Research Article Congestion Control in Wireless Software ...downloads.hindawi.com/journals/misy/2016/3454651.pdf · wireless network. Taking advantages of the applica-bility of Lyapunov-Krasovskii

Research ArticleCongestion Control in Wireless Software DefinedNetworks with Propagation Delay and External Interference:A Robust Control Approach

Xi Hu and Wei Guo

National Key Laboratory of Science and Technology on Communications,University of Electronic Science and Technology of China, Chengdu, China

Correspondence should be addressed to Xi Hu; [email protected]

Received 3 July 2016; Revised 11 October 2016; Accepted 20 October 2016

Academic Editor: Fidel Liberal

Copyright © 2016 X. Hu and W. Guo. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Wireless Software Defined Network (WSDN) presents a new network architecture where the control plane of forwarding devicesis shifted to a centralized controller. It is critical to maximize network throughput and keep the network stable during congestioncontrol. However, stability control is insufficient to achieve these aims in the presence of delay and interference. In this paper,we adopt robust control to tackle these problems. Firstly, an efficient weighted scheduling scheme is proposed to maximize thenetwork throughput. Secondly, a robust control model is presented, which is analyzed by Lyapunov-Krasovskii functionals. Thesufficient conditions are formulated by Linear Matrix Inequalities (LMIs). Finally, a numerical simulation is conducted to indicatethe effectiveness of the proposed scheme.

1. Introduction

In Wireless Software Defined Networks (WSDNs), the con-trol plane of the forwarding devices is shifted to a centralizedcontroller [1–4]. The forwarding devices keep connectedwith the centralized controller for constant monitoring andthe controller may proactively feed control policies to theforwarding devices to keep the network stable, which is morebeneficial to control and manage wireless network [5]. Dueto network congestion caused by excessive data based onnetwork services with the same or lower priorities (known asprecedence order) [6, 7], it is essential to effectively controlcongestion and keep network stable [5, 8], for instance, inthe differentiated services (DiffServ) for Quality of Services(QoS) that is a combination of several qualities or propertiesof a network service. In this case, WSDN cannot achieve themaximal global throughput. Furthermore, it is difficult tomaintain stability of awireless network for a long time. Propa-gation delay and external interference are two key factors thataffect a long time stability control for the stabilityWSDN.Thepresence of propagation delay causes increment of networkcost and unreliability [9], and changing external interferencein wireless environments leads to network instability [10].

The two factors make WSDN more difficult to be achievedby long-time stability controls which are even inadequatefor tackling the restabilizaion problem, and we thus adoptrobust control to solve it. Note that one has to achieve themaximal global throughput via stability control approachesbefore reaching the robustness.

Some existing solutions would prefer to analyze a robustcontrol for network congestion in SDNs [11–13]. Reference[11] focuses on robust network architecture without addi-tional delays by establishing a prototype with the topol-ogy. The authors in [12] modeled network congestion con-trol model with additional time-delay elements to providea prominent QoS based on Transmission Control Proto-col/Active Queue Management (TCP/AQM) networks. In[13], Entity Title Architecture (ETArch) transport model wasproposed to tackle QoS control problem. The prototypesor models are dependently established to analyze networkcongestion control in the current counterpart methods.Unfortunately, these methods cannot robustly control theglobal wireless network congestion for a long time in con-sideration of propagation delay and external interference. Inaddition, they lack sufficient mathematical support.

Hindawi Publishing CorporationMobile Information SystemsVolume 2016, Article ID 3454651, 10 pageshttp://dx.doi.org/10.1155/2016/3454651

2 Mobile Information Systems

In general, the current researches have three cruciallimitations. Firstly, the control policies are not located in thecentralized controller but implemented by flow tables in theforwarding devices. Secondly, propagation delay is seldomconsidered in device-controller pairs during congestion con-trol. Finally, the traditional theories for robust control do notwork well in WSDNs.

In this paper, to obtain the robust control with themaximal global throughput in WSDNs, a new robust controlmodel is proposed by using Lyapunov-Krasovskii functionals[14, 15]. The centralized controller generates control policesand feeds control instructions to the forwarding devices.Thus, the forwarding devices could follow these instructionsto adjust the padding waiting time. By defining the controlledstate as the difference value between the current state and theoptimized state, the robust control problemwith propagationdelay and external interference can be formulated into arobust 𝐻∞ control problem [15]. This robust 𝐻∞ controlproblem is then tackled by Lyapunov-Krasovskii functionals.A theorem is also proposed to give the sufficient conditionsfor the robust 𝐻∞ control in LMIs. Based on the theorem,control policies are proposed. Numerical examples are givento indicate the effectiveness of theoretical analysis.

Ourmain contributions in this paper are listed as follows:(i) Presenting a realistic global control strategy and the

analysis scheme of delay and interference in WSDNs:scheduling scheme currently implemented in thecentralized controller can achieve stability controlthrough the global view of real WSDN with theexternal interference; additionally, the propagationtime-delay is essential in real WSDNs and we definean upper bound of delay as the network propagationdelay and analyze its influences in global view ofWSDN.

(ii) Proposing a general and effective approach to theglobal robust congestion problem, by calculatingoptimized values for network parameters with aweighted fair scheduling scheme, and maintainingthat optimized status via transformation of a closed-loop network model to a normal robust 𝐻∞ controlmodel.

(iii) An interdisciplinary effort is made to construct arobust control strategy by combining the stabilityanalysis theories and congestion control principles ofwireless network. Taking advantages of the applica-bility of Lyapunov-Krasovskii functionals in stabilityanalysis of WSDN, the paper achieves the desiredglobal robust control for network congestion.

The rest of the paper is organized as follows. Section 2briefly introduces the related work. Section 3 proposes thescheduling model and establishes a robust 𝐻∞ controlmodel with propagation delay and external interference indevice-controller pairs. Section 4 introduces the problemformulation and some preliminaries. Section 5 models therobust control by using Lyapunov-Krasovskii functionals andobtains sufficient conditions. In Section 6, simulations aregiven to verify the theoretical analysis. The conclusions aredrawn in Section 7.

2. Related Work

In the recent years, congestion control in WSDNs hasattracted considerable attention. Numerous researches haveoffered stability and robust control algorithms for congestioncontrol in many wireless network scenarios, for example,WSN and VANET, to achieve robust control for networkcongestion.

