Multicast capacity with max min fairness for heterogeneous networks

622 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 22, NO. 2, APRIL 2014

Multicast Capacity With Max-Min Fairnessfor Heterogeneous NetworksYixuan Li, Qiuyu Peng, and Xinbing Wang, Member, IEEE

Abstract—In this paper, we investigate the multicast capacity forstatic ad hoc networks with heterogeneous clusters. We study theeffect of heterogeneous cluster traffic (HCT) on the achievable ca-pacity. HCT means cluster clients are more likely to appear nearthe cluster head instead of being uniformly distributed across thenetwork. Such a property is commonly found in real networks.By adopting max-min fairness, the minimum among all individualmulticast capacities of clusters can be maximized. Since this min-imal individual multicast capacity will not be maximized unlimit-edly, our work focuses on deriving the upper bound of the min-imum individual multicast capacity (we refer it as minimum ca-pacity for simplicity) in HCT, which provides the best performancefor the minimum multicast capacity to attain in the whole net-work. We find that HCT increases minimum capacity for ad hocnetworks. Furthermore, the multicast capacity achieving schemeis provided to justify the derived asymptotic upper bound for theminimum capacity. Our work can generalize various results ob-tained under nonheterogeneous networks in previous literature.

Index Terms—Heterogeneous cluster, multicast capacity, staticclustered network.

I. INTRODUCTION

A WIRELESS network is modeled as a set of nodes thatsend and receive messages over a common wireless

channel. Since the seminal work done by Gupta and Kumar [1],there is significant interest toward the asymptotic capacity ofthe network when the number of nodes grows. The authorsof [1] prove that the per-node capacity1 is ina static network. Later in [2], the capacity result is analyzedunder more general fading channel condition, and a similarresult is given. Then, Franceschetti et al. [3] designed anoptimal routing protocol with capacity achievingvia percolation theory.

Manuscript received October 29, 2012; revised March 16, 2013; acceptedMarch 19, 2013; approved by IEEE/ACM TRANSACTIONS ON NETWORKINGEditor S. Ramasubramanian. Date of publication April 25, 2013; date ofcurrent version April 14, 2014. This work was supported by National Fun-damental Research Grant No. 2011CB302701, NSF China under Grant No.61271219,61202373, Shanghai Basic Research Key Project No. 11JC1405100;the China Ministry of Education Fok Ying Tung Fund under Grant No. 122002,and the China Ministry of Education New Century Excellent Talent underGrant No. NCET-10–0580.The authors are with the Department of Electronic Engineering Shanghai Jiao

Tong University, Shanghai 200240, China (e-mail: [email protected];[email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNET.2013.2255065

1Given two functions and :means ; means

; is equiv-alent to is equivalent to

means and.

The multicast network, which generalizes the above unicastnetwork, received more attention recently, and the estimation ofthe achievable multicast capacity is required in many applica-tions like sensor networks and TV streaming [25]. Li et al. [4]studied the achievable capacity in multicast networks. In theirwork, there are multicast sessions, each comprising onesource and destinations, and they find that the capacityscales as based on the Manhattan Routingscheme. Their results generalize both unicast and broadcast [5]capacity. In [6], Shakkottai et al. studied a different multicastframework where there are multicast sources anddestinations per flow. Their network can support a rate of

for each flow. Later on, the multicast capacity

under Gaussian channel is obtained in [7], [8]. The achievablecapacity in mobile multicast (motioncast) is explored in [9],and optimal mobile multicast capacity is presented in [10],which is a generalization of [11] and [12]. In [13] and [14],multiple-input–multiple-output (MIMO) cooperations are in-troduced to improve multicast capacity.Since nodes in the same multicast session can be treated as

members of a cluster, multicast networks can also be viewedas clustered networks. However, there are few works con-cerning the cluster behavior of multicast networks. Uniformlydistributed cluster (muticast session) traffic is assumed, andcluster heterogeneities are rarely involved in previous works.Actually, most real networks are characterized by variousclustered heterogeneities, and some aspects have already beeninvestigated in unicast networks, which include the following.Spatial Heterogeneity:Wireless nodes are not likely to be uni-

formly distributed across the deployed region in realistic net-works—e.g., wireless users may cluster in urban areas, so thereare less users in suburban areas. Since spontaneous groupingof the nodes around a few attraction points occurs commonlyin wireless network, Alfano et al. [15], [16] extended the ca-pacity scaling to networks with inhomogeneous node density.In their work, nodes are generated according to a specified pointprocess, and they show that the bottleneck is in the node sparseregion because the network capacity is related to the minimumnode intensity.Pattern Heterogeneity: It is likely that there exists more than

one type of traffic pattern in the network, and nodes of the sametraffic pattern constitute a cluster. In [17], Wang et al. studieda unified modeling framework composed of unicast, multicast,broadcast traffic. Later in [18], Ji et al. explored networks com-posed of both unicast and converge-cast traffic and showed thatMIMO cooperation can be applied to increase capacity for bothtypes of traffic. Li et al. [19] dealt with networks containingsome helping nodes for packet delivery. Using this method, thenormal nodes and helping nodes can be viewed as two clusters.

1063-6692 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

LI et al.: MULTICAST CAPACITY WITH MAX-MIN FAIRNESS FOR HETEROGENEOUS NETWORKS 623

The primary incentive motivating us to investigate clusterednetworks is because of its practical use in real life. For instance,in a military battlefield, commanders from different placesmust send requirements through a common wireless channelto their respective soldiers around them. In sensor networks,local schedulers also need to send packets to their adjacentclient sensors. In addition, the study of capacity performance inclustered networks under multicast traffic is still not sufficientyet, and that is another reason why we set the model. Sinceclients of the same data flow are nonuniformly distributed, theopen question is the following:• What are the impacts of heterogeneous traffic on multicastcapacity in clustered networks?

