End-to-End Rate Selection for Opportunistic Reception inMulti-Rate Wireless Networks

Raju KumarDept of Computer Science and Engineering

The Pennsylvania State UniversityUniversity Park, PA 16802

rajukuma@cse.psu.edu

Sharanya EswaranApplied Research

Telcordia Technologies,Piscataway, NJ 08854

seswaran@research.telcordia.com

Thomas La PortaDept of Computer Science and Engineering

The Pennsylvania State UniversityUniversity Park, PA 16802

tlp@cse.psu.edu

Abstract—In this paper we propose an end-to-end algorithm,

called NUM-RS, for jointly selecting link transmission rates and

source rates in a multi-hop multi-rate wireless network. Prior

works on rate selection, including those that explicitly account

for opportunistic reception, perform rate selection on a hop-

by-hop basis, attempting to maximize the throughput on each

link. Our algorithm leverages the Network Utility Maximization

(NUM) framework, thus providing end-to-end semantics for

rate selection and proportional fairness with low overhead. By

using end-to-end semantics NUM-RS considers both source rates

and congestion in the vicinity of links used by a flow when

selecting link rates. Our results show that NUM-RS increasingly

outperforms contemporary hop-by-hop rate selection schemes as

the number of hops in the flows increase. For example, 20%

and 50% of 8-hop flows exhibit performance gains of at least36% and 15%, respectively, in terms of end-to-end throughput.

In some cases, gains of up to 80% can be achieved.

I. INTRODUCTION

IEEE 802.11 [1] is a popular wireless standard used formulti-hop wireless networks. As wireless links exhibit differ-ent and varying channel characteristics, IEEE 802.11(a/b/g)provides multiple transmission rates - ranging from 1Mbpsto 54Mbps from which to choose. A higher transmission rateallows packets to be injected into the medium at a higher rate,but it may also lead to a higher bit error rate depending onthe channel conditions.

Though IEEE 802.11 does not specify an algorithm to selecta transmission rate for a link, or to adapt it as the channelconditions vary, this problem has received considerable atten-tion [7], [13], [22]. The focus of these rate adaptation algo-rithms is to change transmission rate as channel characteristicsvary. For any given channel condition, the component of rateadaptation that selects the transmission rate is referred to asrate selection. To the best of our knowledge, all rate selectionalgorithms operate in a hop-by-hop manner. This allows rateselection to be nimble so that rate adaptation can keep pacewith varying link conditions. For relatively stable wirelessnetworks, we propose a low overhead end-to-end rate selectionscheme that brings significant gains compared to hop-by-hopapproaches.

Traditional routing forwards packets in a hop-by-hop man-ner on a selected route. By virtue of the wireless medium beingbroadcast in nature, some transmissions may be serendip-itously received multiple hops downstream with a non-negligible probability. This phenomenon, referred to as oppor-tunistic reception, occurs frequently in a multi-rate wirelessnetwork as illustrated in Afanasyev et al. [3]. Opportunistic

reception has recently been leveraged by Biswas et al. [6] andChachulski et al. [8], among others.

In this paper, we propose a low overhead end-to-end rateselection algorithm called NUM-RS that accounts for oppor-tunistic reception in a multi-rate wireless network. NUM-RSuses end-to-end semantics to determine both source rates andlink transmission rates. We leverage the Network Utility Max-imization (NUM) framework, proposed by Kelly et al. [14]for wired networks. The incorporation of NUM enables adistributed scheme to determine the rates at which traffic fordifferent flows should be injected onto links while maintainingproportional fairness [14]. It also considers trade-offs of linkoccupancy time with opportunistic reception when settinglink rates. NUM-RS inherits salient attributes of NUM -feasibility, fairness, and the ability to implement a distributed,low overhead protocol. Our contributions include -

• We extend the NUM framework as applied to wirelessnetworks to explicitly take the characteristics of a multi-rate wireless network and opportunistic reception intoaccount and to enable end-to-end rate selection.

• We design a protocol to implement NUM-RS in a wire-less network and analyze its overhead. We conclude thatNUM-RS imposes a marginal increment in overhead ascompared to other NUM based schemes.

Our results show that in networks where source ratesare determined using the NUM framework, existing hop-by-hop approaches for rate selection are sub-optimal. NUM-RSincreasingly outperforms these rate selection schemes as scopefor opportunistic reception increases. For example, NUM-RSbrings over 36% and 15% gains in terms of data deliveredto destinations in 20% and 50% of the flows, respectively,when compared to the best hop-by-hop rate selection approach.Up to 80% gains are also observed in specific instances. Inaddition, NUM-RS outperforms Modrate [3], a rate selectionalgorithm specifically designed for opportunistic reception.

The rest of this work is organized as follows: §II presentsa background on opportunistic reception and §III presentsa background on NUM and its extension to wireless net-works. §IV presents details of NUM-RS. A detailed qualitativeanalysis of the overhead increment imposed by NUM-RS ispresented in §V. Results are presented in §VI. Related workis discussed in §VII and §VIII concludes the paper.

