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Stability and Delay Consideration for Flow Controlover Wireless Networks
Minghua Chen, Alessandro Abate, Avideh Zakhor, Shankar SastryDepartment of Electrical Engineering and Computer Sciences
University of California at Berkeley, CA 94720{minghua, aabate, avz, sastry}@eecs.berkeley.edu
Abstract—In this paper we develop a general frameworkfor the problem of flow control over wireless networks, eval-uate the existing approaches within that framework, andpropose new ones. Significant progress has been made onthe mathematical modeling of flow control for the wired In-ternet, among which Kelly’s contribution is widely acceptedas a standard framework. We extend Kelly’s flow controlframework to the wireless scenario, where the wireless linkis assumed to have a fixed link capacity and a packet lossrate caused by the physical channel errors. In this frame-work, the problem of flow control over wireless can be for-mulated as a convex optimization problem with noisy feed-back. We then propose two new solutions to the problemachieving optimal performance by only modifying the ap-plication layer. The global stability and the delay sensitivityof the schemes are investigated, and verified by numericalresults. Our work advocates the use of multiple connectionsfor flow, or congestion control, over wireless.
Index Terms—Wireless, flow control, congestion control,primal algorithm, global stability, delay stability
I. I NTRODUCTION
TCP has been widely successful on the wired Inter-net since its first implementation by Jacobson [1] in
1988. TCP Reno, the most popular TCP version today,in its congestion avoidance stage, increases its windowssize by one if no packet is lost in the previous round triptime, and halves the windows size otherwise. Thekeyas-sumption TCP relies on is that packet loss is a sign ofcongestion. In wireless networks however, packet losscan also be caused by physical channel errors; thus thecongestion assumption breaks down, resulting in TCP un-derutilizing the wireless bandwidth. Similar observationshold for TCP-friendly schemes, e.g. TCP-Friendly RateControl (TFRC) [2, 6, 25], as they share the same key as-sumption as TCP.
Consequently, there have been a number of efforts toimprove the performance of TCP or TFRC over wire-
This work was supported by NSF grant ANI-9905799, the AFOSRcontract F49620-00-1-0327 and the NSF grant CCR-0225610.
less [8–24]. All these methods either hide end-hosts frompacket loss caused by wireless channel error, or provideend-hosts the ability to distinguish between packet losscaused by congestion, and that caused by wireless chan-nel error. For example, Snoop is a TCP-AWARE linklayer approach which suppresses acknowledgement pack-ets (ACK) from the TCP receiver, and does local retrans-missions when a packet is corrupted by wireless channelerrors [8]. End-to-end statistics can be used to detect con-gestion when a packet is lost [17–24]. For example, by ex-amining trends in the one-way delay variation, one couldinterpret loss as a sign of congestion if one-way delay isincreasing, and a sign of wireless channel error otherwise.The disadvantage of these schemes is that they need mod-ifications to network infrastructure or protocols, makingthem hard to deploy.
Our recent work on MULTFRC [25], modifying onlythe application layer, provides a new approach to improvethe performance of TFRC in wireless networks, and assuch is also applicable to TCP. MULTFRC opens appro-priate number of connections given the observed roundtrip time, so as to fully utilize the wireless bandwidth,while keeping the packet loss rate and round trip time ata minimum. Theoretical analysis on the optimality, NS-2 simulations, and actual experiments evaluating MULT-FRC performance are included in [25]. The main issueleft to be addressed for MULTFRC is its stability. Thisis because in MULTFRC, boththe numberof connectionsandsending rateof each connection are dynamically ad-justed, potentially resulting in network instability.
While the existing approaches provide practical insighton how to improve TCP over wireless performance, it isunclear whether they can easily scale to a network as largeas the Internet. Hence a general framework for flow con-trol over wireless is needed to address both stability andscalability issues prior to any implementation. Specif-ically, answers to the following questions are expectedfrom this analysis:
• How does one re-define the problem of flow control
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over wireless network?• Are there any optimal and robust application layer
schemes that solve the problem?
Recently there has been a great deal of research ac-tivities on decentralized end-to-end network flow controlalgorithms. A widely recognized setting, introduced byKelly et. al. [26], is to associate a utility function witheach flow, a cost function with each resource, and to max-imize the aggregatenetsystem utility function. Under thisframework, flow control schemes can be viewed as algo-rithms to compute the optimal solution to this maximiza-tion problem. Kelly et. al. [26] proposed two complemen-tary flow control algorithms, theprimal and thedual.
In primal algorithms, the users adapt source rates dy-namically based on the prices along the path, and therouters select a static law to determine their prices directlyfrom the arrival rates at the link. The algorithms pro-posed by Kunniyur and Srikant [30], Alpcan and Basar[35], Vinnicombe [34] belong to this class. The stabil-ity issue of various primal algorithms are investigated in[26,32,33,36].
In dual algorithms on the other hand, the routers adaptthe prices dynamically based on the link rates, and theusers select a static law to determine the source rates di-rectly from the prices along the path and the source pa-rameters. The algorithms proposed by Low and Lapsley[27], Paganini et. al. [39], Yaiche et. al. [29] belong tothis class. The stability issue of various dual algorithmsare investigated in [26,39].
There is also another class of algorithms named primal-dual, where both prices and source rates are changed dy-namically by routers and users, respectively. The algo-rithms proposed by Low and Lapsley [27], Kunniyur andSrikant [31], Pagnini et. al [39] belong to this class. Thestability issue of various primal-dual algorithms are inves-tigated in [27,31,37,39].
These frameworks can be used to understand and de-sign the congestion control algorithms and predict theirperformance. Low pointed out that different versions ofTCP and queue management algorithms such as DropTailand RED can be analyzed under the same duality modelwith different utility functions and update functions [28].Kunniyurand and Srikant investigated two algorithms thatcan be used to understand TCP’s behavior [30]. Kellyshowed TCP is a primal like algorithm with packet lossrate as the associated price function [40]. The stabilityfor the system with and without delay, with and withoutdisturbance, are reviewed and developed in [40]. The pa-per also discusses the selection of the TCP parameters, inorder to achieve a scalable robust congestion control.
In this paper, we extend Kelly’s framework in [26] to
the wireless network scenario, where a wireless link is as-sociated with a fixed bandwidth and a fixed packet lossrate caused by the physical channel errors. Specifically,we analyze the performance of the primal algorithm inthe wireless case with packet loss rate as the price. Asufficient and necessary condition for suboptimal systemperformance is derived. We then analyze existing wirelessflow control approaches and propose two efficient optimalapproaches. The stability with and without delay are in-vestigated and numerical results are provided to validateour analysis. While we adopt a particular form of pricefunction in this paper, we do not make any assumptions inthe stability proofs, other than the basic one by Kelly [26]on the price function. Hence our stability results apply toa more general setting.
