Utility Optimal Real-Time MultimediaCommunication in Wireless Mesh Networks

Eren Gurses, Anna N. KimCentre for Quantifiable QoS in Communication Systems

Norwegian University of Science and TechnologyN-7491 Trondheim, Norway

Email: {gurses, annak}@q2s.ntnu.no

Abstract—Supporting real-time multimedia communica-tion over multi-hop wireless mesh networks is a challengingproblem, considering the necessity of intelligent allocationof shared wireless medium amongst different nodes withinthe network while ensuring the desired level of quality ofservice (QoS). Many existing cross-layer design approachesthat are aiming for an integrated operation of the differ-ent protocol layers often either not consider the networkfunctions as a whole or rely on mechanisms that provideglobal network information. There is certainly a need fordeveloping distributed and scalable algorithms for optimalutilization of system resources and providing the end-to-end QoS. In this paper, we utilize the general networkutility maximization (GNUM) framework that tackles thecross-layer design problem in a systematic manner. Morespecifically by incorporating contention based IEEE 802.11MAC protocol, we developed an optimal joint congestion-contention control scheme that maximize network through-put while at the same time providing end-to-end QoS formultimedia traffic. The proposed algorithm distributedlycalculates the optimal solution by means of exchanging pricesbetween the source and the network links, and hence fullyscalable to large networks. Our results reveal the trade-offs between QoS requirement and rate allocation, andultimately provide guidelines for how the medium access andtransport layer parameters should be selected in order toguarantee QoS for the application. The proposed algorithmcan be easily generalized into other multi-hop wireless adhoc networks.

Index Terms—multimedia communication, QoS, multi-hop, wireless mesh networks, convex optimization, dualdecomposition.

I. INTRODUCTION

A wireless mesh network (WMN), as depicted in Fig. 1,consists of a number of wireless stations (nodes) thatcover an area. The nodes communicate with each otherin a multi-path, multi-hop fashion via the wireless links.The wireless channel linking two nodes may be sharedby multiple flows generated by different sources. If wewant to support real-time multimedia communication insuch networks, the following aspects need to be taken intoconsideration: the error prone and possibly time varyingnature of the wireless channel, the delay constraint andbursty bit streams of multimedia applications, and thatthe wireless channel is a shared medium. In addition, weneed to address these issues from a network perspective.

Centre for Quantifiable QoS in Communication Systems, Centre ofExcellence, is appointed by The Research Council of Norway, andfunded by the Research Council, NTNU and Uninett

Since how resources such as bandwidth and power areallocated for each source-destination pair over the wirelesslinks can directly affect the over-all network stability andthroughput [1]. This problem is particularly challengingwhen the WMN is organized in a decentralized manner,where no single node or manager regulates the informationflow based on the network topology and the application.

A decentralized wireless mesh network can also beconsidered as a type of ad hoc wireless networks, wheremuch of the resource allocation problems are tacked usingthe cross-layer design (CLD) approach [2]. By exploitingthe inter-dependency of the existing, isolated OSI pro-tocol layers, CLD has been shown to be successful invarious wireless communication scenarios[3]. We surveya few recent findings that deal with supporting multimediaapplication in a wireless ad hoc network using CLD.And more specifically, we pay attention to the schemesthat involve physical (PHY), medium access (MAC) andtransport layer, which are particularly relevant to our work.

One large group of CLD approaches jointly optimizesthe lower layers (PHY, MAC) with the application layer.For example, in [4], link adaptation combined with fullyscalable video coder and MAC layer retransmissions toguarantee QoS. Others link the transport layer and theapplication layer to a congestion distortion optimization(CoDiO) framework [24], which can be applied for de-termining the optimal packet transmission schedule thatavoids congestion while meeting the decoding deadline.The network queuing model used is the simplified M/M/1model. In [6], a cross-layer rate control scheme wasproposed by estimating the effective throughput for eachsource-destination pair, assuming a homogeneous networktraffic model. This is then integrated into a joint-sourcechannel coding scheme with delay constraint. In summary,these approaches generally deal with only a few layers ofthe protocol stack while either not taking the mid-layers(network, transport layers) into account or making oftensimplified assumptions regarding the network operatingas one system, hence lack the abilities of addressing thedifferent, often conflicting issues in a realistic manner.

