Congestion Control as a Constrained State Regulation Problem
K. KASSARA kassara@facsc-achok.ac.ma
Department of Mathematics, University Hassan II, P.O. Box 5366, Casablanca, Morocco Editor:J. P. Aubin
Received October 3, 2001; Revised December 4, 2001; Accepted January 16, 2002
Abstract.This paper investigates the field of congestion control in networks. It provides a new approach based on viability theory and feedback regulation. Basically it is shown that feedback control laws which lead to congestion avoidance can be derived by a selection procedure of a certain set-valued map. Finally, the paper studies existence of stabilizing congestion feedback by using Lyapunov theory along with some facts of set-valued analysis.
Keywords:congestion avoidance, feedback control, viability theory, stabilizing feedback, Lyapounov functions
1. Introduction and Statement of the Problem
It is widely recognized that the design and control of modern communication networks present challenges of a mathematical, engineering and economic nature. For the Inter- net, in particular, one of the most studied problems involves the notion of congestion, see [5,8–12 and references therein]. This arises whenever packets are lost due to over- load of a resource within the network. Currently, the control of congestion is designed by the Transmission Control Protocol which is implemented in navigators. It proceeds by regulating its sending rates according to an indication of congestion that it can receive, see [5,8].
In this paper, however, the purpose is to present a new mathematically oriented approach for studying the problem by setting it as a state constrained regulation problem. Thereafter we use technics provided by set-valued analysis and viability theory [1,3] to derive feed- back control laws which lead to congestion avoidance. Unlike the papers [5,9,12] which deal with discrete stochastic models, the present study is based on a continuous model consisting of a deterministic controlled system of differential equations which govern the rates of the users of a network, see [10]. However the approach can be extended to other kinds of network models.
In the same context, the paper investigates stability in networks and concerns itself with designing feedback controllers taking into consideration both stability of the network to a given equilibrium state and congestion avoidance in the network. This problem is of great interest as stressed in [5,10] where it is pointed out that users prefer to send constant rates which stabilize the network to an equilibrium which maximize the agregate utility of all the users.
Consider a network which consists of a setJ of resources. A router (or a user) is a nonempty subset ofJ. The set of all possible routes is denoted byN. For eachr∈N let xr(t)be the rate allocated to userrat timet. Then according to [10] thexr’s for allr∈N are subject to the following system of ordinary differential equations
˙ xr =κ
wr−xr
j∈r
µj
, xr 0,
(1)
whereκ >0 and µj =pj
sj,s∈N
xs
(2) withpj standing for a non-negative continuous increasing functions, not identically zero.
Equations (1) correspond to a response by userrwhich encompasses a steady increase of rate proportional towrand a steady decrease at rate proportional to the stream of feedback signals received. It is also supposed that the userrcan smoothly vary the parameterwr by observing a charge per unit flow of
λr =wr
xr
.
This means thatwr can be taken as the control variable in the model described by sys- tem (1). Now we are in a position to state the congestion problem. Suppose that a resource of the network has capacity per time to cope with packets of total sizeνwith any excess of lost. Then we merely have
Congestion ⇐⇒
s∈N
xs > ν.
It follows that congestion of the network can be avoided whenever users are able to adjust their rateswrin such a manner that
s∈N
xs ν. (3)
This leds us to define
Definition 1 Letw:[0, t1(→RN a measurable function. It is said a congestion control if inequality (3) is satisfied on[0, t1(.
The remainder of the paper is organized as follows: Section 2 reviews some basic tools such as contingent cones and viability theory along with facts of set-valued analysis. Sec- tion 3 provides characterizing results for feedback congestion control laws. Section 4 examines stabilizing feedback and congestion avoidance. Finally, Section 5 has some con- cluding remarks.
Notations We denote byRn+the orthant of vectorsx =(x1, . . . , xn)such thatxi 0 for eachi. The usual scalar product of two vectorsxandyis denoted byx;y. The boundary of a subsetDofRnis denoted by bdry(D)and its interior by int(D), whilea+Dstands for the set ofa+xwherexbelongs toD.
