Modelling Communication Networks Frank Kelly University of Cambridge

- Dynamic routing and stability
- Coalitions and global routing
- Broadband traffic and statistical sharing

Richard Gibbens, Philip Hunt, Peter Key, David Songhurst, Martin Whitehead, Stephen Turner

Robert Clayton

- calls arrive randomly, at rate
`a` - resource has
`C`circuits - accepted calls hold a circuit for a random holding time, with unit mean
- blocked calls are lost
- proportion of calls lost is

**Dynamic routing and stability**- Colitions and global routing
- Broadband traffic and statistical sharing

- respond robustly to failures and overloads
- lessed the impact of forecasting errors
- make use of spare capacity in the network
- permit flexible use of network resources

- instability
- complexity

- Complete graph,
`n`nodes - All links have capacity
`C` - Call routed directly if possible; otherwise one randomly chosed alternative route may be tried

- Arrival rate per link
`a` - Capacity per link
`C` - Let
`B`be the blocking probability

Then, as the number of nodes

- An alternatively routed call uses twice as much capacity as a directly routed call
- At moderate loads this causes a loss of efficiency; and, at critical loads, instability
- Can be controlled by
*trunk reservation*:- allow an alternatively routed call onto a link only if there are more than
`s`free circuits `s`is called the*trunk reservation parameter*

- allow an alternatively routed call onto a link only if there are more than

*dotted*links are overloaded*dashed*links have spare capacity- how should alternative routes be chosen?
- note that there are
`n`(`n`-1)(`n`-2) possible choices in an`n`-node network

Two simple rules,

*sticky principle*(for first-tried route), and*last chance principle*(for trunk reservation priority)

Analytical (black) and simulated (grey)

(USA busy hour)

(European busy hour)

Add a road:

- Initially all delays are 83
- After adding road, all delays are 92
- Effect of adding the road is to increase delay for
*all*users - In congested networks, a
*user equilibrium*may be far from a*system equilibrium*

- Dynamic routing and stability
**Coalitions and global routing**- Broadband traffic and statistical sharing

- Let
`N`be set of players - A
*coalition*is a subset`S`of`N` - Which coalitions are likely to be stable against competing, overlapping coalitions?

Define the *value* of a coalition `S` to be:

`V`(`S`) = [cost separate] - [cost with coalition `S`]

Subset S | Separate | Coalition | V(S) | Saving |
---|---|---|---|---|

UK,USA,J A,UK,USA F,J,USA A,F,USA UK,C,J A,UK,C |
13895 12610 6904 5600 3995 3869 |
11134 10406 5609 4801 3199 3127 |
2761 2204 1295 799 796 742 |
20% 17% 19% 14% 20% 19% |

Subset S | Separate | Coalition | V(S) | Saving |
---|---|---|---|---|

A,J,UK,USA F,J,UK,USA A,F,J,USA C,J,UK,USA |
18558 20248 18847 15860 |
13573 16733 15982 13044 |
4985 3515 2865 2816 |
27% 17% 15% 18% |

A,F,J,UK,USA A,C,J,UK,USA |
24990 20667 |
19188 15570 |
5802 5097 |
23% 25% |

The *core* is the set of vectors (`x`(`i`), `i` a player) satisfying:

The *least core* is the smallest epsilon-core

A | F | J | UK | USA | Winner |
---|---|---|---|---|---|

409 | 409 | 2528 | 409 | 2048 | J,USA |

409 | 409 | 2528 | 2048 | 409 | J,UK |

1877 | 409 | 886 | 2221 | 409 | A,UK |

1877 1877 | 409 409 | 2528 2528 | 409 579 |
579 409 | A,J A,J |

429 409 | 409 409 | 886 906 | 3670 3670 |
409 409 | UK UK |

Shapley values | |||||

826 | 415 | 1076 | 1479 | 2003 |

- Dynamic routing and stability
- Coalitions and global routing
**Broadband traffic and statistical sharing**

- Ethernet LAN traffic measurements
- Leland, Taqqu, Willinger and Wilson
- 27 consecutive hours - packet level resolution

15 sources with peak 4 and mean 2.6

*Black* case:
60 sources with peak 1 and mean 0.5

15 sources with peak 4 and mean 1.0

These two cases have

- the
*same*collection of peak rates - the
*same*mean and variance for total load

Statistical sharing will be essential for broadband networks (as it has been for telephone and computer communication networkds)

Two major issues:

- statistical characterization of traffic
- importance of tail probabilities

- effective bandwidth

As `z` increases, the effective bandwidth increases from the *mean* to the *maximum* of `X`(`t`)/`t`

- On/Off source

Effective bandwidth as a function of*mean*and*peak*

- Periodic source, with random phase

Effective bandwidth, as a function of`z`and`t`

- Fractional Gaussian source

- Periodic On/Off source

- The effective bandwidth of a source depends sensitively upon its statistical characteristics
- Often a source may know its peak rate, but not its mean rate
- Can tariffs be designed that perform the dual role of
- conveying information to the network that allows more efficient statistical sharing
- conveying feedback to the source about the congestion it is causing?

Overall aim is to design simple tariff structures, admission controls and routing schemes which permit the entire system, comprising many networks and huge numbers of users, to function coherently.

Issues:

- stability, robustness (traditionally an engineering issue, involving randomness and feedback)
- coordination of players (traditionally an economic issue, involving externalities and incentive compatability)

- Dynamic routing
- Coalitions and global routing
- Broadband traffic and statistical sharing

Further information and references are available at the Clifford Paterson home page