Stability control of network congestion has drawn wide-spread attention and research interests [16, 17]. Existing workhas been proposed to tackle the stability control problem. In[18] an analytic study on the interval service response waspresented to evaluate the performance based on black burstmechanism. A scheduling strategy was proposed to establishtheWSN dynamical model, which adopted traditional stabil-ity control theory to address system state stabilization [19]. In[20], the authors operatedmultiple sensors scheduling simul-taneously, and let these sensors switch randomly accordingto some optimal probability distribution to obtain the bestexpected stable state performance.

Robust control has attracted particular interest in theliterature for the traditional network control system. Basedon the wireless characteristic, robust congestion for networkcongestion has been applied in many wireless networkcircumstances, for example, WSN [21], VANET [22, 23],Queueing Network [24], and WSDN [11, 25] In WSDNs, thecentralized controller had a global view and is responsible forthe control and management of all flow tables at each Open-Flow device [25]. A modified SDN-OpenFlow architecturewith general operation logic, Multi-Protocol Label Switch(MPLS) management logic, and QoS management logic, wasproposed in [11]. In [26] the authors demonstrated the SDNcontrol model in Wireless Sensor and Actor Networks forresource allocation in tasks processing.

Lyapunov-Krasovskii functionals and Linear Matrix In-equalities (LMIs) method are usually proposed to controlnetwork congestion. In [9], the authors provided propagationdelay model for traditional load frequency control based onLinearMatrix Inequalities (LMIs).𝐻∞ is adopted in [15, 27–30] in Network Control Systems (NCSs) to achieve the robustcongestion control.

3. Model and Analysis

Figure 1 shows a typical scenario of aWSDNwith propagationdelay and external interference. The centralized controller isable to collect information from all forwarding devices todeal with network congestion. There are two types of delaysoccurring in thewireless networks: implementation delay andpropagation delay [31]. Comparing to the propagation delay,extremely short implementation delay is ignored and propa-gation delay is considered to analyze the network model, andthey usually are not of the same order. The propagation delayis affected by the control plane transmission.The variability ofinterference in wireless environments leads to network insta-bility [10].The centralized controllermakes global schedulingfor the forwarding devices, and accordingly the influence ofinternal interference is avoided via the prearrangement ofnetwork parameters. The paper thus merely considers the

Mobile Information Systems 3

Centralizedcontroller

Forwardingdevice

WSDN

u(k)

y(k

)

dcd

(k)

ddc

(k)

u(k

)

Figure 1: A typical scenario of WSDN with the propagation delay𝑑dc(𝑘) from the forwarding device to the centralized controller (DC)and the one 𝑑cd(𝑘) from centralized controller to the forwardingdevice (CD).

external interference. There exists an optimized stable statein each forwarding device on stability control for networkcongestion. Based on the scheduling scheme, the controlpolicies in the centralized controller provide network servicesto process the data flows on the forwarding devices.

Our goal is to maximize the global network throughputby stability control and keep the global network parametersstable at this optimal state by robust control with propagationdelay and external interference. The propagation delay isdenoted by 𝑑(𝑘), where 𝑘 is the discrete count number.

The padding waiting time is a key network parameterfor robust control in WSDNs. As shown in Figure 2, theprocessing time is the duration consumed to process anetwork service. A waiting duration may also be introducedby a scheduling scheme to slow down the forwarding device,and further processing of network services is postponed forthis waiting duration. The padding waiting time is definedas the sum of the processing time and the waiting duration.When all forwarding devices are assumed to be at thesame processing speeds, minimizing the sum of the paddingwaiting timemay shorten the total service time of all networkservices and maximize network throughput.

The closed-loop WSDN may be classified into two parts,which are analyzed in the following subsections.

3.1. A Nonpreemptive Scheduling Scheme Design on StabilityControl for Maximizing the Global WSDN Throughput. Anovel nonpreemptive scheduling scheme is proposed, whichcan be prearranged to stabilize the network parameters ofeach forwarding device. In order to tackle network con-gestion problem, in a scheduling problem, one has to findthe minimization of the sum of the padding waiting timeunder the given constraints. Appropriate padding waitingtimes are arranged for each forwarding device.There exists anoptimized scheduling scheme in each forwarding device.The

schemes together implement the stability control for networkcongestion and maximize the global network throughput.

There are 𝑁 end-to-end network transmission services𝐸𝑖, 𝑖 = 1, 2, . . . , 𝑁. Each network transmission service iscomposed of 𝑛𝑖 point-to-point network transmissions 𝑇𝑖𝑗,where 𝑗 = 1, . . . , 𝑛𝑖 and 𝑛𝑖 ≤ 𝑖. The point-to-pointnetwork transmission𝑇𝑖𝑗 obeys a sequential precedence order(QoS/QoE) 𝑇𝑖1 → 𝑇𝑖2 → ⋅ ⋅ ⋅ → 𝑇𝑖𝑛𝑖 that compose thecorresponding service 𝐸𝑖. For convenience, a discretizationof event-based sampling ismade, so the centralized controllercan be considered as an event-based digital controller. Thereare 𝐾 forwarding devices 𝐹1, 𝐹2, . . . , 𝐹𝐾 in the closed-loopWSDN that can process these point-to-point network trans-missions.

Each point-to-point network transmission 𝑇𝑖𝑗 is pro-cessed within a given duration𝑝𝑖(𝑘) > 0with nonpreemptionby a forwarding device, where 𝑘 is the discrete count number.The specific forwarding device is defined as as 𝑢𝑖𝑗(𝑘) ∈{𝐹𝑘, 𝑘 = 1, 2, . . . , 𝐾}. Suppose that each forwarding devicecan process only one𝑇𝑖𝑗 at a time, and there are enough bufferspaces in the forwarding devices to store pending networktransmissions. Once the forwarding device 𝑘 starts to processthe network transmission services in the fixed duration 𝑝𝑖(𝑘),it will not stop until the network transmission services arefinished. Each point-to-point network transmission 𝑇𝑖𝑗 isarranged to a forwarding device 𝐹𝑖𝑗 in the controller fromthe previous configuration, such as the wireless routingalgorithm. The arrangement had been optimized before thetransmission in the ideal network situation. Assume thatthere are only 𝑘 levels of the padding waiting time, which isproportional to weight 𝑤𝑖. Define a scheduling 𝑆 = (𝑆𝑖(𝑘)) asthe vectors of the padding waiting time for all the networktransmission 𝑇𝑖𝑗 in the forwarding devices 𝑢𝑖𝑗(𝑘). Notably, wehave 𝑆𝑖𝑛𝑖(𝑘) = 0, where the destination devices finish the end-to-end transmission without any more padding waiting time.

