Distributed Channel-Assignment and ThroughputControl in Multi-Radio Multi-Channel Wireless

NetworkHang Qiu∗, Hongkun Li†, Xinbing Wang∗, Sen Yang∗, Xin Huang∗ and Yu Cheng†

∗Dept. of Electronic EngineeringShanghai Jiao Tong University, China

Email: {sunny boy, xwang8, twood, h-x}@sjtu.edu.cn†Dept. of Electronic & Computer EngineeringIllinois Institute of Technology, Chicago, USA

Email: {hli55, cheng}@iit.edu

Abstract—In multi-radio multi-channel (MR-MC) wirelessnetwork, the performance can be substantially enhanced byoperating multiple transmissions on multiple radios via differentnon-overlapping channels. However, this improvement requirescareful design on routing, channel assignment, radio arrangementand scheduling to avoid interference conflicts and radio interfacecontention so that it can fully utilize the abundance resourceof radios and channels in MR-MC wireless network. In thispaper, we propose a novel online distributed joint throughputcontrol protocol for routing, resource allocation and scheduling,which is independent to prior knowledge or global informationof the system. We provide a joint design to directly control thelink-radio-channel assignment and routing to achieve a bettersystem performance, by using the new decoupled tuple-basedmodel for MR-MC wireless network. Further, we prove that theproposed algorithm can guarantee a higher constant fraction ofthe maximal capacity of a wireless network system, comparingto some well-known existing distributed approaches. Simulationresults are presented to demonstrate the capacity and stabilityof wireless network systems with offered load using differentresource allocation algorithms and the salient effect of exploitingmultiple radio interfaces and channels.

I. INTRODUCTION

With the rapid development of technology, especially thematurity of transceivers capable of operating on multiplechannels, the idea of providing one communicating node withmultiple radios offers a promising future for enhancing theperformance of multi-channel wireless network. Instead ofoperating on only one channel, each link can exploit multiplenon-overlapping channels over multiple radios, which can notonly carry multiple flows, finish multiple task simultaneouslyand maximize the system capacity, but also largely diminishthe restraint of interference.

However, the realization of this interesting advantage formulti-radio multi-channel (MR-MC) wireless network requiresfor a careful joint design. This design, taking multiple fac-tors into consideration, may correspondingly leads to fourcorrelated sub-problems, including (i) radio assignment: tobalance the burden of each radio of both transmitting nodeand receiving node for each communication link, in order to

use all the radio resource to the most. (ii) channel assign-ment: to choose the operating channels deliberatively for eachcommunication while minimizing co-channel interference. (iii)scheduling: once flows are assigned to their paths, arrange theactivation priority for each communication, while at the meantime considering possible conflicts in the whole system again.(iv) routing: for each flow with a clear source and destination,the routing path is crucial to avoid unnecessary congestiondelay and interference conflict and to improve the systemcapacity. Therefore, to enhance the performance of multi-radiomulti-channel wireless network, we propose a joint design forthose four correlated resource allocation problems.

In our work, we first separate the whole problem intotwo incorporable parts: routing and transmitting. In routingpart, we balance both the queueing delay and transmissiondelay to form a priority list of different flow paths for eachnode to its destination. Based on those lists, we propose adynamic routing protocol to incorporate with the transmittingpart, so that they can cooperate with each other, minimizingtransmission latency and conflicts in the system at the sametime. In transmitting part, we assign data packets to a specificradio-channel pair for its transmission. We integrate radioarrangement, channel assignment and scheduling as a wholeto jointly plan for the assignment of data flows to availableradios and channels. This combination can consider all thepossible conflicts together, including radio interface contentionand co-channel interference, while making one single decision.Consequently, by incorporating both routing and transmittingparts, we tackle the whole problem from the users’ point ofview: given the destination of users’ packets anytime, thisdesign can dynamically choose a path and deliver the packetswith comprehensive consideration of interference, conflictsand end-to-end delay in MR-MC wireless network system.

Traditionally, design for the resource allocation in multi-channel wireless network has been widely researched[6][7][12], and most approaches focused on link-based methodin accordance with the assumption that each communicationnode is only equipped with one transceiver or radio interface.

Although these approaches have been efficient and effectiveto solve the problem in single radio single channel (SR-SC)or multi-channel (SR-MC) wireless network, as simplified asit is, however, it can only deal with the traditional schedulingproblems with the smallest unit as one link. Direct extensionmay not fully utilize the advantage of multi-radio multi-channel wireless network [19], because it cannot split radioand channel assignment apart.

In our work, we consider the smallest unit as one node-radio-channel tuple and use tuple-based model to decouplethe channel and radio assignment. This seperation inspires usto explore more potential scheduling contingencies and to pro-vide more direct control from user to the actual transmissionwith specifically assigned radio and channel. Our distributeddesign is distributed with polynomial computational complex-ity and convenient for implementation. Further, we prove thatour control protocol for the transmitting part can achieve ahigher constant fraction of the maximum capacity under thecontext of MR-MC wireless network system comparing tosome existing work on distributed approach. In fact, our pro-posed distributed protocol can achieve the best performance ofcentralized greedy maximal scheduling algorithms. Moreover,our design can automatically track the changes of burden andnetwork condition on each node and each path, and adjustthe assignment strategy online without any prior knowledge,restraint of the topology or traffic load. Because of this, thedesign is especially suitable for MR-MC wireless network, forits salient ability to exploit the resource of radios. While it isunder the assumption of protocol interference model, it canbe flexible enough to adapt to other interference models. Ourmain contributions are as follows:

• We propose a novel radio-channel pair assignment mech-anism which can guarantee efficiency and fairness simul-taneously.