For a Gâteaux differentiable functionV:D→Rwe recall that
dV (x)y=
∇V (x);y = n i=1
∂V
∂xi
yi for eachy =(y1, . . . , yn)∈Rn,
where dV (·),∇V and∂V /∂xi respectively denote the differential, the gradient operator and the partial derivatives of the functionV.
A set-valued mapQ:D →Rnis said proper ifQ(x)= ∅for everyx. It is said lower semicontinuous if for eachx ∈Dand any sequence of elementsxk ofDconverging tox then for eachy ∈ Q(x), there exists a sequence of elementsyk ∈ Q(xk)that converges toy.
2. Preliminary Results and Definitions
We devote this section to give the main mathematical tools to be used next. LetDbe a non-empty subset of the Euclidean spaceRnwithn1 and define the contingent cone at x∈Das in [3,4] as follows
TD(x)=
y ∈Rn lim inf
h↓0
d(x+hy, D)
h =0
where d(y, D)¯ = infy∈Dy − ¯y for eachy ∈ D. It is useful to note the following properties of the contingent cones:
(a) Ifx ∈int(D)thenTD(x)=Rn. (b) TD(x)is closed for eachx ∈D.
(c) IfDis convex thenTD(x)is convex for eachx ∈D.
(d) IfDis closed and convex then the mapTD(·)fromDto subsets ofRnis lower semi- continuous.
(e) See [1, Proposition 3.1.4]. Let(Li)1in,Mbe a sequence of closed convex subsets satisfying the constraint qualification assumption
0∈int n
i=1
Li−M
(4) and consider the subset
D=
x =(x1, . . . , xn)xi ∈Li ∀iand n i=1
xi ∈M
(5) then we have
TD(x)=
y=(y1, . . . , yn)yi ∈TLi(x)∀iand n i=1
yi ∈TM n
i=1
xi
. (6)
(f) Viability theory.One of the most important properties of contingent subsets is provided by their use in viability theory [1,3]. Specifically letg:Rn → Rn then a subsetD ofRnis said to belocally viableunder system[ ˙x=g(x);t0]if there existst1>0 and a solutionx¯ : [0, t1[→D. Notet1= ∞ifDis compact. Such viability property can be characterized by the tangential condition
g(x)∈TD(x)for eachx∈D (7)
provided that
Dis locally compact andgis continuous onD. (8)
(g) Continuous selections. For a set-valued mapQ:D → Rnthe mappings:D → Rn is called a selection ofQifs(x)∈ Q(x)for everyx. We cite at this opportunity the famous Michael selection theorem [4]: If the mapQhas closed convex values and is lower semicontinuous then for any couple(x0, y0)such thaty0 ∈ Q(x0)the mapQ has a continuous selections which satisfiess(x0)=y0. For a closed convex valued mapQ, the minimal selection consists of takings(x)=πQ(x)(0), whereπ stands for the operator of best approximation. It is of interest to notice that this selection is rarely continuous, see [4]. Also we refer the reader to [1] where other selection procedures are studied.
3. Feedback Congestion Controls
Let the routes of the networkN be denoted 1,2, . . . , nthen the system of differential equations (1) may be rewritten as follows
˙ x=κ
w−σ (x) , x, w∈Rn+, t 0,
(9) whereσ =(σ1, . . . , σn)with
σi(x)=xi n j∈ri
µj, ri theith route, i=1, . . . , n,
x=(x1, . . . , xn)
(10)
andµi as given in (2). In another side inequality (3) which implies congestion can be expressed by the viability condition
x(t)∈Dν for eacht 0,
where Dν =
x =(x1, . . . , xn)∈Rnxi 0∀iand n i=1
xi ν
. (11)
Henceforth by virtue of Definition 1 feedback congestion control lawsw = c(x)may be characterized by the viability of the subsetDν with respect to system (9) in which w=c(x).
Definition 2 The mappingc:Dν → Rn is called a feedback congestion control law for system (9) ifw=c(x)defines a congestion control for allx0∈Dν.
