Proceedings of the European Control Conference 2007
Kos, Greece, July 2-5, 2007
TuB07.2
Control and economics in networking, via optimization:
perspectives from an emerging discipline
Fernando Paganini
Abstract— This paper describes a developing field of
research in which control, networking and economic theory are interacting at an unprecedented level of depth,
over problems of network resource allocation. Networking
provides a large scale application with fully embedded
decentralized control systems, and users with competing
objectives; economics brings market tools to understand
this decentralization of decision makers and objectives;
control theory brings the dynamic viewpoint to fields
accustomed to the equilibrium perspective. The key to a
more unified theory is the language of convex optimization.
We will describe a prototype problem and its current
understanding through a combination of the above viewpoints, and outline other directions of development of this
fertile interaction.
I. INTRODUCTION
Control theory has evolved far beyond its original
motivation in the study of single loop or small-scale
feedback systems, into a rich and diverse mathematical
discipline. To achieve its true impact, this body of
theory must apply to problems where design intuition is
more limited; a clear example are large scale systems.
Two fundamental difficulties appear when attempting
this transition: one, with large-scale comes decentralized
control, a restriction which makes most of our theory
inadequate. Two, as the scale grows, often the number
of design objectives grows as well; again, most of
control theory addresses single objective problems or
those where a few objectives can be easily traded off by
a design engineer that holds global decision power.
Economic theory, on the other hand, has been conceived for studying the interaction of multiple decision
makers and objectives. For many important problems,
market mechanisms in terms of prices and quantities
provide elegant descriptions of the “control laws” of
the individual agents, and their global impact. Its viewpoint is, however, mainly that of equilibrium: finding
a set of variables from which the decentralized agents
would have no incentive to deviate. When it comes to
dynamic systems operating away from their equilibrium,
the understanding is more limited, for the most part
restricted to quasi-static arguments. Another limitation
of economic analysis is that the “network topology” of
agent interactions is rarely described in detail, resorting
instead to simplified settings that yield conclusions of
only general value.
Fernando Paganini is with Universidad ORT, Montevideo, Uruguay.
Email: [email protected]. This work was supported
by PDT-Uruguay, project S/C/IF/54/119, and by AFOSR-US, grant
FA9550-06-1-0511.
ISBN: 978-960-89028-5-5
The telecommunications network (the singular term
being increasingly adequate) is probably the largestscale engineering system ever constructed. Of the multiple embedded control mechanisms at its various layers,
few have been designed with a sound dynamic analysis,
often leading to malfunctions such as oscillations. On
the other hand, a large portion of the technology is
dedicated to solving an economic problem: how to
allocate resources among users. Gone are the days of
telephony where the homogeneity of the application
allowed engineers to allocate resources via “central
planning” imposed on passive users. In the current
competition for bandwidth among highly heterogeneous
and often greedy users, it is difficult to predict the
resulting allocations, let alone correlate them with any
economic rationality.
Can network controls be designed to operate in the
stable regime? Can we rationalize the equilibria arising
from the economy of bandwidth allocation? These broad
questions seem hopelessly ambitious at the scale of
e.g. the Internet; yet, surprising advances have been
reported in recent years to greatly improve our answers.
The mathematical language essential to these advances
is that of convex optimization, which in fact has old
roots in each of the above-mentioned fields. Control has
relied on optimization both for trajectory planning and
feedback design, and has been at the forefront of recent
advances in convex programming; economic theory has
been fashioned in the language of optimization (of
utility, or cost), and tools such as convex duality appear
naturally in price generation; optimization has been used
classically in networking problems such as routing. It is
therefore natural to resort to this common framework to
advance this interdisciplinary research.
In this paper we describe some of these recent advances. Space considerations preclude from an even
remotely comprehensive survey of the literature; we will,
nevertheless, show how various threads of research can
be unified into a common, prototype framework. In Section II we present the scenario and notational framework.
In Section III we show how this setup leads to problems
where the economic viewpoint illuminates the search for
a desired equilibrium and the tools for a decentralized
solution. In Section IV we incorporate dynamic control,
and survey some concrete results on convergence to
these equilibria. In Section V we comment on other
related areas of activity and future research opportunity.
