Roeeediog
of
the
40th
IEEE
Conference
on
Decision
and
Conel
Orbdo,
Florida
USA,
December
2001
Scalable Laws for Stable Network Congestion Control
Fernando
Paganinil
John Doyle2
Steven
Low3
Abstract
This paper discusses flow control in networks, in
which sources control their rates based on feed-
back signals received from the network links,
a
fea-
ture present in current TCP protocols. We de-
velop
a
congestion control system which is arbitrar-
ily scalable, in the sense that its stability is main-
tained for arbitrary network topologies and arbi-
trary amounts of delay. Such a system can be im-
plemented in a decentralized way with information
currently available in networks plus a small amount
of additional signaling.
1
Introduction
Flow control in a communication network like
the Internet concerns the adjustment of individual
source transmission rates so that network resources
are fully utilized, and link capacities are not ex-
ceeded. Existing protocols such as TCP regulate
their transmission rates by
a
congestion window
of
outstanding packets; feedback signals from the net-
work (packet acknowledgment or loss) are used to
dynamically adjust this window to match available
capacity. The additive increase multiplicative de-
crease
(AIMD)
algorithm of TCP Tahoe
[6]
and its
enhancements (Reno, etc.) has performed remark-
ably well as the network scales up several orders of
magnitude. However future growth combined with
the upcoming diversity of traffic and communica-
tion substrates will stretch the limits of this simple
algorithm, calling for
a
more extensive analytical
investigation of the problem.
Exciting progress has been made very recently
tw
wards a theoretical understanding of TCP, both of
its equilibrium, using optimization theory, and its
dynamics, using control theory. Key to these de-
velopments has been to explicitly model the feed-
back signal generated by links and communicated
to sources; in practice these signals correspond to
packet dropping, or packet marking when Explicit
Congestion Notification (ECN) bits are available.
Interpreting these signals as
prices
has allowed for
the rate assignment problem to be cast in
sup-
plyldemand terms [8]; in a related interpretation
[13], prices correspond to Lagrange multipliers of
a certain optimization problem. When applied to
the existing variants of TCP, this framework allows
for the analysis of the resulting equilibria
[12].
In terms of the
dynamics
of TCP, very recently
an analytical model has been developed
[18] that
correlates well with standard simulators, and al-
lows for stability studies
[5]. It is found that TCP
exhibits oscillatory instabilities as delays grow, and
perhaps more surprisingly, as link capacities grow.
This motivates the investieation of new
nrotocols
..
where one could provide guarantees of stability, ro-
bustness to delay and scalability to large capacities
and arbitrary topologies.
Some work in recent years coming from the con-
trol community
(e.g. [I, 19, 161) has investigated
both classical and modern tools in these problems,
however results are mainly confined to single bot-
tleneck networks. For arbitrary networks, the work
on price signals bas allowed for global stability re-
sults
[8,
13, 20, 101, but only when ignoring de-
lays. When considering delays, one is forced to
slow down the control gains to retain stability (see,
e.g.
[8,
13, 211). As noted in [9], this mechanism
is already implicit
in window-based protocols and
it is therefore conceivable that one could obtain
stability for arbitrary delays. In this vein, recent
work in [7, 17,
221 has derived conditions for delay
rohustness for the protocols in
181.
In this paper we pose the objective of finding a
protocol which can be implemented in a decentral-
ized way by sources and routers, and satisfies some
basic objectives: efficient use of network resources
in equilibrium, and local stability for arbitrary ca-
pacities, delays, and routing. These requirements,
laid out in Section 3, lead us to adopt some con-
ditions for the linearized dynamics: integration at
the links, and certain scalings on the gain at sources
and links. In Section
4
we then prove that a system
with these properties will satisfy our requirements,
combining ideas in [7] with methods of
multivari-
able robust control. In Section
5
we show how
'UCLA
Electrical Engineering,
LOS
~~~~l~~, CA
90095.
to implement the source control laws by nonlin-
1594.
Corresponding author.
E~~~I:
paganini@ee.ue~a.edu.
ear functions which around equilibrium provide the
ZControl
and
Dvnamical Svstems. Caltech.