In this paper, nodes in each multicast session are composed ofa cluster, and heterogeneous cluster traffic (HCT) is defined asfollows: Clients of the same cluster (data flow) are likely to bedeployed around a cluster head specified by an inhomogeneousPoisson process (IPP).We describe this clustering behavior witha variable , which describes the extent of HCT.Our asymptotic analysis for multicast capacity is based on

the max-min fairness, through which the minimum among allindividual multicast capacities of clusters can be maximized.The formal definition of max-min fairness will be providedin Section II. Typically, there are more than one criterion inevaluating the efficiency of a certain scheduling policy. Mostprevious literature such as [4], [15], and [16] aims at derivingthe bounds for total throughput. However, maximizing thetotal throughput is sometimes not desirable because it maycause imbalance among individual multicast capacities ofclusters and cannot avoid the abnormal minimum value ofindividual multicast capacity. For example, consider a net-work of 100 clusters, among which 99 clusters have equalrates of 100 Mb/s and one cluster has a rate of 1 Mb/s. Suchabnormal individual multicast rate of 1 Mb/s can be regardedas service outage compared to other large rates and should beimproved if possible. By adopting max-min fairness, such rategap among different clusters can be narrowed, and potentialservice outages can be reduced. The relative unfair individualmulticast capacity of 1 Mb/s caused by traffic congestion orunreasonable rate allocation can be enhanced to a higher value.As we know, this minimal individual multicast capacity will notbe maximized unlimitedly, so there must exist an upper limitthat the minimum capacity cannot exceed. Our paper, beingset upon the max-min metric, focuses on deriving the upperbound of the minimum multicast capacity. Investigating theminimum capacity is meaningful because it provides the bestperformance for the minimum multicast capacity to attain inthe whole network. Note that for the purpose of brevity, we willuse the notion “minimum capacity” to represent the “minimumindividual multicast capacity” throughout the paper.Previously, researchers had established a framework for

multicast rate allocation with max-min fairness in wired net-works [20], [21] . In [20], the authors not only justified themax-min fairness for multicast traffic, but also gave efficientalgorithms to compute the rate allocation vector based on themetric. Nevertheless, providing algorithms is sometimes not astraightforward way of showing the capacity performance ofnetworks, although it does help in practical congestion controlcomputing. Our work, instead, directly shows a quantitativeanalysis and bounds the value of minimum capacity using

explicit formulations rather than iterative computing proce-dures. In other words, the seminar work [20] can help validatethe max-min fairness for multicast traffic; our work then goes astep further to provide a scalable analysis for the best capacityperformance that the minimum capacity can attain. We findthat the minimum capacity increases in HCT because the totaltransmission length is shortened, and we offer a quantitativerelationship between the minimum capacity and .The rest of the paper is organized as follows. In Section II,

we outline some preliminaries of the network and our main re-sults. In Sections III and IV, close forms of the upper bound forminimum capacity with distribution function and distributionvariance are derived, respectively. In Section V, we provide arouting scheme for achieving the upper bound of minimum ca-pacity in the uniform random cluster model. A discussion ofthe results is presented in Section VI. Finally, we conclude thispaper in Section VII.

II. PRELIMINARIES AND MAIN RESULTS

A. Network Topology

We consider extended networks composed ofclusters distributed over a two-dimensional torus re-

gion of edge2 . We specify a homoge-neous Poisson process (HPP) to generate cluster head , whoseposition is denoted by for cluster . Then,generates its cluster members according to an IPP whose inten-sity at is given by , where isthe expected size of the cluster and is the dispersion den-sity function. In order to investigate the impact of HCT on min-imum capacity, we will discuss in detail here what further as-sumptions should be made for both and . Because thedispersion density function determines the client distribu-tion of each multicast session, intuitively, HCT is mainly de-scribed by the characteristics of . In addition, HCS dealswith the cardinality of each cluster, so it is strongly related to

.First, we outline the properties of the dispersion density func-

tion as follows.1) is invariant under both translation and rotationwith respect to , therefore can be rewrittenas , and it is a nonincreasing, nonnegative,bounded, and continuous function with respect to theEuclidean distance .

2) Integration of over the whole torus is equal to1, .

The first property restricts the dispersion density function toa regime that clients are more likely to distribute around thecluster head, that is to say there are more clients around thecluster head and less clients in remote areas with respect to thecluster head. The second property can be derived by defining anonnegative, nonincreasing continuous function such that

and then normalizing it over the whole area

2A cluster dense regime is assumed, which means the distance between ad-jacent clusters tends to 0 when approaches infinity. We will assume

throughout the paper.


Notice that in asymptotic analysis ifwe neglect the factor .Now we need to offer a quantitative value for a given disper-

sion density function to depict its degree of heterogeneity,then we can analyze the relationship between the minimum ca-pacity and the degree of HCT. As we know, the expectation candescribe the average value of a function and in this case, it cor-responds to the average density distribution

Variance can describe the fluctuation level of a functionaround its expectation. In this case, we define the varianceof as distribution variance , which can be utilized todepict HCT

We omit the term due to the wraparound property of atorus, and it can liberate us from the border effect. In case of uni-form traffic, , therefore , and larger hetero-geneity results in larger . Finally, we specify a special pointprocess as a uniform cluster random model (UCRM) whose dis-persion density function is as follows:

otherwise

where is defined as cluster radius. It means

clients of each cluster are randomly and uniformly distributed ina disk of radius centered at its cluster head. We prove that thistopology can achieve the largest value of the minimum capacitygiven a fixed in Section IV.We classify these clusters into super clusters ( ) based

on their cluster size, in order to explain the effect of clustersize. For each cluster , its size

, where is an increasing sequence over and theclusters in possess the largest size. In addition, some furtherassumptions are the following.1) .2) There are clusters in , and the other

clusters are randomly allocated to.

The second assumption indicates that the number of clusterswith size is the same order as the total number of clusters.Although it is somewhat restrictive here, it is constructive whenwe design our capacity achieving scheme and allows us to gainimportant insight into the structure of the problem.Under the above assumptions, cluster clients are likely to be

distributed near the cluster head, and the cluster size conformsto a Poisson distribution with rate . In addition, there are

clusters in for applying the Chernoffbound. In order to simplify our analysis, we take both and

as the cluster size and as the number of clustersin for . Such simplification does not influenceour results in order sense. Fig. 1 is an example of the networktopology.

Fig. 1. Demonstration of network topology. Nodes in the same cluster are la-beled with the same number. H and C represent head (source) and clients (des-tination), respectively.

Fig. 2. Demonstration of two successful transmissions.