• We design a new distributed dynamic scheduling algo-rithm for MR-MC wireless network under tuple-basedmodel. Combined with radio-channel pair assignmentmechanism, we prove our design for the whole transmit-ting part can achieve a higher fraction of the maximumsystem capacity.

• To incorporate with the transmitting part, we establish adynamic distributed routing protocol, which can balancetransmission latency with conflicts and interference in thewhole wireless network system.

• We evaluate the capacity and stability of wireless networksystems with varied offered load and different resourceallocation algorithms. Simulation results demonstrate theperformance of our design out of joint consideration andthe salient effect of exploiting multiple radio interfacesand channels.

The rest of the paper is organized as follows, SectionII reviews some other existing works in related field. Insection III, we introduce our system model in detail. SectionIV will introduce our new distributed and provable efficientjoint control protocol, incorporated with a novel distributed

dynamic scheduling algorithm. And in section V, we willcombine a novel dynamic distributed routing protocol withthe whole system design to complete joint consideration ofrouting, resource allocation and scheduling. After evaluatingthe performance of all these approaches in different networktopology through simulation in section VI, we will concludethis paper in section VII.

II. RELATED WORK

The throughput control problem in wireless network iswidely researched while only some of these literatures arefocused on MR-MC wireless network. As stated before, wesplit this complicate problem into several sub-problems. Corre-spondingly, our work is related to some prior work in resourceallocation, scheduling, routing and some existing joint design.We summarize these works below and point out their relationswith our design.

Centralized Channel Assignment and Scheduling:Channel assignment and scheduling algorithms are gen-erally categorized into two classes: centralized and dis-tributed. Centralized channel assignment and scheduling[16][24][10][6][3][9][13][8] aims to find the global best sched-ule with complete information of network topology, offeredload, link capacity and so on. The focus of these work variesfrom low complexity [3], joint routing and scheduling design[16][23], and cutting infrastructure cost [24]. Our work isinspired by the server-side greedy scheduling design of S.Bodas etal. in [3] and the consideration of the tradeoff betweenthroughput and end-to-end delay in [13][8]. By designing in adistributed fashion, our work remains convenient for practicalimplementation and flexible to adapt to change and variationof wireless network condition.

Distributed Channel Assignment and Scheduling: Fordistributed channel assignment and scheduling, there are alsoplenty of work and design but only a few in a practical anddynamic fashion [25][19]. The distributed scheduling part ofour design is related to the work of X. Wu et al. [7], whichdevelops a distributed greedy scheduling algorithm with aprovable lower bound and upper bound. And B. Ko et al.[25] proposed a distributed channel assignment mechanism inMR-MC 802.11 mesh network. Their design in decouplingthe channel selection decision from the data forwarding isquite similar with the two-stage queueing mechanism in [19].Building on this foundation, our work provides a joint consid-eration for channel assignment and scheduling, incorporatingthree different stages from user to the actual transmission: flowrouting, radio arrangement and channel assignment.

Some Existing Joint Design: The throughput controlof MR-MC wireless network posed a set of sub-problems,the cooperation of some incorporable problem solutions ofrouting, resource allocation and scheduling may yield betterperformance. For some existing joint design, M. Alicherryet al. [16] proposed a joint channel assignment and routingprotocol which is a centralized and offline solution. Also tojointly solve the resource allocation problem, S. Merlin et al.[10] developed a throughput optimization framework under

Fig. 1: Tuple Link Model. We split one link into several tuplelinks. Each tuple link is connecting source and destinationnode with one fixed operating channel and different butdefinite tranmitting radio and receiving radio.

link-based model, which includes congestion control, channelallocation and scheduling algorithm. A. H. Mohsenian-Rad etal. [23] added routing into their consideration and transformedthe problem into a joint linear mixed-integer optimizationproblem. Still, these solutions are centralized and inconvenientfor practical implementation. To design a distributed jointthroughput control protocol, X. Lin et al. [19] provides severaldesign on scheduling algorithms and joint channel assignmentwith provable performance. They use two-stage queueingmechanism to take channel diversity into account becauselink-based method [14][1] often overlooked different radiosand channels by considering them as one link unit. Note thatdifferent channels have varied capacity due to backgroundnoise and interference change with time and location. Anddifferent radios may have different amount of backloggedpackets. Careful arrangement may lead to less conflicts andbetter performance. Therefore, H. Li et al [20] proposedthe tuple model to form another distributed joint throughputoptimization framework for MR-MC wireless network. Thisframework, inspired by the tool of multi-dimensional conflictgraph developed in [21], completely decouples a link, andmakes the design of simultaneous radio arrangement andchannel assignment in this paper possible. Building on thesefoundations, we propose a practical joint design includingchannel assignment, radio arrangement, scheduling and rout-ing in a distributed and dynamic fashion, with comprehensiveconsideration of channel diversity, radio burden imbalance andtransmission latency in MR-MC wireless network system.

III. SYSTEM MODEL

Consider a common multi-radio multi-channel multi-hopwireless network with N nodes and multiple links L. Thetransmissions in this network are denoted as flow set F ={f1, f2, ...fF }. To form a more general model, there is nolimits to the identity of radio-interfaces that each node isequipped with, or to the available channels in this network.This may apply to most common wireless networks.

Fig. 2: Network Model. We use the tuple based model toform the whole wireless network system. Note that differentnode may have different number of radio interfaces and variednumber of available channels.