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II. A PROTOTYPE SCENARIO
Links
We describe now the problem of controlling the
distribution of traffic in an idealized network. The
description, taken from [18], includes what is called
rate or congestion control from traffic sources, and the
control of routing by network nodes; as such, it would
be suitable for describing the transport and network
layers of a TCP/IP network like the Internet. It could
be considered, however, a subproblem in a more general
resource allocation problem for a network in which other
aspects (e.g. the physical layer) are also dynamically
controlled.
Consider a network made up of a set of nodes N , and
a set of directed links L between them. Nodes, denoted
by the indices i and j, can be sources or destinations of
packets, or intermediate router nodes. We describe the
links either by a single index l, or by the directed pair
(i, j) of nodes they connect.
The network supports various traffic flows between
source-destination pairs of nodes. We use the index k ∈
K to denote an individual flow or “commodity”, and
s(k), d(k) denote respectively the corresponding source
and destination nodes, unique for each k. In general,
traffic could follow multiple paths between source and
destination: we thus introduce the following variables:
k
• x , external rate of commodity k (e.g. in packets
per second) entering the network at the source;
k
• yl , rate of commodity k through link l;
k
• xi , total rate of commodity k coming into node i.
At the source node, we have
xks(k) = xk ,
(1)
which assumes no commodity k traffic loops back to the
source. We write the natural flow balance equations
X
k
, j 6= s(k),
(2)
yi,j
xkj =
Telecommunication links are mainly characterized by
the bandwidth they can transport. There are two main
ways to specify this:
• A capacity cl in packets/second that can be transported. This could be fixed or subject to variations.
• An increasing barrier function φl (yl ) that specifies
a cost associated with transporting rate yl . This
could be seen as a soft way to enforce a capacity
constraint, or a model for the delay in the link
queue, which will be perceived even before yl
reaches the capacity limit, due to stochastic effects.
Routers
Routing takes place at the internal nodes i ∈ N ; each
must decide on which of its outgoing links (i, j) ∈ L it
will forward incoming packets of commodity k. At the
level of flows, we can model this process through
k
k
yi,j
= αi,j
xki ,
(i, j) ∈ L,
(5)
k
based on the routing fractions or “split ratios” αi,j
satisfying
X
k
k
αi,j
≥ 0,
αi,j
= 1.
(i,j)∈L
Some restrictions could be imposed on split ratios:
• Destination-based routing. Here we define split
d
ratios αi,j
depending only on the destination d, i.e.
d(k)
k
αi,j
= αi,j .
k
Single-path routing. Here αi,j
’s are integers, therefore only nonzero for one outgoing link per node.
Both of the above restrictions are the norm in the
current IP routing protocol. The first will be assumed
henceforth.
•
(i,j)∈L
xki
=
X
k
yi,j
,
i 6= d(k).
(3)
(i,j)∈L
The total rate on link l is given by
X
yl =
ylk .
(4)
k
The above definitions apply generically to all the
problems under consideration. To complete the picture,
one must indicate how sources, links, and routers will
be modeled.
Sources
Traffic sources are basically of two types:
k
• Inelastic sources have a fixed, assigned rate x0 that
the network must accommodate.
• Elastic sources can adapt their rate to the current
circumstances; they should be regulated to exploit,
but not exceed, the available bandwidth.
It is sometimes useful to infer from the above an
overall relationship between the vector x = {xk } of
input source rates, and the vector y = {yl } of total
k
link rates. Given a set of split ratios αi,j
, under some
mild assumptions on connectivity, one can derive from
equations (1-4) and (5), a linear relationship of the form
y = Rα x.
(6)
Rα is called the routing matrix, and generalizes the case
of single-path routing studied in the literature; in that
special case it has elements
½
1 if source k uses link l
.
Rlk =
0 otherwise
Based on this general scenario, we move on in the
following section to a description of different resource
allocation problems, and how they can be addressed
through a combination of control, economics, and optimization.