B. Transmission Protocol

All wireless transceivers can communicate over a commonchannel with limited bandwidth . We adopt the protocolmodel for interference proposed in [1]. In each time-slot, asender can successfully transmit at b/s to a destinationwhen the Euclidean distance between any other active trans-mitters and is larger than , where is theEuclidean distance between and ; is a positive constantindependent of the position of , and it specifies a guardzone for a successful transmission. Note that in broadcast cases,the interference radius is defined as times the lengthbetween the furthest nodes and the source. In this paper, weassume that the minimal transmission range is denoted by ,and we will prove that nearest neighbor transmission is alsodominant. Fig. 2 illustrates two concurrent transmissions.

C. Traffic Model

Amulticast traffic pattern is assumed where each cluster headgenerates data flows to their respective clients, e.g., in Fig. 1.The one-to-many data flow in can be modeled as a multi-cast tree spanning 1 head and clients. In [4], Euclideanminimal spanning tree (EMST) is employed to bound the lengthof transmission for each multicast session in nonclustered net-works with uniform node distribution. We employ new tech-niques to study heterogenous clustered networks, which gener-alizes uniform cases. Let denote the EMST for ,and the definition is given as follows.


Definition of EMTS: Assume that cluster head generatesits cluster members according to anIPP. The EMST for in two-dimensional space connectspoints using lines such that thetotal Euclidean length of all the lines is minimized and anypoint can be reached from any other by following the lines.Note that the communication between any source–destina-

tion (SD) pairs can also go through multiple relays from otherclusters.Since the network topology defined above is fundamentally

related to both the number of clusters and the network phys-ical extension , we hope to find out how the minimum capacityasymptotically scales. Now we give the definition of capacity.Definition of Asymptotic Capacity: Let de-

note the individual multicast capacity for cluster . The mul-ticast rate vector consists of theindividual multicast capacity of all clusters, and it is also iden-tified as network capacity. The minimum multicast capacity isdefined as , which is the min-imal achievable data rate in these clusters. Then,is defined as the asymptotic minimum capacity if there existconstants , such that

Let denote the number of data units already generatedin that have not yet been delivered to all of its members attime . Then, must guarantee a nonbacklog network, whichmeans .

D. Max-Min Fairness

A feasible multicast rate vector is max-min fair if and onlyif an increase of any individual multicast capacity within thedomain of feasible allocations must be at the cost of a decreaseof some already smaller individual multicast capacity. Formally,we have the following.Definition of Max-Min Fairness [22]: A multicast rate

vector is max-min fair if and only if, for any other feasiblemulticast rate vector , if the individual multicast capacity ofcluster satisfies , then there must exists some clustersuch that and .In [20], the authors justified the properties of max-min fair-

ness for multicast service in a wired network. In their model, therate allocation for different multicast sessions was also denotedby a rate vector, in which an element represents the individualmulticast capacity for one cluster. The properties can be adaptedto the wireless networks since our mathematical model is al-most the same as the single-rate multicast service model shownin [20]. That is to say, max-min fairness can be applied in mul-ticast traffic to enhance the minimum capacity. Based on this,our major concern is to find out how large can the minimum ca-pacity attain at most. The rest of the paper will focus on this ,i.e., the minimum capacity. Since the main purpose of our paperlies in deriving the upper limit for minimum capacity, we omitthe detailed proof of the max-min fairness properties to avoidredundancy.

E. Mathematical Notations

Throughout our paper, we denote as the number of hopsrequired for transmitting bit to all clients in . is the lengthof transmission of bit in its th hop. isthe number of nodes that can overhear a packet during a trans-mission of bit in its th hop. is the circular regioncentered at with radius . is the radius of the influential re-gion centered at the head. Nodes outside the influential regioncannot act as relays for that cluster. is the total Euclideanlength of a multicast tree , which can be calculated by sum-ming up the Euclidean length of all edges belonging to the tree.

F. Useful Known Results

Throughout the paper, the following two known results willbe used for proving theorems. In particular, the Chernoff boundis applied to bound the probability of sums of independent vari-ables, and the Hölder’s inequality will be consulted for derivingthe bound for minimum capacity.Lemma 2.1: Chernoff bound: Given a Poisson random vari-

able with expectation , for any , the Chernoff boundis given by

(1)

Lemma 2.2: Hölder’s inequality: If is a measurable subsetof with the Lebesgue measure, and and are measurablereal- or complex-valued functions on , then Holder inequalityis

(2)

where .

G. Main Results

Our main results are summarized as follows.• Given the dispersion density function , the upperbound of minimum capacity is given as follows:

where is some constant.• HCT increases the upper bound of minimum capacity ,and a universal relationship between and is obtainedas

III. UPPER BOUND TO MINIMUM CAPACITY

In this section, we will provide an upper bound of the min-imum capacity, which is a crucial step for deriving the relation-ship between the capacity and the distribution variance. First,we will prove several key results inherent to the static networkwith multicast traffic. There are some tradeoffs that must betolerated, among which are the number of hops, transmission


range, and limited radio resources. Therefore, a thorough com-prehension of the implicit relationships among them is construc-tive for deriving the upper bound of the achievable capacity.It consumes radio resources to forward a bit to relays

or destinations. The following lemma captures the tradeoffsamong number of hops, transmission range, and limited radioresources.Lemma 3.1: Constraint of Protocol Model: Under the pro-

tocol model, the following inequality must be held for anyrouting scheme when the simulation time is sufficientlylarge:

(3)

Proof: When is sufficiently large, the total number of bitscommunicated from the head to its clients in cluster is .Assume two pairs and are active inthe given time-slot, then according to the transmission protocolmodel

which is derived in [1]. Thus, disks of radius times the trans-mission radius centered at the receiver can be viewed as an “ex-clusion region” that rejects transmitters from other data flows.Such a property also holds for broadcast that the transmissionrange is defined as the furthest node that can receive the packets.Let be the transmission radius centered at the receiver for theth hop of bit , and be the overlap area between the “exclu-sion region” of bit ’s th hop and the deployed region . Notethat at least a quarter of the disk is within the given region (theextreme situation happens when the node is near the peripheryof the region). Then

Therefore, radio resources can be viewed as the limited band-width times the simulation time and the network area ,such that the following inequality is satisfied:

The above inequality is a basic requirement for multihoptransmission type, and the cooperative MIMO as in [14] isnot considered here. For simplicity, we define the area of theexclusion region consumed by one bit in a certain hop as asingle unit of transmission resources. The left side sums upall the transmission resources consumed by each hop of eachbit in each cluster . Since at most bits can be carriedin time with bandwidth , for area , the upper boundof total transmission resource is . Thisinequality cannot only be used in our heterogeneous case, butalso in homogeneous case as well since it is derived regardlessof node distribution. In a static network, there is a minimaltransmission range to guarantee full network connectivity. Innetworks with uniform node distribution. It is proved that the

transmission range is sufficient for networkconnectivity w.h.p. in [1]. Now we need to characterize afeasible transmission range , below which the transmissionis not possible in our framework. In other words, we want toderive such that the number of nodes in is nonzeroall the time.Lemma 3.2: Let represent the number of nodes within

the transmission range when a node wants to transmit itspackets, then when .