In [7], a synergic optimization over application, net-work, MAC and PHY layer was proposed by assuming anoverlay network infrastructure where conditions of linksand topology information are conveyed. The proposedframework offers to define the optimal modulation at PHY,

1-4244-0981-0/07/$25.00 ©2007 IEEE.

Fig. 1. A generic wireless mesh network

the optimal retry limit at MAC, optimal route to the re-ceiver and the optimal packet scheduling at the applicationlayer. The authors also concluded that to guarantee QoS,utilization of network information gathered through theoverlay network is of great importance. Clearly, use ofsuch feedback information becomes impractical as thenetwork scale increases. Complexity of the optimizationalgorithm then also becomes significant as it operates ona per packet basis. Nevertheless, the importance of morenetwork oriented optimization is well emphasized.

The network utility optimization (NUM) framework be-gan its application in communication networks by meansof designing congestion control protocols [8]. The networkutility function of each source (user) is a function ofresource allocated to them. The optimality of the resourceallocation algorithm can be ensured by maximizing thesum of the utility functions of all sources. Under thisframework, Chiang et al proposed a mathematical theorythat treats layering as a “optimization decomposition”(LOD) problem [9]. Fundamentally different from theother ad hoc cross-layer design, the LOD provides asystematic approach to joint optimization of the protocollayers. Modeling the network traffic as fluid flows fromeach source, the operation of an entire protocol stack isformulated as a generalized network utility maximization(GNUM) problem. Each layer then becomes subproblemafter decomposition while the interactions between layersare abstracted as optimization variables. More importantly,the GNUM problem can be solved in a distributed manner.It essentially eliminates the need of overview of networkinformation and can be scaled up to larger network sizesuch as densely populated sensor networks. There hasbeen series of work that utilizes this approach [1],[10]-[13]. For example, joint power control and congestion con-trol in [1], simultaneous routing and resource allocation in[12], joint congestion control, routing and scheduling in[13] to name just a few. Concerning joint congestion andcontention control, a probabilistic random access MACmodel was applied in distributed optimization of a classof utility functions [14]; while in [11], a deterministicapproximation of the MAC layer was used in the NUMframework. However, the delay constraint associated withmultimedia applications are not yet addressed in these

studies. And this is precisely the problem we are tryingto solve in this paper using GNUM. In particular, weinvestigate joint optimization of MAC and transport layerin WMN, where the average end-to-end delay functionof the multimedia application incorporates the DistributedCoordination Function (DCF) based 802.11 contentionprotocol. We would like to emphasize that the wirelessmesh network and the 802.11 MAC protocol are the exam-ple systems that the problem is presented in. The proposedalgorithm can be applied in other multi-path multi-hopwireless networks that support delay constrained applica-tions.

Our contribution in this paper is three-fold. First, wederived a delay model for 802.11 without assuming sat-uration conditions. Second, by using this delay model,we provide the proper mathematical formulation of theGNUM problem for joint congestion-contention controlwith delay constraint in WMN. Finally, we devise adistributed and scalable algorithm which optimizes thetransport and MAC layer parameters such that the QoScan be guaranteed to the application.

The paper is organized as follows. We first present theabstract WMN topology in Section II, along with systemparameters. This is followed by a brief description inSection II-A on error control mechanisms that involvesPHY and application layer to ensure distortion due to lostor corrupted packets are minimized. We then move on tothe MAC layer issues in Section II-B with detailed delayanalysis. The complete GNUM formulation is presentedSection III. The solutions and simulation results are pre-sented in Section III-B and IV respectively, followed byin depth discussions in Section V. We then conclude thepaper and give an outlook of possible future work.

II. SYSTEM MODEL

We consider a sample WMN with topology as depictedin Fig. 2, which constitutes 8 wireless mesh nodes. Thereare a set of S sources communicating over l ∈ L wirelesslinks. We denote the flow rate of each source s as xs.Some flows such as xA experience multiple hops, whileflows like xB are transmitted over single hop.

Given such communication scenario, we want to de-termine the necessary means that a delay constrainedmultimedia flow can be supported successfully. We beginour analysis by identifying the factors contribute to lossesin wireless multimedia communication. This is followedby a detailed description of how they are handled on thedifferent protocol layers.