Above all, the following arrangement variables is intro-duced to propose a mathematical description

𝑞𝑖 (𝑘) = {{{1 if 𝑇𝑖𝑗 is arranged in 𝑢𝑖𝑗 (𝑘)0 otherwise. (1)

For each arrangement 𝑞(𝑘) = (𝑞𝑖(𝑘)), we aim at minimiz-ing the sumof the paddingwaiting time.When all forwardingdevices have the same processing speeds, minimizing thesum of the padding waiting time may shorten total servicetime of all network services andmaximize the global networkthroughput.Thus, the optimized problem for stability controlcan be formulated as follows:

min𝑘∈N

∑𝑖

𝑆𝑖 (𝑘)s.t. 𝑆𝑖 (𝑘) + ∑

𝑚∈𝑢𝑖𝑚(𝑘)

𝑞𝑚 (𝑘) (𝑝𝑚 (𝑘) + 𝑆𝑚 (𝑘))≥ 𝑆𝑖 (𝑘 + 1) (𝑖 ≥ 𝑚 = 1, 2, . . . , 𝑛𝑖 − 1) ,𝑆1𝑤1 = ⋅ ⋅ ⋅ = 𝑆𝑖𝑤𝑖 = ⋅ ⋅ ⋅ = 𝑆𝑁𝑤𝑁 ,


Network services

Forwardingdevice

Forwarding device

Schedulingscheme

Centralizedcontroller

Padding waiting time

Processing time Waiting duration

Figure 2: The padding waiting time for robust control in WSDNs.

∑𝑘∈𝑢𝑖𝑗(𝑘)

𝑞𝑖 (𝑘) ≤ ∑𝑘∈𝑇𝑖𝑗

𝑞𝑖 (𝑘) = 1,∑𝑖

𝑆𝑖𝑛𝑖 (𝑘) = 0,𝑞𝑖 (𝑘) ∈ {0, 1} ,𝑆𝑖 (𝑘) > 0.

(2)

The padding waiting times can be arranged under theconstraints before transmission. At the beginning, the opti-mized problemofminimizing the sumof the paddingwaitingtime can be easily solved and the arrangement proportionis uncomplicated calculated as 𝑆(𝑘). At this time, whenthe optimized state 𝑆(𝑘) is calculated and prearranged, thepadding waiting time of each forwarding device is stable onstability control at the criterion state 𝑆(𝑘).

However, propagation delay and external interferencein wireless environments may cause the network unstable.Therefore, the robust control problem is considered to main-tain network stability in the presence of propagation delayand external interference.

3.2. Robust Control Model with Propagation Delay and Exter-nal Interference. Briefly speaking, when a network trans-mission service enters the WSDN or it is generated in aforwarding device, it is queued in the buffer and waits forprocessed and transmitted. If the communication mediumbecomes free, the scheduling scheme could be nonpreemp-tively implemented for establishing an end-to-end path in thecentralized controller after receiving the advertisements fromthe forwarding devices. The controller designs the controlpolicy by means of the scheduling scheme and then sendscontrol instructions to adjust the padding waiting time ineach forwarding device.

With propagation delay and external interference, thecurrent state 𝑆𝑖(𝑘) in each forwarding device pursues theprearranged ideal state 𝑆𝑖(𝑘) (referring to Section 3.1) byrobust control, that is, 𝑆1(𝑘) → 𝑆1(𝑘), . . . , 𝑆𝑖(𝑘) → 𝑆𝑖(𝑘), . . . ,𝑆𝑁(𝑘) → 𝑆𝑁(𝑘). Let 𝑥𝑖(𝑘) = 𝑆𝑖(𝑘) − 𝑆𝑖(𝑘) be the error stateand lim𝑘→∞‖𝑆𝑖(𝑘) − 𝑆𝑖(𝑘)‖ = 0 for 𝑖 = 1, . . . , 𝑁.

According to analyzing the propagation delay in theclosed-loop WSDN, the propagation delay from forwardingdevice to the centralized controller (DC) and the reversed one(CD) are defined as 𝑑dc(𝑘) and 𝑑cd(𝑘), respectively. Supposethat the DC delay 𝑑dc(𝑘) is observable, and the forwardingdevice receives the feedbackmessage from the controller withtheCDdelay𝑑cd(𝑘) in Figure 1. Denote𝑑(𝑘) = 𝑑dc(𝑘)+𝑑cd(𝑘).

The forwarding devices continually adjust their paddingwaiting times following the control instructions. Thus, thenetwork services in the forwarding devices constantly achievenonpreemptive scheduling bymeans of the advertisements (apacket-in message).

First, the forwarding devices advertise the error stateof the network service 𝑥(𝑘) = (𝑥𝑖(𝑘)) to the controller.With the packet-in message, the controller calculates 𝑥(𝑘),and the controller classifies the global information of theerror state and generates a control policy 𝑢(𝑘) to keepingthe padding waiting time stable in Figure 1. The controlpolicy needs to restabilize the padding waiting time in thepresence of propagation delay and external interference. Thecontroller makes appropriate adjustments of the weightedmatrix 𝐵𝑢 ∈ R𝑛×𝑛. Finally, the controller sends a packet-outmessage, which indicates that the flow originated the packet-in message and has been implemented associated with thecontrol instruction. The closed-loop WSDN is accomplishedand modeled with propagation delay as

𝑥 (𝑘 + 1) = 𝐴𝑥 (𝑘) + 𝐵𝑢𝑢 (𝑘) , (3)

where 𝐴 is the parameters represented the network featuresthat are nonnegative constant matrices with appropriatedimensions.