Our paper differs from Chiang’s work [41] and Chiangand Bell’s work [42] in the models applied for wirelessnetwork. We model the wireless link with a fixed capac-ity and a packet loss rate caused by channel error. Thisassumption is made by taking into account the separa-tion between physical, transport and application layers. Incontrast [41] and [42] break down the layering abstrac-tion, and allow end users and middle nodes to change thewireless transmission power in order to vary the capac-ity and interference of the wireless links. As such [41]and [42] define a more general optimization problem thanours. Another key difference is that [41] and [42] assumethe timescale of rate control to be large enough to achievethe wireless channel capacity, and that all packets are suc-cessfully protected from channel error. In contrast, weassume the timescale of rate control to be on the orderof round trip time so that wireless channel error can stillcause packet loss.
This paper is structured as follows. Section II includesproblem formulation. Then two approaches addressingthe problem together with global stability and delay sta-bility issues are discussed in Sections III and IV. SectionV includes discussions and future work.
II. PROBLEM FORMULATION
A. Overview of flow control framework and a primal al-gorithm
Consider a network with a setJ of resources, and letCj
be the finite capacity of resourcej, for j ∈ J . In practice,the resources in the network are its links. Let aroute rbe a non-empty subset ofJ , and denoteR to be the set ofpossible routes. Setajr = 1 if j ∈ r, so that resourcejlies on router, and setajr = 0 otherwise. This definesa 0-1 routing matrixA = (ajr, j ∈ J, r ∈ R), indicatingthe connectivity of the network.
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Associate a router with a user, i.e. a pair of sender andreceiver, and assume users behave independently; further-more, assume a ratexr allocated to userr results in utilityUr(xr). Ur(xr) is assumed to be an increasing, strictlyconcave, and continuously differentiable function ofxr
over the rangexr ≥ 0, corresponding to the definition ofelastic traffic in [43].
Let U = (Ur(·), r ∈ R), andx = (xr, r ∈ R). Assumeutilities are additive, so that the aggregate utility of theentire system is
∑r∈R Ur(xr). Define the cost incurred
at link j asPj(·). The flow control problem under a de-terministic fluid model, first introduced by Kelly et. al.[26] and later refined in [40], is a concave optimizationproblem maximizing the net utility:
max∑
r∈R
Ur(xr)−∑
j∈J
Pj
∑
s:j∈s
xs
, (1)
where the cost functionPj(·) for using resourcej, similarto the one introduced in [30], is
Pj(y) =∫ y
0pj(z) dz. (2)
pj(z) is called the price function and is required to bea non-negative, continuous, increasing function and notidentically zero. With these assumptions onpj(z), thefunction Pj(y) is strictly convex. One common pricefunction used in practice is the packet loss rate1, whichis zero for no congestion, and concavely increases whenthere is congestion, e.g.
pj(y) =(y − Cj)+
y, (3)
whereCj is the capacity of linkj, andy denotes the ag-gregate rate passing through linkj. Fig. 1.(a) shows anexample ofpj(·) as packet loss rate on a link with capacityof 1000 bps.
The end-to-end packet loss rate for userr is1 − ∏
j∈r pj(∑
s:j∈s xs), which is approximately∑j∈r pj(
∑s:j∈s xs) whenpj(
∑s:j∈s xs) is small. In the
rest of this paper, we assume price function to be packetloss rate as shown in (3), and to be small enough so that∑
j∈r pj(∑
s:j∈s xs) is the end-to-end packet loss rate forany userr.
For the logarithmic utility function Ur(xr) =wo
r log xr, wherewor is a weight and can be understood
as pay per unit time, Kelly et. al. in [26] show that both
1Even though the packet loss rate here is not a strictly increasingfunction, for simplicity we assume it is. In practice, this assumptionmakes little difference.
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y (bps) − aggregate arriving rate
p j(y)
packet loss rate as a function of aggregate link rate
(a)
1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
y (bps) − aggregate arriving ratep j(y
)
packet loss rate as a function of aggregate link rate and channel error rate
(b)
Fig. 1. Examples of the price function as packet loss rate on a linkwith capacity 1000 bps.
primal and dual algorithms can be used to solve the op-timization problem (1). In the rest of this paper, we willfocus on the primal algorithm only, as it results in a dis-tributed end-to-end based TCP-like behavior, and closerto what is currently deployed in pratice. In particular, weconsider the following primal algorithm [26] :
d
dtxr(t) = kr
wo
r − xr(t)∑
j∈r
µj(t)
, r ∈ R (4)
with kr being a positive scale factor or the step size affect-ing the adaptation rate; and
µj(t) = pj
∑
s:j∈s
xs(t)
. (5)
Equation (4) describes the time evolution ofxr, i.e. thesource rate of userr. Equation (5) indicates the genera-tion of congestion signal at linkj in terms of a conges-tion price functionpj(·). The congestion signals gener-ated along the links on userr’s route are summed up andthen fed back to the userr as the indicators of networkcongestion. Therefore, user exploits only aggregate infor-mation along its path. In practice users obtain feedback
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prices by observing packet loss, ECN marking, or the in-crease in round trip time; the system in (4) corresponds toTCP-like additive increase, multiplicative decrease algo-rithm. Equation (5) refers to packet dropping or markingalgorithm deployed on routers, e.g. RED [45] and ECN[46].
Observe that the source parameterwor is proportional
to the number of connections opened by userr. To seethat, assumewo
r is set towir in (4) to control the one con-
nection’s rate when userr opens one connection. Userr openingN − 1 additional connections can be thoughtof addingN − 1 additional users along the same path asuserr, and associating the same source parameterwi
r tocontrol their rate. Thus, the aggregate rate of theseN con-nections,xr, is the sum of theseN users’ rates, and is ad-justed in a way shown in (4) but withwo
r = N ·wir. Hence
the source parameterwor is proportional to the number of
connections opened by userr, and changing number ofconnections is equivalent to adjustingwr proportionally.
Kelly has also shown the entire system to be globallyasymptotically stable using a Lyapunov function, and lo-cally robust to source rate and price disturbances. Kellyet. al. [26], Johari and Tan [32], Vinnicombe [33] studydelay sensitivity in order to derive conditions under whichthe entire system is robust to delay. For a network, globalasymptotical stability assures that given any starting ratewith associated network congestion situation, the rateswill eventually converge to unique finite values, avoidingcongestion collapse. The robustness to delay and distur-bance implies that delays and disturbances do not affectthe stability of the system, i.e. the rates still converge re-gardless of end users receiving feedbacks from networkwith heterogeneous delays.
If the end users deploy the algorithm in (4) and the linksin networks deploy the algorithm in (5), the unique, glob-ally asymptotically stable rates of the entire network, de-noted byxo = (xo
r, r ∈ R), is given by
xor =
wor∑
j∈r pj
(∑s:j∈s xo
s
) , r ∈ R; (6)
This unique solution is also optimal in the sense that thenetwork bottlenecks are fully utilized, the total net utilityis maximized, and the users are proportionally fair to eachother [26].
B. Flow control over wireless
For the wireless network, we assume some of the linksare associated with not only a fixed capacity but also apacket loss rate caused by the physical channel errors.Thus the end-to-end packet loss rate, i.e. the price sensing
by end-hosts via end-to-end measurement of the packetloss, now becomes noisy. In the presence of noise, flowcontrol algorithms still aim to address the same optimiza-tion problem shown in (1), and the optimal solution is stillxo.