A. Network Model

To completely reflect the diversity and randomness betweeneach node, radio and channel, we use node-radio-channel tuple[20] to represent a specific link tuned on a specific channel andtransmitting packets from its starting node to its destination.Let Mi denote the number of radio interfaces that node i isequipped with, and assume that there is C channels available,the original link between two nodes l(i,j) is transformedinto MiMjC tuple links, denoted as tuple link set Tl(i,j) ={t111, t112, ..., tMiMjC} for link l(i.j) (Figure 1). Thus, if thereis more than one tuple link t ∈ Tl(i,j) is scheduled, that isequivalent to say the link l(i,j) is scheduled. Moreover, thistuple link model decouples the link transmission into tuple linktransmissions so that simultaneous transmission using differentchannel within one link can be explicitly reflected by theirtransmitting and receiving radio. Consequently, all the linksin the whole network system are transformed into tuple linkset T = {t1, t2, ...tT } where T =

∑Ll=l1

Tl.For each and every tuple link t, we further define their

affiliation, i.e. we denote its transmitting and receiving nodeas Tnt and Rnt, transmitting and receiving radio as Trt andRrt, the link of the tuple link set it belongs to as l(t). Notethat each tuple link is itself representing a specific channel,we thus define their unique transmission rate rt over theirown channel. The accumulated number of packets waiting tobe transmitted is denoted as ηt for one particular tuple link tand ql for the whole link. All these definitions are summarizedin Table I for the convenience of reference, their usage will bediscussed in detail later in this literature. The whole networkmodel is shown in Figure 2.

B. Traffic Pattern

The traffic in the wireless network may vary with time,location. Some are modeled approximately to follow somestationary of ergodic process. We view all the traffic in thewireless network as several multi-hop flows, with one fixedpath from its origin to its destination. We assume that it has

TABLE I: Model DefinitionsN The node set {1, 2, ...N}.T The tuple link set {t1, t2, ...tT }.L The link set {l1, l2, ...lL}.Mi The number of radio interfaces of node i.

Tl(i,j) The tuple link set within link l(i,j) {t111, t112, ..., tMiMjC}.l(t) The link of tuple link set that t belongs to.C The channel set {c1, c2, ...cC}F The flow set {f1, f2, ...fF }.λfi Offered load for flow fi.Tnt The transmitting node of tuple link t.Rnt The receiving node of tuple link t.Trt The transmitting node-radio pair of tuple link t.Rrt The receiving node-radio pair of tuple link t.It Tuple link set that has co-channel interference with tuple link t.

E(i) Tuple link set that is originating or terminating at node-radio pair i.qli The queue length of link liηti The queue length of tuple link ti

a constant packets injecting rate λf at the source node in atime slot. Further, we denote Hl

f as the routing flag for link land flow f . If the flow f passes through link l, the flag willbe true, otherwise it is false. Once the flow is established, thepath remains unchanged until the flow is terminated. Hence,with the initiation and expiration of multiple flows, the modelcan accord closely with the real wireless network system.

C. Interference Model

In this section, we will introduce the interference modelbetween tuple links according to protocol model [5], and thedefinition of interference degree.

1) Co-channel Interference: For the convenience of findingthe tuple links with co-channel interference, we define It asthe tuple link set that has co-channel interference with tuplelink t. In other words, tuple links in set It are all within theinterference range according to protocol model and they areoperating on the same channel as tuple link t. Note that thisset It includes all the tuple links in Tl(t) that use the samechannel.

2) Radio Interface Contention: In this model, we assumethat one radio can only be tuned for one specific channel,though it can switch channels dynamically. Similarly, wefurther define E(i) as the tuple link set that is originatingor terminating at node-radio pair i. Naturally, if tuple link t isscheduled, then all tuple links in set E(Trt) and set E(Rrt)will be blocked.

3) Interference Degree: Interference degree K(l) is definedin the link-based model as the maximum number of links thatcould operate simultaneously without interference accordingto protocol model if link l is scheduled. And the interferencedegree K is the maximum number of all K(l). As shownbefore, any one scheduled tuple link t ∈ Tl(i,j) is equivalentto the link l(i,j) being scheduled. So, we will continue usingthis definition to evaluate the performance of our proposedalgorithm.

D. Efficiency evaluation

We adopt the definition of efficiency ratio [19] to addressthe efficiency of the proposed algorithm. If given a networktopology with fixed routing condition, there is a protocolwhich can stabilize the wireless network system at offered

load vector−→λ , and it reaches the optimal capacity region,

then the efficiency ratio is the maximum factor α that theproposed algorithm can stabilize the system at offered loadα−→λ . In short, the proposed algorithm can achieve a constant

fraction α of the maximum throughput.

IV. DISTRIBUTED THROUGHPUT CONTROL DESIGN WITHPROVABLE EFFICIENCY IN MR-MC WIRELESS NETWORK

In this section, we introduce a novel distributed tuple-basedthroughput control protocol with provable efficiency underthe context of MR-MC wireless network. We incorporatescheduling, channel assignment and radio balance manage-ment together to provide data packets with one-stop service.

A. Distributed Dynamic Scheduling

In preparation for the whole throughput control protocol,we propose a novel distributed dynamic scheduling algorithm.We first generalize some straight forward extension of existinglow complexity scheduling algorithm, and then introduce ourdynamic scheduling algorithm building on that foundation.

Traditional greedy scheduling algorithm can be directly ex-tended into tuple-based model, which generally follows threesteps: First, choose the tuple link with the largest tuple-queue-length-weighted tuple-rate ηtrt to be scheduled. Second, crossout all the tuple links that is within the interference rangeand interfering with the chosen tuple link. Those includes(i) all tuple links that has the same transmitting or receivingradio, (ii) all tuple links operating on the same channel thathas the same transmitting or receiving node, (iii) all tuplelinks operating on the same channel that do not have sametransmitting or receiving node but are within the interferencerange. Finally, repeat the first two steps until the schedulingprocess ends when all tuple links has been either selected orcrossed out due to two types of interference. A possible finalscheduling result is shown in Figure 3.