Thus, the control command 𝑢(𝑘) = 𝐾𝑥(𝑘𝑙), 𝐾 ∈ R𝑛×𝑛

denotes robust control strength, and the control policies areconsidered as

𝑥 (𝑘 + 1) = 𝐴𝑥 (𝑘) + 𝐵𝑢𝑢 (𝑘) ,𝑢 (𝑘) = 𝐾𝑥 (𝑘𝑙) ,

𝑘 ∈ [𝑘𝑙 + 𝑑 (𝑘) , 𝑘𝑙 + 𝑑 (𝑘 + 1)] ,(4)


where 𝑑(𝑘) = 𝑑dc(𝑘) + 𝑑cd(𝑘), and 𝑢(𝑘) is the controlinstructions and 𝑘𝑙 is the discrete count number. Consider thepropagation delay and rewrite 𝑢(𝑘) as𝑢 (𝑘𝑙) = 𝑢 (𝑘 − 𝑑 (𝑘 − 𝑘𝑙)) = 𝑢 (𝑘 − 𝑑 (𝑘))

= 𝐾𝑥 (𝑘 − 𝑑 (𝑘)) ,𝑘 ∈ [𝑘𝑙 + 𝑑 (𝑘) , 𝑘𝑙 + 𝑑 (𝑘 + 1)] .

(5)

The external interference in this paper is defined asthe interference of the duration. Therefore, the externalinterference is considered as a kind of additive interference,which may lengthen the padding waiting times in the for-warding devices.Then, the closed-loop networkmodel can beformulated into a robust𝐻∞ control model. Simultaneously,the closed-loop WSDN (4) adds the external interferencepart.

𝑥 (𝑘 + 1) = 𝐴𝑥 (𝑘) + 𝐵𝑢𝐾𝑥 (𝑘 − 𝑑 (𝑘)) + 𝐵𝑤𝑤 (𝑘) , (6)

where 𝐵𝑤 is the weight of external interference that isnonnegative constant matrices with appropriate dimensions.For convenience, we assume the external interference islimited energy and duration.

Thus, the robust𝐻∞ controlmodel of the error state withpropagation delay and external interference is formulated inthe closed-loop WSDN.

4. Problem Formulations ofRobust 𝐻∞ Control

According to the term (4), the robust 𝐻∞ control model ofthe error state with propagation delay and external interfer-ence in the closed-loop WSDN is described by

𝑥 (𝑘 + 1) = 𝐴𝑥 (𝑘) + 𝐵𝑢𝑢 (𝑘) + 𝐵𝑤𝑤 (𝑘) ,𝑢 (𝑘) = 𝐾𝑥 (𝑘 − 𝑑 (𝑘)) ,𝑧 (𝑘) = 𝐼𝑥 (𝑘) ,

(7)

where 𝑥(𝑘) = [𝑥1(𝑘), . . . , 𝑥𝑛(𝑘)]𝑇 ∈ R𝑛 is the error statewhich denotes the varying value of the padding waitingtime between the current state and the ideal state, 𝑢(𝑘) =[𝑢1(𝑘), . . . , 𝑢𝑛(𝑘)]𝑇 ∈ R𝑛 is the control instruction, 𝑤(𝑘) =[𝑤1(𝑘), . . . , 𝑤𝑛(𝑘)]𝑇 ∈ R𝑛 is the external interference oflimited energy and duration with covariance matrix 𝑤 andhas expectation zero, 𝑧(𝑘) = [𝑧1(𝑘), . . . , 𝑧𝑛(𝑘)]𝑇 as a measure-ment is the output of the robust control, and 𝑘 is the discretecount number. Define

𝐴 = (𝑎𝑖𝑗)𝑛×𝑛 = {{{𝑎𝑖𝑗 = 0, if 𝑖 < 𝑗0 ≤ 𝑎𝑖𝑗 ≤ 1, if 𝑖 ≥ 𝑗,

𝐵𝑢 = (𝑏𝑖𝑗)𝑛×𝑛 ,𝐵𝑤 = diag {𝑏1, ��2, . . . , ��𝑛} ,

(8)

and the constant matrix 𝐼 ∈ R𝑛×𝑛. As the aforementionedanalysis, 𝑢(𝑘) = 𝐾𝑥(𝑘 − 𝑑(𝑘)), 𝐾 ∈ R𝑛×𝑛 denotes robustcontrol strength. 𝑑(𝑘) denotes the propagation delay.

Definition 1 (see [29]). For given two positives 𝑑𝑚, 𝑑𝑀 >0, the closed-loop WSDN (7) is stable with the externalinterference 𝑤(𝑘). 𝑑𝑚, 𝑑𝑀 represented its upper and lowerbounds, respectively, which satisfies 0 < 𝑑𝑚 < 𝑑(𝑘) ≤ 𝑑𝑀.Definition 2. The closed-loop WSDN (7) is said to stable,if there exists a state feedback control instruction 𝑢(𝑘) =𝐾𝑥(𝑘−𝑑(𝑘)),𝐾 ∈ R𝑛×𝑛.Thus,𝑢(𝑘) is said to the robust controlpolicies.

Lemma 3 (Schur complement [32]). Given constant matrices𝑃, 𝑄, 𝑅, where 𝑃𝑇 = 𝑃 and 𝑄𝑇 = 𝑄, and then the LMI[ 𝑃 𝑅𝑅𝑇 −𝑄 ] < 0 is equivalent to the following condition: 𝑄 > 0,𝑃 + 𝑅𝑄−1𝑅𝑇 < 0.5. Criteria of Robust 𝐻∞ Control

In the following, let𝐴 = 𝐴−𝐼, 𝐵 = 𝐵𝑢𝐾. Rewrite closed-loopWSDN of the error state (7) into a more compact form as

𝑦 (𝑘) = 𝐴𝑥 (𝑘) + 𝐵𝑥 (𝑘 − 𝑑 (𝑘)) + 𝐵𝑤𝑤 (𝑘) ,𝑧 (𝑘) = 𝐼𝑥 (𝑘) , (9)

where 𝑦(𝑘) = 𝑥(𝑘 + 1) − 𝑥(𝑘) is the difference state, and𝑦 (𝑘) = [𝑥1 (𝑘 + 1) − 𝑥1 (𝑘) , 𝑥2 (𝑘 + 1)

− 𝑥2 (𝑘) , . . . , 𝑥𝑛 (𝑘 + 1) − 𝑥𝑛 (𝑘)]𝑇 . (10)

Theorem 4. Consider the robust 𝐻∞ control model of theerror state with propagation delay and external interferencein the closed-loop WSDN (9). Given the external interferenceattenuation level 𝛾 and positive integers 𝑑𝑚, 𝑑𝑀 (0 < 𝑑𝑚 <𝑑𝑀). The closed-loop WSDN achieves robust 𝐻∞ control, ifthere exist appropriate dimension symmetric positive definitematrices 𝑃 > 0, 𝑄 > 0, 𝑅 > 0, and 𝑆 > 0, 𝑋 = [𝑋11 𝑋12∗ 𝑋22