Next we are in the need to define the following map onDν
Fν(x)=.
Rn+ if
n i=1
xi < ν,
w∈Rn+ n i=1
wi n
i=1
σi(x)
if n i=1
xi =ν.
(12)
Then we are in a position to show the following result.
THEOREM3 Letc:Dν → Rnbe a continuous mapping then it is a feedback congestion control law iff it is a selection ofFν.
Proof: First note that the subsetDν is compact, therefore condition (8) holds and hence we can use (7) to characterize the feedback congestion control law cby the following tangential condition
κ
c(x)−σ (x)
∈TDν(x) for eachx ∈Dν
and sinceTDν(x)is a cone for eachxit follows that
c(x)−σ (x)∈TDν(x) for eachx ∈Dν, (13)
hencecis a feedback congestion control law iff it is a selection of the map F (x)=.
σ (x)+TDν(x)
∩Rn+ for eachx ∈Dν. (14)
The remainder of the proof aims to show thatFν =F. In fact, we can rewriteDνas follows Dν =
x =(x1, . . . , xn)xi ∈R+∀iand n i=1
xi ∈ ]−∞, ν]
.
Then we can use notee)of Section 2 withLi =R+andM = ]−∞, ν]. We remark that constraint qualification assumption (4) is satisfied in this case. We thereby get
TDν(x)=
y=(y1, . . . , yn)yi ∈TR+(xi)∀iand n
1
yi ∈T]−∞,ν]
n
i=1
xi
Now applying a direct calculus of the contingent subsets in the last expression yields TR+(ξ )=
R ifξ >0, R+ ifξ =0 and
T]−∞,ν](ξ )=
R ifξ < ν, R− ifξ =ν.
It follows that for eachx∈bdry(Dν)we have
TDν(x)=
y|yi 0 fori∈I (x)
if n i=1
xi < νandI (x)= ∅,
yyi 0 fori∈I (x)and n i=1
yi 0
if n i=1
xi =νandI (x)= ∅,
y n i=1
yi 0
if n i=1
xi =νandI (x)= ∅, where
I (x) .
= {i|xi =0} for eachx ∈Dν and we have taken into consideration that
bdry(Dν)=
x∈DνI (x)= ∅or n
i=1
xi =νandI (x)= ∅
.
Of course, as forx ∈ int(Dν)we haveTDν(x)=Rn. Consequently, by using (14) with TDνas above and noting from (10) thatσi(x)=0 onI (x)we getFν =Fending the proof
of the theorem.
It is convenient to note that such a congestion control law does effectively exist. It suffices to take
ci(x) .
=λiσi(x) for eachx ∈Dν with 0λi 1 for eachi.
We can then easily verify that it provides a continuous selection of the mapFν.
4. Stabilizing Feedback and Congestion Avoidance
In this section we take the issue of whether it is possible to take into consideration both stability in the network and congestion avoidance. Specifically the purpose is to solve the following problem for a given statee∈Dν:
Pe,s: Find a feedback controlce:Dν →Rnsuch that:
(i) ce(e)=σ (e), ‘equilibrium condition’, (ii) xe(t, x0)∈Dνfor eacht, x0∈Dν, ‘congestion avoidance’, (iii) lim
t→∞xe(t, x0)=e, for eachx0∈Dν, ‘asymptotic stability’,
where for eachx0∈Dν,xe(t, x0)denotes the solution of system (9) withw =ce(x). Of course, according to Section 3, it can be seen that ProblemPe,sreduces to seek a continuous selection of the mapFν which satisfies (i) and (iii). Although existence of a continuous law satisfying (i) and (ii) may hold out of Michael’s Selection Theorem, see note (g) of Section 2, global asymptotic stability condition (iii) may require additional conditions.
Indeed we can use a technic based on Lyapounov theory as in [3]. LetV , W:Dν →Rbe two non-negative functions satisfying
V is continuously differentiable and
W is continuous andW (x)=0 iffx=e. (15)
Then consider the following maps for everyx ∈Dν: Sν,e(x) .