Channel 1

Channel 2

Channel 3

Channel 4

Channel 5

Fig. 3: Possible scheduling result according to interferencemodel in Section III. Note that we assume there are 3 radiosand 5 channels available.

It has been proved that in SR-SC wireless network, centralizedgreedy maximal scheduling can achieve an efficiency ratio of1/κ, and in MR-MC wireless network, 1/(κ+2). For detailsof the proof, we refer to the proposal in [19], which is similar.

Although the greedy scheduling algorithm has a provedperformance, it is still highly centralized and inconvenient for

practical implementation. By introducing the greedy thoughtinto a distributed fashion as a regional optimization, we intendto develop a practical distributed dynamic scheduling algo-rithm for MR-MC wireless network. It requires only regionalinformation and simple computation. Similar with greedyscheduling, we still use tuple-queue-length-weighted tuple-rateηtrt to gauge the transmitting ability for each tuple link inthe tuple-based model. Distributed Dynamic Scheduling willfollows:

• For each transmitting link to make the regional bestdecision, the transmitting node will choose relative strongtuple links without any co-channel interference or radiointerface conflict. Within one link, the decion is quite sim-ilar with centralized greedy maximal scheduling, which isto choose the tuple link with largest tuple-queue-length-weighted tuple-rate ηtrt. Other interfering tuple links willalso be crossed out after each tuple link is selected.

• All links make the decision concurrently in one decisionturn. Each link can only select no more than one tuplelink for one link at each turn iteratively until there is nomore tuple links left. In other words, the iteration stopswhen all tuple links are either scheduled or removed afterseveral decision turns.

These two steps above make one scheduling iteration andthis iteration will continue to repeat occasionally during thewhole transmitting process. The reason to select no more thanone tuple link at each turn is that it can balance the wholesystem and select stronger tuple links. Otherwise, some strongtuple links may be crossed out due to excessive selection ofweak tuple links within one link. To be more flexible andself-adaptable, the whole system will iterate the two steps (i.e.scheduling iteration) above occasionally. During the iteration,the system will check if selected tuple link is strongest (withlargest ηtrt) among all the tuple links operating all the samechannel within the interference range. If not so, substitute theselected tuple link and refresh the removed tuple link list atthe same time.

The scheduling iteration can bring three advantages: (i)The system can automatically recognize the distribution ofdata flow and adapt to different flow changes. (ii) Due to thelack of global information, especially topology, the concurrentdecision may result in limited throughput and regional con-flicts and congestion. However, the iteration can diminish thiscircumstance and solve regional congestion because it makessecond choice after comparison between selected and removedtuple links. (iii) Through multiple iterations and comparisons,strong tuple links with less conflict cost will be graduallypreserved, regional greedy effect will gradually be diminishedto achieve better global performance.

B. Joint Design with Resource Allocation

Greed isn’t always the best thing in the world. Can wedesign a better protocol to achieve higher performance? Thefact that all those greedy protocols put temporary benefitahead of fairness of the whole system may leave some spacefor improvement. If we add a backlog management unit, the

scheduling for the whole system will be less blind in thatdecisions are based on reasonable regional distribution ofoffered load and received packets. Meanwhile, as mentionedin Section II, extra attention for radio/channel diversity mayfurther enhance the performance.

To solve the problem of backlog management with fullconsideration of channel contention and radio balance, weprovide data with one-stop service: directly control the datapackets and assign them to its ultimate transmitting port. Inother words, after synthesizing limited information about thecurrent network system, we make better decisions for whichradio to transmit, which channel to transmit over, which radioto receive simultaneously. For ease of exposition, we assumeeach data flow has only one fixed path. The whole DistributedTuple-based Throughput Control Protocol follows:

• STEP I: In this step, each link li will decide which tuplelink and how much work load distributed to that tuplelink from the link’s packets queue qli . If let xt(τ) to bethe number of packets link l assign to tuple link t at timeslot τ , then the assignment rule follows Equation (1).

xt(τ) =

rt , if δqlrt ≥∑k∈It

ηk

rk+

∑k∈E(Trt)

ηk

rk+

∑k∈E(Rrt)

ηk

rk

0 , else.

(1)

where the left side of the inequality can be interpreted asthe transmitting ability of tuple link, while the three itemson the right side representing the contention cost fromthe co-channel interference, radio interface contention onthe transmitter side and receiver side, respectively. δ is apositive constant carefully chosen for each tuple link. Itmay vary in different topology to balance the difference.After this step of assignment, the link queue ql willbe drained min

{ql,

∑t∈Tl

xt(τ)}

packets out into thequeues of its tuple links. The design of this assignmentwill be further explained later. Figure 4 illustrates thepackets assignment process.

• STEP II: Update all the link queues for each linkto prepare for the next time slot communication.As stated in Step 1, link queue ql will be drainedmin

{ql,

∑t∈Tl

xt(τ)}

packets out. Meanwhile, after onetime slot of transmission, each queue will simultaneouslyreceive the packets from its source, and if it is the sourcenode of flow fi, will generate λfi packets. Under stabletransmission, we assume each node in any flow path fiwill receive λfi packets on average in each time slot,while nodes that are no longer in any flow path will not.So the evolution of ql will follow Equation (2).

ql(τ+1) = ql(τ)−min

{ql(τ),

∑t∈Tl

xt(τ)

}+

fF∑f=f1

Hlf (τ)λf

(2)where

Offered Load

Received Packets

Link Queue

Radio 1

Radio 2

Radio R

Radio 2

Radio R

C1

C2

Cn

C1

C1

C2

C2

Cn

Cn

Fig. 4: Packets Assignment Process. We assign backlogged data packets in link queue to its tuple with definite operationchannel and radio. According to equation (1), the existing packets in tuple link queue will influence the choice for next packet,so the burden of each tuple link will be dynamically balanced to avoid resistance of conflict. Thus the balance between radiosand channels will largely enhance the whole performance of the system.