Assuming each link has a packet loss rate due to phys-ical channel errors, denoted byεj ≥ 0, the noisy pricefunction, denoted byνj(·), is the total packet loss rate as-sociate with linkj as a combination ofεj andpj(·):
νj(t) = qj
∑
s:j∈s
xs(t)
= pj
∑
s:j∈s
xs(t)
+
1− pj
∑
s:j∈s
xs(t)
εj
≥ pj
∑
s:j∈s
xs(t)
, (7)
Fig. 1(b) shows an example ofqj(·) as packet loss rate ona link with capacity of 1000 bps,ε = 0.05 andpj(·) shownin Fig. 1(a). When the wireless link is not congested,q(·)is ε since all the packet loss is caused by channel error;when the link is congested and packets are dropped at therouter,q(·) gradually increases, similar top(·) in Fig. 1(a).
With this noisy price, userr adjusts its rate based on thenoisy feedback from the network:
d
dtxr(t) = kr
wo
r − xr(t)∑
j∈r
νj(t)
, r ∈ R. (8)
Observe thatνj(t) is also a non-negative, continuous, in-creasing function, and not identically zero; hence follow-ing a similar analysis in [26], the system converges to anew stable pointx∗ = (x∗r, r ∈ R):
x∗r =wo
r∑j∈r qj
(∑s:j∈s x∗s
) , r ∈ R. (9)
Comparing the stable pointx∗ in (9) with xo in (6),which is the optimal solution for the optimization problemshown in Equation (1), we havex∗r < xo
r if∑
j∈r εj > 0.Since the optimal solution for problem in (1) is unique, theequilibrium pointx∗ is then a suboptimal solution. Theanalysis shows that if the packet loss rate caused by chan-nel error is not zero, then the system (8)-(7) converges to asuboptimal equilibrium point, with non-maximal net util-ity, and hence the network bottleneck could be underuti-lized. This defines the flow control problem over wireless,which we try to address in this paper.
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Although the primal algorithms (8) and (4) are not ex-actly the same as TCP in practice, their behaviors are sim-ilar and they face the same problem in wireless networks.Both of them treat the noise in the packet loss rate asthe sign of congestion and reduce the rates unnecessar-ily. Hence, it is likely that the solution and analysis forthe problem of primal algorithms over wireless networkcould give insights for designing solutions for the prob-lem of TCP over wireless network. In fact, the approacheswe propose in this paper have strong relation with MULT-FRC proposed in [25], both of which advocate the use ofmultiple connections for flow control over wireless.
C. Existing approaches
Existing approaches [8–24] employ a number of prac-tical schemes to provide userr with the correct price∑
j∈r µj(t), or its estimated value. End users can thenmodify the source rate control law to react to
∑j∈r µj(t)
rather than to the noisy one∑
j∈r νj(t). Hence in mostexisting approaches the problem of flow control over wire-less is reduced to the problem of flow control over wirednetworks.
Our previous approach [25], on the other hand, adjustswr to solve the problem. The intuition is clearly seen from(9), in that if the numeratorwo
r is changed to a largervalue then it is possible to preservex∗r = xo
r. This isa new approach and can be widely deployed in practice,since changingwr is proportional to the number of con-nections in practice, adjustingwr can be implemented byadjusting number of connections correspondingly, requir-ing only modifications to the application layer.
Motivated by our previous approach [25], we proposetwo new approaches to wireless flow control based on ad-justingwr; and hence both of them are end-to-end appli-cation layer based schemes and require no modification toeither the network infrastructure or to the network proto-cols. In the first approach,wr is instantaneously adjustedusing a static function with respect toνj(t) andµj(t); inthe second approach,wr is gradually adjusted by a dy-namic update law.
In both approaches, we assume the end user to haveaccess to both the correct price function
∑j∈r µj(t) and
the noisy one∑
j∈r νj(t). In practice, users can get anestimate of the end-to-end packet loss rate
∑j∈r νj(t)
by carrying out end-to-end measurement; it is also pos-sible to apply schemes from [17–24] to obtain an esti-mate of the packet loss rate caused by congestion only, i.e.∑
j∈r µj(t). We will revisit this assumption in the discus-sion in Section V. Our goal is to achieve the optimal totalnet utility with no modification to network infrastructure,requiring control law for source rate of userr to be (8).
III. STATIC UPDATE SYSTEM
Let the source parameter for userr be a function oftime, denoted bywr(t). In this systemwr(t) is adjustedusing a static law with respect toνj(t) andµj(t) as fol-lows:
wr(t) = wor
∑j∈r νj(t)∑j∈r µj(t)
, (10)
and source rate for userr then is given by:
d
dtxr(t) = kr
wr(t)− xr(t)
∑
j∈r
νj(t)
. (11)
The intuition behind this approach is that when noise issignificant andνj(t) is large, we increasewr(t) in (10),so as to compensate forxr(t)
∑j∈r νj(t) term in (11), and
hence to properly controlxr(t). It is easy to see the equi-librium point for the system (11)-(10) isxo, by setting allthe derivatives ofxr(t) to be zero; hence the optimal so-lution is the equilibrium point. From both a control anda practical point of view, it is of interest to investigate theglobal stability and delay stability of the system, as wewill do in the later subsections.
A. Global stability
Put simply, a globally, asymptotically stable systemconverges to its equilibrium point regardless of its initialstarting point and the size of the system; for our problem,global stability ensures that in an arbitrarily large networkwith arbitrary number of users, any user ratexr, r ∈ Rconverges to the optimalxo
r, r ∈ R, and the network isnot overwhelmed; this means the rates are appropriatelycontrolled according to the network congestion conditionssuch that bottlenecks are fully utilized, at the same time asavoiding congestion collapse (Section I in [1]). Motivatedby the work in [26], we show here the global stability ofsystem (11)-(10)-(7) through Lyapunov arguments.
Theorem 1:System (11)-(10)-(7) is globally asymptot-ically stable with the following Lyapunov function:
V (x) =∑
r∈R
wor log xr −
∑
j∈J
∫ ∑s:j∈s xs
0pj(y)dy. (12)
All trajectories converge to the equilibrium pointxo in (6)that maximizesV (x).
Proof: Refer to Appendix A.
B. Delay stability
It is also important to investigate whether our proposedcontrol system is stable, i.e. the rate converge toxo
r, when
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there is delay between the entities that generate the feed-back signals, e.g.νj(t) andµj(t), and the controllers, e.g.the senders, that receive the feedback signals. It is difficultto prove global stability in the presence of delay. Instead,we investigate the delay stability locally, i.e. in the regionaround equilibrium pointxo, enabling us to make use ofarguments in the frequency domain. In the presence ofdelay, the system (11)-(10)-(7) can be re-written as
d
dtxr(t) = krw
or
∑j∈r νj(t− d2(j, r))∑j∈r µj(t− d2(j, r))
−krxr(t− Tr)∑
j∈r
νj(t− d2(j, r))(13)
and
µj(t) = pj
∑
s:j∈s
xs(t− d1(j, s))
, (14)
νj(t) = qj
∑
s:j∈s
xs(t− d1(j, s))
, (15)
andd1(j, r) + d2(j, r) = Tr ∀r ∈ R, (16)
whered1(j, r) is the forward delay from sender of routerto link j, andd2(j, r) is the return delay from the linkj to the sender of router. Tr hence is the round triptime on router. Here we assumeTr is fixed, as arguedin [26] and [32], at least for the time scales we are in-terested in. The following theorem provides a sufficientcondition upon each end user for the entire network to berobust to heterogeneous delay, i.e. differentTr values fordifferent usersr.