Typically, multimedia sources generally can tolerate acertain degree of errors. This is due to the fact that thehuman perception has limited precision and that errorconcealment can be applied at the decoder side for betterquality. However, robust source coding alone is not suffi-cient to combat the distortions in wireless channels, wherepackets are vulnerable to both bit and packet level errors,and may experience different error patterns within a largerange of error rates. In addition, packets arriving beyondtheir decoding deadline are also interpreted as losses.

One way of conducting error control is to differentiatethe two types of losses, i.e. packets arrived on time butdiscarded due to bit errors and packets arriving late or notarrive at all (dropped due to buffer overflow for example).If we can ensure bit errors are minimized, then propermechanism can be applied to ensure a timely delivery,and vice versa. Such approach has the advantage of nothaving a single layer carrying all the load of error control,like that conducted in an end-to-end manner. In addition,different network elements can contribute to the overallpacket delay. It is very difficult to analyze and influencethem as one. We therefore first tackle the bit error problemby link layer adaptation schemes. Losses due to delay andnetwork congestion can then be minimized by implement-ing our proposed price based joint congestion-contentioncontrol algorithm. Doing so will remove the need forimplementing complex algorithms at the application layer[17] [18] [19] [20].

A. PHY Layer Error Control

Link adaptation at the PHY layer is the most commonand powerful tool to tackle with the bit errors in the wire-less channel at the PHY layer. We follow the techniqueproposed in [15] which basically adjusts the tansmitterpower to achieve the optimal signal-to-noise ratio (SNR)level γ∗ in order to maximize the link throughput underan average power constraint Pavg . Although we do notpropose a new algorithm for the PHY layer, some entitiesoptimized in this layer, such as link capacity cl andMAC packet size B, are later required by the cross-layeroptimization algorithm proposed in Section III.

The link adaptation in [15] is done as follows. Giventhe MAC layer packet of size B bits and constellationsize b bits/symbol. There is Bs = B/b symbols perpacket. Now, for a MAC layer frame of size Bf ≥ Bs,instead of transmitting for the entire duration of the MACframe, we activate the transmitter only for a portion ofBf . At the same time, we adjust the power level toBf/Bs times higher the Pavg . This power adjustmentratio Bf/Bs corresponds to the ratio between the optimalsymbol SNR γ∗ and the observed symbol SNR γ, whereγ∗ = (Bf/Bs)Pavg

N0Rs. N0 and Rs are the noise spectral

density and symbol rate before adaption respectively.One important implication from the link adaptation is

that the resulting optimized transmitter power levels arewhat being used for calculating the information theoreticcapacity at the link layer. This capacity, combines with theMAC layer protocol, determines the network’s capacityregion. For the rest of the paper we refer to the informationtheoretic link capacities for link l as cl as given in (1),where kl is the code rate applied on link l.

cl =

{klbRs

γlγ∗l

, for γlγ∗l≤ 1

klbRs , for γlγ∗l> 1 (1)

B. Delay Analysis for the MAC Layer

Having ensured that bit errors can be minimized usinglink adaption, we can now move on to identify the delaycomponents.

A: 1 → 4 → 6 → 7B: 4

D: 5 → 6C: 3 → 2

xAxB

xC

xD

Fig. 2. A sample wireless ad hoc network topology

Fig. 3. Conflict graph corresponding to the topology

In wireless networks, in addition to the propagationdelay, there are two factors that effect the end-to-enddelay. The first factor is the network layer queueing delaydue to the mismatch between the effective link capacityand the total data rate traversing that link. If a propercongestion control algorithm is in place and the sourcesare adjusting their rates accordingly, then the networklayer queuing delay element can be effectively minimized.The second factor of delay results from medium access,which is determined at the MAC layer. In this section wewill first define the parameters that determine the MAClayer operations and then provide a delay analysis basedon these parameters.

