network to send your voice messages. Often the voice messages need to go over many hops to get to the receiver; for example if you call a friend in Melbourne from your mobile in Brisbane, then the route might send your messages along these hops:

1. Your mobile to the nearest mobile phone tower
2. Tower to the Brisbane exchange
3. Brisbane exchange to the Sydney exchange
4. Sydney exchange to the Melbourne exchange
5. Melbourne exchange to your friend’s nearest exchange
6. Friend’s exchange to your friend’s house

There is a limitation to this system though; each of these hops (called links) can only handle so many phone calls at the same time. So if the Brisbane-Sydney link is too congested then either the phone call will have to take another route (perhaps through Canberra) or cannot be connected at all. The goal of Teletraffic theory is to create models for these networks, so that you know the likelihood that a call cannot be connected due to congestion. This likelihood is called the route blocking probability and it depends on the network structure and also on the expected rate of traffic along the different telephone routes.

The basic model used for this system is the fixed-route circuit-switched loss network which gives the required blocking probabilities, but there is a problem. Actually calculating the probabilities on a computer is not feasible for any large network (number of routes >= 100). This is why the bulk of Teletraffic theory concerns approximations and simulations. Perhaps the most influential approximation is the Erlang Fixed Point Approximation (EFPA).

Instead of looking at a network with J links, the EFPA splits the problem so that it considers J networks each with one link. This effectively adds an assumption to the model which is definitely not true in the real world, one that says the links block independently, yet remarkably the EFPA still gives extremely accurate blocking probabilities. Even better, EFPA takes little computer time to run.

There are other remarkable properties of this approximation. For a given network configuration, if the supply and demand for each route increases at a similar rate then the error in the EFPA reduces to zero. So starting with a flawed model for a telecommunications network and growing supply and demand (which does happen over time) then the approximation gives you the exact blocking probabilities.

Modern Teletraffic theory hopes to produce more complicated models to better handle situations such as: new forms of traffic, packet-switched networks, backup routes and dynamic routing. More effective telecommunication infrastructure will help all persons and industries in this new information era.

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