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III. RESOURCE ALLOCATION PROBLEMS
The control challenge is the real-time regulation of
rates and routes so that users can take advantage of
the available bandwidth resources. We highlight the
following aspects of the problem:
• Decentralized control. By construction, and scale,
there can be no global authority: the individual systems (sources, routers) must regulate themselves.
• Multiple objectives. Each flow k cares about the
rate it can obtain.
• Decentralized information. No entity knows the
network state. Moreover, the telecommunication
network itself (the controlled system) must provide
the feedback channel.
From a control engineer’s perspective, this scenario
appears unfamiliar and daunting: the scale is large, the
information and control restrictions are severe, even the
objective is dubious. Here is where economic theory
comes in. Clearly, the above problem of allocating scarce
bandwidth resources is of an economic nature; we thus
turn to such models and tools to formulate the design
specifications and to seek a solution architecture.
A first contribution of economic theory is the
modeling of elastic traffic sources, as suggested by
[11]. These obtain a utility from the assigned rate xk ,
modeled through an increasing, concave function utility
function Uk (xk ), in some commonly agreed units
of “numeraire” [15]. Based on this notation, welfare
economics has studied how the individual preferences
can be aggregated into a collective “social welfare”.
One commonly used aggregation is the following.
Problem 1 (WELFARE): Applies to elastic sources,
arbitrary routing. Maximize
X
Uk (xk )
(7)
k
subject to link capacity constraints yl ≤ cl , and flow
balance constraints (1),(2),(3),(4).
Note that for concave utility functions, the above is a
convex program, since constraints are linear. Therefore
this has a well-behaved optimum. In the above, routing
was left as a degree of freedom, and in particular this
allows for traffic to be split via multiple paths.
If, instead, one assumes routing is fixed, given by a
routing matrix Rα , we can formulate the following:
2 (WELFARE, fixed routing): Maximize
PProblem
k
U
(x
),
subject to link capacity constraints
k k
Rα x ≤ c.
Here flow balance is already imposed by the routing
matrix. Note that we also have a convex program.
For single-path routing, this was labeled the “System”
problem in [11]. One could also consider routing to be
a degree of freedom, but forced to be single path: here
convexity is broken, see [24].
In the above problems, link capacities are represented
in a fixed, inelastic way, whereas sources are elastic. The
opposite situation can also be considered: here, sources
are assumed to have fixed demands xk0 , but the routing
can be chosen to alleviate congestion in the network,
expressed via elastic cost functions φl (yl ) as introduced
before. The resulting optimization is:
Problem 3 (TRAFFIC ENGINEERING): Applies to
inelastic sources, variable routing. Minimize
X
φl (yl )
(8)
l
subject to demand constraints xk ≥ xk0 , and flow
balance constraints (1),(2),(3),(4).
The above problem has a long history, originating
in transportation networks ([20] and references therein).
A reasonable cost function in that context is φ(yl ) =
yl dl (yl ), where dl (yl ) denotes delay or latency in link
l; then (8) represents the total traffic (packets, or cars)
stored in the network. The classical literature has studied
the “Wardrop equilbrium” [26] resulting from selfish
routing decisions by traffic units (drivers), based on the
delay they experience. This equilibrium is known [20]
to be arbitrarily inefficient with respect to the social
optimum of Problem 3.
In telecommunication networks, where routing decisions are made by routers, a network operator who
knows the “traffic matrix” of point-to-point demands
can attempt to match the optimal traffic engineering
allocation, see e.g. [21].
It is natural to consider as well a combination of
elastic sources and elastic link costs.
Problem 4 (SURPLUS): Applies to elastic sources,
arbitrary routing. Maximize
X
X
S :=
Uk (xk ) −
φl (yl )
(9)
k
l
subject to flow balance constraints (1),(2),(3),(4).
The above convex program from [18] combines the
utility maximization of (7) with the cost minimization
of (8), i.e. congestion control with traffic engineering. In
economic terms, the quantity S of (9) is the aggregate
surplus (see [15]), a natural object of optimization. To
be meaningful, utility and cost must be expressed in
the same units of “numeraire” (relevant to all entities in
the network). We will assume cost functions grow fast
enough with yl so that surplus is upper bounded, and
Problem 4 has a finite optimum.
