Computers and Electrical Engineering 39 (2013) 1758–1766

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Computers and Electrical Engineering

journal homepage: www.elsevier .com/ locate /compeleceng

Convergence rate control for distributed multi-hop wirelessmesh networks q

Yi Chen a, Ge Gao b,⇑, Ming Liu a, Huaxiong Yao a, Jinglei Guo a

a School of Computer Science, Huazhong Normal University, 430079 Wuhan, Chinab National Engineering Research Center for Multimedia Software, Wuhan University, 430072 Wuhan, China

a r t i c l e i n f o a b s t r a c t

Article history:Available online 23 December 2012

0045-7906/$ - see front matter � 2012 Elsevier Ltdhttp://dx.doi.org/10.1016/j.compeleceng.2012.12.00

q Reviews processed and recommended for publi⇑ Corresponding author.

E-mail address: chenyi30@mail.ccnu.edu.cn (G. G

Control message delay is a major factor affecting distributed network utility maximization(NUM) in practical wireless mesh networks, and the inherent delay of the control responseleads to severe oscillations and slow convergence. However, existing NUM techniqueseither fail to pay attention to the convergence rate or achieve suboptimal results. Thispaper proposes a novel technique to solve the reverse impact of inherent control delayin the distributed NUM of wireless mesh networks with noisy feedback by overcomingthe slow convergence induced by control delay in conjunction with feedback noise. Aself-tuning algorithm reduces the number of required iterations for convergence while alsoproviding an optimum value in distributed networks, and substantive simulations provethis improved convergence performance within a distributed multi-hop wireless mesh net-work. A theoretical explanation built from the viewpoint of control theory is also put for-ward to describe the model.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, using a utility function approach to maximizing resource allocation has attracted increasing attention andinterest for different kinds of networks. Where network control is carried out in a distributed manner, the utility functionserves as a measure of user satisfaction and fairness, using local information by estimation to achieve individual optimality.However, implementing a distributed solution requires a correlating communication among the network elements providinginformation exchange and feedback. Due to the potential congestion of and the random access methods used by wirelessnetworks, message delays in wireless networks are not negligible. That is, any control solution is out of date when a longdelivery time for a message occurs. Inexact adjustment values caused by slow control responses can cause packets to out-number the network capacity or cause underutilization of the channels. Values describing the shifting state of permittedpackets are defined by oscillations, which again lead to packets being dropped or channels being underutilized. Thus, oscil-lations harmful to the efficiency of the network will occur without careful adjustment of these factors.

This scenario gives rise to the following important questions that must be addressed for wireless mesh networks:

� How can we formulate effective constraints for the utility function of wireless mesh networks that, by their nature, expe-rience control delay?� Is there an existing convergence for distributed equations through which we can resolve the performance impact caused

by this control delay?

. All rights reserved.3

cation to Editor-in-Chief by Associate Editor Dr. Mehdi Shadaram.

ao).

Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766 1759

� In the case of oscillations induced by long delays that hinder convergence, are there any robust methods to accelerate therate of convergence?

The reminder of the paper is organized into four sections. Section 2 reviews related works that have studied wirelessmesh networks using network utility maximization. Section 3 describes a wireless mesh network and presents a theoreticalanalysis of the new control model. Simulation results are presented in Section 4 to visualize the proposed algorithm and Sec-tions 5 concludes the paper.

2. Related work

Wireless mesh networks are a promising technology with the potential to provide low-cost Internet access to wideareas—such as entire cities or rural areas—and enable different types of applications and clients to access the Internet.Fig. 1 illustrates an example of a wireless mesh network.

Recent work on network utility maximization (NUM) originates from the seminal work of Kelly et al. [1], who used aneconomical method to solve resource contention in wireline networks. In this method, network resources are impartiallyallocated among all users and adjusted using shadow prices. It is worth noting that Kelly’s work [2] assumed the existenceof convergence. More research from other groups has built upon this original work, combining cross layer messages withcongestion control [3,4].