Hlf (τ) =

1 , if flow f passes through link l.

0 , otherwise.(3)

• STEP III: After the tuple links’ queue has been assigned,the scheduling process will begin. To get a better per-formance, we use Distributed Dynamic Scheduling de-scribed in Section IV-A to finally choose the strong tuplelinks to transmit and avoid both co-channel interferenceand radio-interface contention. As shown before, eachtuple link which is backlogged (i.e. ηt ≥ rt) should atleast hold one of the two conditions: either selected totransmit or crossed out due to co-channel interference orradio interface conflict. As the final schedule result comesout, the queue of scheduled tuple link t will be drained rtpackets out. Recall that link queue ql has assigned xt(τ)packets into tuple link queue ηt, the evolution of tuplelink queue ηt will follow Equation (4).

ηt(τ + 1) = ηt(τ)− rtSt(τ) + xt(τ) (4)

where

St(τ) =

1 , if tuple link t is scheduled.

0 , otherwise.(5)

Remark: The design of this algorithm is to help the sys-tem operate more efficiently by using those tuple links withstronger transmitting abilities and lower contention cost. It canbalance the whole system to avoid congestion as well. Specif-ically, if tuple link t is scheduled, then due to the co-channelinterference, it will render the congestion of those tuple linkswithin its interference range using the same channel, whichcan be represented as the quantity

∑k∈It

ηk

rk. Meanwhile,

those tuple links will occupy one radio on each side of the link.Hence, those tuple links using the same radios will be blocked.The cost can be represented as the quantity

∑k∈E(Trt)

ηk

rkand

∑k∈E(Rrt)

ηk

rkfor the transmitting radio and receiving

radio, respectively. Thus, on one side, from the view of oneparticular link, it will assign packets to its tuple links only ifthe transmitting ability is weighted more than the contentioncost if the assigned tuple link is scheduled. So it can guaranteethe fairness without impairing efficiency. On the other side,from the view of the whole system, if any congestion happensin a particular link, the increase of ql will also increase theprobability of assigning packets to less competitive tuple links.With this design, such less crowded links will automaticallyraise their priority of assignment and consequently balance thetransmission of the whole system.

The following gives the efficiency of the Distributed Tuple-based Throughput Control Algorithm and its proof.

Theorem 1. Given the routing matrix [Hlf ] of a wireless

network system, and assume that each flow will only have onefixed path, the proposed distributed tuple-based algorithm canachieve the efficiency ratio of 1/K, where K is the interferencedegree.

Proof: We define the Lyapunov function as Equation (6) toestablish and verify the stability.

V (−→q (τ),−→η (τ)) = Vq(−→q (τ)) + Vη(

−→η (τ)) (6)

where

Vq(−→q (τ)) =

lL∑l=l1

δq2l2

(7)

Vη(−→η (τ)) =

tT∑t=t1

ηt2rt

∑k∈It

ηkrk

+∑

k∈E(Trt)

ηkrk

+∑

k∈E(Rrt)

ηkrk

(8)

Then we can define the lyapunov drift as the sum of thedrift of Equation (7) and Equation (8) .

∆V (−→q (τ),−→η (τ)) = ∆Vq(−→q (τ)) + ∆Vη(

−→η (τ)) (9)

The drift of Vq(−→q (τ)) is:

∆Vq(−→q (τ))

=

lL∑l=l1

δ

2

[ql(τ + 1)2 − ql(τ)

2]

=

lL∑l=l1

δ

{ql(τ)

[ql(τ + 1)− ql(τ)

]+

[ql(τ + 1)− ql(τ)

]22

}

≤lL∑

l=l1

δql(τ)

[ fF∑f=f1

Hlf (τ)λf −

∑t∈Tl

xt(τ)

]+ C (10)

where C is a positive constant to bound the quan-

titylL∑

l=l1

δ2

[ql(τ + 1) − ql(τ)

]2, which can be chosen as[∑fF

f=f1Hl

f (τ)λf+∑t∈Tl

xt(τ)

]2where link l is the one which

maximizes this quantity.The drift of Vη(

−→η (τ)) is:

∆Vη(−→η (τ))

=

tT∑t=t1

ηt2rt

{ ∑k∈It

[xk(τ)

rk− Sk(τ)

]+

∑k∈E(Trt)

[xk(τ)

rk− Sk(τ)

]+

∑k∈E(Rrt)

[xk(τ)

rk− Sk(τ)

]}(11)

The proof of the theorem can be interpreted as if somescheduling algorithm can stabilize the system at offered loadK−→λ , then the proposed algorithm can stabilize the system

at the offered load−→λ . Since the offered load K

−→λ should

stabilize the system, we assume that there exists an equivalentassignment yt ≤ rt for tuple link t, and the followinginequality should hold:

(1 + ϵ)2KfF∑

f=f1

Hlfλf ≤

∑t∈Tl

yt, ∀l (12)

Considering the co-channel interference and the definitionof interference degree K, i.e. within the interference range, thenumber of scheduled tuple link using the same channel canbe no more than K. Thus, for any tuple link t:∑

k∈It

ykrk

≤ K (13)

Considering the radio-interface contention, for any tuplelink t that is scheduled: ∑

k∈E(Trt)

ykrk

= 0 (14)

∑k∈E(Rrt)

ykrk

= 0 (15)

To prove the theorem, we will only have to prove thestability under the offered load

−→λ . We can let the assignment

x′t =

yt

(1+ϵ)K , then

(1 + ϵ)

fF∑f=f1

Hlfλf ≤

∑t∈Tl

x′t, ∀l (16)

According to the definition of interference degree, andnoting that

∑k∈E(Trt)

x′k

rk= 0 and

∑k∈E(Rrt)

x′k

rk= 0, so we have:

(1 + ϵ)(∑k∈It

x′k

rk+

∑k∈E(Trt)

x′k

rk+

∑k∈E(Rrt)

x′k

rk)

=(1 + ϵ)∑k∈It

x′k

rk

≤1 (17)

Finally, the drift is bounded by:

E

[V (−→q (τ + 1),−→η (τ + 1))− V (−→q (τ),−→η (τ))|−→q (τ),−→η (τ)

]≤

lL∑l=l1

δql(τ)[ fF∑f=f1

Hlf (τ)λf −

∑t∈Tl

xt(τ)]

+

tT∑t=t1

ηt2rt

{ ∑k∈It

[xk(τ)

rk− Sk(τ)

]+

∑k∈E(Trt)

[xk(τ)

rk− Sk(τ)

]+

∑k∈E(Rrt)

[xk(τ)

rk− Sk(τ)

]}+ C

=

lL∑l=l1

δql(τ)[ fF∑f=f1

Hlf (τ)λf −

∑t∈Tl

x′t

]+

lL∑l=l1

δql(τ)[∑t∈Tl

x′t −

∑t∈Tl

xt(τ)]

+

tT∑t=t1

ηt2rt

{ ∑k∈It

[x′k

rk− Sk(τ) +

xk(τ)

rk− x′

k

rk

]+

∑k∈E(Trt)

[x′k

rk− Sk(τ) +

xk(τ)

rk− x′

k

rk

]+

∑k∈E(Rrt)

[x′k

rk− Sk(τ) +

xk(τ)

rk− x′

k

rk

]}+ C

Due to the Distributed Dynamic Scheduling in STEP III,within the interference range, there is at least one tuplelink scheduled,

∑k∈It

Sk ≥ 1. Considering the inequalities

(14)(15)(16)(17), the drift is further bounded by:

E

[V (−→q (τ + 1),−→η (τ + 1))

− V (−→q (τ),−→η (τ)) | −→q (τ),−→η (τ))

]≤− ϵ

[ lL∑l=l1

δql(τ)

fF∑f=f1

Hlf (τ)λf +

tT∑t=t1

ηt2rt

∑k∈It

x′k

rk

]

+

lL∑l=l1

∑t∈Tl

(x′t − xt(τ))

rt

{δql(τ)rt−[ ∑

k∈It

ηk(τ)

rk+

∑k∈E(Trt)

ηk(τ)

rk+

∑k∈E(Rrt)

ηk(τ)

rk

]}+ C (18)

By the proposed algorithm, the scheduled tuple link t isallocated with xt = rt packets while x′

t ≤ rt. The assignmentprinciple guarantees that the weighted transmission ability islarger than the contention cost.

δql(τ)rt ≥[ ∑k∈It

ηk(τ)

rk+

∑k∈E(Trt)

ηk(τ)

rk+

∑k∈E(Rrt)

ηk(τ)

rk

](19)

So the whole third term is non-positive. Consequently, wederive the lyapunov drift which is proved to be negative whenql and ηt increase. The network stability then follows [4].

Combining scheduling and resource allocation, we formu-late the whole joint design for transmitting part of MR-MCwireless system in Algorithm 1.

V. JOINT ROUTING AND RESOURCE ALLOCATION DESIGNIN MR-MC WIRELESS NETWORK

In this section, we propose the routing part to incorporatewith the transmission part in Section IV to finish the completedesign. We aim to provide a joint control protocol for commu-nicating nodes to guide flows with less interference confliction,less transmission latency and better system performance. InTuple-based Throughput Control Protocol (Transmission Part),we once assume each flow has only one fixed path, and viewmultipath with same source and destination as different flows.Hence, the distribution of flows into multipath is another sub-problem to solve. We intend to design an online distributedrouting algorithm so that the whole joint design can beimplemented completely distributed.

The goal of the routing part in the joint design is tofind multiple paths from source to destination, compute theselection cost for each path and choose the right path for com-munication. Similar with ad hoc on-demand distance vectorrouting (AODV) in ad hoc wireless network, the source nodewill broadcast request all the way through to the destination,

Algorithm 1 Distributed Tuple-based Throughput ControlAlgorithm

1: Initialization2: PHASE 1: Resource Allocation and Data Preparation3: for each link li do4: Update the link queue with generated and received

packets from last time slot.5: for each tuple link t ∈ Tli do6: if δqlrt ≥

∑k∈It

ηk

rk+

∑k∈E(Trt)

ηk

rk+

∑k∈E(Rrt)

ηk

rkthen

7: Assign rt packets to tuple link t.8: end if9: end for

10: end for11: PHASE 2: Scheduling and Transmission12: if schedule times out or congestion happens then13: (Scheduling iteration)14: while there is tuple links not selected or crossed out

do15: for each backlogged link li do16: Tuple link with largest ηtrt without conflicts.17: Refresh cross-out tuple links table.18: for each tuple link t ∈ Tli do19: ηt(τ + 1) = ηt(τ)− rtSt(τ) + xt(τ)20: end for21: end for22: end while23: end if

and at the meantime, get an additional knowledge of each nodeon the path for path burden computation. To be specific, theselection cost for link l is

∑t∈Tl

ηt

rt+ γql, where

∑t∈Tl

ηt

rtcan be interpreted as transmission burden and γql as queueingdelay. γ is a positive constant to balance the influence betweenqueueing delay and transmission delay. Thus, the whole selec-tion cost for a possible path P is:∑

l∈P

(∑t∈Tl

ηtrt

+ γql) (20)

The routing procedure is as follows. First, the sourcenode broadcasts its routing request to its one-hop neighbournodes. The request contains the information of source node,destination node, relay nodes on the path, sequence numberand the sum of selection cost on each link along the path.Then, each relay node adds the cost for next hop into theselection cost, updates the relay nodes array, and fowardsthe routing message through that hop to the next node. Notethat in this scenario, they delete loop route and preserve eacheffective route with different selection cost. Finally, when thedestination node receives request, it will send back reply withall the information of the particular path that the requestmessage has been through. Reply will be unicasted throughthe reverse path to the source node.