Theorem 2:The system (13)-(14) is locally stable if
kr
∑j∈r qj∑j∈r pj
∑
j∈r
pj +∑
j∈r
p′j∑
s:j∈s
xos
<
π
2Tr∀r ∈ R,
(17)wherepj , qj is the value ofpj(·) andqj(·) evaluated at theequilibrium point respectively;p′j is the derivative ofpj(·)evaluated at the equilibrium point.
Proof: Refer to Appendix B.Interpretingkr as the speed userr adapt its ratexr, similarto the results in [32] and [33], Theorem 2 implies that userr should adapt its rate in the time interval determined byround trip timeTr. This is becausekr is bounded by anumber inversely proportional toTr. This confirms thegeneral intuition pointed out in [1], [32] and [33] that endusers in a network should change their rate at the timescaleat least on the order of its round trip time.
It is interesting to compare the delay stability conditionfor wireless scenario, i.e. Theorem 2 in Equation (17), tothe one for wired case wherepj(·) = qj(·), also shown in[33], namely:
kr
∑
j∈r
pj +∑
j∈r
p′j∑
s:j∈s
xos
<
π
2Tr∀r ∈ R. (18)
As seen, the conditions for userr in wired and wireless
case only differ by a route-dependent constant∑
j∈r qj∑j∈r pj
,
which is larger than one whenpj(·) and qj(·) take theforms in (3) and (5) respectively. Hence upper bound onkr in wireless case, given by (17), is smaller than that inwired case. Intuitively this can be explained as follows:since in wireless case bothwr(t) and x(t) are adjustedbased on the feedback prices , the system is more sen-sitive to obsolete prices caused by delays in the system;hence the system should adjust in a more cautious way,resulting in smaller upper bound on step sizekr.
C. Numerical results
In this subsection, we perform simulations using MAT-LAB to verify the results of our analysis on the static up-date system.
1) Simulations for Global Stability:The topology isshown in Fig. 2; there are three users in the nine nodenetwork with two wireless links in it. The wireless linkis represented by dashed line and is associated with a pairof parameters: capacityC and the packet loss rate causedby channel errorpw. The setting for two wireless linksareC1 = 6000 bps, p1
w = 0.01, andC2 = 10000 bps,p2
w = 0.02. End users apply (11) to control their sourceratexr, and (10) to control the source parameterwr; eachlink measures the aggregate incoming rate as the sum ofthe user rates, and computes the pricesp(·) andq(·) us-ing (3) and (7) respectively; in practice this correspondsto packet dropping at the bottleneck router and on thewireless links. In our simulations, the sum of the prices,i.e.
∑j∈r pj(·) and
∑j∈r qj(·), are assumed bo be known
to user; in practice this corresponds to users carrying outend-to-end measurements to estimate
∑j∈r qj(·), i.e. the
end-to-end packet loss rate, and∑
j∈r pj(·), i.e. packetloss rate caused only by congestion.
We setwr to be 500 when the price∑
j∈r µj(t) is zero,in order to preventwr(t) computed by (10) from becom-ing unbounded. In the context of the approach in [25], thiscan be understood as setting an upper limit on the numberof connections one application can open. The parameterskr, r = 1, 2, 3 is set to 1;wo
1, wo2 andwo
3 are set to be 50,40, 30 respectively, facilitating differentiating the conver-gence processes of users.
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The convergence ofxr, wr, packet loss rates caused byboth channel error and congestion i.e.q1 andq2, packetloss rates caused by congestion only i.e.p1 andp2, andthe total net utility are shown in Fig. 3. As seen, the ratesand the net utility converge nicely2, and the wireless linksget fully utilized. As expected,wr(t) changes instantly assoon as the wireless links becomes fully utilized. This be-havior is not necessarily desirable in practice. Sincewr(t)is proportional to the number of connections in practice,it might cause the number of connections to change dra-matically, resulting in a great deal of control overhead be-tween end users; moreover, small stochastic disturbanceon the source rate or the feedback prices might result inunnecessary fluctuation on the number of connections.
Fig. 2. Simulation topology.
2) Simulations for Delay Stability:To show that thesystem is robust to delay when conditions in Theorem 2are met, we investigate a simple scenario where two usersshare a wireless link with capacity of6000bps and packetloss ratepw = 0.03. Users 1 and 2 have round trip timesof T1 = 1 second, andT2 = 1.6 seconds respectively. Theparameters are selected as in Table I:kr is selected to be
TABLE IPARAMETERS FOR SIMULATION FOR STATIC UPDATE SYSTEM.
para. value req. bound onkr para. valuek1 0.35 0.367 wo
1 30k2 0.2 0.229 wo
2 60
slightly smaller than the required upper bounds from The-orem 2; the quantities such asp1 andp′1 are obtained byfirst solving the optimization problem in Equation (1) nu-merically in order to compute the equilibrium point, andthen evaluating the functionsp1(·) andp′1(·) at the equilib-rium point. We choose differentwo
r , r = 1, 2 to facilitatedifferentiating between the convergence processes of twousers.
Convergence ofxr, wr, packet loss rate caused by bothchannel error and congestion i.e.q1, packet loss rate
2Here we start with initial conditionx1 = x2 = x3 = 0; similarconvergence observations hold for other random initial conditions.
caused by congestion only i.e.p1, and the total net util-ity are shown in Fig. 4. As seen,x1 andx2 converge tothe optimal values around 4000 bps and 2000 bps respec-tively; the net utility also asymptotically converges.p1, q1
andwr initially oscillate and then converge. It is interest-ing to investigate the frequency of the oscillation, and thereason behind it, as it could provide insight to the transientbehavior of the system, and indicate ways to suppress theoscillation. We plan to investigate this in the future.
IV. DYNAMIC UPDATE SYSTEM
The approach in Section III requireswr(t) to change in-stantaneously, resulting in rapid changes inwr(t) as seenin Fig. 3(b); in practice it is desirable to adjustwr(t) grad-ually so that (a) the number of connections is also adjustedslowly, and (b) decrease sensitivity to measurement errorin packet loss rate. In this section, we consider the morecomplex case wherewr(t) is dynamically updated, as fol-lows: ∀r = 1, 2, . . . , R,
d
dtwr(t) = cr
(wo
r − wr(t)
∑j∈r pj(
∑s:j∈s xs(t))∑
j∈r qj(∑
s:j∈s xs(t))
),
(19)wherepj(·) and qj(·) are defined in (3) and (7) respec-tively with source rate for userr changing as (11). It iseasy to see that the equilibrium point for the system (11)-(19) isxo.