However, there are fundamental differences when applying the existing theories to wireless networks. Multi-hop wire-less networks have a longer response time than 1-hop networks or wired networks due to channel interference and randomaccess. In order to effectively utilize channel spectrum, AL-Fuquha et al. [5] presented a scheme focused on cooperativemobility and channel selection. To enhance network utilization, a unified framework including spectrum sensing and chan-nel selection was proposed in [6]. Researchers studying WiMAX mesh networks plan to incorporate the cognitive radios intoIEEE 802.16h [7], while bandwidth allocation in wireless networks, jointing routing, optimization, and delay problems, arediscussed in [8–10].

With regard to wireless networks—and in particular wireless mesh networks—the impact of random noise from the pri-mary users in wireless mesh networks was systematically studied in [11] using a sub-gradient distributed algorithm. In addi-tion, there is stochastic noise due to estimator errors, as discussed in [12]. Although noisy feedback is a characteristic ofdistributed equations, there is an inherent control delay that hinges heavily on the performance of the distributed network.Besides feedback noise, many researchers have also studied multi-hop delay. Hou and Kumar [13] studied the assignation ofthroughputs in order to maximize the quality of service (QoS) requirements of clients based on end-to-end delay constraintsin a NUM framework. Huang and Krishnamurthy [14] adjusted the transmission rate in a cognitive radio using random delaythreshold policies. Most papers that try to provide tighter control of the actual queuing delays [15,16] based on NUM do soby having the delay meet a prior requirement.

The aim of our work is to optimize the performance of wireless mesh networks by accelerating the convergence rate sub-ject to feedback noises and control delay. The most obvious difference between our work and previous studies is that pre-vious works have only guaranteed convergence or end-to-end delay.

3. System model

We first consider a wireless mesh work composed of edge routers, relay routers, and a gateway router acting as the back-bone of the network. The edge routers collect traffic from clients and send it to the gateway router via the relay routers. Theclients consist of primary users and common users. There are jFj unicast sessions in the mesh network, i.e. all sessions areunicast. Every session f(f 2 F) has a pair of source and destination nodes, namely S(f) to D(f), so we can use session f and user f

Internet

mesh router

mesh client

Fig. 1. Architecture of a wireless mesh network.

1760 Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766

interchangeably. We denote the available acyclic paths from S(f) to D(f) by Cf, where a path is one possible route connecting

S(f) and D(f). In this way, the k-th path is represented by Ckf Ck

f 2 Cf

� �, which needs xk

f xkf 2 xf

� �bandwidth. The integral flow

rate xf of session f is given by

xf ¼XjCf j

k¼1

xkf ð1Þ

subject to xf � If and where If = [0, IM,f] is the bounded transmitted rate of flow f.We can describe our system algorithm of distributed equations for NUM in terms of stochastic estimator errors and con-

trol delay. The multiprotocol label switching technique is a multipath routing protocol that determines routing paths in ad-vance, in which the multiple path sets are either manually preconfigured or obtained through signaling mechanisms. Inwireless networks, aggregated source rates are estimated in the intermediate router due to random access, which meansthe price of the link is affected by estimator errors. In addition, the update rate is not timely due to feedback packet loss,so the resulting improper data rates deteriorate the performance of the network. Finally, the stochastic control delay isnon-ignorable, which can allow overdue new rates; in other words, the control is not timely. Because of these factors, ifthe sum of the aggregated flows does not match the link capacity, the system may become unstable.

For each link q 2 Q, let gq(t) be the aggregated flow through link q at time t, where the aggregated flow needs skf time to

reach link q from the source node. Thus, the real system’s aggregated flow at link q can be described by

gqðtÞ ¼X

f2qðKÞ

Xk2Cf

xkf t þ sk

f

� �þ Egq

ðtÞ 8q 2 Q ð2Þ

where q(K) is the session routing data set through link q. Let skf be the delay that flow xk

f needs to reach link q and EgqðtÞ be

the estimation error of the aggregated flow gq(t) through the link.We can define {x⁄} as an optimal operation point fixed on a definite network condition, and define g�q to be the optimal

aggregated flow in linkq. The bias of the aggregated flow through link q is described as

eqðtÞ ¼ ðcq � gqðtÞÞ � cq � g�q� �

ð3Þ

eDðtÞ ¼ ðcq � gqðtÞÞ � ðcq � gqðt � 1ÞÞ ¼ eqðtÞ � eqðt � 1Þ ð4Þ

Note that as the speed of eq(t) ? 0 increases, convergence performance improves. We can borrow a concept from controltheory, and introduce two variables to characterize the dynamic state. The first variable is eq(t), which is the error at timet for the dynamic state. The other variable is eD(t), which is the differential of the error at time t. From the view of controltheory, implementing a proportional-differential (PD) algorithm can overcome any negative influence due to system delay.