Proof: In our cluster dense regime, when

based on Cher-noff bound and Riemann sum (one can refer to [15, Theorem 1]).Similarly, the lower bound of is as .Then, we come to case. Note that the node

density is upper-bounded by an HPP with rate ,and the probability that the number of nodes inside the diskexceeding is

During the above derivation, we used the Lagrange form ofthe remainder term and . Therefore, when

, w.h.p. Note that is aprerequisite for any transmissions, and we complete our proof.

Therefore, we know a necessary condition for the transmis-

sion range is . Now we need to char-

acterize the total length for transmitting a bit . In [4],they prove that nearest neighbor transmission can achieve op-timal multicast capacity in uniform traffic distribution. They ob-tain that the number of hops required is for destinations.The situation is complicated here since traffic is not uniformlydistributed across the network. Clients in the samemulticast ses-sion are more clustered around the head, and a larger transmis-sion range can cover more clients in a cluster when the transmis-sion happens near the cluster head (source) than other places.Therefore, it is unknown whether the many more destinationsinvolved in a larger transmission range can compensate for thesacrifice of radio resources. Unfortunately, the following anal-ysis reveals that nearest neighbor transmission is also optimalin HCT.In [4], EMST is investigated in multicast traffic, and it can

help us obtain the upper bound of minimum capacity in non-heterogeneous networks. To obtain the results under heteroge-neous cluster traffic, we first introduce the following lemma[24, Theorem 1] as follows.Lemma 3.3: If is the density of the probability function

for picking points, then for large and , the size of theEMST is approximately , whereis a constant depending only on the dimension .Our case corresponds to and such that

It reveals that the length of the Euclidean minimum mul-ticast spanning tree constitutes two terms, the square root of


Fig. 3. Demonstration of the two conditions for MTTL.

the number of nodes and . Note that weeliminate the constant for simplicity. Intuitively, the min-imum length for connecting all the nodes is the minimum totaltransmission length (MTTL). For a given graph, MTTL can becounted by summing all the irredundant transmission lengthswith extra intermediate nodes and edges. Although MTTL issuperficially similar to the EMST problem, the length of MTTLis far less than EMST for the following reasons.• Larger transmission range can cover more than one node,but only the length of the longest SD pair is counted. For in-stance, if a node broadcast its message to all the other nodesin one time, the MTTL is at most . Therefore MTTL isrelated to the transmission range .

• Nodes from other clusters can act as relays to help forwardinformation. We must consider the impact of relays on theMTTL.

Fig. 3 illustrates two examples of the above two questions,respectively. In order to answer the first question (derive therelationship between the transmission range and MTTL), weconstruct a new concept, -simplified cluster . Algorithm 1illustrates how to generate given and the dispersiondensity function .

Algorithm 1: Generation of -simplified EMST from EMST.Input: , Output:

1: Specify two point sets , .2: Randomly label each node in with numbers

.3: Choose nodes with the smallest label number in .4: Add the chosen nodes to and discard all the nodeswithin in , where is the position of thechosen node.

5: Back to step 3 until no node is left in , .

Let denote the multicast tree spanning . Thelength of each branch in is larger than , and all theabandoned nodes are within a distance from the nodes in. according to the definition

of EMST. After the thinning process following Lemma 3.3, wespecify two regions according to . Let denote thepoint intensity after Algorithm 1.• Dense Region : Nodes in this region are populous,and we specify a radius

for this circular region because is invariant underrotations. After the thinning process,on the basis of Chernoff bound.

• Sparse Region : Nodes in this regionare relatively sparse such that there are at most a constantnumber of nodes in if . After the thinningprocess, the node density is at least a constant fraction ofthe original density. Therefore, .

Lemma 3.4: There exists a constant such thatcan be lower-bounded as

Proof: The length of EMST is determined by the pointintensity according to Lemma 3.3, thus there exists a constant

, such that

where is also a constant. Note that our result holds even whenor is empty.

Now we come to the second question. The following lemmatells us that is at most times larger thanMTTL when relays are utilized, which is proved in [23].Lemma 3.5: Given nodes , any multicast tree spanning

these nodes (may be using some additional relay nodes) willhave a Euclidean length of at least , where

and is the EMST spanning .Then, we obtain the lower bound of MTTL for as

Based on the above inequality, we obtain a lower bound ofas

(4)

Note that what we are interested in is the order sense ofthe result, therefore we directly regard as the transmissionlength. Now we will investigate the upper bound of minimumcapacity based on the previous analysis of the restrictionsimposed by the network. The results obtained here are somefundamental limits that cannot be violated by any routingprotocols.Theorem 3.1: Under the assumptions of the proposed wire-

less networks, the following tradeoffs must be satisfied by allscheduling policies:

(5)

where and is a constant.


Proof: Using the Cauchy–Schwartz inequality and (3), wehave

(6)

Because is the minimal transmission range in the network,the following inequality is satisfied:

and substitute it into inequality (6), we can obtain that

(7)

On the other hand, we will get a lower bound on the left sideof the above inequality. First, we divide clusters into two sets, as

Based on (4) and the fact that

given in Lemma 3.3, we obtain that

if , there exists a constant , such that

(8)

where

Then, by substituting (7) into (8), we can obtain

(9)

According to the definition of minimum capacity

(10)

Recalling the previous assumption that the number of clustersin with size is the same order with total cluster num-bers , we have . Hence, thereexists a constant such that

(11)

Combining (9) and (10), we obtain

Else if

(12)

Substituting (7) and (11) into (12), we obtain

(13)

During the above derivation, is a necessary con-dition to guarantee network connectivity, where is a con-stant. After comparing the results in the two cases, the only thing


required to do is to prove that there exists a constant suchthat

Recalling Lemma 3.4, it is equivalent to prove

Note that and , and

there exists a constant that can meet the above inequality andsatisfies (5).Theorem 3.2: The achievable minimum capacity in our

network is upper-bounded as follows:

(14)

Proof: Since we assume that the number of clusters withsize is in the same order with total number of clusters ,we have

Combining with Theorem 3.1, the term can be eliminated.