Finally, we could also restrict the above problem to
fixed routing:
Problem 5 (SURPLUS, fixed routing.): Maximize
X
X
S :=
Uk (xk ) −
φl (yl ), subject to y = Rα x.
k
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l
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Remark: Other references [11], [23] formulate
optimization problems in terms of rate variables
for each end-to-end path through the network.
This is mathematically equivalent, but of a greater
dimensionality since there is an exponential number of
paths. For reasons of scalability, we prefer a formulation
with variables which are far fewer and which have local
meaning to either sources or routers.
The formulation of the above optimization problems
helps reduce the multi-objective nature of the resource
allocation. It does not yet, however, deal with the
decentralized control aspect of the problem. Economics
comes again to aid us here, by offering market mechanisms for decentralized resource allocation, through the
introduction of prices.
For each of the scarce network resources, the communication links l ∈ L, define a scalar variable pl ,
price per unit of flow, which measures its degree of
congestion, or scarcity. There are different ways to
define such prices, tailored to the solution of each of
the optimization problems, as described below. For the
moment, we assume prices are well defined and depend
only on the total traffic yl through the link; there is no
“service differentiation” between commodities.
Link prices introduce a “congestion currency” which
can be propagated across the network, and used by
individual nodes for the control of rates and routes. To
keep these control laws decentralized and simple, each
node should generate local variables that summarize
the congestion state of paths available to it, for each
given destination. Specifically, define node prices qid ,
i ∈ N , representing the congestion price from node i to
destination d, using the current routing patterns.
Node prices are thus chosen to satisfy
qdd = 0,
X
d
αi,j
[pi,j + qjd ],
qid =
i 6= d.
(10)
(i,j)∈L
Given link prices pi,j , it is shown through similar
arguments as those in [10] that the above equations have
unique solutions for qid , provided that the split ratios αd
have a path from every node to the destination. Furthermore, under these conditions an iterative procedure
in which each node updates its prices to the right-hand
side of (10), based on information from neighbors, and
communicates its result to its neighbors, will converge
to the desired node prices. We do not model this process
here. At the source node of commodity k, the node
price summarizes the congestion cost of the network.
We denote it by
d(k)
q k := qs(k) .
It can be shown that for fixed routing splits, the vectors
q = {q k } and p = {pl } of source and link prices satisfy
the transpose equation
T
q = Rα
p.
The following basic lemma relates the price and
flow variables; it constitutes a “conservation of money”
transacted for each commodity. For a proof see [18].
Lemma 1: For each commodity k,
X
xk q k =
ylk pl .
l∈L
IV. DECENTRALIZED CONTROL AT
DIFFERENT TIME-SCALES
The economic viewpoint has given both a definition of
desirable equilibrium points for resource allocation, and
a set of variables with local meaning to each node, which
can be used to control them. It remains to specify the
decentralized control laws: how to adapt prices pl , rates
k
so that the system reaches
xk , and routing ratios αi,j
one of these desirable points. The following dynamic
version of our previous Lemma will be useful in our
investigation.
Lemma 2: For each commodity k,
X
X
d(k)
d(k)
ẋk q k =
ẏlk pl −
xki α̇i,j [pi,j + qj ], (12)
l∈L
k k
x q̇ =
X
l∈L
(i,j)∈L
ylk ṗl
+
X
d(k)
d(k)
xki α̇i,j [pi,j + qj
]. (13)
(i,j)∈L
While in theory all these variables could vary at any
speed, implementation considerations introduce different
time-scales into the problem. These are discussed below.
A. The fast time-scale: congestion control
In the Internet, traffic sources rely on feedback from
acknowledgement packets to control their rates; this
means they can adapt as quickly as a few network
round-trip times, in the order of seconds. Routing
changes, based on explicit communications between
routers, propagate more slowly, in the order of a minute.
It is therefore of interest to study the situation of fixed
routes, but adaptive rates. Problem 2 and Problem 5 are
relevant to this situation.
We now present two well-established decentralized
control structures to solve these problems. The following
notation is used extensively:
½
w, if w > 0 or z > 0;
+
[w]z :=
0
otherwise.