Therefore, we can define the primary problem P1 in a dynamic state to overcome the ill effects of oscillations induced bycontrol delay

P1: 0 1

max

Xf2F

Uf

Xk2Cf

xkf

@ A ð5Þ

Subject toXk2Cf

xkf 6 IM;f ð6Þ

gqðtÞ 6 cq 8q 2 Q ð7Þeq ¼ 0 ð8ÞeDðtÞ ¼ eqðtÞ � eqðt � 1Þ ¼ 0 ð9ÞeqðtÞ ¼ ðcq � gqðtÞÞ � cq � g�q

� �ð10Þ

c¼q Wq1T

logð1þ KcqÞ ð11Þ

We then construct the Lagrangian function by adopting a distributed PD algorithm to reduce the convergence time

eLðx; k;lq;lDÞ ¼Xf2F

Uf

Xk2Cf

xkf ðtÞ

0@

1AþX

f2F

kf IM;f �Xk2Cf

xkf ðtÞ

0@

1AþX

q2Q

ðlqeqðtÞ þ lDeDðtÞÞ

¼Xf2F

Uf

Xk2Cf

xkf ðtÞ

0@

1Aþ kf IM;f �

Xk2Cf

xkf ðtÞ

0@

1A

0@

1AþX

q2Q

lq cq �X

f2qðKÞ

Xk2Cf xk

fðtÞ

0B@

1CA

þXq2Q

lDððcq � gqðtÞÞ � ðcq � gqðt � 1ÞÞÞ �Xq2Q

lq cq � g�q� �

Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766 1761

Note that the solution of the Lagrangian function eLðx; k;lq;lDÞ is the optimum solution of (5)–(11).The dual problem of P1can be written asD1:

mink;lq ;lDP0

Dðx; k;lq;lDÞ ð12Þ

Note that the control effect worsens when a distributed control method is implemented in the presence of noisy feedbackand within the inherent characteristics of wireless access systems. Therefore, we must explore these distributed equations toovercome this issue. Let gf ðtÞ; gqðtÞ, and gDðtÞ be the corresponding estimators and ef, eq, and eD be the respective step sizes.The stochastic distribution version of the primal-dual algorithm can thus be written as

kf ðt þ 1Þ ¼ ½kf ðtÞ � ef ðtÞgf ðtÞ�10 ð13Þlqðt þ 1Þ ¼ ½lqðtÞ � eqðtÞgqðtÞ�10 ð14ÞlDðt þ 1Þ ¼ ½lDðtÞ � eDðtÞgDðtÞ�10 ð15Þ

We can get the estimator from the following network of equations

gf ðtÞ ¼ gf ðtÞ þ Ebf ðtÞ þ Emf ðtÞ ð16ÞgqðtÞ ¼ gqðtÞ þ EbqðtÞ þ EmqðtÞ ð17ÞgDðtÞ ¼ gDðtÞ þ EbDðtÞ þ EmDðtÞ ð18Þ

where Ebf(t) is the biased estimation error of gf(t), Ebq(t) is the biased estimation error of gq(t), and EbD(t) is the biased estima-tion error of gD (t) = eq(t) � eq(t � 1), with Emf(t), Emq(t), and Emq (t) the respective corresponding martingale difference noises.

The source node updates according to

xkf ðt þ 1Þ ¼ xk

f ðtÞ þ erðtÞgrðtÞh iIM;f

0ð19Þ

where grðtÞ is the corresponding estimator of gr(t)

grðtÞ ¼ grðtÞ þ EbrðtÞ þ EmrðtÞ ð20Þ

grðtÞ ¼@Uf

@xf� kf �

Xq2Qk

f

ðlq þ lDÞ ð21Þ

Here, Ebr(t) is the estimation error of gr(t) and Emr(t) is the martingale difference noise.