=
y∈TDν(x)|dV (x)y−W (x)
(16) and
Fν,e(x) .
=
w∈Rn+|κ
w−σ (x)
∈Sν,e(x)
=
σ (x)+ 1 κSν,e(x)
∩Rn+. (17) Hence it is not hard to show the following result.
PROPOSITION 4 Any continuous selection of the mapFν,e which satisfies condition (i) provides a solution of ProblemPe,s.
Proof: Letce:Dν → Rn be a selection of the mapFν,ewhich satisfiesce(e) =σ (e) then system (9) withw=ce(x)has a solutionx¯:[0,∞[→Dν. In addition, we have
κdV (x)
ce(x)−σ (x)
−W (x) for eachx∈Dν.
Then we can argue as in [3, Section 4.5] to conclude that(V , W )stands for a Lyapounov pair and thereforex¯e(t, x0)→east → ∞for everyx0. This shows thatceis a solution
of ProblemPe,s.
Although Proposition 4 acquaints on the way how to get a stabilizing feedback which solves ProblemPe,s, existence of such solution may fail and thereby it deserves to be studied. We first begin by proving the following lemma.
LEMMA5 Assume that the functionV satisfies
∀x∈Dν, ∃y ∈TDν(x) such thatdV (x)y <0 (18) then the mapSν,e(·)is proper and lower semicontinuous.
Proof: First we show that the mapSν,e(·)is proper. For that purpose letx ∈ Dν and consideryas in (18) then it merely can be seen that the vector
¯
y =αy, whereα > W (x)
−dV (x)y, is satisfying the following expression
¯
y ∈TDν(x) and dV (x)y¯+W (x) <0 (19)
and then it belongs toSν,e(x)and therebySν,e(x)= ∅.
To show that the mapSν,e(·)is lower semicontinous onDνwe use [4, Lemma 4.2] which claims that it is equivalent to show that the function
:x∈Dν →d
z, Sν,e(x)2
is upper semicontinuous for eachz∈Rn. Indeed givenz∈Rnandx ∈Dν then we have
(x)= min
y∈TDν (x) dV (x)y+W (x)0
z−y2 (20)
We observe that existence of an elementy¯as in Equation (19) provides the Slater condition for the convex constrained optimization problem (20), see, for instance, [7, Theorem 6.7]
yielding the formula
(x)=sup
λ0
inf
y∈TDν(x)
z−y2+λ
dV (x)y+W (x)
for eachx∈Dν (21) Now let(xn)n be a sequence inDν, which converges tox and giveny ∈ TDν(x). Since TDν(·)is lower semicontinuous onDν, see note (d) of Section 3 there exists a sequence yn∈TDν(xn)which converges toy. It follows that
y∈TinfDν(xn)
z−y2+λ
dV (xn)y+W (xn) z−yn2+λ
dV (xn)yn+W (xn)
(22) for eachλ0 andn∈N. However, by conditon (15) we get
dV (xn)yn+W (xn)→dV (x)y+W (x) asn→ ∞. Consequently by passing to the lim sup in (22) we obtain
lim sup
n→∞ inf
y∈TDν(xn)
z−y2+λ
dV (xn)y+W (xn) z−y2+λ
dV (x)y+W (x) for eachλ0 andy ∈TDν(x). It follows that
lim sup
n→∞ inf
y∈TDν(xn)
z−y2+λ
dV (xn)y+W (xn)
inf
y∈TDν(x)
z−y2+λ
dV (x)y+W (x)
for eachλ0. Next, by writing (xn)by Equation (21) and noting that lim sup
n→∞ sup
λ0
[·]sup
λ0
lim sup
n→∞ [·]
we obtain the desired inequality lim sup
n→∞ (xn) (x)
ending the proof of the lemma.
THEOREM6 Assume that the functionV satisfies
∀x∈Dν, ∃y ∈TDν(x) such that
∇V (x);y <0 (23)
then ProblemPe,shas a solution.