The composite system built upon the equation for theratesxr in (11) and forwr in (19), similar to the staticupdate system in Section III, is a nonlinear, coupled, andmultivariable. Also, note that two equations are not ex-actly symmetrical even though they might appear to beso.
A. Global stability with two time-scale approach
In this subsection we will show that the dynamic updatesystem, explicitly defined through (11) and (19), is en-dowed with some stability properties. To begin with, wemake the following key observation:the dynamics cor-responding to (11) and (19) evolve in two different timescales; the first in a faster one, while the second in aslower one. This assumption is not merely an abstraction;rather, it reflects a physical reality thatwr(t), interpretedas the number of connections, is in general adjusted in amuch slower time scale than source ratexr(t).
After vectorizing the relation (11) and (19), we can then
8
rewrite the coupled equations, as the following pair34:
εx(t) = kw(t) − diag(kx(t))AT (p ◦Ax(t)) = F(x);(20)
w(t) = G(w, x). (21)
Here ε is a very small number which we introduceto reframe our current setting with the classical small-perturbation framework, for instance, [26] and [40]; thischange can be trivially obtained through rescaling of theconstant vectorial parameterk. Sincew changes muchslower thanx, we can think ofkw to be a fixed parameterwithin Equation (20). In this case5, the following holds:
Theorem 3:For the overall system (20)-(21), featuringprice functions that are functions of only the aggregaterates, e.g. (3) and (7), the following is true:
• Equation (20) has a region of equilibrium pointsF(x) = 0 that identifies a manifoldw = h(x);
• Within this manifold, Equation (21), also known asthe reduced system, has a unique global equilibriumwhich is the solution ofG(h(x), x) = 0;
• The functionsF, G, h and their partial derivatives arebounded near the global equilibrium;
• The equilibrium manifold for the boundary-layersystem6 is exponentially stable;
• The global equilibrium of the reduced systemw =∂h∂x x = G(x, h(x)) is asymptotically stable;
Hence, by the stability of singular disturbance non-linearsystem in [47] and [48], there exists anε∗ such that,∀ε ≤ε∗, the global equilibrium point of the composite systemis asymptotically stable.
Proof: Refer to Appendix C.Similar to the interpretation to Theorem 1 in Section
III-A, Theorem 3 implies system defined in (11) and (19),under the assumption the source ratexr(t) is adjustedmuch faster than the source parameterwr(t), i.e. cr ¿ kr,converges to its equilibrium point regardless of its initialstarting point and the size of the system; for our prob-lem, again it ensures that in an arbitrarily large networkwith arbitrary number of users, any user ratexr, r ∈ Rwill converge to the optimalxo
r, r ∈ R, and the networkis not overwhelmed. That is, the rates are appropriatelycontrolled according to the network congestion conditionsuch that the bottlenecks in the network are fully utilized,yet congestion collapse is avoided.
3From now on, bold letters will denote vectorial quantities.4The vectorskw andkx are built through the componentwise prod-
ucts of the vectorsk, w andk, x respectively.5For simplicity, we shall consider the simplified casec = 1.6This system is obtained through a rescaling of time,τ = t/ε and
letting ε → 0.
B. Simulations
In this subsection, we perform simulations using MAT-LAB to verify the results of our analysis on the dynamicupdate system.
1) Simulations for Global stability:The topology, net-work settings and parameter are exactly the same as thosein the simulations for global stability of static update sys-tem in Section III-C.1. The only difference is that in thissimulation end users apply (19), rather than (10) in Sec-tion III-C.1, to control the source parameterwr. To meetthe two timescale assumption made in Theorem 3, wemake the update frequency ofwr to be 50 times slowerthan the one ofxr; this has the similar effect as makingcr
50 times smaller thankr. In practice, this is reflected bythe requirement that the number of connections needs tochange much slower than the source rates.
Convergence ofxr, wr, packet loss rate caused by bothchannel error and congestion i.e.q1 andq2, packet lossrate caused by congestion only i.e.p1 andp2, and the totalnet utility are shown in Fig. 5. As seen, the rates and thenet utility converge nicely7, and the wireless link get fullyutilized; we also notice that the converged rates are ex-actly the same as the ones in Section III-C.1. We observethatwr changes slowly, as compared to the simulation re-sults in Section III-C.1 for the static update system. Thisbehavior is desirable in practice, as it implies the numberof connections, which is proportional towr, varies slowly,and is not sensitive to stochastic disturbances in the net-work.
V. D ISCUSSION AND CONCLUSION
In this paper we have defined the problem of flowcontrol over wireless to correspond to net utility maxi-mization under noisy feedback. We have proposed twoschemes to solve the problem. Both of them change thesource parameterwr only, which corresponds to changingthe number of connections in practice. The first approachapplies a static update law to changewr, while the secondapproach applies a dynamic update law. Both systems areshown to be globally stable. The condition for delay sta-bility on the static update system is also provided. Numer-ical results are provided to verify the analysis. In practice,the evolution ofwr as a function of time can be inter-preted as the evolution of the number of connections as afunction of time. This is becausewr can be thought of asbeing proportional to the number of connections. As such,a sender opens a number of connections, whereby the ratefor each connection obeys Equation (8) with awr chosen
7Here we start with initialx1 = x2 = x3 = 0; the similar conver-gence observations hold for other initial conditions.
9
as a constant corresponding to one connection only, andthe aggregate rate of all connections obeys Equation (11)with aggregatewr(t) evolving as in Equation (10) or (19).
This work provides insight on how to improve the per-formance of TCP or TCP-friendly schemes over wireless,by only modifying the application layer to open multipleconnections. Although we adopt a particular form of pricefunction in the paper, the stability results actually holdmore generally, with only the assumption that the pricefunction is non-negative, continuous, increasing functionand not identically zero.
One observation to make is that although the two pro-posed schemes are designed to address the flow controlover wireless problem, they can also address the wiredone. To see that, note that in wired network we haveεj = 0, j ∈ J , i.e. no noise in the price function. Hencewe can interpret it as a special case for both static and dy-namic systems withqj(·) = pj(·), as implied from (7).Then all the analysis and conclusions follow, i.e. both thestatic update and the dynamic update system converge tothe equilibrium pointxo globally, and the static updatesystem is locally robust to delay under certain conditions.This implies one can have the same solution for flow con-trol over both wired and wireless networks.
Even though in this paper we assume userr to haveaccess to both
∑j∈r pj(·) and
∑j∈r qj(·), only the ratio
between them is needed to controlwr(t) in both the staticupdate system in Section III and the dynamic update sys-tem in Section IV. The ratio
∑j∈r pj(·)/
∑j∈r qj(·) is
a kind of nondecreasing route-congestion indicator func-tion in the sense that it is zero when there is no congestionalong router, and takes non-zero value when there is con-gestion along router. Hence, in practice it is possible toestimate this ratio by measuring the congestion status ofthe route, or using any non-decreasing congestion indica-tion function that is zero when there is no congestion andis less than or equal to one when there is congestion. Insome sense, this is what MULTFRC does in [25] in or-der to control the number of connections. Specifically,MULTFRC estimates the queuing delay along the routeand infers congestion based on whether the queueing de-lay is larger than a dynamic threshold, and controls thenumber of connections based on this inferred congestionstatus. It is of great interest to investigate how to estimatethe ratio, how robust the proposed schemes are to the esti-mation error, and what performance of a congestion indi-cator function is as compared to the ratio. This is part ofour future work.
