## Clifford Paterson Lecture 1995

Modelling Communication Networks Frank Kelly University of Cambridge

### Overview

• 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

#### Erlang's Formula

• 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

#### Repacking

> C frequencies in total: reuse possible in non-adjacent cells

### Overview

• Dynamic routing and stability
• Colitions and global routing
• Broadband traffic and statistical sharing

#### Dynamic routing

Aims:
• 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
Problems:
• instability
• complexity

#### Alternative routing

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

#### Alternative routing

• Arrival rate per link a
• Let B be the blocking probability

Then, as the number of nodes n grows, the blocking probability B approaches a solution of

#### Trunk reservation

• 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

#### Traffic mismatches

• 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, trunk reservation and sticky random reset, allow DAR to act as a randomized parallel algorithm, able to solve dynamically a complex maximum flow problem.

#### Sticky routing

Two principles:
• sticky principle (for first-tried route), and
• last chance principle (for trunk reservation priority)

### Node failure

Analytical (black) and simulated (grey)

(USA busy hour)

#### Performance - morning

(European busy hour)

• Initially all delays are 83
• 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

### Overview

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

#### Coalitions

• 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]

#### 3 player coalitions

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%

#### 4,5 player coalitios

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

Recall N is the set of players, and V(S) is the value of the voalition formed by a subset S.

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

#### The least core

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

The least core is the smallest epsilon-core

#### Extreme points of the least core

N={A,F,J,UK,USA}

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

### Overview

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

#### Example: LAN traffic

• Ethernet LAN traffic measurements
• Leland, Taqqu, Willinger and Wilson
• 27 consecutive hours - packet level resolution
Graphics courtesy of Walter Willinger)
28 hours: Time unit - 100 seconds
3 hours: Time unit - 10 seconds
17 minutes: Time unit - 1 second>
2 minutes: Time unit - 0.1 seconds
10 seconds: Time unit - 0.01 seconds

#### Statistical multiplexing

Dashed case: 60 sources with peak 1 and mean 0.1
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

#### Tail probabilities

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
Approack:
• effective bandwidth

#### Effective bandwidth

Let X(t) be the amount of work that is produced by a source in an interval of length t. Define the effective bandwith of the source to be:

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

### Examples

• 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

#### Effective bandwidth and tariffs

• 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?

#### Tariff:

##### peak rate 10 Mb/s

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)
Distinctions disappearing!

### Overview

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

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