As stated earlier, we consider the contention based802.11 MAC protocol using DCF which enables theexclusive use of the conflicting wireless links. To easeunderstanding of our analysis and to visualize the con-tending links of the network, we use a conflict graphshown in Figure 3 Note that this graph is not requiredby the algorithm proposed in the following section. Foreach link, it is sufficient to gather only the local contentioninformation within 2-hops distance by means of messagepassing. In the conflict graph, we label each wireless linkof a given topology (Fig.2) with a unique number andrepresent it as a link node. Each line connecting the twolink nodes in the conflict graph means these two linksinterfere with each other. All link nodes that are connectedto each other form a subgraph that is referred to as aclique. A maximal clique is a complete subgraph that isnot contained in any other complete subgraph.

Following the DCF algorithm, we define a parameter althat determines the medium access probability on link l.This parameter can be interpreted as the probability thatlink l captures the medium for transmission among the

other contending links within a maximal clique n in theconflict graph where L′(n) denotes the set of links inclique n. We define a conflict matrix F with entries fnlas in (2) and dimension N×L where N and L denote thetotal number of maximal cliques and links in the networkrespectively.

fnl ={

1 , for l ∈ L′(n)0 , otherwise (2)

For each maximal clique n, the sum of medium accessprobabilities of links that conflict should satisfy the in-equality

∑l∈L′(n) fnlal ≤ εn, where εn ∈ [0, 1] denotes

the usable portion of a channel after excluding the effectof collisions. εn = 1 corresponds to the case of perfectscheduling where there is no collision. For IEEE 802.11it is shown in in [21] that εn ≈ 0.85 and it can keep thesame value for a large number of stations when RTS/CTSsignaling is used.

The total delay due to the use of this MAC protocol onlink l is then the sum of propagation delay d(p)

l , mediumaccess delay d

(m)l and link layer queueing delay d

(q)l as

given in (3). The details of the delay analysis are providedin Appendix A, including calculation of d(m)

l and d(q)l .

dl(x, al) = d(p)l + d

(m)l (al) + d

(q)l (x, al) (3)

By simply using equations (18) and (20) from Ap-pendix A, the average delay dl(x, al) on link l can bewritten as:

dl(x, al) = d(p)l +

(1+∑s rlsxsclal

)(I + T + ∆(1− al)2

al

),

(4)where rls indicates if the source s uses link l: rls = 1is true and 0 otherwise; and T is the packet transmissiontime, ∆ is the slot duration.

III. CROSS-LAYER DESIGN USING GENERALIZEDNETWORK UTILITY MAXIMIZATION

We introduce the GNUM framework in this sectionfollowed by the joint congestion-contention control proto-col. First, a few words about the network utility functionUs(·). This function can take on different form, whichin turn represents different form of fairness in the net-work. The basic requirement is that the utility functionis continuous, differentiable and strictly concave. We optfor Us(xs) = log xs which corresponds to proportionalfairness, as seen in [1], [13].

A. GNUM Forumulation

Given the WMN topology in 2. We formulate theGNUM problem as in the equations (5)-(9). The overallutility of the network is maximized in terms of the utilitiesUs(xs) of each source s ∈ S under the set of constraints.The first constraint denotes the restrictions on source ratesx = [x1, ..., xS ]T given that the routing matrix is Rand average MAC layer link capacities are C a. Thelink access matrix is defined as C = diag(c1, ..., cL)

and entries of R is given by rls = 1 if source s useslink l and rls = 0 otherwise . The second constraintprovides a bound t = [t1, ..., tS ]T on the end-to-enddelay of each source where d(a) = [d1(a1), ..., dL(aL)]T

represents the average delay of links in the network.For the sake of completeness, we refer the end-to-enddelay vector of all sources as τ = RT d(a) where eachentry τs gives the forward trip time of packets fromsource s to their destination. Finally the third constraintis the restatement of the inequality on medium accessparameters a = [a1, ..., aL]T given in Section II-B whereε = [ε1, ..., εN ].

maxx,a

∑s

Us(xs) (5)

s. t. R x ≤ C a (6)

RT d(x, a) ≤ t (7)F a ≤ ε (8)xs ≥ 0 and 0 ≤ al ≤ 1 (9)

This set of constraints essentially take care the packetlosses both due to congestion and late arrivals. With thefirst constraint in (6), sources are restricted to operate ata rate that their sum on each link is lower than the link’sMAC layer capacity and as a result packet losses due tocongestion are minimized. With the second constraint in(7), network parameters a and x are optimized such thatthe end-to-end delay of each source will conform to theapplication imposed delay requirement. The solution ofthe above constrained optimization problem will maximizethe network throughput while minimizing the loss due todropped packets and late arrivals.