Future work also includes the condition for the dynamicupdate system to be delay stable, and a closer investiga-tion of today’s TCP, to provide more precise guidance on
how to design a practical scheme; for example apply ouranalysis to the TCP model in [40]. It is also interesting toput the practical MULTFRC scheme, as proposed in [25],into the theoretical framework, and to arrive at guidelinesfor improvements.
REFERENCES[1] Jacobson V, ”Congestion avoidance and control”,Proceedings of ACM SIGCOMM, 1998:314-329.[2] Sally Floyd, Mark Handley, Jitendra Padhye and Joerg Widmer, ”Equation-Based Congestion Con-
trol for Unicast Applications”,Proc. ACM SIGCOMM 2000, Aug. 2000, pp. 43 - 56[3] S. Floyd and K. Fall, ”Promoting the Use of End-to-End Congestion Control in the Internet”,
IEEE/ACM Transactions on Networking, Aug. 1999[4] Jitendra Padhye, Victor Firoiu, Don Towsley, and Jim Kurose, ”Modeling TCP Throughput: A
Simple Model and its Empirical Validation”,ACM SIGCOMM 98, pp. 303 - 314[5] W. Tan and A. Zakhor, ”Real-Time Internet Video Using Error Resilent Scalable Compression and
TCP-Friendly Transport Protocol”,IEEE Transactions on Multimedia, June 1999, vol. 1, no. 2, pp.172-186
[6] J. Mahdavi and S. Floyd, ”TCP-Friendly unicast rate-based flow control”,Technical note sent toend2end-interest mailing list, http://www.psc.edu/networking/papers/tcpfriendly.html, Jan. 1997
[7] M. Mathis, J. Semke, J. Mahdavi, T. Ott, ”The macroscopic behavior of the TCP congestionavoidance Algorithm”,CCR, July 1997, vol. 27, No. 3
[8] H. Balakrishnan, V. Padmanabhan, S. Seshan, R. Katz, ”A comparison of mechanisms for improv-ing TCP performance over wireless links”,ACM SIGCOMM’96, pp.256 - 269
[9] Wenqing Ding, Jamalipour A, ”A new explicit loss notification with acknowledgment for wirelessTCP”, 12th IEEE International Symposium on Personal, Indoor and Mobile Radio Communica-tions, Part vol.1, 2001, pp.B-65-9 vol.1. Piscataway, NJ, USA.
[10] Ratnam K, Matta I, ” WTCP: an efficient mechanism for improving wireless access to TCP ser-vices”, International Journal of Communication Systems, vol.16, no.1, Feb. 2003, pp.47-62. Pub-lisher: Wiley, UK.
[11] Rendon J, Casadevall F, Carrasco J, ”Wireless TCP proposals with proxy servers in the GPRSnetwork”, 13th IEEE International Symposium on Personal Indoor and Mobile Radio Communi-cations, Part vol.3, 2002, pp.1156-60 vol.3. Piscataway, NJ, USA.
[12] H. Balakrishnan and R. Katz, ”Explicit Loss Notification and Wireless Web Performance”,IEEEGlobecom Internet Mini-Conference, Nov. 1998
[13] Yaling Yang, Honghai Zhang, Kravets R, ”Channel quality based adaptation of TCP with lossdiscrimination”,IEEE Global Telecommunications Conference, Part vol.2, 2002, pp.2026-30 vol.2.Piscataway, NJ, USA.
[14] Chiasserini CF, Meo M, ”Improving TCP over wireless through adaptive link layer setting”,IEEEGlobal Telecommunications Conference, Part vol.3, 2001, pp.1766-70 vol.3. Piscataway, NJ, USA.
[15] Cobb JA, Agrawal P, ”Congestion or corruption? A strategy for efficient wireless TCP sessions”,Proceedings IEEE Symposium on Computers and Communications, 1995, pp.262-8. Los Alamitos,CA, USA.
[16] Jin-Hee Choi, See-Hwan Yoo, Chuck Yoo, ”A flow control scheme based on buffer state for wire-less TCP”,4th International Workshop on Mobile and Wireless Communications Network, 2002,pp.592-6. Piscataway, NJ, USA.
[17] S. Biaz, N. H. Vaidya, ”Distinguishing congestion losses from wireless transmission losses: a neg-ative result”,Computer Communications and Networks, 1998
[18] S. Biaz, N. H. Vaidya, ”Discriminating congestion loss from wireless losses using inter-arrival timesat the receiver”,Proc. of IEEE Symposium on Application-specific System and Software Engr. andTechn., pp. 10-17, Richardson, TX, Mar 1999
[19] N. Samaraweera, ”Non-congestion packet loss detection for TCP error recovery using wirelesslinks”, in IEE Proceedings of Communications, 146(4), pp. 222C230, Aug 1999.
[20] Y. Tobe, Y. Tamura, A. Molano, S. Ghosh, and H. Tokuda, ”Achieving moderate fairness for UDPflows by pathstatus classification,” inProc. 25th Annual IEEE Conf. on Local Computer Networks(LCN 2000), pp. 252C61, Tampa, FL, Nov 2000.
[21] S. Cen, P.C. Cosman, and G.M. Voelker, ” End-to-end differentiation of congestion and wirelesslosses”,Proc. Multimedia Computing and Networking (MMCN) conf. 2002, pp. 1-15, San Jose,CA, Jan 23-25, 2002
[22] Sinha P, Venkitaraman N, Nandagopal T, Sivakumar R, Bharghavan V, ”A wireless transmissioncontrol protocol for CDPD”, IEEE Wireless Communications and Networking Conference, Partvol.2, 1999, pp.953-7 vol.2. Piscataway, NJ, USA.
[23] Sinha P, Nandagopal T, Venkitaraman N, Sivakumar R, Bharghavan V, ”WTCP: a reliable transportprotocol for wireless wide-area networks”,Wireless Networks, vol.8, no.2-3, 2002, pp.301-16.Netherlands.
[24] Jae-Joon Lee, Fang Liu, Kuo C-CJ, ”End-to-end wireless TCP with non-congestion packet lossdetection and handling”,Proceedings of Spie - the International Society for Optical Engineering,vol.5100, 2003, pp.104-13. USA.
[25] M. Chen and A. Zakhor, ”Rate Control for Streaming Video over Wireless”,Proceeding of Infocom2004, Hongkong, China, March, 2004.