With information of different path and varied selection cost,the source node will decide which and how much packets it

Fig. 5: Simulation Topology. This figure is a randomly gen-erated topology with three identified user and one of theirpossible flow paths. With this topology, we carry out threetype of simulation in terms of capacity, stability and multi-radio improvement.

should assign to each path in the priority of selection cost ofthat path. The decision rule follows:

xp(τ) =

min

{Qfi ,M max

t∈Tp start

rt}

,

if δQfi ≥∑l∈P

(∑t∈Tl

ηt

rt+ γql)

0 , else.

(21)

where xp(τ) is the amount of packets that source nodeallocates to path p, and Tp start is the tuple link set of thestart link of path p. Qfi is the total packets of flow fi waitingto be transmitted at source node. Therefore, if the constraintis not satisfied, which means the selection cost is larger thanthe backlogged pressure, then assigning packets to these pathswill not yield any gain and these paths will be abandoned untilworse congestion occurs.

Algorithm 2 Joint Routing, Scheduling and Resource Alloca-tion Design for MR-MC Wireless Network

1: Initialization2: for each node i do3: for each flow passes through node i do4: Check each flow path from i to its destination5: if δQfi ≥

∑l∈P

(∑t∈Tl

ηt

rt+ γql) then

6: Assign min{Qfi ,M max

t∈Tp start

rt}

to ql

7: end if8: end for9: end for

10: Carry out Distributed Tuple-based Throughput ControlAlgorithm (Algorithm 1)

Moreover, to be more flexible and adaptable, every nodecan be the source node when deciding which route to transmit.Thus, if a relay node has multiple paths to the destination node,

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

200

400

600

800

1000

1200

1400

Normalized offered load

Ave

rage

que

ue b

ackl

og

DGMSTDTCCGMS

Fig. 6: Comparison of different scheduling algorithm andproposed tuple-based throughput control protocol in singleradio scenario. Although the advantage of doing multitaskhas not been used in single radio scenario, our proposeddistributed tuple-based throughput control protocol still canachieve almost 75% of the centralized scheduling algorithms.

it can re-assign packets with same destination on differentlink queues so that it brings more flexibility to deal withpossible confliction and congestion with other flows. In fact,the assignment will actually decide which route to transmit thepackets. We use this routing protocol online and adjust the flowpath flexibly according to real network condition. Hence, oursingle fixed path algorithm can adapt to multiflow multipathscenario. Combined with Tuple-based Throughput ControlProtocol in Section IV, we build a joint design (Algorithm2) for routing, channel assignment, radio arrangement andscheduling which can automatically adapt to different changesand vibrations in MR-MC wireless network.

VI. PERFORMANCE EVALUATION

In this section, we will evaluate the performance of Dis-tributed Tuple-based Throughput Control Protocol with otherexisting algorithms under tuple-based model. The simulation isbased on a random topology in Figure 5. There are 10 nodesand 18 links. Due to multiple radio interfaces, links can becompletely duplex. We carry out 4 set of simulation regardingto system capacity, system stability, advantage of exploitingmultiple radio interfaces and dynamic routing.

A. System Capacity

To simulate and compare different scheduling approacheswith our proposed throughput control protocol, we first testthe system performance in a single radio scenario (Figure 6).Under tuple-based model, every link is decoupled into severaltuple links as shown before. The activation of each tuplelink will follow the interference model described in SectionIII. Without losing generality, we assume the normalizedtransmission range and interference range of each node is0.3 and 0.6, respectively. We assume that there is 3 available

0 200 400 600 800 10000

10

20

30

40

50

60

Time Slot

Sou

rce

node

que

ue b

ackl

og

Offered load = 0.2Offered load = 0.3Offered load = 0.4

0 200 400 600 800 10000

10

20

30

40

Time Slot

Ave

rage

que

ue b

ackl

og

Offered load = 0.2Offered load = 0.3Offered load = 0.4

Fig. 7: Stability of source node and average queue backlog.The stability of source node queue and average queue backlogcan reflect the capacity of the system. If the queues remainstable, it shows that the system capacity is large enough tohold the offered load.

channels, and the tuple link capacity is uniformly selected inthe range [0,1] in each time slot to reflect the diversity betweeneach link-radio-channel tuple. To test the proposed tuple-based distributed throughput control (TDTC) protocol withcentralize greedy maximal scheduling (CGMS) and distributedgreedy maximal scheduling (DGMS), we randomly choosethree crossed flow path, and simulate the network system underdifferent offered load (Figure 5). Figure 6 shows the averagequeue backlog with different offered load at the 60000th timeslot, which can be regarded as infinity in this simulation. Thebacklog can be viewed as the congestion of data packets due tolack of transmitting ability or the block by interference, whichcan then reflect the performance of system control. Meanwhile,the sudden growth in Figure 6 can be seen as the limit andboundary of system capacity region.

As shown in the simulation result (Figure 6), our proposeddistributed tuple-based throughput control protocol can largelyenhance the system performance, and achieve almost 75% ofthe centralized scheduling algorithms while our design remainsin a distributive fashion. The analysis of simulation resultshows that the proposed distributed tuple-based throughputcontrol can largely improve the capacity of the system. Itavoids the worst drawback of greedy algorithms which isrelative blindness at each decision that it cannot consider theproblem comprehensively. While link-based greedy maximalscheduling has loss on capacity, tuple-based model will evenaugment the loss, though it take radio and channel diversityinto consideration. In SR-SC wireless network, the blindnessis only for different links, while in MR-MC wireless network,greed also blinds different channels and different radio inter-faces because decision is made for smaller unit. On contrary,combined with comparison and backlog management, ourprovably efficient tuple-based throughput control protocol canavoid this blindness to a great extent by careful design ofresource allocation strategy.