1) Primal congestion control: In this proposal originating in [11], source rates are updated by
,
ẋk = κ(xk )[Uk′ (xk ) − q k ]+
xk
(14)
where κ(xk ) > 0. Other variants are given in [22], with
the common feature that they seek the equilibrium where
the source maximizes its “consumer surplus” Uk (xk ) −
q k xk , for given aggregate prices q k .
Links, in turn, generate prices as marginal costs:
(11)
835
pl := φ′l (yl ).
(15)
TuB07.2
Combined with (6) and (11), this generates a set
of closed loop equations, for which we can show the
following (for a proof sketch, refer to Theorem 6 below):
Theorem 3: Under (6) (11), (14-15), for fixed α’s, the
system converges globally to a solution of Problem 5.
2) Dual congestion control: This proposal from [14]
attempts to solve Problem 2 in a distributed way, through
its Lagrangian dual
Lα (p, x) =
X
Uk (xk ) +
X
[Uk (xk ) − q x ] +
k
=
X
pl (cl − yl )
l
k k
k
X
pl cl
(16)
l
Note that prices pl appear as Lagrange multipliers, and
we have invoked Lemma 1. Convex duality implies the
optimum of Problem 2 is
Ψα := min Wα (p) := min[max Lα (p, x)].
p
p
x
(17)
The dependence on routing α is made explicit for later
purposes. The dual dynamics is based on following
gradient in the dual optimization,
ṗl = γl (pl )[yl − cl ]+
pl ,
(18)
for some γl (pl ) > 0. Based on the received price q k ,
the sources instantaneously maximize Uk (xk ) − q k xk ,
i.e. choose the rate that satisfies
Uk′ (xk ) = q k , or xk = 0 and Uk′ (xk ) < q k .
(19)
Theorem 4: The dual dynamics (6), (11), (18), (19)
solves Problem 2.
Proof: Compute the derivative of W (α, p) over trajectories; since x is instantaneously maximizing in (17), the
Envelope Theorem (see [15]) gives
X
X
∂L
W˙ α =
ṗ = −
q̇ k xk +
ṗl cl .
∂p
k
l
d
Now invoke (13) with α̇i,j
= 0, and obtain
W˙ α =
X
ṗl (−yl + cl )
X
γl [yl − cl ]+
pl (yl − cl ) ≤ 0.
f
y(s) = Rα
(s)x(s),
f
f
where the dynamic matrix Rα
(s) has entries rl,k e−τl,k s .
Here rl,k is the previous static entry, but we have now
included the pure, forward delay between source k and
link l. Similarly, for the feedback path one can write
b
b
q(s) = [Rα
(s)]T p(s), with entries rl,k e−τl,k s .
Despite the linear simplification, analyzing stability
remains a formidable task, since one seeks this property over arbitrary network topologies and parameters;
nevertheless, for the case of single path routing, general
answers have been given. In this case, sources expef
b
rience a single round-trip delay τk = τl,k
+ τl,k
, and
multivariable stability analysis techniques can be used
to reduce this problem to classical single loop analysis
of delays. The conclusion is that not all primal or dual
laws will yield delay-stability, but some can be crafted
to do so; see [22], [17] and references therein. These
references also study “primal-dual” algorithms which
offer better control in the solution of these problems.
Recently, progress has been made in tackling the
full nonlinear, delay-differential global stability problem.
For simple network topologies, it is possible to prove
stability conditions which are close to those of the linear
analysis. More conservative conditions are required for
to obtain results in arbitrary networks. Some references
are [22], [25], [19].
B. The slow time-scale: adaptive routing
l
=−
3) Delay effects: The previous results on the global
properties of congestion control are very powerful: in
a completely decentralized way through propagating
prices, sources and links are able to solve a global optimization over an arbitrary network topology. However,
they have a “quasi-static” flavor, common in economic
theory, in which differential equations represent gradual
adaptation at unspecified speed. When attempting to
operate these systems quickly, a previously ignored
aspect appears: the inevitable delays of feedback carried
out across a large, distributed network.
Here is where control theory comes in: how quickly
can this feedback loop be controlled and remain stable?