[26] F. P. Kelly, A. Maulloo, D. Tan, ”Rate control for communication networks: shadow prices, pro-portional fairness, and stability”,Journal of the Operationl Research Society, vol. 49, pp. 237-252,1998
[27] S. H. Low, D. E. Lapsley, ”Optimzation flow control I: Basic algorithm and convergenece”,IEEE/ACM Trans. on Networking, pp. 861-875, December 1999
[28] S. H. Low, ”A Duality Model of TCP and Queue Management Algorithms”,IEEE/ACM Trans. onNetworking, August 2003
[29] H. Yaiche, R. R. Mazumdar and C. Rosenberg, ”A game-theoretic framework for bandwidth allo-cation and pricing in broadband network”,IEEE/ACM Trans. on Networking, pp. 667-678, October2000
[30] S. Kunniyur and R. Srikant, ”End-to-end congestion control: utility functions, random losses andECN mars”,Proceeding of IEEE Infocom, Tel Aviv, Israel, March 2000
[31] S. Kunniyur and R. Srikant, ”A time scale decomposition approach to adaptive ECN marking”,Proceeding of IEEE Infocom, pp. 1330-1339, March 2001
[32] R. Johari, D. Tan,End-to-End congestion control for the Internet: delays and stability, IEEE/ACMTrans. on Networking, vol. 9, no.6, 818-832, December 2001
[33] G. Vinnicombe, ”On the stability of end-to-end congestion control for the Internet”,University ofCambridge Technical Report CUED/F-INFENG/TR.398, http://www.eng.cam.ac.uk/ gv, 2001
[34] G. Vinnicombe, ”Robust congestion control for the Internet”,University of Cambridge TechnicalReport, http://www.eng.cam.ac.uk/ gv, 2002
[35] Tansu Alpcan and Tamer Basar, ”A game-theoretic framework for congestion control in generaltopology networks”,Proc. of the 41st IEEE Conference on Decision and Control, pp. 1218-1224,Las Vegas, NV, December 2002
10
[36] Tansu Alpcan and Tamer Basar, ”Global Stability Analysis of End-to-End Congestion ControlSchemes for General Topology Networks with Delay”,IEEE Conference on Decision and Control,Dec 9-12, 2003; Maui, Hawaii
[37] Z. Wang, F. Paganini, ”Global Stability with time-delay of a primal-dual congestion control”,toappear in 2003 CDC, Maui, HI.
[38] S. Kunniyur and R. Srikant, End-To-End Congestion Control: Utility Functions, Random Lossesand ECN Marks”,To appear in IEEE/ACM Transactions on Networking, Oct. 2003
[39] F. Paganini, Z. Wang, J. C. Doyle, S. H. Low, ”Congestion control for high performance, stabilityand fairness in general network”,to appear in ACM/IEEE Trans. on Networking
[40] F. Kelly, ”Fairness and stability of end-to-end congestion control”,European Journal of Control,9 pp. 159-176, 2003
[41] M. Chiang, ”To layer or not to layer: balancing transport and physical layers in wireless multihopnetworks”Proceeding of Infocom 2004, Hongkong, China, March, 2004.
[42] M. Chiang, J. Bell ”Balancing supply and demand of bandwidth in wireless cellular networks:utility maximization over powers and rates”Proceeding of Infocom 2004, Hongkong, China,March, 2004.
[43] S. Shenker, ”Fundamental design issues for the future Internet”,IEEE Journal of Selected Areason Communications, vol. 13, pp.1176-1188, 1995
[44] C. A. Desoer, Y. T. Yang, ”On the generalized Nyquist stabiliity criterion”,IEEE Trans. on Auto-matic Control, 25:187-196, 1980
[45] S. Floyd and V. Jacobson, ”Random early detection gateways for congestion avoidance”,IEEE/ACM Trans. on Networking, 1(4):397-413, August 1993
[46] S. Floyd, ”TCP and Explicit Congestion Notification”,ACM Computer Communication Review,24(5), October 1994
[47] Shankar Sastry, ”Non Linear Systems, Analysis, Stability and Control”,Springer Verlag, 1999
[48] Hassan Khalil, ”Nonlinear Systems (3rd edition)”,Prentice Hall, 2001
APPENDIX
A. Proof for Theorem 1Proof: To seeV (x) is a Lyapunov function, first it is concave
with respect toxr, then note
∂V (x)
∂xr=
wor
xr−
∑j∈r
pj
( ∑s:j∈s
xs
),
thus finally
d
dtV (x(t)) =
∑r∈R
∂V (x)
∂xr· d
dtxr(t)
=∑r∈R
kr
∑j∈r qj
(∑s:j∈s xs(t)
)
∑j∈r pj
(∑s:j∈s xs(t)
) ·
(wo
r − xr(t)∑j∈r
pj
( ∑s:j∈s
xs(t)
))2
≥ 0.
HenceV (x) is strictly increasing witht, unlessx(t) = xo. The func-tion V (x) is thus a Lyapunov function of the system. The unique valuexo maximizingV (x) is a stable point of the system, to which all tra-jectories converge.
B. Proof for Theorem 2Proof: To analysis the delay stability locally, letxr(t) = xo
r +yr(t), and linearize the system equations around the equilibriumxo
r =wo
r/∑
j∈r pj , we have:
d
dtyr(t) = −kryr(t− Tr)
∑j∈r
qj − krxor
∑j∈r qj∑j∈r pj
·∑j∈r
∑s:j∈s
ys (t− d1(j, s)− d2(j, r)) p′j .
Take the Laplacian transform and carry out the simplification usingmatrix form, we have
sY (s) = −diag{kr
∑j∈r qj∑j∈r pj
} · diag{e−sTr} · diag{xor} ·
{diag{(xo
r)−1}diag{
∑j∈r
pj}+ M(s)
}Y (s), (22)
whereY (s) = (y1(s), y2(s), . . . , yn(s))T is a n × 1 vector withnbeing the number of users, andM(s) = (mrq(s))n×n is a n × nmatrix with:
mrq =∑
j∈q∩r
p′j · e−s[d1(j,q)−d1(j,r)].
Comparing above (22) to (6) in [33], we can see they only different
with a diagonal matrixdiag{ ∑
j∈r qj∑j∈r pj
}. Thus the local stability in-
vestigated here is exactly the same as the one in [33], different by onlya constant diagonal matrix. Hence the rest of the proof follows the ideaand logic flow in [33], and the results follow.
C. Proof for Theorem 3Proof: First, it is key to realize that a system of the form (11)
has a unique equilibrium point, due to the structural assumptions onthe functionsqj and because of the positivity of thexr ’s. Kunniyurand Srikant, [31], have shown that a system of this form, and with thesame assumption on the structure of its components, is exponentiallystable8.
Now, given the equilibrium manifoldw = h(x), or more explicitlyw = diag(x)AT q9, we focus on the stability properties of the reducedorder systemw = ∂h
∂x x = G(x, h(x)). In the passing, it is againimportant to underline that the reduced system, similar to the boundarylayer one, in either of its expressions will accept a unique equilibrium.
One observation to make here about the manifold before we go onis that for anyx on the manifold, none of its elements is zero, result-ing in diag(x) to be full rank. This is because anyx on the manifoldis nothing but the unique optimal solution for the concave optimiza-tion problem in (11) with correspondingw = h(x); hence none oftheir element should be zero by the concavity of the utilities functionit maximizes.
