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

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
  

A telephone network

Repacking

Cellular radio network

Repacking constraints

Queuing network

Overview

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

Dynamic routing

• 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

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
• Capacity per link C
• Let B be the blocking probability

Instability and Hysteresis

Bistability

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

Minimal loss

Traffic mismatches

• 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

Dual-parented network

Sticky routing

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

Relative loss

Random link failures

Node failure

Analytical (black) and simulated (grey)

International access network

Performance - afternoon

(USA busy hour)

Performance - morning

(European busy hour)

Road traffic example

Braess' Paradox

• 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

Implied costs

Overview

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

Global routing

Traffic profiles

Alternative routes

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

4,5 player coalitios

The core

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

The least core

The least core is the smallest epsilon-core

Extreme points of the least 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

Overflow

Capacity savings

Transit flows

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

Statistical multiplexing

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

Distribution of total load

Tail probabilities

Two major issues:

• statistical characterization of traffic
• importance of tail probabilities

• effective bandwidth

Effective bandwidth

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 2 Mb/s

Tariff:

peak rate 10 Mb/s

Issues:

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

Overview