B. Joint Congestion-Contention Control

The above introduced GNUM problem can be solveddistributedly by the means of dual decomposition, pro-vided all the constraints are convex functions. For sources, the sum of the link delays (i.e. end-to-end delay)τs =

∑l rlsdl(x, al) is however, not a convex function.

Fortunately, we can bypass the problem by making thefollowing approximation. Consider the common valuesgiven in [22]. In (4), the slot duration is then ∆ = 50µ sec.∆ becomes negligible compared to the sum of packettransmission time, T ≈ 3500µ sec and distributed inter-frame space (DIFS) time I = 2∆+SIFS = 128µ sec forshort interframe space SIFS = 28µ sec, while (1−al) isnever greater than 1. Hence the second term of the productin (4) can be approximated as I+T

al. On the other hand,

due to the flow constraint in (6), aggregate flow on linkl can never exceed link’s capacity,

∑s rlsxs ≤ clal and

should be operated close to the capacity due to the utilitymaximization. Then the first term of the product in (4)can be approximated (especially for congested nodes) as1 +

Ps rlsxsclal

≈ 2 for the maximum utility case. Doingso, we can approximate the link delay function as in (11)while satisfying the convexity requirements over a.

D(λ, µ, ν) = maxxs,al≥0

L(λ, µ, ν, x, a) (10)

= maxxs,al≥0

∑s

Us(xs)− λT (Rx− Ca)− µT (RT d(a)− t)− νT (Fa− ε)

= maxxs,al≥0

∑s

{Us(xs)−

∑

l

λlrlsxs

}+∑

l

{λlalcl −

∑n

νnfnlal

}−∑s

∑

l

µsrlsdl(al) +∑s

µsts +∑n

νnεn

dl(al) ≈ d(p)l + 2

(I + T

al

)(11)

We can now proceed with dual decomposition. First, werewrite the GNUM problem in (6)-(9) as an unconstraineddual problem in terms of the Lagrangian L(λ, µ, ν, x, a)using λ, µ and ν. The resulting dual problem becomes:

minλ,µ,ν

D(λ, µ, ν), (12)

s. t. λ, µ, ν ≥ 0

where the corresponding dual function is given in (10).When solving the dual problem, the dual parameters

λ, µ,ν that minimize the dual function D(λ, µ, ν) can becalculated by fixing primal parameters x and a. The primalparameters are determined by fixing the dual parametersλ, µ, ν. This iterative algorithm calculates the global opti-mal primal variables by the means of exchanging priceinformation (the dual variables) between the links andthe sources. The resulting algorithm is a distributed jointcongestion-contention protocol.

Here we show the calculation of the various pricesλ, µ, ν using the gradient algorithm as an example, sincethe dual function D(λ, µ, ν) is continuous and differen-tiable. We have λ(t+ 1) = λ(t)−βλ∇λD(λ, µ, ν) where∇λ = [∂/∂λ1...∂/∂λL]T . Other dual parameters ν andµ can be similarly calculated by using gradient vectors∇ν and ∇µ. The congestion price λl for each link l,contention price νn for each maximal clique n and end-to-end delay price µs for each source are calculated as in(13)-(14) and (15) respectively.

λl(t+ 1) = λl(t)− βλ(alcl −∑s

rlsxs) (13)

νn(t+ 1) = νn(t)− βν(εn −∑

l

fnlal) (14)

µs(t+ 1) = µs(t)− βµ(ts −∑

l

rlsdl(al)) (15)

Expressions to calculate x and a that maximize La-grangian L(λ, µ, ν, x, a) by using prices λ, µ, ν is pro-vided in Appendix B. Rates are calculated as xs =U ′s−1(∑l rlsλl) by sources while channel access probabil-

ities are found by using 2δa3l−Xl(λl, ν)a2

l−Yl(µ) = 0 forXl(λl, ν) = λlcl−

∑n νnfnl and Yl(µ) =

∑s µsrls2(I+

T ) (see Appendix B).We summarize the distributed algorithms that are per-

formed at each source s and each link l in Table I. Inthe source algorithm, congestion control is done at each

TABLE IDISTRIBUTED JOINT CONGESTION-CONTENTION CONTROL

ALGORITHM FOR REAL-TIME MULTIMEDIA COMMUNICATION

Source Algorithm:Step S0 : Each source s initializes µs, xs = 0 for 0 ≤ xs ≤ xmax

and µs ≥ 0.

