Side_1_360

where φadenotes the average size (in bits) of a
requested file in aggregate a. The first factor in
equation (14) represents the number of packets
that must be transmitted and the second factor
represents the number of times each packet is
transmitted before it is successfully received.
Inserting equations (13) and (14) into equation
(12) we notice that the average number srof
TCP flows in progress on path ris constant

sa= νaφa/ μacc.

Note, however, that the request completion time
will depend on the performance of the LSP,
hence the satisfied demand, i.e. the session
throughput per time unit, will increase the better
the performance, and vice versa.

The total rate at which packets are offered to a
TCP connection is adjusted in response to the
congestion encountered along the route. We
therefore adjust the offered load ρto match the
user packet rate λas

(15)

Equations (6) through (15) define a system of
two simultaneous non-linear equations

ρr= fr(ρr, ρr')

ρr'= fr'(ρr, ρr')

which can be solved numerically by applying
Newton’s method with numerically computed
Jacobians.

The revenue function corresponding to the ob-
jective function (2) when ais a TCP aggregate is
defined similarly, but with the “hard” blocking
probability E(⋅,⋅) enforced by CAC replaced by
the “soft” blocking probability reflected in the
attraction factor α, hence

F(a, xa) = θaνaφaαa (16)

where θais the revenue per bit, νaφais the num-
ber of bits offered per time unit and αa= αa(xa)
is the fraction of bits carried under configuration
(xa).

Note that in more elaborate models of TCP and
possibly also in models of higher layer applica-
tion protocols, the blocking factor αmay include
“hard” events related to protocols. In TCP e.g., a
retransmission time-out failure will occur when
repeated losses and time-outs have forced the
value of the time-out to its maximum value 64
seconds.

5.1.2 Extension to Multi-Path Routing

The model can readily be extended to the case

where several LSPs are used for each TCP

aggregate. Let there be Ksuch LSPs and let

superscript (k) refer to the kthLSP. Let the frac-

tion of the traffic carried on LSP kbe denoted by

φ(k)and assume that traffic is balanced to yield

equal loads on all paths

Equal loads (and equal buffer memories) result

in equal packet loss probabilities eand equal

queues qfor all paths. An average user is

assumed to see an average path which behaves

like a weighted sum of the individual paths.

Re-writing equation (8) for the average packet

transmission time results in

(17)

since the scaling factors φ(k)which apply to γ

cancel out. Rewriting equation (17) as yr(k)=

Ar/μ(rk)and letting μr= ΣKk=1μr(k), the average

packet transmission time xis

(18)

which implies that the net effect of distributing

transmission capacity between Kequally loaded

paths is that the transmission time is scaled by K.

Finally, the average window sizes w, user packet

rates λ, and number of transfers in progress sare

obtained for multiple paths as above.

5.2 An Application

Consider the network [16] presented in Figure 5,

which is a fictitious representation of the core

NSF ATM backbone consisting of 8 nodes con-

nected by 20 uni-directional links. The transmis-

sion capacity of each link is 5,624 units (1 unit

equals 64 kbits/sec). The double lines between

nodes 3 and 4 and nodes 7 and 8 indicate that

two uni-directional links connect these nodes.

The network carries two TCP services which are

aimed at domestic and business users respec-

tively. The two categories differ in terms of the

speed of their access links and the sizes of their

requests. For domestic users we have μacc= 48

kbps and φ= 30 kbytes, while for business users

ρr=

saλrpusr+sa′λr′(1−er′)pack

μr

.

φ(k)=

λ(k)

∑K

k=1λ(k)

=

μ(k)

∑K

k=1μ(k)

.

y(rk)=

γrpusr+γr′(1−er′)pack

γr+γr′(1−er′)

1

μ(rk)

yr=

∑K

k=1

α(k)

Ar

μ(rk)

=

∑K

k=1

μ(rk)

μr

Ar

μ(rk)

=

∑K

k=1

Ar

μr

=

KAr

μr