IV. UPPER BOUND OF MINIMUM CAPACITY WITHDISTRIBUTION VARIANCE CONSTRAINED

In Section III, we have completed our analysis of the relation-ship between the upper bound of minimum capacity and thedispersion density function . However, we cannot tell whatthe impact is on the minimum capacity. Therefore, a quantita-tive value is needed to describe the degree of heterogeneityof each dispersion density function . There are two reasonsfor studying their relationship. First, we cannot see whether het-erogeneity increases or decreases minimum capacity basedon (14). is a good indicator of the degree of heterogeneity.Second, we can predict the achievable minimum capacity justby knowing the distribution variance. The exact point processis not needed using this model. This is useful because it is oftendifficult to obtain the dispersion density function.Recalling the definition of , there are various dispersion

density functions satisfied given a fixed . We will showhow to find the point process that achieve the largest value ofminimum capacity.Theorem 4.1: Given the distribution variance , the min-

imum capacity is bounded as follows:

(15)

The respect dispersion function is identical to .According to Theorem 3.1, a smaller results

in a larger capacity. Therefore we will derive the minimumgiven a fixed .

Theorem 4.2: Define a real variable function. Then, we can prove that can minimize

among all with distribution variance .Proof: We will refer to Hölder’s inequality for the proof of

this theorem as follows.Let , and we can obtain that

Therefore, our minimization problem is transformed asfollows:

s.t.

for all

We substitute , , , andinto (2). Then, we can obtain the following inequality

in space :

Note that is a monotonously decreasing function and theinequality will become an equation when

otherwise

which means that can achieve the minimal value of.

Recalling Theorem 3.2, we complete the proof ofTheorem 4.1. In this case, the node distribution conformsexactly to the proposed UCRM. In this model, cluster membersare uniformly and randomly distributed in a disk of radiuscentered at the cluster head. In Section V, we will provide thecapacity achieving scheme to approach this bound and verifythe upper bound of minimum capacity is achievable in ordersense.

V. CAPACITY ACHIEVING SCHEME FOR UNIFORM CLUSTERRANDOM MODEL

We have already derived the upper bound of minimum ca-pacity when two types of heterogeneities are involved. In thissection, wewill provide a capacity achieving scheme for UCRMbased on percolation theory. We prove that the minimum ca-pacity achieved by our scheme matches the asymptotic upperbounds when the number of sessions is large enough.We find that only when the length of the multicast spanning

tree is on the same order with , the individualmulticast capacity can approach the theoretical upper bound inorder sense. Intuitively, when the degree of HCT is high enough,the nodes in each cluster are only overlapped with a constantnumber of clusters, which corresponds to . We


denote such a case as trivial cluster overlapping. On the con-trary, when the degree of HCT is relatively low, .We denote such a case as fully cluster overlapping. The networktopology between these two extreme cases is denoted by partialcluster overlapping. The capacity achieving scheme for uniformcases has been developed in [4], and we discuss the other twocases in this section.

A. When

In this case, . There are at

most a constant number of clusters inside for ,and a simple TDMA scheme can achieve capacity foreach cluster.

B. When

In this case, . All the nodes

are fully overlapped, and we find that the achievable minimumcapacity is identical to the uniform case.

C. When

In this case, , and the traffic in eachcluster is not so aggregated because is relatively smaller.In [3], an information highway is proposed to approach thecapacity upper bound for unicast nonclustered networks basedon percolation theory, and we apply this concept to our routingscheme. Before we outline the definition of an informationhighway, some useful lemmas are provided.Lemma 5.1: Let denote the number of nodes within

a disk of radius . Then, if the cluster radius, the following inequality is satisfied:

Proof: Let denote the overlapping area of twocircles of radius , with centers of distance away, and

denotes number of nodes within radius . Note that thedistribution of cluster clients is HPP within a circle of radius ,thus we can obtain

During the above derivation, we utilize the following notationsto simplify our calculations:

and

The overlapping area of two circles is smaller than the area ofthe small circle, therefore

Then, we substitute it into and obtain

Lemma 5.2: There exists a constant such that if we partitionthe square into cells with equal side length , theprobability that at least one node resides in a cell is larger than

.Proof: Each cell of edge length contains a disk

of radius such that the probability that at least one noderesides in a cell can be lower-bounded as follows:

Now we outline the definition of an information highway:Choose large enough so that , andwe equally partition the square into cells of edge .

Thus, there are cells, and each cell in th rowand th column is denoted by . is open if it contains atleast one node. A horizontal (vertical) cross path is defined as aset of open cells that cross from left to right (top to bottom).The gray cell in Fig. 4 is an example of percolation path, whichrepresents a wireless backbone that is used to multihop dataacross the network[3]. According to [3, Theorem 5], if the prob-ability that a cell contains at least one node is larger than , thereare w.h.p. disjoint paths crossing from left to right (topto bottom). A set containing these horizontal and ver-tical paths are called information highways. Note that the cellscomposed of the highway are called percolated cells, and thenodes are called representing nodes. The following propertiesare proved in [3] via percolation theory.• In each horizontal (vertical) rectangle of size3

, there are at least hor-izontal (vertical) highway paths w.h.p. It indicates thereare disjoint crossing paths from left to right andtop to bottom, respectively. (Theorem 5)

• The length of each crossing path is bounded by .

3 and is some constant, and is to make

an integer.


Fig. 4. Demonstration of routing protocol.

• The distance between two adjacent horizontal (vertical)paths is at most .

• There exists a spatial and temporal scheme that canachieve throughput on the highway. It means thateach cell composing the highway can be considered linkedby optical wires to its neighbors with bandwidth .(Theorem 3)

The information highway is an infinitely large componentsuch that each node in the deployed region can connect to itwithin a hop of length . Now we need to constructa multicast spanning tree for to route its packets.Lemma 5.3: The Prim’s algorithm is utilized to construct afor cluster , and we prove that .Proof: Prim’s Algorithm: Initially, each node is a separate

part, then we iteratively find the shortest edge to compose alager part until one part is left. Each member in is confinedin a disk of radius . We utilize a square of edge to coverthe whole circle. At each th ( ) step, there are

parts remaining. We equally partition the squareinto cells with edge length , and

there exists at least one cell that contains more than two parts,which means the shortest edge connecting two parts in the thstep is at most . Therefore, the upper bound of

is

(16)

Based on the above analysis, a capacity routing scheme isprovided in Algorithm 2.