Step S1 : Each source calculates its rate by xs = U ′s−1(Pl rlsλl).

For proportional fairness it uses Us(xs) = log(xs), hence:

xs = 1.X

l

rlsλl

Distribute information on rate xs and delay price µs tolinks l ∈ L(s). Go to Step S3 if xs converges.

Step S2 : Each source calculates delay price µs by solving equation:

µs(t+ 1) = µs(t)− βµ(ts −X

l

rlsdl(al))

If xs and µs not converged go to Step S1, else continue.

Step S3 : If λl and dl(a) provided by links are different frompreviously received ones go to Step S1, else wait in StepS3.

Link Algorithm:Step L0 : Each link l initializes λl, νn, al = 0 for 0 ≤ al ≤ 1 and

λl, νn ≥ 0.

Step L1 : Calculate congestion prices λl for each link l by usingxs, s ∈ S(l). and contention prices νn for each cliquen by the member links l ∈ L′(n) with the exchange oflocal contention information al, l ∈ L′(n) as follows:

λl(t+ 1) = λl(t)− βλ(alcl −Xs

rlsxs)

νn(t+ 1) = νn(t)− βν(εn −X

l

fnlal)

Feed back link prices λl to sources s ∈ S(l).

Step L2 : Each link calculates al by solving the following equa-tion for Xl(λl, ν) = λlcl −

Pn νnfnl and Yl(µ) =P

s µsrls2(I + T ) which are given by ( 21).

2δa3l −Xl(λl, ν)a2

l − Yl(µ) = 0

If al, λl and νn not converged go to Step L1, else continue.

Step L3 : If xs and µs provided by sources are different frompreviously received ones go to Step L1, else wait in StepL3.

source s by means of adjusting the source rate xs asgiven in step S1. While updating xs, each source s usescongestion prices λl generated for links l traversed by thatsource which is given in step L1 of the link algorithmin Table I. Furthermore, each source calculates the delayprice µs for it’s end-to-end traffic and communicates theprice value back to the links on its route as given in stepS2 of source algorithm. This delay price generation is thekey functionality of the proposed algorithm. In addition

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Fig. 4. Source rate xs and channel access al parameters when (a) there is no delay constraint, (b) there is end-to-end delay constraint of ts = 100msec. for source s = A

to congestion price λl, the links calculate contention priceνn for each maximum clique n that they are the memberof, by means of local exchange of their channel accessprobabilities al. For each maximal clique n, links in thatclique calculate the contention price as given in step L1of link algorithm. Finally in step L2 of link algorithm,the optimal channel access probability a is found bymaking use of the prices λ, µ and ν, which results inthe desired contention control policy. These algorithmsare executed asynchronously and distributedly, until thenetwork’s contention and congestion parameters a and xconverge.

IV. SIMULATION AND DISCUSSION

The distributed algorithms are simulated in Matlab fora network described by the topology in Fig. 2 wherethere are L = 7 links and S = 4 sources with thecorresponding conflict graph shown in Fig. 3. Note that, asemphasized in Section II-B, links do not need the globalconflict graph information. Rather, it is enough for eachlink to identify the relevant maximal cliques which can beachieved by means of simple local message passing withinat most 2-hop distance. In simulations we set a generallink capacity cl = 1 Mbps, MAC packet size B = 400bytes and T = 3500µs, ∆ = 50µs, I = 128µs according

to [22] and εn ≈ 0.85 by referring to the results in[21]. We set the parameters of joint congestion-contentioncontrol algorithm as βλ = 0.1, βµ = 7.2, δ = 0.8 andβν = 2δ/(ON) for O = 4 and N = 3. Note that thebound 0 < βν < 4δ/(ON) is found by the convergenceanalysis of the gradient algorithm [23] where O and Nrespectively denote the size of maximal cliques and thelargest number of cliques that contain the same link.