Pasadena.
correct dvnamic linearization. and in Section 6 we
CA
91125.
give some comments on packet-level implementa-
3Computer Science
&
Electrical Engineering, Caltech,
Pasadena, CA
91125.
tion. Conclusions are given in Section
7.
0-7803-7061-9/01/$10.00
O
2001
IEEE
185
2
Problem Formulation
We are concerned with a system of
L
communi-
cation links shared by a set of
S
sources. For each
link
1
we have: the capacity CI, the aggregate rate
yl of all flows through the link, and the price sig-
nal
pi. For each source
i
we have the source rate
xi,
and the aggregate price qi of all links used by
source
i.
We will work with
pow
models, i.e. re-
gard these variables as deterministic, non-negative
real-valued. We use vector notation to collect the
above variables across all links or sources; thus we
definec,y,p~R~,andz,q~R~.
We now model the network, including the propa-
gation delays, in terms of the relationships (in the
Laplace domain, denotes transpose)
~(s)
=
Rf(~)z(s), (1)
q(s)
=
Rb(~)~p(s).
(2)
Here Rf and Rb are the delayed forward and back-
ward routing matrices, defined by
q
=
qo
+
6q. Assuming the set of bottlenecks is un-
changed by this small perturbation,
6pr is only non-
zero for bottleneck links. Therefore for the local
analysis to follow, we can write the reduced model
where the matrices
R,,
Rb,
and the vectors 615,
6g
are obtained by eliminating the rows correspond-
ing to non-bottleneck links. For simplicity in the
sequel, we drop the "bar" notation and assume the
routing matrices such
as
(3)
refer only to bottleneck
links.
A subtlety arises when employing the linear equa-
tion (5) for incremental flows: a bottleneck link
includes effectively a saturation nonlinearity in its
outgoing flow, preventing an increase
62 in one of
its source flows from propagating to downstream
links. We eliminate this issue by assuming that the
target rate
cr for each link is slightly lower than
its actual capacity (a "virtual" capacity, see
[ll]).
This is also desirable since it leads to empty equi-
eCr!l8
if source
i
uses link
1
[Rf(s)h,i
=
,
(3)
librium queues.
0 otherwise
We will also assume that the matrix R
=
Rt(0)
=
,.
,
and similarly for Rb(s) with
rll
replaced by
rtl.
Rb(0) is of full row rank. This means that there are
link
flows
are
obtained
by
aggregation,
with
no algebraic constraints between bottleneck link
respective delays
rll
,
of source flows using the link;
flows, ruling out, for instance, the situation where
prices seen by sources are aggregations, with delays
all flows through one link also go through another.
rtI, of link prices used by the source. We define the
Note,
that
Our
refer
now
to
total round trip time (RTT) by source,
bottlenecks only, and typically in the above sce-
nario only one of the links would be a bottleneck;
r;
=
~b
+
~f
(4)
SO
our assumption is quite generic
r,l
1.1'
This quantity is available to sources in real time.
3
Control objectives and
a
proposed law
What remains to be specified is: (i) How the links
fix their prices based on link utilization; (ii) how
We now proceed to lay out a series of
objectives
the sources fix their rates based on their aggregate
for the feedback control laws in Purely local (lin-
price. These operations are up to the designer, hut
earized)
terms. These will lead us to conjecture a
have
a
main restriction: both must be
decentml.
candidate local control law, which we argue is the
ized.
For instance the source rate
z,
can only de-
simplest that can satisfy our requirements, and at
pend on the
aggregate price
qi.
~h~
the same time allows for the required decentralized
objective
of
this feedback is
to
allow
for
flows
to
implementation. In the next section we will prove
track ,-hanging conditions in
traffic
demand, link
that it actually achieves our objectives.
capacity, routing, etc. Key design considerations
A first objective is that the target capacity
cr is
are thus to ensure dynamic stability and
regula-
matched at equilibrium; this calls for an integra-
tion of these systems around equilibria that satisfy
tor in the feedback loop. As will become clear in
desirable static properties.
the next section, to have stability we must perform
l-his
paper
is mainly concerned with the dy.
the integration at the lower-dimensional end of the
namic properties around a given equilibrium
point
problem, i.e. at the links. So we write
so, yo,po, qo. For more discussion on equilibria see
Section
5,
but for now we only require that yo1
5
el
PI
=
86~1
(link capacities are not exceeded), and that non-
as
in
[13],
i.e. prices integrating excess capacity.
bottleneck links (those where
yo1
<
cr) have a price
The constant
@I
will be chosen later.
pol
=
0.