Algorithm 2: Capacity Achieving Scheme for UCRM

1: Access Point Mapping: Establish mappings andfor each node . Horizontally divide the

square into slices of size . Thenthere are at least paths in each slice. Denoteeach path in the slice as .We further divide each slice into subslices ofsize each. If node is in the th subslice,

denotes the percolated cell on with the sameordinate. Mapping of is the dual of , and ismapped by applying the above algorithm to vertical paths.

2: Medium Access in Highway: Each representing nodecan be active for a constant portion of time in a cellpartitioned network based on in [11, Lemma 6]. Thereforethere exists a spatial and temporal accessing policy suchthat each representing node can deliver bits to itsadjacent representing node as [3].

3: Routing Protocol: A multicast spanning tree isconstructed in Lemma 5.3. Each time-slot is divided intothree mini-slot and corresponds to three phases as inFig. 4. For each branch in linking nodes and ,the three phases are as follows:• • Uplink: drains its data to .• • Highway: This phase corresponds to step 2 andutilizes multihop transmission along the horizontal pathfrom to the intersection with the vertical pathin which resides then forwards to .

• • Downlink: downloads the data from .

According to Algorithm 2, the average number of nodes apercolated cell has to serve is . The next lemma illustratesthe minimum achievable data rate in the uplink and downlinkphases.Lemma 5.4: In the uplink and downlink phase, a data rate of

can be sustained for each transmission.Proof: According to Algorithm 2, for each node ,

both and is upper-bounded by. It means that if we equally divide the region

into subsquares of size ,and must reside in the same cell. The same is

true for and . Then, according to [11, Lemma 6],each cell can be allocated a constant time to be active withinthe mini-slot. According to Lemma 3.2, there are at most

nodes in each cell, which means there are atmost pair. Thus, each pair can beallocated fraction of a time-slotfor transmission.Then, we begin to analyze the second phase. Although the

highway can be virtually considered as a wired network withbandwidth , each path must help relay data for many clus-ters. Therefore, the allocated radio resources for a cluster is lim-ited. In the following part, we will study the maximum numberof clusters a percolated cell can serve.


Fig. 5. Demonstration of influential range and a division of a percolation path.

Lemma 5.5: Each percolated cell cannot relay data for clusterwith head at a distance of away from the

cell, therefore .Proof: We refer to Fig. 5 for the proof. Each path is

constrained within a strip of width . Thus,

is upper-bounded by . Recalling thatthe radius of the disk is , we know the farthest cellthat can be used is the black cell, which is at a distance

away from the kernel.Each branch in the spanning tree is regarded as an SD pair

linking two nodes. The length of it determines how many hopsshould be used on the highway.Lemma 5.6: Assume that the length of an SD pair is , then

the number of hops required is ,where is a constant.

Proof: Denote the abscissa and ordinate of a pointas and . Assume and are two representing points intwo percolated cells on the same horizontal path. Letdenote the number of hops required for transmission from To. We can prove that there exists a constant , such that

(17)

where . Now we divide the path intosubpaths as in the second figure in Fig. 5, and

denotes the length of the thsubpath. Then, the proof of (17) is as follows:

which is also identical to prove

For each path on the highway system, the length of the path is onthe order of according to its property. Thus, there exists aconstant , such that the length of every path is upper-bounded

by , which means . Taking the expected valueon both sides, we can obtain

Then, we can obtain

Taking and note that each hopcan forward the packet of distance on the highway,we complete the proof of (17). The vertical path is the dual ofthe horizontal case. Now assume that a sender at generatestraffic to a receiver at a distance away. Let be the in-tersection of the horizontal and vertical path in whichand reside. According to the properties of percolationpaths, every percolation path is constrained within a strip ofwidth such that

Then, the number of hops required for transmission with nodesaway is upper-bounded by

Lemma 5.7: Assume that cell is within a distance from’s head. Then, if , the probability that is

utilized to transmit for cluster is upper-bounded as

(18)

Proof: There are at leastpercolated cells within the disk of radius . Follow the Primalgorithm used in Lemma 5.3 to construct a spanning tree. Let

be the indicator whether cell is used in the th step and


be the probability that is used in the whole process. Then,based on Lemmas 5.5 and 5.6

The above lemma provides an upper bound on the probabilitythat a multicast flow from a cluster is routed via the cell whenit is within the influential range of that cluster. Let us now studythe total number of times that the cell is used by our capacityachieving scheme. We know that if .

Let denote the prob-ability that a cell is used for if and is withinthe influential range of ’s head. For each cluster ,clients are randomly and independently distributed in a disk ofradius . Each pair generated in the Prim’s Algorithm isalso a random process. Then, we will apply Vapnik–Chervo-nenkis theorem to prove our results, and the VC-dimension of amulticast spanning tree is according to [4].Theorem 5.1: (VC-Theorem): If is a set of finite VC-di-

mension VC- , and is a sequence ofi.i.d. random variables with common probability distribution ,then for every

when

Theorem 5.2: For each , let andfor , then the following inequality should

be satisfied based on VC-Theorem:

where is the number of flows using by clusters be-longing to and is a set of sequences approaching 0when n goes to infinity. Equation (21) should be satisfied when

, and (22) should be satisfied when .Proof: The method to construct multicast trees are i.i.d.

variables within a disk of radius . Then, the multicast treescan be viewed as i.i.d variables. Therefore, we can use theVC-Theorem. Recalling that, given a cell , the probability that

a multicast tree will cross it is at most when thecell is within its influential range . Hence, the whole set ofthe cell

when

where is the VC-dimension. Then, substitute (18)into it and note that , and we obtain

(19)

Now let and , and for inequality (19)to be true

(20)

is required. In this case, denotes the number of clusters be-longing to within the disk of radius . Ap-plying the same technique as Lemma 3.2, we can obtain that

if . Therefore, when

, if

(21)

(20) is satisfied. Substitute into (19), and we com-plete the proof when . A similar technique is em-ployed for if the following condition is satisfied:

(22)

The above constraint of means that only when is suffi-ciently large, the number of flows over a certain cell can beupper-bounded. In addition, we find that if the number of flowsover in is upper-bounded by , is also an upper boundof the number of flows in because is anincreasing sequence over .Lemma 5.8: Given a cluster radius , the data

rate that the highway can sustain for is denoted by . When

when

(23)

Proof: In case of , there are at most

clusters within a disk of radius , thereforeonly a small portion of super clusters may have members


inside the disk. The number of flows through each cell isupper-bounded by w.h.p according to Theorem 5.2.