] ≥0, and appropriate dimension matrices 𝑁1, 𝑁2, so that thefollowing conditions in (11) hold:

Ξ = [[[𝑋11 𝑋12 𝑁1∗ 𝑋22 𝑁2∗ ∗ 𝑅

]]]

≥ 0,

Ω =[[[[[[[[[[[[[

𝜔11 𝜔12 𝑃𝐵𝑤 𝐴𝑇 𝑑𝑀𝐴𝑇 𝐼∗ 𝜔22 0 𝐵𝑇 𝑑𝑀𝐵𝑇 0∗ ∗ −𝛾2𝐼 𝐵𝑇𝑤 𝑑𝑀𝐵𝑇𝑤 0∗ ∗ ∗ −𝑃−1 0 0∗ ∗ ∗ ∗ −𝑑𝑀𝑅−1 0∗ ∗ ∗ ∗ ∗ −𝐼

]]]]]]]]]]]]]

< 0,(11)


where

𝜔11 = 𝑃𝐴 + 𝐴𝑇𝑃 + 𝑄 + (𝑑𝑀 − 𝑑𝑚 + 1) 𝑆 + 𝑁1 + 𝑁𝑇1+ 𝑑𝑀𝑋11,

𝜔12 = 𝑃𝐵 − 𝑁1 + 𝑁𝑇2 + 𝑑𝑀𝑋12,𝜔22 = −𝑄 − 𝑆 − 𝑁2 − 𝑁𝑇2 + 𝑑𝑀𝑋22.

(12)

Proof. We firstly define

𝑦 (𝑙) = 𝑥 (𝑙 + 1) − 𝑥 (𝑙) . (13)

Then, we obtain

𝑥 (𝑘 + 1) = 𝑥 (𝑘) + 𝑦 (𝑘) ,𝑥 (𝑘) − 𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑘−1∑

𝑙=𝑘−𝑑(𝑘)

𝑦 (𝑙) = 0. (14)

In the closed-loop network, the Lyapunov-Krasovskiifunctionals can be expressed by

𝑉 (𝑘) = 4∑𝑛=1

𝑉𝑛 (𝑘) ,𝑉1 (𝑘) = 𝑥𝑇 (𝑘) 𝑃𝑥 (𝑘) ,𝑉2 (𝑘) = 𝑘−1∑

𝑗=𝑘−𝑑(𝑘)

𝑥𝑇 (𝑗)𝑄𝑥 (𝑗) ,

𝑉3 (𝑘) = 0∑𝜃=−𝑑𝑀+1

𝑘−1∑𝑗=𝑘−1+𝜃

𝑦𝑇 (𝑗) 𝑅𝑦 (𝑗) ,

𝑉4 (𝑘) = −𝑑𝑚+1∑𝑗=−𝑑𝑀+1

𝑘−1∑𝑙=𝑘+𝑗−1

𝑥𝑇 (𝑙) 𝑆𝑥 (𝑙) ,

(15)

where 𝑃 = 𝑃𝑇 > 0, 𝑄 = 𝑄𝑇 > 0, 𝑅 = 𝑅𝑇 > 0, and 𝑆 = 𝑆𝑇 >0 are positive definite symmetric matrices. Define △𝑉(𝑘) =𝑉(𝑘 + 1) − 𝑉(𝑘); thus△𝑉1 (𝑘) = 𝑥𝑇 (𝑘 + 1) 𝑃𝑥 (𝑘 + 1) − 𝑥𝑇 (𝑘) 𝑃𝑥 (𝑘)

= 2𝑥𝑇 (𝑘) 𝑃𝑦 (𝑘) + 𝑦𝑇 (𝑘) 𝑃𝑦 (𝑘) ,△𝑉2 (𝑘) = 𝑘∑

𝑗=𝑘−𝑑(𝑘)+1

𝑥𝑇 (𝑗)𝑄𝑥 (𝑗)

− 𝑘−1∑𝑗=𝑘−𝑑(𝑘)

𝑥𝑇 (𝑗)𝑄𝑥 (𝑗)= 𝑥𝑇 (𝑘) 𝑄𝑥 (𝑘)

− 𝑥𝑇 (𝑘 − 𝑑 (𝑘)) 𝑄𝑥 (𝑘 − 𝑑 (𝑘)) ,

△𝑉3 (𝑘) = 𝑑𝑀𝑦𝑇 (𝑘) 𝑅𝑦 (𝑘) − 𝑘−1∑𝑙=𝑘−𝑑𝑀

𝑦 (𝑙) 𝑅𝑦 (𝑙)

≤ 𝑑𝑀𝑦𝑇 (𝑘) 𝑅𝑦 (𝑘) − 𝑘−1∑𝑙=𝑘−𝑑(𝑘)

𝑦𝑇 (𝑙) 𝑅𝑦 (𝑙) ,△𝑉4 (𝑘) = (𝑑𝑀 − 𝑑𝑚 + 1) 𝑥𝑇 (𝑘) 𝑆𝑥 (𝑘)

− 𝑘−𝑑𝑚∑𝑙=𝑘−𝑑𝑀

𝑥𝑇 (𝑙) 𝑆𝑥 (𝑙)≤ (𝑑𝑀 − 𝑑𝑚 + 1) 𝑥𝑇 (𝑘) 𝑆𝑥 (𝑘)

− 𝑥𝑇 (𝑘 − 𝑑 (𝑘)) 𝑆𝑥 (𝑘 − 𝑑 (𝑘)) .(16)

For any appropriate dimension matrix 𝑁𝑖 (𝑖 = 1, 2), wehave

0 = 2 [𝑥𝑇 (𝑘)𝑁1 + 𝑥𝑇 (𝑘 − 𝑑 (𝑘))𝑁2]× [𝑥 (𝑘) − 𝑥 (𝑘 − 𝑑 (𝑘)) − 𝑘−1∑

𝑙=𝑘−𝑑(𝑡)

𝑦 (𝑙)] . (17)

For an appropriate dimension matrix 𝑋 = [𝑋11 𝑋12∗ 𝑋22] ≥ 0,

we get

0 ≤ 𝑘−1∑𝑙=𝑘−𝑑𝑀

𝜁𝑇1 (𝑘)𝑋𝜁1 (𝑘) − 𝑘−1∑𝑙=𝑘−𝑑(𝑘)

