Side_1_360

nected. An infinite1)population of users request
file downloads of e.g. web pages from servers.
Both users and servers have direct access to
ELRs of an MPLS network. The bandwidth of
the access links is limited on the user side and
infinite on the server side. No losses occur on
the access links. The capacity of the servers is
also assumed to be infinite.

Consider two ELRs iand econnected by an LSP
rfrom ito eand an LSP r'from eto i. A server
Ais attached to router iand a user ais attached
to router e. Figure 3 illustrates the packet flow
when a user arequests service from server A:
data packets of average length pusrare trans-
ferred from the server Ato the user avia LSP r
and acknowledgement packets of average length
packare returned from the user ato server Avia
LSP r'. The situation when a user battached to
the router irequests service from a server B
attached to the router eis also illustrated in Fig-
ure 3. Each LSP rand r'may thus carry both
data and acknowledgement packets.

We model an edge router transmitting packets
on an LSP rby an M/M/1/Kqueue which is
characterised by its transmission rate μrbits per
second and the maximum number K– 1 of pack-
ets that it can store in its buffer memory. The
packets in this model represent the weighted
average of the data and acknowledgement pack-
ets – see equations (7) and (15) below. With ref-
erence to Figure 4, let γrdenote the rate at which
servers offer data packets to LSP rand let γr'
denote the rate at which servers offer data pack-
ets to LSP r'. Let er'denote the loss probability
for packets on path r'. Then γr'(1 – er') denotes
the rate at which users offer acknowledgement
packets to LSP rto acknowledge the data pack-
ets that they have successfully received from
servers.

The performance of each LSP rwill be defined
by a fixed point equation in a multi-dimensional
space which includes the packet loss probability
e, the load ρoffered to the LSP, the round trip
time t, the user window size w, the packet rate λ
per TCP flow and the number sof TCP flows in
progress. Since packets flow in both directions
we are looking for a fixed point which consists
of two sets {e,ρ,t,w,λ,s}: one for LSP rand one
for LSP r'. The same reasoning, mutatis mutan-
dis, applies to acknowledgements on LSP r'.

The first stepin the derivation of the fixed point
equation is to derive an expression for the round
trip time, which is the average time taken for a
data packet to be transmitted from the server to

the user and an acknowledgement packet to be

returned from the user to the server. We begin

by observing that in terms of the M/M/1/K

model the probability ethat a packet will be

lost is related to the offered load ras

(6)

The rate γat which data packets are offered to

the path can be computed from the equation

(7)

The average time yrto transmit a packet is then

given by

(8)

The average number qof packets in the system

is given by the M/M/1/Kformula

(9)

Figure 4 Model of two edge

routers connected by LSPs

1)Throughout this section “infinite” denotes a value which is large enough for its precise value to be

irrelevant.

Figure 3 Packet flow in the

forward and reverse directions

A

i

b

a

e

B

dataA

r

ackb r ́

dataB

acka

acka

dataB

ackb

dataA

A:γr

i

e

b:γr ́ (1-er ́)

B:γr ́

a:γr(1-er)

er ́

er

r

r ́

μr

μr ́

er=

ρKr

∑K

k=0ρ

kr=

{ 1 −ρr

1 −ρKr+1ρ

K

r ρr
=1

1

K+1 ρr=1.

ρr=

γrpusr+γr′(1−er′)pack

μr

.

ρr=

γrpusr+γr′(1−er′)pack

μr

.

qr=

∑K

k=0

k

ρkr

∑K

k=0ρkr

=

{

ρr

1 −ρr

1 −(K+1)ρKr+KρKr+1

K^1 −ρKr+1 ρr
=1

2 ρr=1.