Theorem 1. The distributed equations of (13)–(15) and (19) converge to an optimal solution in the steady state.

Proof. First define xkf ð1Þ as the optimal value in the steady state, then

xkf ðt þ 1Þ � xk

f ðtÞ ¼ erðtÞgrðtÞ ð22Þ

Put (21) into (22) and apply the final value from (13)–(15) and (19) in the steady state, respectively. Throwing off the squarebrackets in (19), we get

xkf ðt þ 1Þ � xk

f ðtÞ ¼ erðtÞgrðtÞ ð23Þ

Put grðtÞ ¼ grðtÞ þ EbrðtÞ þ EmrðtÞ into (23), which becomes

xkf ðt þ 1Þ � xk

f ðtÞ ¼ erðtÞ½grðtÞ þ EbrðtÞ þ EmrðtÞ� ð24Þ

where Ebr(t) is the estimation error of gr(t), with Emr(t) the martingale difference noise. We then apply the final-valuetheorem from control theory to (23)

xkf ð1Þ¼ lim

s!0sxk

f ðsÞ

¼ lims!0

s1s erðsÞ

@Uf

@xfðsÞ�kf ðsÞ�

Xq2Qk

f

ðlqðsÞþlDðsÞÞ

264

375

¼ lims!0

s1s erðsÞ

@Uf

@xfðsÞ�kf ðsÞ�

Xq2Qk

f

lqðsÞ

264

375

þlims!0

erðsÞXq2Qk

f

lDðsÞ

¼ xkf ð1Þþ lim

s!0erðsÞ

Xq2Qk

f

lDðsÞ

1762 Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766

Note that when t ?1, then lD(t) = 0 and we get xkf ð1Þ ¼ xk

f ð1Þ. This means that the distributed equations of (13)–(15) and(19) aim to overcome the ill effects of stochastic noise and delays, converging to a steady optimum of the distributed equa-tions. Therefore, we achieve a maximization of network utility. h

Theorem 2. The new distribution equations of (13), (14), (15) and (19) accelerate the rate of convergence.

Proof. Since the source transmission data rate primarily depends on every link’s price through its end-to-end path, weanalyze how the price influences the rate of convergence. The optimal aggregated flow in link q is g�q, assuming every link

has such an optimal flow. If the capacity of link q is c�q, we get a steady state link value of g�q � c�q� �

, and if the network

environment does not change, the state value will remain as this link value. Therefore, we can investigate how to affect a

system approaching this g�q � c�q� �

steady state value.

Consider the sub-gradient Eqs. (13)–(15), where @Uf

@xfhas a definite boundary when xf is far from zero. Note that the partial

differential increases slightly if the source packet rate is near its target and its influence on the transmission rate is nearlyconstant. In addition, kf is equal to 0 when the network is under a medium or heavy load. For simplicity, let us denote@Uf

@xf� kf þ Ebr þ Emr using Im, where Im represents the generalized noise in the wireless system.If we apply a Laplace transformation to (19), we get

Xkf ðsÞ ¼

1ser Im�

Xq2Qk

f

ðlqðsÞ þ lDðsÞÞ

264

375 ð25Þ

The transfer block diagram of the control system is illustrated in Fig. 2. For simplicity, the terms of the addictive gauss whitenoise can be ignored when analyzing the steady state because the equations converge to an optimum solution. We can thenopen the square bracket and apply a Laplace transformation to (14) and (15) to get lqðsÞ ¼ �eq

EðsÞs and lD (s) = �eDE(s), which

we put into (19)

Xkf ðsÞ ¼

1ser

Xq2Qk

f

eq

sþ eD

� �EðsÞ ¼ 1

s2 er

Xq2Qk

f

ðeq þ eDsÞE ðsÞ

If we assume all of the wireless links have the same values of eq and eD, we get

Xkf ðsÞeEðsÞ ¼

erðeq þ eDsÞs2 ð26Þ

where eEðsÞ ¼Pq2QkfEðsÞ.