𝜁𝑇1 (𝑘)𝑋𝜁1 (𝑘)

= 𝑑𝑀𝜁𝑇1 (𝑘)𝑋𝜁1 (𝑘) − 𝑘−1∑𝑙=𝑘−𝑑(𝑡)

𝜁𝑇1 (𝑘)𝑋𝜁1 (𝑘) .(18)

Thus, from (16) to (18), we have

△𝑉 (𝑘) ≤ 𝜁𝑇2 (𝑘) {Λ + Π𝑇1 (𝑃 + 𝑑𝑀𝑅)Π1} 𝜁2 (𝑘)− 𝑘−1∑𝑙=𝑘−𝑑(𝑘)

𝜁𝑇3 (𝑘, 𝑙) Ξ𝜁3 (𝑘, 𝑙)+ 𝛾2𝑤𝑇 (𝑘) 𝑤 (𝑘) ,

(19)

with

𝜁1 (𝑘) = [𝑥𝑇 (𝑘) 𝑥𝑇 (𝑘 − 𝑑 (𝑘))]𝑇 ,𝜁2 (𝑘) = [𝑥𝑇 (𝑘) 𝑥𝑇 (𝑘 − 𝑑 (𝑘)) 𝑤𝑇 (𝑘)]𝑇 ,

𝜁3 (𝑘, 𝑙) = [𝑥𝑇 (𝑘) 𝑥𝑇 (𝑘 − 𝑑 (𝑘)) 𝑦𝑇 (𝑙)]𝑇 ,

Λ = [[[[

𝜑11 𝜑12 𝑃𝐵𝑤∗ 𝜑22 0∗ ∗ −𝛾2𝐼

]]]]

,

Π1 = [𝐴 𝐵 𝐵𝑤]𝑇 ,Π2 = [𝐼 0 0] .

(20)


Defining Θ = Λ + Π𝑇𝑃Π + 𝑑𝑀Π𝑇𝑅Π and using Schurcomplement lemma (Lemma 3), the LMIs in (19) can makeinequalities Θ < 0 true. Then there exists a positive scalar𝜀 > 0 such thatΘ < 𝜀𝐼 < 0. Therefore, it follows that Δ𝑉(𝑘) ≤−𝜀‖𝑥(𝑘)‖2 < 0 ∀𝑥(𝑘) = 0.

Considering 𝑤(𝑡) = 0 and Schur complement lemma,following the inequalities (11), we have

△ 𝑉 (𝑘) + 𝑧𝑇 (𝑘) 𝑧 (𝑘) − 𝛾2𝑤𝑇 (𝑘) 𝑤 (𝑘)≤ 𝜉𝑇 (𝑘)Ω𝜉 (𝑘) < 0. (21)

Sum 𝑘 from 0 to∞with the initialization of𝑉(0) = 0; wecan obtain

∞∑𝑘=0

{𝑧𝑇 (𝑘) 𝑧 (𝑘) − 𝛾2𝑤𝑇 (𝑘) 𝑤 (𝑘)} < 0. (22)

Based on the Lyapunov-Krasovskii functionals, the robust𝐻∞ control model of the error state with propagationdelay and external interference in the closed-loop WSDNcan achieve robust 𝐻∞ control 𝐽 < 0 with desired 𝐻∞performance index ‖𝑇𝑤𝑧(𝑧)‖∞ < 𝛾 following (11).

The proof is complete.

The parameters of desired robust 𝐻∞ control can beobtained through LMI in MATLAB, and the performanceevaluations are given through SIMULINK in MATLAB.

6. Simulation Results

In this section, a numerical simulation is conducted toindicate the effectiveness of the proposed scheme in WSDNsand the control policies are designed given inTheorem 4.

In actual WSDNs, QoS mechanism is the typical instanceas the precedence order of network service contained inte-grated service and differentiated service. Usually, there areeight priority levels in the QoS mechanism, defined from 0to 7 with 0 being the highest. Denote 𝑥𝑖(𝑘), 𝑖 = 0, 1, . . . , 7,as these eight priority levels in WSDNs. Consider the robust𝐻∞ control model (9) with different network parameters toclearly demonstrate different QoS control policies in the cen-tralized controller as follows: 𝐵𝑤 = diag{0.2, 0.1, 0.31, 0.51,0.11, 0.21, 0.31, 0.41} and𝐴

=

[[[[[[[[[[[[[[[[[[

0.22 0 0 0 0 0 0 00.36 0.15 0 0 0 0 0 00.65 0 0.3 0 0 0 0 00.1 0 0.51 0.22 0 0 0 00.63 0.14 0 0.25 0.33 0 0 00.36 0 0.25 0 0 0.47 0 00.36 0.94 0.74 0.51 0 0.65 0.45 00.31 0.26 0 0.59 0 0.3 0.1 0.14

]]]]]]]]]]]]]]]]]]

, (23)

2 4 6 8 10 120k

Variation of y1(k)Variation of y2(k)Variation of y3(k)Variation of y4(k)


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

yi(k

),i=

0,1,

...,7

Figure 3: The variations of 𝑦𝑖(𝑘) (at eight priority levels) with theinitial state 𝑦𝑖(0) = 0.5, 𝑖 = 0, 1, . . . , 7, in the presence of propagationdelay and external interference.

where 𝐴 denotes the relationship between 𝑥(𝑘 + 1) and 𝑥(𝑘).Define

𝐴 = (𝑎𝑖𝑗) = {{{𝑎𝑖𝑗 = 0, if 𝑖 < 𝑗0 ≤ 𝑎𝑖𝑗 ≤ 1, if 𝑖 ≥ 𝑗 (24)

that means the precedence orders. In actual wireless networkenvironment, before starting its execution, the flow withlower priority needs to wait for finishing the completions ofall flow queues with nonlower priorities in the forwardingdevice 𝑘.The padding waiting time consists of the probabilityweight of every nonlower priority.

6.1. Effectiveness Verification of the Proposed Scheme inWSDNs. According to Theorem 4, there exists a feasiblesolution to LMIs (11). We use the different state 𝑦𝑖(𝑘) = 𝑥𝑖(𝑘+1)−𝑥𝑖(𝑘) to reflect the approaching results of error state.Thatis, 𝑦𝑖(𝑡) means the error state of the padding waiting timebetween adjacent time intervals.