Given the difficulty of delay-differential equations, a
generally used approach is to study the delay-stability of
the linearized dynamics around the global equilibrium.
In particular, to replace the static relationship (6) by the
equation (in Laplace transforms)
(20)
l
Thus W (α, p), decreases over trajectories to a point
where every link satisfies yl = cl , or yl < cl and pl = 0.
These conditions only occur at a saddle point of (17);
from duality, the equilibrium rates solve Problem 2.
At the slow end of the spectrum, consider a network,
perhaps a backbone of the Internet, that handles large
aggregates of traffic: from this vantage point, traffic
demand is often viewed as constant, or varying very
slowly in time, e.g. according to the daily cycle of usage
(hours). The question of how to optimally route this
load is of the form of Problem 3. For a moderate-sized
network, this could be solved offline and imposed on
routers, or solved online by some form of distributed
router adaptation.
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If one adds the restriction of single path routing,
common in current IP networks, then complications
appear; from the point of view of Problem 3, this added
restriction makes the problem non-convex, and therefore
solution algorithms go into heuristics. As for adaptive
routing, under single-path routing it leads to “route flap”
oscillations where traffic is switched between routes as
congestion alternates between them. These were empirically very common in the early Arpanet, and can also
be studied theoretically (see [3], [24]).
These issues make more attractive the use of multipath routing, under which Problem 3 is convex. In
this regard, the classical paper [10] proposes a method
d
per destination node are
in which traffic splits αi,j
adapted in real-time. Abstracting from the algorithm of
[10] the main requirements on route adaptation, we can
d
state the following conditions for the vector {α̇i,j
}j of
derivatives, for each destination d and node i [18]:
d
• The vector {α̇i,j } is a function of the vectors of
d
current ratios {αi,j
} and the prices {pi,j + qjd }.
d
• {α̇i,j } should be negatively correlated with the
route prices, and maintain node balance:
X
d
α̇i,j
(pi,j + qjd ) ≤ 0
(21)
(i,j)∈L
X
d
=0
α̇i,j
not decouple cleanly; at the routing time-scale, an elastic
source appears to adapt instantaneously, which is not
the same as an inelastic demand that does not adapt at
all. The reason traffic engineers in a backbone can get
away with an inelastic model, and even “overprovision”
capacity with respect to demand, is that they are dealing
with aggregates of sources bottlenecked elsewhere in the
network, typically in the access loop. Now, as technology creates faster access pipes, this separation breaks
down and one is forced to consider route adaptation
under congestion control. This has motivated us in [18]
to study the combination of routing and congestion
control dynamics. Problem 1 and Problem 4 are relevant
to resource allocation in this scenario.
Our first result concerns the combination of primal
congestion control with adaptive routing.
Theorem 6: Under (14-15), and our assumptions on
the adaptation of α, the system converges globally to a
solution of Problem 4.
Proof sketch: We take the derivative of the surplus
along system trajectories,
X
X
φ′l (yl )ẏl
Ṡ =
Uk′ (xk )ẋk −
=
(i,j)∈L
•
X
[Uk′ (xk )
X
κ(xk )[Uk′ (xk ) − q k ][Uk′ (xk ) −
k
− q ]ẋk +
k
d
} = 0, and this
Equality in (21) occurs only if {α̇i,j
happens only if for each (i, j) ∈ L we have
=
−
(23)
In other words, split ratios per node only settle
when prices of routes that carry traffic have equalized (and thus are equal to the node price) and the
remaining unused routes have higher price.
Theorem 5: Under the above assumptions on the
adaptation of α, marginal cost pricing pl := φ′l (yl ) as in
(15), and xk ≡ xk0 , the system converges globally to a
solution of Problem 3.
The proof is similar to Theorem 6 below.
C. Across time-scales: combined congestion control and
adaptive routing
We have described two very different time-scales of
network resource allocation, typically treated separately
in networking research. Indeed, there are entire communities that operate with either picture in their minds:
static supply of capacity, or static demand for bandwidth. When expressing both problems in the common
language of economic theory and optimization, and a
common notation, we are aiming at reducing this barrier
which is often artificial.