Therefore, the end result of the new equations from the viewpoint of control theory is to add a new zero point s = �eq/eDto the original system in a closed loop transfer function. The zero point has the effect of accelerating the rate convergence,where the distance of the zero point to the origin determines the speed of convergence and the new relationship between thestep sizes of eq and eD determines the placement of the zero point. Therefore, the new method can accelerate the rate ofconvergence by choosing appropriate zero points in the left complex plane. We will now demonstrate this method throughsimulations. h

x00

x0kmax

xfmax0

xfmaxkmax

cq*

s

-+

-+

Imcq+1

*

-

-

εn(s)/sxf

k

xfk

++

s+

+

eq(t)

eq+1(t)

Fig. 2. The transfer block diagram of the control system.

Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766 1763

4. Performance

In this section, we illustrate the convergence performance of our set of distributed equations via a series of numericalexamples.

4.1. Basic network structure

We first consider a cognitive wireless mesh network, as depicted in Fig. 3, consisting of four edge routers denoted by S1,S2, S3, and S4 as the source nodes. All of the edge routers collect data from primary users, and then transmit the packets togateway node G1 via relay wireless routers R1, R2, and R3. We also assume that the edge routers detect all feasible paths andalso predetermine the paths before transmitting the packets using route protocol. The available paths for the simulation arepresented in Table 1.

We first examine the convergence performance of the distributed algorithm when the link is impacted by feedback noiseinduced by estimator errors. For the simulation, the common available band was 10 MHz, the transmit power was 3 mW,and the received power was inversely proportional to the square of the distance. Table 2 provides the rate for each pathto achieve convergence after running the algorithm.

Fig. 4a illustrates the trajectories of the rate variables f1: {f11, f12, f13}, Fig. 4b shows the trajectories of the rate variablesf2: {f21, f22, f23}, Fig. 4c shows the trajectories of the rate variables f3: {f31, f32, f33}, and Fig. 4d gives the simulation resultsof the rate variables f4: {f41, f42, f43}. Fig. 5 depicts the simulation results of the utility of different flows. The figure showsthat the fluctuations in rate and utility are so significant that 80 iterations are required to reach the equilibrium point. Thisresult reveals that the noises in the link play an important role in the convergence of the distributed equations.

S1 S2 S3 S4

G1

R1R2 R3

Internet

Fig. 3. A simulated three-tier architecture.

Table 1Paths between the edge routers and the gateway router for a cognitive wireless mesh network.

f1 f 11 : S1! R1! G1 f3 f 1

3 : S3! R2! R1! G1

f 21 : S1! R1! R2! G1 f 2

3 : S3! R2! G1

f 31 : S1! R1! R2! R3! G1 f 3

3 : S3! R3! G1

f2 f 12 : S2! R1! G1 f4 f 1

4 : S4! R3! R2! R1! G1

f 22 : S2! R2! G1 f 2

4 : S4! R3! R2! G1

f 32 : S2! R2! R3! G1 f 3

4 : S4! R3! G1

Table 2Convergence rates for each path.

f1 f 11 : 15:4 f3 f 1

3 : 7:6

f 21 : 5:3 f 2

3 : 6:4

f 31 : 4:6 f 3

3 : 3:7

f2 f 12 : 7:8 f4 f 1

4 : 4:8

f 22 : 5:7 f 2

4 : 5:2

f 32 : 3:5 f 3

4 : 17:0

Fig. 4. Convergence of the rate variables (Mbps).

Fig. 5. Convergence of the utility function.

Fig. 6. Convergence of the rate variables (Mbps).

1764 Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766

Fig. 6 illustrates the trajectories of the rate variables of f1, f2, f3, and f4, while Fig. 7 depicts the simulation results of theutility of different flows after we apply distributed equations to overcome the challenge of inherent stochastic control delay.For this simulation, we chose s = �eq/eD = �4.5 as the system zero point. We first compare the rate and utility of every flowusing distributed PD equations to the rates achieved using only stochastically distributed equations. Note that the optimumsfor both rate and utility are the same, which means both algorithms can converge to this value. Moreover, we also observethat the number of iterations of f11 drops from 80 to 40, reducing the convergence time by 50%. The reason for this increasein the speed of convergence is that we implement the PD algorithm at the source nodes; a method which is equivalent inview of control theory to adding a new zero point of s = �4.5 to the system transfer function. Thus, these results show thatour distributed algorithm accelerates the rate of convergence.