We first look at the no-delay constraint case, by se-lecting a sufficiently large value of ts = 500 msec. InFig. 4(a), we show the algorithm’s convergence behaviourand the resulting values for x and a. The resulting sourcerates that maximize the network’s overall utility are givenon the left in Fig. 4(a), where Source A gets the lowestbandwidth according to the proportional fairness criteriaimposed by Us(xs). The corresponding channel accessprobabilities are given on the right in Fig. 4(a) wherelinks 1 and 7 used by only source A get the lowest sharewithin their maximal cliques. Due to these low channelaccess probabilities of links 1 and 7 and number of hopsin the end-to-end route of source A, the end-to-end delayτs is around 250 msec as given in Fig. 5(a) which is 4-10times higher than the delay of other sources.

If we apply the proposed algorithm for a more restricteddelay constraint, like ts = 100 msec, the resulting x and

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Fig. 5. Resulting end-to-end delay τs for joint congestion-contention control algorithm (a) without delay constraint (i.e. ts = 500 msec) and (b)with delay constraint of ts = 100 msec for s = A where τs < ts.

a parameters, network’s total utility and even the fairnesscriteria between sources change. In order to illustrate theeffect of the algorithm, we keep ts = 500 msec for allsources s 6= A and set the tA = 100 msec, which isthe multimedia source with a tighter delay constraint. Wefirst observe the change in channel access probabilities al,as shown in Fig. 4(b). Here, al of links 1 and 7, whichmade a major contribution to the delay of τA = 250 msecin the previous no-delay constraint case, dramaticallyincreased from values < 0.1 to range 0.25 − 0.3. Thisincrease, in turn, reduces the channel access probabilitiesof other links that are contending with link 1 and 7within the relevant maximal cliques in the contentiongraph. Especially, the channel access probabilities of links2, 3, 5 which are not relevant to multimedia source A,are decreased significantly, in order to meet the real-timerequirements of source A.

Due to these changes in channel access probabilities a,the allocated source rates x are effected. The results areshown in Fig. 4(b). The source rates xs are decreased fors = B,C,D while it is increased for source s = A by thecongestion control mechanism of the joint optimizationalgorithm in order to meet the new channel capacitiesdetermined by a.

Finally in Fig. 5, it can be observed that the end-to-enddelay of source A is reduced from 250 msec to ≈ 100msec, which is the expected outcome of the applied jointcongestion-contention control algorithm.

V. CONCLUSIONS AND FUTURE WORK

In this work we proposed a systematic GNUM basedcross-layer design approach for the joint optimizationof transport and MAC layers in order to maximize thenetwork’s overall throughput while at the same timeproviding end-to-end QoS for multimedia traffic. Theproblem is formulated by specifically addressing a WMNsetting with stations using IEEE 802.11 protocol and usingthe corresponding average delay for DCIF. The algorithmis also applicable to other multi-hop ad hoc networkingscenarios since the MAC layer is optimized with respect to

an abstract parameter a which corresponds to the channelaccess probability after channel collision is resolved (inour case by using CSMA/CA with RTS/CTS) rather thana hardwired parameter of a specific MAC protocol. Finallythe proposed algorithm is fully distributed and scalable.The simulation results show that optimization can beperformed iteratively based on local link information andconverge to the network optimum congestion-contentioncontrol parameters, which maximize the system through-put while satisfying application’s QoS requirements.

As a future work, one immediate next step should be toadd the optimal routing functionality to the proposed jointcongestion-contention control algorithm. Furthermore inthis paper we used link adaptation techniques based ononly power adjustment in order to provide QoS. Bettertrade-offs should be investigated between power and chan-nel code rate adaptation. Finally in this work, in order tomeet the delay requirements we do not take into accountthe effect of priority queueing on the performance of theproposed algorithm which will definitely have a significantimpact on the optimized congestion and contention controlparameters.

APPENDIX A

In 802.11 the DCF (Distributed Coordination Function)simply describes the distributed algorithm to fairly sharethe channel and resolves the collision problem by usingRTS/CTS signalling and defining a backoff phase. Maingoal of backoff phase is to minimize the probability ofcollision which happens after the channel is released bythe capturing node and then nodes start to contend tocapture it. In our delay analysis we use the standardbinary exponential backoff (BEB) algorithm which selectsthe backoff time time from a geometric distribution withparameter p of a p-persistence algorithm [22].