The next, main objective is to have local dynamic
Now consider a small perturbation around equi-
stability for arbitrary network delays, link
capaci-
librium;
x
=
so
+
62, y
=
yo
+
6y,
p
=
pa
+
bp, ties and routing topologies.
186
Regarding delays, consider first the case of a sin-
gle link and source. The link integrator, plus the
network delay will yield a term
e-"/s in the loop
transfer function, which leads to instability
as
7
grows, unless the gain is made a function of
T.
In-
deed, introducing
a
gain
l/r
in the loop (specifi-
cally at the source, which measures RTT), gives a
loop transfer function
-,,
e
-
7s
(7)
which is scale-invariant: namely, its frequency re-
sponse is a function of
0
=
TW, so Nyquist plots
for all values of
r
would fall on a single curve
r,
depicted below. If the overall gain is kept under
control, then we can have stability for all
r.
Fur-
ther, closed loop time responses for different
7's
would be the same except for scale.
Summarizing the above requirements in their
simplest possible form, we propose the source con-
trol law (between
69, and bz;) to be the static gain
where
a;
is a parameter. The sign provides nega-
tive feedback. As link control, we take
an
integrator
with gain normalized by capacity,
1
bpi
=
-6~1.
Cl
s
(9)
With this normalization, the price corresponds to a
(virtual) queueing delay at the link, similar to the
case of TCP Vegas
[3,
141.
4
Linear stability results
I
9~
I
I
-.
.
-~
0.
Figure
1:
Nyquist plot of eJ8/j0
Going now to the general network case, we will
extend the above by including a gain
l/ri
at each
source. In the multivariable setting, however, we
must also compensate for the effect of the routing
matrices
Rf, Re; intuitively, as more links partici-
pate in the feedback the gain must be appropriately
scaled down. The difficulty is implementing this in
a
decentralized fashion, without access to the global
routing information.
One source of information that can be exploited
is that at equilibrium, the aggregate source rates
add up to capacity, so
Rzo
=
c. This motivates
1
the following heuristic: introduce (i) a gain
,8l
=-
9
at each link; and (ii) a gain zo; at each source, in
Figure
2:
Overall feedback loop
With the source and link controllers described
above, we proceed to study the linearized stability
of the closed loop. We express the overall,
multi-
variable feedback loop in the classical configuration
of Figure
2,
with open loop transfer function
L(S)
=
R~(S)KRT(S)C$, (10)
where
4
is the identity matrix of size
L,
and
1
X
=
diag(n;),
C
=
diag(-).
4
Note that there are no unstable pole/zero cancel-
lations within
L(s); the proposition below provides
stability conditions for such multivariable loops
with integral control.
Proposition
1
Consider a standard unity feed-
I
back loop, with L(s)
=
yF(s);. Suppose:
addition to the
117;
factor.
In
the case of
a
single
link, and possibly many sources, this gives a loop
(9
F(s) is analytic in Re(s)
>
0, and IIF(s)II
5
transfer function of
0
in Re(s)
2
0.
zoi
e-j,w
L(jw)
=
--,
(ii) F(0) has strictly positive eigenualues
CI
riw
i
-.
~.--
(iii) For all
7
E
(O,l],
-1
is not an eigenvalue of
which is a convex combination of points of the form
L(~w),
w
#
0.
(7);
looking at the figure, it follows that this convex
combination will remain stable by a Nyquist argu-
Then the closed loop is stable for ally
E
(0,1].
ment. In the multiple link case, we are still left with
the need of bounding the gain of the backward path In essence, the above conditions are a "nominal"
Rb;
an analogous strategy is to introduce a gain
&
stability requirement for small y, that says that
at each source,
Mi being the number of bottleneck we have strictly negative feedback of enough rank
links in the source's path. For a discussion on its
to stabilize all the integrators, and
a
"robustness"
implementation, see Section
5.
argument that says we can perform a homotopy to
187
y
=
1
without bifurcating into instability. Details
of the proof are omitted for brevity.
