Side_1_360

The round trip time tof a data packet can now

be calculated as

(10)

where τis the propagation delay. The terms in

the above equation correspond respectively to a

data packet being queued at the forward LSP r,

transmitted on the forward path, propagated to

the far end, and transmitted on the access link

followed by an acknowledgement packet being

transmitted on the access link, queued at the

backward LSP r', transmitted on the backward

path and propagated to the near end.

The second stepis to compute the rate λof data

packets per TCP flow. We do this by first deriv-

ing an equation which relates the user window

size wto the packet loss probability eby consid-

ering the evolution of the window size over

time, both of which are modelled as discrete in

terms of packets and round trip times respec-

tively. Given the window size and the round

trip time, we can compute the rate λ.

For a TCP Reno connection performing conges-

tion avoidance and for which all losses are

detected by triple acknowledgements such that

fast recovery and fast retransmit apply, we may

write

This equation states that w(n) packets are trans-

mitted during the nthtime interval: the window

size will be incremented by one in the next inter-

val if all of the transmissions are successful and

halved otherwise. The probability of an increase

is (1 – e)w(n)and the probability of a decrease

1 – (1 – e)w(n). A steady state average may now

be obtained by letting n→ , which leads to

Finally, a lower limit of w 1 and an upper limit

of wwmaxare applied. The lower limit models

the protocol while the upper limit can be set to

account for restrictions imposed by the receiver.

Note, however, that the user window size w

refers to an average, hence the bounds are not

strict.

The rate λof data packets per individual TCP

flow is related to the user window size wand the

round trip time tas

(11)

This equation states that one window size of data

is sent during one round trip time after which

any lost packet is re-transmitted until it is suc-

cessful. Again a limit λr λmaxis applied which

can be set to account for the access rate

μacc/pusr.

The third stepin the derivation of the fixed

point equation is to compute an improved esti-

mate of the load ρroffered to LSP rbased on the

performance according to Equation (6) – (11).

Several TCP flows may be in progress in paral-

lel. Let υadenote the total file transfer request

rate (requests per second) on an LSP rsupport-

ing aggregate a. The average number saof TCP

flows in progress is related to the request rate υa

as

(12)

where αais an attraction factor which reflects

the performance experienced by the users, and

nais the total number of packets (including re-

transmissions) transmitted in order to satisfy a

request. The equation follows from Little’s

result where ανis the arrival rate to a system

andn/λis the time spent in the system.

The attraction factor αis intended to model the

relationship between the performance of a net-

work and the traffic offered to it. The rationale

for this is that users tend to request more web

pages the smaller the presentation delay, and

vice versa. The presentation delay is in turn

related to queuing, transmission, propagation

and loss recovery. Given the constraints imposed

by user access links, it is suggested to represent

presentation delay as being proportional to the

throughput rate λ(1 – e) of un-errored packets

normalised by the maximum rate μacc/ pusrat

which users can extract packets from the access

link. Assuming a linear relationship between

presentation delay and attraction we obtain

(13)

Equation (13) implies that full attraction α= 1

occurs when the bottleneck has been shifted

from the MPLS network to the user access.

The transmission factor is obtained as

(14)

≤

tr=qryr+

pusr

μr

+τ+

pusr

μacc

+

pack

μacc

+qr′xr′+

pack

μr′

+τ

wr(n+1)=(wr(n) + 1)(1−er)wr(n)

+

wr(n)

2

(

1 −(1−er)wr(n)

)

.

∞

wr=(wr+ 1)(1−er)wr+wr
2
(1−(1−er)wr).

≥

≤

λr=

wr

tr+trer/(1−er)

.

sa=αaυa

na

λr

αa=

λr(1−er)

μacc/pusr

.

na=

φa

pusr

1

1 −er