B. System Stability

This set of simulation takes a close look on the stabilityof the network system with different offered load using theproposed throughput control protocol. We intend to focus oneach time slot when the average queue backlog and source

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

200

400

600

800

1000

1200

1400

Normalized offered load

Ave

rage

que

ue b

ackl

og

DGMS with 1 radio 3 channelsDGMS with 3 radios 5 channelsTDTC with 1 radio 3 channelsTDTC with 3 radios 5 channels

Fig. 8: System performance with different resource supply.This figure shows the salient effect of exploiting radio re-source. With multiple radios, our proposed protocol can con-trol the balance of radio burden to further enhance systemcapacity.

node backlog of each flow changes. Figure 7 shows thevibrations of the average queue backlog and queue backlogof source node of one flow as time passes by. If a MR-MCwireless network system is stable with certain offered load,the average queue backlog should maintain around a constantvalue with a little vibration within tolerance. Otherwise, theaverage queue backlog will soar to infinity as time grows toinfinity. Similarly, the queue vibration of the source nodesof different flows can also reflect the stability of the MR-MC wireless network system. In other words, if a MR-MCwireless network system is stable with certain offered load,there should be no congestion for each and every flow inthe network system. So, if a flow is successfully deliveredto its destination node consistently, the queue backlog of thesource node should maintain around a constant value with alittle vibration within tolerance as well. Otherwise, the queuebacklog will rise up to infinity as time grows to infinity. Asshown in Figure 7, simulated in the topology in Figure 5 with 1radio and 3 channels, the system can be stable at normalizedoffered load 0.2, while offered load 0.3 might stabilize thesystem in a longer time and offered load 0.4 might not stabilizethe system, at least not in 1000 time slots.

C. Multiple Radio Interface Management

To see the effect of multiple radio interface management, wesimulate and compare the system performance with differentsupply of resource in MR-MC wireless network to verifythe advantage of radio arrangement and channel assignment.In Figure 8, we plot the change of average queue backlogwith different offered load and different resource. From thesudden growth point described earlier, we can view the systemcapacity with different available radio interfaces. The addi-tion of radio interfaces not only diminishes the interference,but also makes multi-path flow possible, both of which can

0 200 400 600 800 10000

20

40

60

80

100

120

140

160

180

Time Slot

Ave

rage

que

ue b

ackl

og

Routing path(load=0.8)Routing path(load=1)Routing path(load=1.2)Fixed path(load=0.8)Fixed path(load=1)Fixed path(load=1.2)

Fig. 9: Comparison of system average queue backlog betweendynamic routing and fixed path approach with different offeredload. Each node is equipped with 3 radios and 5 channels.

largely improve the system capacity. Centralized and dis-tributed greedy scheduling cannot take channel diversity andradio burden imbalance into account, so that their control overchannel assignment and radio arrangement is not ideal. Thegreedy choice at each step may lead to interference conflictsand radio interface contention with other strong tuple links,while our proposed throughput control protocol can balancethe cost and carry out a better resource allocation strategy.As shown in Figure 8, the addition of radio interfaces andnon-overlapping channels do not improve much under greedymaximal scheduling algorithms. The enhancement is quitelimited because the assignment of data packets in greedymaximal scheduling is almost blind so that the schedulingitself may increase the possibility of queueing congestionand fierce interference conflicts. However, under our proposedthroughput control protocol, the abundance of resource is fullyutilized and with triple radio interfaces and 2 more channels,the system performance has gained a huge improvement.

D. Dynamic Routing

We use the same topology with same flow sources anddestinations as in Figure 5, while adding the dynamic routingfunction into our simulation. The user can use multi-flowmulti-path to transmit their packets now. We plot the change ofaverage queue backlog with different offered load to comparethe performance of our routing function with one-fixed pathapproach in Figure 9. Note that multi-path simulation areunder the multi-radio scenario. We provide each node with3 available radio interfaces and 5 non-overlapping channels.Comparing to one fixed path approach, dynamic routing caneffectively enhance the performance. Through simulation wesee system throughput control with routing path can holdoffered load 1, and even almost stablize the system underoffered load 1.2, while fixed path approach might not stablizeat offered load 0.8, let alone offered load at 1 or 1.2. Whenoffered at 1.2, the average queue length increases almost

linearly with time, which means the system transmitting abilityunder this approach is far too weak to hold the offered load.

VII. CONCLUSION

In this paper, we introduce a distributed dynamic jointdesign on resource allocation including channel assignment,radio arrangement, scheduling and routing for multi-radiomulti-channel wireless network. We use tuple-based model todecouple links into several link-radio-channel tuples to care-fully consider channel diversity and radio burden balance sothat the whole system can fully utilize the abundance of radioand channel resource in MR-MC wireless network to improveits performance. We have proved that our distributed tuple-based throughput control protocol can achieve an efficientratio of 1/κ. Combined with routing based on synthesizingall limited information that a distributed node can get, weprovide data packets with one-stop-service control to guidethe flow with less interference conflicts and better systemperformance. Simulation result evaluated and verified betterperformance of proposed control protocol compared with otherexisting algorithm and natural extension of greedy schedulingalgorithms. It also shows our protocol is suitable for MR-MC wireless network in that it effectively exploit and managemultiple radio interfaces and transmit with reasonable routingto enhance the system performance.

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