Indeed, even in the Internet where routing tables vary
slowly relatively to congestion control, the problem does
X
q k ẋk −
k
X
pl ẏl
l
q k ]+
xk
k
either qid = pi,j + qjd ,
d
or αi,j
= 0 and qid < pi,j + qjd .
l
k
(22)
X X
d(k)
d(k)
xki α̇i,j (pi,j + qj
),
k (i,j)∈L
where we have invoked (14) and (12). Both of the above
sums are non-negative, using (21); so we conclude that
Ṡ ≥ 0, the surplus increases along trajectories.
The remainder of the proof concerns the study of the
set {(x, α) : Ṡ = 0}, which allows us to conclude
stability via the Lasalle invariance principle [12]. This
study has some subtleties, see [18].
Remark: the same argument applies to Theorem 3 under
fixed routing, or Theorem 5 under fixed demands. As
such, it provides an elegant unification of the two threads
of research.
Now, for networks with slow adaptation of routes
relatively to rates, like the Internet, it may be more
interesting to model elastic demand as instantaneously
satisfying (19). Under marginal cost pricing, the resulting route adaptation still solves Problem 4.
If, instead, we consider the “dual” price dynamics
(18), an equilibrium point is the solution to Problem 1.
Establishing that this equilibrium is globally attractive
is still work in progress; at the time of writing, we
can prove the following fact [18]: if route adaptation
occurs at a slower time-scale than price dynamics,
there is indeed global convergence. When, instead, both
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dynamics occur at the same speed, we have found simulations which show oscillations, but only in degenerate
networks; we conjecture that global convergence holds
in generic cases, but the question remains open.
To understand this issue, refer back to the proof of
Theorem 4. When routing splits α are not constant, then
route adaptation contributes a positive term to Ẇ in (20),
so its dynamic behavior over time is inconclusive, and
it cannot be used for establishing convergence. This is
inevitable: a sensible adaptation of the split ratios tries
to maximize Ψα in (17), to its maximum over α which
is the solution of Problem 1. Its contribution to the
Lagrangian is therefore in the increasing direction.
The result in [18] is obtained assuming link prices
have stabilized before we change our routing:
Theorem 7: For each set of split ratios α, define
Ψ(α) by (17), and assume prices and rates take
instantaneously their saddle point values. Updating
α through (21-22), Ψ(α) converges to its global
maximum, the solution to Problem 2.
Proof sketch: We compute the derivative Ψ̇ over trajectories satisfying (21-22), where for the current α(t)
prices instantaneously minimize W (α, p) , and rates
instantaneously maximize L(α, p, x) for the given split
ratios and prices. The Envelope Theorem implies that
we can take derivatives directly on the Lagrangian for
fixed prices and rates; using (16) and (13), we have
X
Ψ̇ = −
xk q̇ k
k
=−
X X
d(k)
d(k)
xki α̇i,j [pi,j + qj
] ≥ 0.
k (i,j)∈L
The remainder of the proof invokes again Lasalle’s
principle for the study of the points where Ψ̇ = 0.
V. RELATED WORK
In this section we give highlights on other related
areas of research which relate to the above problems,
and which also indicate areas of progress in this interdisciplinary field.
A. Stochastic effects
The preceding work has modeled traffic in terms of
rates, as though it were a fluid flow. This approach
has proven more successful in modeling these large
networks than classical queueing theory, which studied
in more detail the effects of random packet arrivals. The
are least two advantages of the fluid flow approach: one,
the higher level of aggregation avoids the complexity of
keeping track of a large number of queues; two, it makes
it easier to model non-stationary effects that arise from
elastic sources.
This does not mean, however, that stochastic issues
disappear from the problem. Some papers (e.g., [11])
include stochastic perturbations in the form of Brownian
noise to model aggregate packet level effects. More
importantly, a factor that the fluid models ignore is the
finite duration of traffic flows. This is naturally modeled
in terms of a stochastic process of arrival of flows to
the network (e.g., download requests), and their size
distribution. This has prompted a line of research to
study the interaction between resource allocation of rate
using price feedback, with the flow arrival process; see
for instance [4].