Therefore, in our FDM network, and thehighway can sustain a data rate of .

When , the number of clusters belonging to

is and , which means eachis allocated bandwidth and

is allocated bandwidth for transmission in the highway.We specify a as follows:

(24)

Then, we know the number of flows across each percolationcell can be upper-bounded by for clustersbelonging to an arbitrary super cluster .Therefore

For , the number of flows crossinga cell is also upper-bounded by , which isthe same as that of clusters belonging to and in this case

.Recalling Lemma 5.4, we know the bottleneck is due to data

delivery on the highway when . Let , de-note the data rate of the uplink and downlink, respectively, and

. The achievable upper bound ofminimum capacity for the whole network is therefore

Therefore, our scheme approaches the same maximum ca-pacity as (15) in order sense if we substitute

into and .

VI. DISCUSSION

Motivated by how max-min fairness framework is applicablefor multicast traffic[20] and the minimum multicast capacitycan be efficiently improved, we hope to find out the upperlimit for the minimum capacity to attain. Investigating theminimum capacity can provide insights into the worst situationof the whole network, and solving the problem for minimumcapacity can significantly benefit the whole network perfor-mance. Different from most previous literature, which namelyemphasized on maximizing the total throughput (or per-clusterthroughput), our paper chooses another aspect in evaluatingthe network performance. By combining the max-min metricand heterogeneity property, we provide a novel perspectiveconcerning the problem of network capacity.Our results can generalize various results obtained in a non-

heterogeneous network like [3], [4], and [6]–[8]. Here, we high-light the feature in our clustered network, heterogeneous clustertraffic, which is characterized in real networks with multicast

Fig. 6. Relationship between minimum capacity and ( scale). In re-gion 2, the minimum capacity increases with . means a nonhetero-geneous network with the same achievable capacity as prior works.

traffic patterns. Moreover, our results for networks with HCTheterogeneous traffic can be effectively applied to large-scalenetworks such as wireless sensor networks [26].HCT characterizes a network property that clients of each

multicast session are not uniformly distributed across the net-work. Previous works are insufficient to estimate the achievablecapacity. Indeed, (5) is a precise formula for the achievable ca-pacity. However, we cannot decide whether HCT increases theminimum capacity because there is no criteria to judge the levelof heterogeneity for a dispersion density function. By intro-ducing the variable distribution variance , we offer a quan-titative description of the extent of heterogeneous traffic. Wefind that if the traffic of a cluster is not uniformly disseminatedacross the region as is assumed in prior works, the minimumcapacity increases.Fig. 6 illustrates the relationship between and derived

from (15). In region 1, HCT is relatively slight, and each clusteris full overlap with other clusters. Fully overlapping indicatesthat in each small region with area , there are

clusters whose members have approximatelyequal probability to reside in . Therefore, it is much like theuniform distributed cases, and the minimum capacity is not im-proved. In region 2, HCT begins to influence the network per-formance by increasing the maximized capacity. Here, clustersare partially overlapped; it indicates that in each small region

with area , there are nearly clusterswhose members have approximately equal probability to residein . In this case, , and is closely related to . Theimplicit reason for an increased minimum capacity is that eachrelay only needs to deliver packets for a smaller portion of theclusters compared to the previous one. In region 3, trivial over-lappingmeans in each small region with area ,there are at most clusters whose members have approxi-mately equal probability to reside in . Therefore, each relayonly needs to deliver packets for a constant number of clusters,and the achievable capacity tends toward .

VII. CONCLUSION AND FUTURE WORK

In this paper, we study the effect of heterogeneity on theasymptotic multicast capacity in clustered networks based onmax-min fairness. Our contributions aremainly divided into twoparts. First, we find that heterogeneous cluster traffic increasesthe minimum capacity among all the clusters. Through ana-lyzing the fundamental constraints of wireless networks fromglobal and local aspects, a quantitative relationship is provided


between minimum capacity and distribution variance .This is the first time this relationship is used to describe hetero-geneity in literature. The effect of heterogeneity is also analyzedin uniform cluster random model, which is the optimal networklayout given a fixed .There are several tasks for us to complete in the future.

First, the cluster dense model should be studied. The clusterdense model indicates that the distance between cluster headsapproaches 0 when . Second, the dispersion densityfunction for each cluster is assumed identical. It wouldbe interesting to study networks with clusters of different .Third, we mainly discusses the heterogeneous cluster traffic,while other aspects of heterogeneity such as heterogeneouscluster size (HCS) is not sufficiently investigated. If the clustersare formed with more diversity in size, it is meaningful to seewhether HCS can increase the capacity or not.

REFERENCES

[1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEETrans. Inf. Theory, vol. 46, no. 2, pp. 388–404, Mar. 2000.

[2] L. L. Xie and P. R. Kumar, “A network information theory for wirelesscommunication: Scaling laws and optimal operation,” IEEE Trans. Inf.Theory, vol. 50, no. 5, pp. 748–767, May 2004.

[3] M. Franceschetti, O. Dousse, D. N C. Tse, and P. Thiran, “Closingthe gap in the capacity of random wireless networks via percolationtheory,” IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1009–1018, Mar.2007.

[4] X.-Y. Li, S.-J. Tang, and O. Frieder, “Multicast capacity for largescale wireless ad hoc networks,” in Proc. ACM Mobicom, 2007, pp.266–277.

[5] A. Keshavarz-Haddad, V. Ribeiro, and R. Riedi, “Broadcast capacityin multihop wireless networks,” in Proc. ACMMobicom, Los Angeles,CA, USA, 2006, pp. 239–250.

[6] S. Shakkottai, X. Liu, and R. Srikant, “The multicast capacity of largemultihop ad hoc networks,” in Proc. ACM Mobihoc, Montreal, QC,Canada, 2007, pp. 247–255.