First let us calculate the Jacobian ofh. Let us distinguish amongtwo cases:
• ∂hi∂xk
= xk
∑j∈ri∩rk
∂qj
∂xk= xkAT
(·,i)∂q
∂xkA(·,k), if i 6= k;
• ∂hi∂xk
=∑
j∈rkqj(·) + xk
∑j∈ri∩rk
∂qj
∂xi= AT
(·,k)q(·) +
xkAT(·,k)
∂q∂xi
A(·,i), if i = k.Writing down the Jacobian, we obtain:
∂h∂x
=
∂h1∂x1
∂h1∂x2
... ∂h1∂xR
∂h2∂x1
∂h2∂x2
... ...... ...
. . ....
∂hR∂x1
... ... ∂hR∂xR
= diag(AT(·,i)q(·)) +
x1AT(·,1)
(∂q
∂x1A(·,1)
)... x1A
T(·,1)
(∂q
∂xRA(·,R)
)
x2AT(·,2)
(∂q
∂x1A(·,1)
)... ...
.... . .
...
xRAT(·,R)
(∂q
∂x1A(·,1)
)... xRAT
(·,R)
(∂q
∂xRA(·,R)
)
.
At this point, let us make an observation on the properties of thefunctionsq; the scalability requirement for the network requests the
8Actually, what the referred paper shows is ”semi-global exponentialstability”, which in our case reduces to exponential stability, due to theconstant feature of a parameter which is instead varying in the quotedcase.
9In the following within this proof, for the sake of clarity, we shallomit the composition operator within the functionsp andq.
11
price functions of each resource to be dependent on the “aggregaterates” along that link10. Therefore, the following always holds:
∀k = 1, . . . , J :∂qk
∂xi=
∂qk(yk)
∂yk
∂yk
∂xi=
∂qk
∂yk, ∀i = 1, . . . , R,
whereyk =∑
k:k∈s xs is the aggregate rate passing through linkk.
Hence we can denotenk ≡ ∂qk∂yk
, ∀k ∈ J.Now, observing in the passing how the rates are distributed along
the rows of the second matrix, it is possible to express the Jacobian as:
∂h∂x
= diag(AT(·,i)q(·)) + diag(x)AT diag(n)A
= diag(x){
diag(x−1AT(·,i)q(·)) + AT diag(n) A
}.
The expression inside the brackets, denoted byD for later conve-nience, is the sum of a positive definite matrix, the first term, and apositive semi-definite one. HenceD is a positive definite matrix, andhas full rank. It is then multiplied by a diagonal term which is also fullrank. This allows us to claim that the Jacobian ofh is always full rank.
After proper simplifications, the reduced system can then be ex-pressed as the following:
x =(∂h
∂x
)−1
w =(∂h
∂x
)−1(wo − diag(x)AT p
), (23)
Drawing inspiration from Kelly, [40], a Lyapunov function for thesystem is:
V(x) =∑r∈R
wor log xr −
∑j∈J
∫ ∑s:j∈S xs
0
pj(y)dy. (24)
The vector of partial derivatives is:
[∂V∂x
]=
[(wor
xr−
∑j∈r
pj(∑
s:j∈S
xs))
r=1,...,R
]T
=(wo − diag(x)AT p
) · diag(x−1);
its componentwise zeros correspond to the equilibria in every directionof the dynamical system. Furthermore,
d
dtV(x(t)) =
[∂V∂x
]T
x
=(wo − diag(x)AT p
) · diag(x−1)D−1diag(x−1) ·(wo − diag(x)AT p
);
from here, it should be clear that the derivative of the Lyapunov func-tion along trajectories of the system is always a positive number unlessxr = xo
r = wor/
∑j∈r pj . This is becausediag(x−1)D−1diag(x−1)
is nothing but a positive definite matrix. The bottom line is thatthereduced order system is asymptotically stable. Therefore, followingthe arguments in [47] and [48] for the stability of singular disturbancenon-linear system, the composite system shown in (11) and (19) isasymptotically stable.
10This way, no need to compute the single rates at each link is re-quired, but just to tally up the total “flow” through it. That way, wemanage to have information over all the network with just knowledgeof the links’ situation.
0 50 100 150 200 250 300 350 400 450 5000
2000
4000
6000
8000
10000
12000
number of iterations
sour
ce r
ates
xr
x1x2x3x1+x2x1+x3
(a)
0 50 100 150 200 250 300 350 400 450 50050
100
150
200
250
300
350
400
450
500
number of iterations
wr
w1
w2
w3
(b)
0 50 100 150 200 250 300 350 400 450 5000
0.005
0.01
0.015
0.02
0.025
0.03
number of iterations
pack
et lo
ss r
ate
on w
irele
ss li
nks
q1
q2
p1
p2
(c)
0 50 100 150 200 250 300 350 400 450 5000
100
200
300
400
500
600
700
800
900
1000
number of iterations
tota
l net
util
ity
(d)
Fig. 3. Static update system: convergence of (a) ratesxr(t), r =1, 2, 3, (b) wr(t), r = 1, 2, 3, (c) packet loss ratepj(·) andqj(·), j =1, 2, and (d) net utility, with initial rate set to 0.
12
0 200 400 600 800 1000 1200 1400 1600 1800 20000
1000
2000
3000
4000
5000
6000
7000
number of iterations
sour
ce r
ates
xr
x1x2x1+x2
(a)
0 200 400 600 800 1000 1200 1400 1600 1800 20000
50
100
150
200
250
300
350
400
450
500
number of iterations
wr
w1
w2
(b)
0 200 400 600 800 1000 1200 1400 1600 1800 20000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
number of iterations
pack
et lo
ss r
ate
on w
irele
ss li
nks
q1
p1
(c)
0 200 400 600 800 1000 1200 1400 1600 1800 2000685
690
695
700
705
710
715
720
725
730
number of iterations
tota
l net
util
ity
(d)
Fig. 4. Static update system with delay: convergence of (a) ratesxr(t), r = 1, 2, (b) wr(t), r = 1, 2, (c) packet loss ratep1(·) andq1(·), and (d) net utility, with initial rate set to 0.
0 1000 2000 3000 4000 5000 60000
2000
4000
6000
8000
10000
12000
number of iterations
sour
ce r
ates
xr
x1x2x3x1+x2x1+x3
(a)
0 1000 2000 3000 4000 5000 600050
100
150
200
number of iterations
wr
w1
w2
w3
(b)
0 1000 2000 3000 4000 5000 60000
0.005
0.01
0.015
0.02
0.025
0.03
0.035
number of iterations
pack
et lo
ss r
ate
on w
irele
ss li
nks
q1
q2
p1
p2
(c)
0 1000 2000 3000 4000 5000 60000
100
200
300
400
500
600
700
800
900
1000
number of iterations
tota
l net
util
ity
(d)
Fig. 5. Dynamic update system: convergence of (a) ratesxr(t), r =1, 2, 3, (b) wr(t), r = 1, 2, 3, (c) packet loss ratepj(·) andqj(·), j =1, 2, and (d) net utility, with initial rate set to 0.