Fig. 7. Convergence of the utility function..

0

5

10

15

20

25

30

f1 f2 f3flows

time(s)

original

PD

(a) The convergence time of the delivery rate (original method vs. PD method)

0%

10%

20%

30%

40%

50%

60%

f1 f2 f3flows

ratio

(b) Improvement of the error ratio of the delivery rate

Fig. 8. The convergence time and the error ratio of the delivery rate affected by delay.

Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766 1765

4.2. Randomly deployed network structure

We next studied the performance of our method at a larger scale, simulating 20 nodes deployed over a 500 � 500 m2 re-gion in the network using a randomly generated topology. Fig. 8 shows the results when a randomly distributed hop-to-hopdelay of between 20 ms and 50 ms is added to the system.

Fig. 8 shows the convergence time of the delivery rate compared to the convergence time using the original method de-scribed in [11]. In this paper, the error ratio of the delivery rate is defined by the period from the start point to the point whenthe system enters a steady state. The figure also demonstrates the improvement in the error ratio for our PD method com-pared with the original method. These results lend themselves to the following observations. First, the PD resource allocationalgorithm outperforms the original algorithm in terms of delay. This result is due to of the fact that the PD algorithm re-trieves a quantity that predicts a real error allocation of bandwidth at a slightly later time. This slightly longer lead timein the error allocation allows for determining a more suitable transmission rate for networks with some delay.

1766 Y. Chen et al. / Computers and Electrical Engineering 39 (2013) 1758–1766

5. Conclusion

In this paper, we presented a proportional-differential algorithm combining nonlinear programming and automatic con-trol theory to diminish oscillations in distributed wireless mesh networks. We showed that the method is comparable toexisting distributed schemes for multi-hop mesh networks in two QoS aspects: (i) rate convergence time, and (ii) conver-gence time for network utility. These virtues of the method exist since the proportional-differential algorithm exploitsthe error feedback and the error’s tendency to adapt each source nodes’ rate for multi-hop distributed mesh networks.Our theoretical analysis used control theory to explain why our algorithm can speed up the convergence rate, and provedit can converge to an optimal steady state value. We see our work as a first step in the investigation of using dynamic con-vergence rates for NUM in conjunction with control theory in wireless mesh networks to overcome performance issuescaused by feedback noise and inherent delay.

Acknowledgments

This work was supported by the National Science Foundation of China under Grant (No. 61202470), the Self-DeterminedResearch Funds of CCNU from the Colleges’ Basic Research and Operation of MOE(Nos. CCNU11A02008 and CCNU10A02008),the Initial Scientific Research Fund of CCNU(Nos. 1200050487 and 1200050486), the National Science Foundation Key Projectof China under Grant (No. 60832002), Important National Science & Technology Specific Project of China under Grant (No.2010ZX03004-003-03).

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Yi Chen is an associate professor in School of Computer Science, Huazhong Normal University. She received the B.S. degree and the M.S. degree fromHuazhong University of Science and Technology, China, in 1995 and 1999, respectively. She received her Ph.D. degree in School of Computer Science fromWuhan University, China, in 2011. Her research interests include the wireless networks and multimedia communication.

Ge Gao is currently an associate professor in Wuhan University. He received the Ph.D. degree from Wuhan University, China, in 2002. His research interestsinclude wireless multimedia networks and digital communications.

Ming Liu is a professor in School of Computer Science, Huazhong Normal University. His research interests include electrical techniques and practicalcommunication systems.

Huaxiong Yao received the Bachelor and Ph.D. degree from the department of Electronics and Information Engineering, Huazhong University of Science andTechnology, China, in 2001 and 2007 respectively. He is currently an associate professor in School of Computer Science, Huazhong Normal University. Hisresearch interests include bioinformatics, wireless network and fiber network.

Jinglei Guo received the Ph.D. degree from Wuhan University, China, in 2011. She is currently an associate professor in School of Computer Science,Huazhong Normal University. Her research interests include computer science and artificial intelligence.