[7] S. Li, Y. Liu, and X.-Y Li, “Capacity of large scale wireless networksunder Gaussian channel model,” in Proc. ACM MobiCom, San Fran-cisco, CA, USA, Sep. 2008, pp. 140–151.

[8] C.Wang, X.-Y Li, C. Jiang, S. Tang, Y. Liu, and J. Zhao, “Scaling lawson multicast capacity of large scale wireless networks,” in Proc. IEEEINFOCOM, Rio de Janeiro, Brazil, Apr. 2009, pp. 1863–1871.

[9] C. Hu, X. Wang, and F. Wu, “Motioncast: On the capacity and delaytradeoffs,” in Proc. ACM MobiHoc, New Orleans, LA, USA, May2009, pp. 289–298.

[10] S. Zhou and L. Ying, “On delay constrained multicast capacity of large-scale mobile ad-hoc networks,” in Proc. IEEE INFOCOMMini-Conf.,San Diego, CA, USA, Mar. 2010, pp. 1–5.

[11] X. Lin and N.B. Shroff, “The fundamental capacity-delay tradeoff inlarge mobile wireless networks,” in Proc. IEEE MedHoc, Bodrum,Turkey, Jun. 27–30, 2004.

[12] L. Ying, S. Yang, and R. Srikant, “Optimal delay-throughput tradeoffsin mobile Ad Hoc networks,” IEEE Trans. Inf. Theory, vol. 54, no. 9,pp. 4119–4143, Sep. 2008.

[13] U. Niesen, P. Gupta, and D. Shah, “The multicast capacity region oflarge wireless networks,” in Proc. IEEE INFOCOM, Rio de Janeiro,Brazil, Apr. 2009, pp. 1881–1889.

[14] C. Hu, X. Wang, D. Nie, and J. Zhao, “Multicast scaling laws withhierarchical cooperation,” in Proc. IEEE INFOCOM, San Deigo, CA,USA, 2010, pp. 1–9.

[15] G. Alfano, M. Garetto, and E. Leonardi, “Capacity scaling of wirelessnetworks with inhomogeneous node density: Upper bounds,” IEEE J.Sel. Areas Commun., vol. 27, no. 7, pp. 1147–1157, Sep. 2009.

[16] G. Alfano, M. Garetto, and E. Leonardi, “Capacity scaling of wirelessnetworks with inhomogeneous node density: Lower bounds,” in Proc.IEEE INFOCOM, Rio de Janeiro, Brazil, Apr. 2009, pp. 1890–1898.

[17] Z. Wang, H. R. Sadjadpour, and J. J. Garcia-Luna-Aceves, “A unifyingperspective on the capacity of wireless ad hoc networks,” inProc. IEEEINFOCOM, Phoenix, AZ, USA, Apr. 2008, pp. 211–215.

[18] M. Ji, Z. Wang, H. Sadjadpour, and J. J. Garcia-Luna-Aceves, “Thecapacity of ad hoc networks with heterogeneous traffic using coopera-tion,” in Proc. IEEE INFOCOM, San Diego, CA, USA, Mar. 2010, pp.1–9.

[19] P. Li and Y. Fang, “The capacity of heterogeneous wireless networks,”in Proc. IEEE INFOCOM, San Diego, CA, USA, Mar. 2010, pp. 1–9.

[20] H. Y. Tzeng and K. Siu, “On max-min fair congestion control for mul-ticast ABR service in ATM,” IEEE J. Sel. Areas Commun., vol. 15, no.3, pp. 545–556, Apr. 1997.

[21] D. Rubenstein, J. Kurose, and D. Towsley, “The impact of multicastlayering on network fairness,” IEEE/ACM Trans. Netw., vol. 10, no. 2,pp. 169–182, Apr. 2002.

[22] D. Bertsekas and R. Gallager, Data Networks. Englewood Cliffs,New Jersey: Prentice-Hall, 1992.

[23] D.-Z. Du and F.-K. Hwang, “A proof of the Gilbert-Pollak conjectureon the Steiner ratio,” Algorithmica, vol. 7, no. 2,3, pp. 121–135, 1992.

[24] J. M. Steele, “Growth rates of Euclidean minimal spanning trees withpower weighted edges,” Ann. Probab., vol. 16, no. 4, pp. 1767–1787,Oct. 1988.

[25] Q. Peng, X. Wang, and H. Tang, “Heterogeneity increases multicastcapacity in clustered network,” in Proc. IEEE INFOCOM, Shanghai,China, 2011, pp. 703–711.

[26] Z. Li, Y. Liu, M. Li , J. Wang, and Z. Cao, “Exploiting ubiquitous datacollection for mobile users in wireless sensor networks,” IEEE Trans.Parallel Distrib. Syst., vol. 24, no. 2, pp. 312–326, Feb. 2013.

Yixuan Li is currently pursuing the B.E. degreein electronic engineering at Shanghai Jiao TongUniversity, Shanghai, China, working with Prof.Xinbing Wang at the Institute of Wireless Commu-nication Technology.Her research interests are in the area of scaling

laws analysis and performance evaluation in wirelessnetworks.

Qiuyu Peng received the B.E. degree in electronic engineering from ShanghaiJiao Tong University, Shanghai, China.His research interests are in the area of asymptotic analysis of capacity and

connectivity in wireless ad hoc networks.

Xinbing Wang (M’06) received the B.S. degreein automation (with honors) from Shanghai JiaoTong University, Shanghai, China, in 1998, theM.S. degree in computer science and technologyfrom Tsinghua University, Beijing, China, in 2001,and the Ph.D. degree with a major in electrical andcomputer engineering and minor in mathematicsfrom North Carolina State University, Raleigh, NC,USA, in 2006.Currently, he is a faculty member with the Depart-

ment of Electronic Engineering, Shanghai Jiao TongUniversity. His research interests include the scaling law of wireless networksand cognitive radio.Dr.Wang has been anAssociate Editor for the IEEE/ACMTRANSACTIONS ON

NETWORKING and IEEE TRANSACTIONS ONMOBILE COMPUTING and a memberof the Technical Program Committees of several conferences including ACMMobiCom 2012, ACM MobiHoc 2012, and IEEE INFOCOM 2009–2013.

Multicast capacity with max min fairness for heterogeneous networks

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