Applying this to the
L(s) in (lo), we take
F(S)
=
R~
(s)KR:(s)c;
we will later add the scaling
7.
Note that (i) is
automatically satisfied. For (ii), note that
eig(F(0))
=
eig(Ci
RKR~C~),
where R
=
Rf (0)
=
Rb(0) is the static routing ma-
trix. Assuming R has full row rank, condition (ii)
will hold. Here we see
the importance of putting
the integrators at the links (the lower dimensional
portion).
If,
instead, we tried to integrate at the
sources, the resulting feedback matrix at DC would
never have enough rank to stabilize the (larger)
number of integrators.
What remains is to establish (iii). For this pur-
pose we bring in some more notation:
Also,
Rb(s)
=
Rf (-~)diag(e-~<~) follows from
(4),
where R?(s)
=
RF(-s) is the adjoint system. We
incorporate this notation to rewrite
L(s), for
s
# 0,
in the more convenient form
L(s)
=
Rr(s)XoMA(s)Rt(s)C. (11)
We now tackle the robustness argument.
Theorem
2
Consider an equilibrium point where
rates match capacity,
i.e. c
=
Rf (0)xo. Let
a,
<
5
and the delays be arbitrary. Then with L(s) as in
(11),
-1
?!
eig(L(iw)), w # 0.
Proof: Since nonzero eigenvalues are invariant un-
der commutation, and also many of the factors in
(11) are diagonal, we observe that
Claim:
This amounts to bounding the spectral radius
Any induced norm will do, but if we use the
1,-
induced (max-row-sum) norm, we find that
1
IICRf (jw)Xollm-in,j
=
max- le-'~lJwxoil
i
uses
I
1
=max-
C
xoi=1;
i
uses
I
note we are dealing with bottlenecks.
Also the
norm
llMRjll is equal to
1,
because each row con-
tains exactly
Mi elements of magnitude 1/Mi. So
p(P)
<
1
as
claimed. Indeed, p(P)
=
1
at w
=
0,
the eigenvector being the vector of all ones.
Now suppose
-1
E
eig(P(jw)A(jw)) for some
w.
We thus have a vector u, lul
=
1
such that
y
=Au, u
=
-Py.
Now
u*y
=
u*Au
=
i
is a convex combination of the {Xi}, which are
points in the curve
r
of Figure 1, scaled by
ai
<
5.
It is clear from the figure that such convex combi-
nations and scaling cannot reach any point in the
half-line (-w,
-11. However, we also have
using (12). So u'y
E
(-w,
-11,
a contradiction.
Remark
1
Some elements of the proof, in particu-
lar the use of
1, induced norms to prove a spectrnl
radius bound, are inspired by the work of [7] for
the control laws in
181. More recently [22] has ex-
tended the stability argument for the laws in
[8] in
a parallel fashion to our work.
Theorem 2 establishes (iii) in Proposition 1; note
that scaling down by
y
is equivalent to making the
a;
smaller. To summarize, we have:
Theorem
3
Let Rf (s), Ra(s) denote the routing
matrices of sources in relation to the bottleneck
links. Suppose
Rf (0)
=
Rb(0) has full row rank,
and that
ai
<
f.
Then the system with source con-
troller gain (8) and link controller (9) is linearly
stable for arbitrary delays and link capacities.
We have modeled a law consisting only of the
integrator and delay dynamics, but otherwise with
only gains at links and sources. Could this stability
argument extend to include additional dynamics?
In this regard, a first comment is that there
can be no more pure integrators. Otherwise the
Nyquist plot in Figure
1
would branch towards
-oo,
and convex combinations of such points could
reach the critical point.
In
particular, the second
order link laws used in
REM
[2] would not qualify.
Given this, and the requirement of scale-invariance
to delay, all additional dynamic terms should be lo-
cated at the sources, so that they can he "clocked"
at the rate of the round-trip time. For instance a
term
A
could be added to the source controller;
restrictions on would he required.