B. Other layers and wireless networks
It is standard in networking to view the overall
communications task in terms of layers of functionality,
that interact with each other vertically within one unit
of equipment, each layer communicating horizontally
with the corresponding layer of another unit. In these
terms, congestion control belongs in the transport layer
and runs in end-systems (sources/destinations), routing
belongs in the network layer, running in all nodes. While
this philosophy prescribes keeping layers as separate
as possible, resource allocation problems like the ones
we have studied often involve multiple layers. To what
extent these can be decoupled is an issue that is best
studied through the optimization framework, as argued
in [6]. More specifically, as we have shown the language
and tools of economics are central to this analysis.
Wireless networks are an important area in which
lower layers of the protocol stack are subject to contention, since users share a medium subject to radio
interference. One approach to resolving this contention
is to schedule transmissions so that they don’t interfere; this multiple-access (MAC) layer technique makes
scheduling a resource allocation problem than can be
treated within the utility maximization standpoint (see
[13] and references therein).
Another strategy is to allow users to interfere but
adapt their physical layer parameters (power, modulation, coding) to share the scarce channel resource.
Here the characteristics of sharing go beyond the simple
economic picture of a divisible good (capacity) allocated
between users: each user’s utility depends in a complex way on the competitors’ decisions; for instance,
it depends on the signal-to-interference ratio that is a
nonlinear function of everyone’s powers. The natural
tool to study this interaction is game theory, see e.g.
[2].
In some scenarios it is possible to pose a full, crosslayer optimization involving transport, network, and
physical layer parameters. This is of particular interest
for “ad-hoc” networks, made up of units which are
simultaneously sources, sinks, and routers of information. In these scenarios, the time-scale separations we
previously invoked for the Internet do not apply. This
subject is still under active development, some recent
references are [27], [28], [7], [5].
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R EFERENCES
C. Pricing and the real economy of networks
In Sections III and IV we described a “virtual economy” of bandwidth, in which physical quantities and
prices are updated in real time to reach equilibrium
points offering some rationality from the point of view
of resource allocation. It remains as yet uncoupled from
the “real economy” of networks, where agents buy and
sell bandwidth for real money, at the far slower timescale of human economic decisions. Recently there has
been interest, however, [8], in further connecting both
aspects; we motivate this through a few examples.
A first example: in current Internet congestion control, a source’s responsiveness to congestion prices is
implicit in the TCP protocol, to which most users adhere.
There is, however, an alternative UDP protocol with no
congestion control at all; what prevents a generalized
greedy attitude which would lead to collapse? A partial
answer is that with bottlenecks at the access loop, the
misbehaving user is currently penalized itself; however,
access technology is accelerating dramatically; at some
point down the line, relying on user collaboration might
be impossible without real economic incentives.
A second issue is the need to differentiate quality of
service (QoS) between different flows, to make possible
the “convergence” over a single network of data, voice
and video. The failure of attempts to widely deploy such
differentiation in the Internet has no doubt contributed
to delaying the often announced convergence. But there
are also economic reasons: keeping networks separate
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different for the same amounts of bandwidth [16]. Now
that convergence appears inevitable (with the onslaught
of voice over IP, etc.), the economic viability of the
industry may require for technologies, and pricing strategies, that make possible such differentiation.
A final example occurs at the network layer. We have
described routing optimization by an entity which cares
about global congestion; the picture is more complicated
for a network made of competing service providers [1].
In particular, routing between these entities are commonly dominated by commercial considerations, rather
than overall “congestion welfare”. Solutions which rely
on announcements of congestion prices (e.g., our qid ’s)
are subject to manipulation. One way to ensure truthful
announcements is to introduce real economic transfers,
as studied by the theory of economic “mechanism design” [15]. A recent application to routing is [9].
VI. C ONCLUSIONS
This paper has described in some detail a framework
that unifies different lines of work in congestion control
and multipath routing. This was done partly for its own
sake, but also to give a concrete instantiation of the interdisciplinary field emerging between control, economics,
and optimization, applied to networking. We anticipate
that the future will bring many more developments in
this exciting area.
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