Suppose the control strength 𝐾 = −0.8𝐼, the controlpolicy 𝐵𝑢 = 𝐴, and the initial conditions are 0.5 and 1,respectively. Two scenarios with the different initial statesare considered to make a comparison. Suppose that a stepfunction represents the external interference with limitedenergy and duration to make the simulation tractable.

Notably, the error states may increase or decrease basedon the different initial state. However, the difference states𝑦(𝑘) can maintain system stable in the presence of propa-gation delay and external inferences, as shown in Figures3 and 4. The different initial states still make convergencein the presence of the step external interference, whichrepresents that all error states 𝑥(𝑘) reach an agreement


0

0.2

0.4

0.6

0.8

1

1.2

2 4 6 8 10 120k



yi(k

),i=

0,1,

...,7

Figure 4: The variations of 𝑦𝑖(𝑘) (at eight priority levels) with theinitial state 𝑦𝑖(0) = 1, 𝑖 = 0, 1, . . . , 7, in the presence of propagationdelay and external interference.

by the centralized controlling. Thus, this simulation canbe conducted to indicate the effectiveness of the proposedscheme in WSDNs.

The final convergence of error states in Figures 3 and4 shows that the robust control has stabilized the statedivergence aroused by the propagation delay and externalinterference, which occurred beyond the stable initial statusconducted by traditional stability control. In other words,the robust control has successfully enhanced the effect of thestabilization by traditional stability control.

6.2. Design of Control Policy on Robust 𝐻∞ Control inWSDNs. This section introduces the design of the controlpolicy based on the proposed scheme in WSDNs. We selectthe intermediate priority 𝑖 = 4 in the simulation. Figures 5and 6 show the variations of the error state under the differentcontrol strength 𝐾. The initial state is 𝑦4(0) = 0.5 (Figure 5),while 𝑦4(0) = 1 (Figure 6). Compared with the differentinitial states and control strength𝐾, it is notable that a tightercontrol results in the smaller width.

Therefore, the appropriate adjustments of QoS controlpolicy can easily be designed in the controller. The controlpolicy can be designed to control the width measurementunder different initial error states.

7. Conclusion

This paper have adopted robust control to tackle the problemsof maximizing network throughput and keeping the networkstable during congestion control in WSDNs. Firstly, anefficient nonpreemptive scheduling scheme has been pro-posed to maximize the global network throughput. Secondly,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y4(k

)

K = −0.8I

2 4 6 8 10 120k

K = −I

K = −0.5I

Figure 5: The variations of 𝑦4(𝑘) with the initial state 𝑦4(0) = 0.5and different control strength𝐾 in the presence of propagation delayand external interference.

K = −0.5IK = −0.8IK = −I

2 4 6 8 10 120k

0

0.2

0.4

0.6

0.8

1

1.2

y4(k

)

Figure 6:The variations of 𝑦4(𝑘) with the initial state 𝑦4(0) = 1 anddifferent control strength𝐾 in the presence of propagation delay andexternal interference.

a robust control model with propagation delay and externalinterference is presented by using Lyapunov-Krasovskii func-tionals. The sufficient conditions have been formulated byLMIs. Finally, the numerical simulation has been conductedto indicate the effectiveness of the proposed scheme.

Future studies should explore the impact of implemen-tation delays and stochastic external interference on theperformance of robust control for network congestion inWSDNs. Delays are ubiquitous in wireless networks while


frequently causing stability problems. The approach that wehave presented would be extended to develop more complexalgorithms and be applied in other wireless fields.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work is supported by the 863 project (Grant no.2014AA01A701); the National Natural Science Foundation ofChina (Grant nos. 61271168, 61471104).

References

[1] S. Agarwal, M. Kodialam, and T. V. Lakshman, “Traffic engi-neering in software defined networks,” in Proceedings of the32nd IEEE Conference on Computer Communications (IEEEINFOCOM ’13), pp. 2211–2219, Turin, Italy, April 2013.

[2] L. F. Muller, R. R. Oliveira, M. C. Luizelli, L. P. Gaspary, andM. P. Barcellos, “Survivor: an enhanced controller placementstrategy for improving SDN survivability,” in Proceedings of theIEEEGlobal Communications Conference (GLOBECOM ’14), pp.1909–1915, IEEE, Austin, Tex, USA, December 2014.

[3] Z. Y. Su, T. Wang, Y. Xia, and M. Hamdi, “FlowCover: low-cost flow monitoring scheme in software defined networks,”in Proceedings of the IEEE Global Communications Conference(GLOBECOM ’14), pp. 1956–1961, Austin, Tex, USA, December2014.

[4] N. McKeown, T. Anderson, H. Balakrishnan et al., “OpenFlow:enabling innovation in campus networks,” ACM SIGCOMMComputer Communication Review, vol. 38, no. 2, pp. 69–74,2008.

[5] S. Tomovic, N. Prasad, and I. Radusinovic, “SDN controlframework for QoS provisioning,” in Proceedings of the 22ndTelecommunications Forum Telfor (TELFOR ’14), pp. 111–114,Belgrade, Serbia, November 2014.

[6] V. Esmaeelzadeh, E. S. Hosseini, R. Berangi, and O. B. Akan,“Modeling of rate-based congestion control schemes in cogni-tive radio sensor networks,” Ad Hoc Networks, vol. 36, pp. 177–188, 2016.

[7] T. Feng, J. Bi, and K. Wang, “Allocation and scheduling ofnetwork resource for multiple control applications in SDN,”China Communications, vol. 12, no. 6, Article ID 7122483, pp.85–95, 2015.

[8] M. Caria, A. Jukan, and M. Hoffmann, “A performance studyof network migration to SDN-enabled Traffic Engineering,”in Proceedings of the IEEE Global Communications Conference(GLOBECOM ’13), pp. 1391–1396, Atlanta, Ga, USA, December2013.

[9] X. F. Yu and K. Tomsovic, “Application of linear matrix inequal-ities for load frequency control with communication delays,”IEEETransactions on Power Systems, vol. 19, no. 3, pp. 1508–1515,2004.

[10] L. Yang, X.-P. Guan, C.-N. Long, and X.-Y. Luo, “Feedbackstabilization over wireless network using adaptive codedmodu-lation,” International Journal of Automation andComputing, vol.5, no. 4, pp. 381–388, 2008.

