Side_1_360

on the LSP. The transmission capacity of each

physical link is 10,000 units. The 25 O-D pairs

that carry low traffic are connected by one trans-

mission link, the 37 O-D pairs that carry more

traffic are connected by two links, 8 O-D pairs

are connected by three links and 1 O-D pair is

connected by four links. The physical links con-

tain 1,270,000 units of transmission capacity.

The traffic load offered to each aggregate ais

ρa= 1.0. The revenue rates

ensure that narrowband connections do not

exclude broadband connections from service.

The XFG algorithm requires 7 minutes of execu-

tion on a Pentium III 1.5 GHz processor to com-

pute the solution. The 1,270,000 units of physi-

cal transmission capacity are used to configure

13,232 LSPs. With reference to Table 1, 148,839

units of bandwidth are configured on 426

(71×6) single link LSPs (direct routes) and

1,121,161 units of link bandwidth are used

to configure 151,212 units of bandwidth on

12,806 multi-link LSPs (transit routes).

The lengths of the LSPs and their bandwidth

assignments are shown in Table 1. Let urdenote

the un-normalised lengthof an LSP rwhich is

equal to the number of links in the route. Let nr

= ur– mrdenote the normalised lengthof an

LSP r,where mris the number of links in the

shortest LSP connecting the O-D pair. A route r

with normalised length nr= 0 is thus the shortest

route connecting the O-D pair, and a route r

with normalised length n(r) = 1 is thus one link

longer than the shortest route connecting the

O-D pair. 74 % of the LSPs are constructed on

the shortest and the shortest-but-one paths con-

necting the O-D pairs: 91 % of the bandwidth is

configured on these shortest LSPs.

5 An Objective Function

for TCP

The LSP design problem presented in Section

4.2.1 – 4.2.3 and the application in Section 4.3

made use of the Erlang-B formula as an objec-

tive function. This approach is suited to service

classes such as voice, for which CAC and equiv-

alent bandwidth are applicable. However, it does

not work for other service classes such as data

which typically are connectionless (hence CAC

does not apply) and adapt their transmission

rates to the congestion encountered (hence

equivalent bandwidth does not apply). The dom-

inating protocol for such services is TCP. In this

section we present an objective function which

can be used to take the main features of TCP

into account when finding and capacitating opti-

mal LSPs. The work is inspired by [17] and also

related to, e.g. [18, 19, 20].

5.1 A Simple Model of TCP

Traffic Performance

This section presents a simple model [21] of

TCP Reno [22, 23] that is restricted to (i) single

path transfers, (ii) bulk data transfers for which

the initial slow start phase of TCP can be

neglected, and (iii) low loss systems for which

time-outs can be neglected. The model is based

on first order approximations and mean field

theory, where all relevant parameters can be

described by their averages and these averages

apply to all flows in an aggregate.

The assumptions (i) – (iii) as well as the consid-

erable modelling simplifications below clearly

suggest that the numerical results obtained may

be questioned. The point is, however, not to pro-

vide detailed quantitativeresults but to show

how TCP traffic can be included from a qualita-

tivepoint of view. This is an important point,

because the very nature of TCP traffic is differ-

ent from traditional services in terms of e.g. the

best effort concept and the ability to adapt to

congestion.

Work in progress includes a more elaborate TCP

model which also accounts for short file trans-

fers, slow start and time-outs. The queuing/loss

model is also elaborated upon.

5.1.1 A Single Path Model

Consider a uni-directional path between two

edge routers to which users and servers are con-

Route length Bandwidths LSPs

Ci C'i Li L'i

0 284,074 0 8,591 0

1 11,885 148,839 1,191 426

2 10,753 21,781 1,099 718

3 7,527 24,859 812 1,056

4 5,050 24,212 552 1,253

5 2,824 22,244 426 1,359

6 1,806 17,586 230 1,303

7 1,023 13,472 163 1,170

8 544 10,920 79 1,082

9 321 8,946 50 998

10 78 7,192 18 877

>10 81 25,915 21 2,990

Total 325,966 325,966 13,232 13,232

Table 1 Distribution of the
allocated bandwidth Cion
LSPs of normalised length i;
distribution of the allocated
bandwidth C'ion LSPs of un-
normalised length i; distribu-
tion of the normalised Liand
un-normalised L'iLSP lengths

θa=

⎧

⎨

⎩

1 s(a)=1, 2 , 3 , 4

2 s(a)=5

4 s(a)=6