188
5
Implementation
and
the
equilibrium
structure
We have presented
a
linearized control law with
some desirable stability properties. We now discuss
how to embed such linear control laws in a global,
nonlinear control scheme whose equilibrium would
linearize as required.
For control at the links, since the gains are con-
stant it is straightforward to implement the price
dynamics
as
Therefore prices integrate excess capacity in
a
normalized way, and are saturated to be always
non-negative. At equilibrium, bottlenecks with
nonzero price will have
y~o
=
c1 as required. Non-
bottlenecks with
ylo
<
c~ will have zero price.
For the sources, the problem is somewhat more
involved since the linearized gains depend on the
equilibrium rate
so;, and also depend on parame-
ters
T;
and
Mi.
Let us assume those parameters are
constant for the moment, and focus on the mapping
between
q, and
xi.
The simplest thing to try is
a
static, strictly decreasing function
z;
=
f,(q,).
Indeed this choice of source and link laws makes
this control
a
special case of the class of algorithms
proposed in
[13]. These algorithms are interpreted
as carrying out a distributed computation to max-
imize the aggregate source utility
xi
U;(xi), sub-
ject to capacity constraints,
y,
<
cr for all
1.
Here
the strictly concave utility function
U,(.)
represents
the source's demand for rate, and is related to
f;
by
Ui
=
(f;)-'. It follows from the work in [13]
that for arbitrary strictly concave utility functions,
these laws have an unique equilibrium and are sta-
ble in the absence of delay.
The question we ad-
dress here is what constraints our conditions for
linear stability in the presence of delay impose on
f;, and thus on the allowable utility functions.
Now given
an
equilibrium point xo, qo, let us im-
pose the linearization requirement
for some
0
<
a;
<
7712,
as needed in
(8).
Let us as-
sume initially that
a;
is constant. Then the above
differential equation can be solved analytically, and
gives the static source control law
aiqi
--
zi
=
fi(qi)
:=
x,,,,;
e
MIG.
Here
z,,,;
is
a
maximum rate parameter, which
can vary for each source, and can also depend on
Mi,
T;
(but not on q,). So we find that an exponen-
tial backoffof source rates
as
a function of aggregate
price, can provide the desired control law, together
with the link control in (13).
The corresponding
utility function (for which
f;
=
(17:)-')
is
Are these exponential laws the only possible
choice? Among static laws, there is a further degree
of freedom in letting the parameter
a;
be a func-
tion of the operating point. In general, we would
allow any mapping
xi
=
f,(qi) that satisfies the
differential inequality
Essentially, the requirement is that the slope of the
source rate function (the "elasticity" in source de-
mand) decreases with delay
T;; in other words, the
longer the delay, the slower the source control must
respond, in order to avoid instability. Similar con-
siderations apply to the number of bottlenecks
M,.
We could also consider certain dynamic control
laws at the source; it can be shown, however, that
under our stability restrictions, this does not add
any further degrees of freedom to the equilibrium
structure that can be obtained from static laws.
Finally, we emphasize that while the above im-
plementations will behave as required around equi-
librium, we have not proved global convergence to
equilibrium in the nonlinear case.
6
Signaling requirements
We briefly discuss here the information needed at
sources and links to implement our dynamic laws,
and the resulting communication requirements.
Links generate prices by integrating the excess
flow
yl
-
c1 with respect to the virtual capacity;
this is easily implemented by maintaining a "virtual
queue" variable.
Sources must have access to the round-trip time
T,, which can be obtained by timing packets and
their acknowledgments, and two variables from the
network: the aggregate price
q;, and the number
of bottlenecks M,. To communicate
q,, the tech-
nique of random exponential marking
[Z]
can be
used. Here, an
ECN
bit would be marked at each
link
1 with probability
1
-
+-*I,
where
+
>
1
is
a global constant. Assuming independence, the
overall probability that a packet from source
i
gets
marked is (see
[Z])
1
-
+-q<
and therefore q; can be estimated from marking
statistics. Alternatively packet dropping can be
189