[11] F. Lopez-Rodrıguez and D. R. Campelo, “A robust SDN net-work architecture for service providers,” in Proceedings of the

IEEEGlobal Communications Conference (GLOBECOM ’14), pp.1903–1908, December 2014.

[12] H. Chibana, M. Yoshino, M. Tadokoro, D. Murayama, K.Suzuki, and R. Kubo, “Disturbance-observer-based activequeue management with time delay using software-definednetworking controller,” in Proceedings of the 41st Annual Confer-ence of the IEEE Industrial Electronics Society (IECON ’15), pp.001049–001054, Yokohama, Japan, November 2015.

[13] J. Castillo, A. Neto, F. Silva et al., “Additions to the ETArchcontrol plane to support multimedia QoS-guaranteed contenttransport over OpenFlow-enabled SDN future internet sys-tems,” in Proceedings of the IEEE Globecom Workshops (GCWkshps ’14), pp. 172–177, IEEE, 2014.

[14] Y. S. Moon, P. Park, W. H. Kwon, and Y. S. Lee, “Delay-dependent robust stabilization of uncertain state-delayed sys-tems,” International Journal of Control, vol. 74, no. 14, pp. 1447–1455, 2001.

[15] D. Yue, Q.-L. Han, and J. Lam, “Network-based robust 𝐻∞control of systems with uncertainty,” Automatica, vol. 41, no. 6,pp. 999–1007, 2005.

[16] R. Johari and D. K. H. Tan, “End-to-end congestion control forthe internet: delays and stability,” IEEE/ACM Transactions onNetworking, vol. 9, no. 6, pp. 818–832, 2001.

[17] G. Bansal and J. B. Kenney, “Achieving weighted-fairnessinmessage rate-based congestion control for DSRC systems,” inProceedings of the IEEE 5th International Symposium on Wire-less Vehicular Communications (WiVeC ’13), pp. 1–5, Dresden,Germany, June 2013.

[18] N. Boughanmi, Y.-Q. Song, and E. Rondeau, “Priority andadaptive QoS mechanism for wireless networked control sys-tems using IEEE 802.15.4,” in Proceedings of the 36th AnnualConference of the IEEE Industrial Electronics Society (IECON’10), pp. 2134–2141, IEEE, Glendale, Calif, USA, November 2010.

[19] C. X. Qu, H. S. Li, and W. Chen, “Virtual queue based dis-tributed data traffic scheduling in WSNs for distributed voltagecontrol in smart micro-grid,” in Proceedings of the 10th IEEEInternational Conference on Intelligent Sensors, Sensor Networksand Information Processing (ISSNIP ’15), pp. 1–6, Singapore,April 2015.

[20] V. Gupta, T. H. Chung, B. Hassibi, and R. M. Murray, “Ona stochastic sensor selection algorithm with applications insensor scheduling and sensor coverage,”Automatica, vol. 42, no.2, pp. 251–260, 2006.

[21] C. Sergiou, V. Vassiliou, and A. Paphitis, “Hierarchical TreeAlternative Path (HTAP) algorithm for congestion control inwireless sensor networks,” Ad Hoc Networks, vol. 11, no. 1, pp.257–272, 2013.

[22] S. Djahel andY.Ghamri-Doudane, “A robust congestion controlscheme for fast and reliable dissemination of safety messages inVANETs,” in Proceedings of the IEEE Wireless Communicationsand Networking Conference (WCNC ’12), pp. 2264–2269, Paris,France, April 2012.

[23] X. Shen, X. Cheng, R. Zhang, B. Jiao, and Y. Yang, “Distributedcongestion control approaches for the ieee 802.11p vehicularnetworks,” IEEE Intelligent Transportation Systems Magazine,vol. 5, no. 4, pp. 50–61, 2013.

[24] R. Pedarsani, J.Walrand, and Y. Zhong, “Robust scheduling andcongestion control for flexible queueing networks,” in Proceed-ings of the International Conference on Computing, Networkingand Communications (ICNC ’14), pp. 467–471, February 2014.

[25] L.-H. Huang, H.-J. Hung, C.-C. Lin, and D.-N. Yang, “Scal-able and bandwidth-efficient multicast for software-defined


networks,” in Proceedings of the IEEE Global CommunicationsConference (GLOBECOM ’14), pp. 1890–1896, IEEE, Austin, Tex,USA, December 2014.

[26] W. Li, D. Liu, B. Zhu, X.Wei,W. Xiao, and L. Yang, “SDN controlmodel for intelligent task execution in wireless sensor and actornetworks,” in Proceedings of the IEEE 83rd Vehicular TechnologyConference (VTC Spring ’16), pp. 1–5, Nanjing, China, May 2016.

[27] F. Rasool and S. K. Ngaung, “Robust 𝐻∞ dynamic outputfeedback control of networked control systems with congestioncontrol,” International Journal of Systems Science, vol. 46, no. 10,pp. 1854–1864, 2015.

[28] F. Rasool, S. K. Nguang, and C.-M. Lin, “Robust 𝐻∞ statefeedback control of networked control systems with congestioncontrol,” Circuits, Systems, and Signal Processing, vol. 32, no. 6,pp. 2761–2781, 2013.

[29] T. Wang, T. Li, X. Yang, and S. Fei, “Cluster synchronization fordelayed Lur’e dynamical networks based on pinning control,”Neurocomputing, vol. 83, pp. 72–82, 2012.

[30] Q. Song and Z. Zhao, “Cluster, local and complete synchro-nization in coupled neural networks with mixed delays andnonlinear coupling,” Neural Computing and Applications, vol.24, no. 5, pp. 1101–1113, 2014.

[31] X. Lin, L. Hong, and R. Jingyu, “Research on predictivecontrol of stochastic network-induced delay for network controlsystems,” in Proceedings of the in Chinese Control and DecisionConference (CCDC ’11), pp. 929–932, Mianyang, China, May2011.

[32] S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, LinearMatrix Inequalities in System and Control Theory, vol. 15 ofSIAMStudies in AppliedMathematics, Society for Industrial andApplied Mathematics, Philadelphia, Pa, USA, 1994.

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Research Article Congestion Control in Wireless Software ...downloads.hindawi.com/journals/misy/2016/3454651.pdf · wireless network. Taking advantages of the applica-bility of Lyapunov-Krasovskii