In the standard operation, a node m that try to accesslink l continuously listens (carrier sense) the channelexcept the period that it captures the channel for transmis-sion. During the listening phase, upon sensing the channelis idle, it waits for a period of I as DIFS (distributed

inter-frame space). If no other node captures the channelduring DIFS period I , node m starts transmission andcaptures the channel throughout a packet transmissiontime of T . If another node captures the channel duringDIFS of node m, then the node m listens to the channelidle. As soon as it detects channel idle, it starts the backoffphase. Backoff time is divided into slots of duration ∆and measured by a backoff slot counter which is initiallyset to backoff time/∆ and decremented at each idle slot.During the backoff period, another node may capture thechannel, then the node m stops count down of the backoffcounter until the channel is idle again. Transmission ofnode m starts only when the counter is zero and theaverage backoff slot count is given as (1−p)/p where theaverage backoff period B(p) of p-persistence algorithm issimply denoted as

B(p) = ∆(1− p)/p (16)

The reader should note that, apart from other studiesin literature focusing on saturation conditions [21], weallow 802.11 not necessarily to operate in backoff mode.That means, a newly generated MAC packet can sense thechannel idle with some probability. Actually this is exactlythe situation if transport layer flow is optimized togetherwith the MAC layer. Given the above DCF algorithmand channel access probability al on link l, averagemedium access delay d(m)

l (al) can be written as in (17).The first term denotes the average delay for observingthe medium idle for transmission over link l. Then, thesummation denotes the total delay that includes the npacket transmission time I+T on other conflicting links,the average backoff time for link l and the transmissiontime I + T for link l after it captures the medium.

d(m)l (al)=al(I+T )+

∞∑n=1

al(1−al)n((n+1)(I+T )+B(al))

(17)After manipulating (17) and replacing B(al), the aver-

age medium access delay is denoted as follows.

d(m)l (al)= al(I + T )

∞∑n=0

(1− al)n+alB(al)∞∑n=1

(1− al)n

+al(I + T )∞∑n=1

n(1− al)n

= (I + T ) +(1− al)al

(I + T ) + (1− al)B(al)

=I + T + ∆(1− al)2

al(18)

The medium access delay d(m)l (al) causes an extra

queueing delay of d(q)l (al) that occur at the link layer.

The amount of the average link layer queueing delay isdetermined by the amount of data accumulated duringd

(m)l (al) divided by the service rate of the link layer as

given in (19).

d(q)l =

(d

(m)l (al)

∑s

rlsxs)/clal (19)

The total delay on link l is given as dl(x, al) = d(p)l +

d(m)l (al)+d

(q)l (x, al) in (3) and can be rewritten as follows

by replacing d(q)l .

dl(x, al) = d(p)l +

(1 +

∑s rlsxsclal

)d

(m)l (al) (20)

APPENDIX B

The Lagrangian L(λ, µ, ν, x, a) in (10) is seperable overx and a after approximating link delay with dl(al) as in(11). We maximize over xs by ∂

∂xsL(.) = 0 and found

xs = U ′−1(∑l rlsλl) where U ′(xs) = ∂

∂xsUs(xs). The

calculation of a is trickier since the primal variable adisappears in the ∂

∂alL(.) = 0 while conducting max-

imization on links. We therefore introduce a quadraticregulating term −∑l δa

2l to the Lagrangian in order to

guarantee the concavity of it over a. In the followingequations, al can be calculated by finding the roots of∂∂al

L(λ, µ, ν, x, a) − ∑k δa2k = 0 after replacing the

Lagrangian given in (10).

λlcl−∑n

νnfnl− ∂

∂al

∑s

∑

k

µsrksdk(ak)−∑

k

δ2a2k = 0

(λlcl −

∑n

νnfnl

)+

(∑s µsrls2(I + T )

)

a2l

= 2δal

After making the following variable changes, such thatXl(λl, ν) = λlcl−

∑n νnfnl and Yl(µ) =

∑s µsrls2(I+

T ), channel access parameter al is given as the roots ofthe following polynomial.

Xl(λl, ν) + Yl(µ)/a2l = 2δal

2δa3l −Xl(λl, ν)a2

l − Yl(µ) = 0 (21)

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