Proof: According to Proposition 4 we have to show that the mapFν,eadmits a continuous selection ce which satisfies condition (i). By observing (17) it is equivalent to seek a continuous selectionse of the mapSe of Equation (16) which satisfies se(e) = 0 (then ce =σ +κse). This carries us to use Michael’s Selection Theorem as stated in note (g) of Section 2. In fact, we first note out of conditions (15) and (16) thatSν,e has closed convex values and 0∈ Sν,e(e). In addition, due to Lemma 5, it is lower semicontinuous.
Consequently it possesses a continuous selectionsewhich satisfiesse(e)=0.
Note that determination of a feedback stabilizing congestion law only depends upon the way to do a continuous selection procedure of the mapFν,e. We have to mention that neither Michael’s Selection Theorem nor minimal selection, see note (g) of Section 2, may be convenient, because the first is not easy to be implemented and the second does not provide a continuous selection in general. Nevertheless we may, for instance, consider the following selection procedure which is inspired by Equation (19). LetAν be the map defined as follows
Aν(x) .
=
y∈TDν(x)|
∇V (x);y <0
for eachx∈Dν
and suppose that it has a continuous selectionyathen a continuous selection of the mapSe may be
se(x)=α(x)ya(x) for eachx ∈Dν, whereαstands for a continuous function satisfying
α(x) > W (x)
−∇V (x);ya(x) for eachx ∈Dν.
5. Concluding Remarks
In this note we have shown how the problem of congestion control in networks can be set as a state constrained regulation problem in the framework of viability theory. A basic consequence is that feedback control laws which involve congestion avoidance are given by a continuous selection procedure of the regulation map. It is of interest to stress what follows:
– While it is assumed that rates of users are governed by a system of differential equations, the approach may potentially be extended to more general types of network models providing that corresponding viability results are available. For instance, this would be the case for discrete and/or stochastic models [2] which is definitively candidate for future investigation.
– An interesting advantage of the viability approach is that congestion avoidance and sta- bilizing feedback may hold simultaneousely. This is due to a sophisticated argument which combines Lyapounov theory and set-valued analysis.
– It is already pointed out that wide-spread of the Internet will entail developing of pricing mechanisms in view to dispatch resources of the networks to users, see [8–10]. It must be remarked that the problem is of an economical nature. An important open problem, we believe, is that how to examine the question out of these references in the context of this paper.
References
1. Aubin, J. P.,Dynamic Economic Theory: A Viability Approach, Springer: Berlin, 1997.
2. Aubin, J. P. and Da Prato, G., “The viability theorem for stochastic differential inclusions,”Stochastic Anal.
Appl., Vol. 16, pp. 1–15, 1998.
3. Clarke, F. H., Ledyaev, Y. S., Stern, R. J. and Wolenski,Nonsmooth Analysis and Control Theory, Springer:
New York, 1998.
4. Deimling, V.,Multivalued Differential Equations, Walter de Gruyter: Berlin, 1992.
5. Gibbens, R. J. and Kelly, F. P., “Resource pricing and evolution of congestion control,”Automatica, Vol. 35, pp. 1969–1985, 1999.
6. Jacobson, V., “Congestion avoidance and control,” inProc. of ACM SIGCOMM’88, pp. 314–329, 1988.
7. Jahn, J.,Introduction to the Theory of Nonlinear Optimization, Springer: New York, 1983.
8. Johari, R., “Mathematical modeling and control of Internet congestion,”SIAM News, Vol. 33, March 2000.
9. Kelly, F. P., “Charging and rate control of elastic traffic,”European Trans. Telecommun., Vol. 8, pp. 33–37, 1997.
10. Kelly, F. P., Maullo, A. K. and Tan, D. K. H., “Rate control in communication networks: Shadow prices, proportional fairness and stability,”J. Oper. Res. Soc., Vol. 49, pp. 237–252, 1998.
11. Odlyzco, A. M., “The current state and likely evolution of the Internet,” inProc. of Globecom’99, IEEE, pp. 1869–1875, 1999.
12. Shenker, S., “Fundamental design issues for the future Internet,”IEEE J. Selected Areas Commun., Vol. 13, pp. 1176–1188, 1996.