II. OPPORTUNISTIC RECEPTIONWe first provide a background on opportunistic reception

(cf. §II-A) and its primary example - ExOR [6] (cf. §II-B). We

2011 8th Annual IEEE Communications Society Conference on Sensor, Mesh and Ad Hoc Communications and Networks

978-1-4577-0093-4/11/$26.00 ©2011 IEEE 601

then present the details of the general opportunistic receptionframework that we employ in this work (cf. §II-C).

A. OverviewDelivering data in wireless networks using traditional rout-

ing requires a route discovery mechanism based on a metriclike ETX [10], ETT [5], etc. Hence the next-hop node forall packets is determined during the route discovery process.This selection of the next-hop node in a route attempts tostrike a balance between the geographical distance covered bya transmission and the probability of its reception at the next-hop node. As a result, nodes other than the next-hop node mayreceive a transmission with non-negligible probability.

Consider the example shown in Fig. 1 where the route fromS to D uses nodes R1 and R2 while R3 is an off-route node.The links are annotated with their respective packet error rates(PER). Conventional routing fails to leverage the broadcastnature of the wireless medium in the following two ways. First,routing does not leverage off-route nodes that may correctlyreceive a packet that is not received by any on-route node -e.g., R1, R2, and D may not receive the transmission of S

but R3 may receive it - this occurs with probability 0.06 inthe example in Fig. 1. Routing requires any off-route node(e.g., R3) to discard these packets. So in spite of the fact thatthe packet may have made some progress toward the eventualdestination, it is discarded altogether and retransmitted.

Second, routing does not leverage the possibility that apacket may be received multiple hops downstream - e.g., R2

may receive the transmission of S. Traditional routing mayor may not allow the reception of this packet multiple hopsdownstream. Even if this packet is received and forwardedby this opportunistic receiver, other upstream nodes (e.g., R1)may potentially receive and forward the same packet as well.Subsequently, preventing the forwarding of multiple copies ofan opportunistically received packet poses a problem.

B. ExOR [6]To resolve these issues, ExOR defers the determination

of the next-hop node for a packet until after it has beentransmitted. For a batch of packets, the source node adds allnodes in the path in a forwarder list ordered based on distancefrom the destination in terms of the ETX [10] metric. After thesource node has transmitted all packets in a batch, the highestpriority node in the forwarder list forwards all the packets thatit has received. A batch map is included in each transmittedpacket. For each packet, this batch map stores the highestpriority node that may have received the packet. Based on thebatch maps received, a node updates its local information andforwards only those packets which have not been forwarded byhigher priority forwarder nodes. Essentially, the mechanismsof ExOR are designed to eliminate unnecessary duplicatetransmissions and collisions while ensuring that each packetis forwarded by the highest priority node in the forwarder listto have received the packet.

C. Generalized Opportunistic Reception FrameworkThe emphasis of this work is to propose a rate selection

algorithm for opportunistic reception in general. Hence we

S R1

R2 D0.2

0.80.95

R3

0.6

0 0

Fig. 1. Opportunistic reception example - Links are annotated with PERemulate a generalized approach to opportunistic reception.This allows our rate selection approach to be applicableto different opportunistic reception algorithms [6], [8], [20].Among all nodes that receive a transmission, the node that isnearest to the destination in terms of ETT [5] (a more suitablemetric for multi-rate wireless networks than ETX [10]) isselected to forward the packet next.

Afanasyev et al. [3] limits the nodes that can be a part ofthe forwarder list to those that belong to the route selected bythe routing scheme. This restriction is referred to as “on-pathoverhearing”. We adopt the same restriction in this work.

Opportunistic reception also implies that if any forwardernode receives the transmission, link layer retransmissions arenot required. Like other opportunistic reception schemes [8],we disable link layer retransmissions for data packets in thiswork. To prevent the issue of the packet being lost at allforwarder list nodes while the transmitter is left unaware, IEEE802.11’s ACK mechanism is retained.

In wireless networks the balance between spatial reuse ofmedium and opportunistic reception is an issue. To preventhidden terminal problems from eroding opportunistic recep-tion, we use RTS/CTS in this work - like Choudhury et al. [9].A downstream node that is farthest from the destination interms of ETT may be chosen as the target node to transmitthe CTS so as to minimize the impact on spatial reuse andstill avoid most hidden terminal problems.

III. NETWORK UTILITY MAXIMIZATION (NUM)The NUM approach proposed by Kelly et al. [14] was

designed for a wired network. We use and extend the no-tation used therein in this work. We summarize Kelly et al.’sapproach in §III-A. We then summarize extension of NUM towireless networks by Xue et al. [23] in §III-B.A. NUM for Wired Networks

Let J be the set of links in the network with Cj being thecapacity of link j ∈ J . Each flow r is associated with a routewhich is a subset of J such that the links in r are connected.Let the set of flows be denoted by R. Let A be a matrix suchthat Ajr is 0 if flow r does not use link j and 1 otherwise.Let the amount of resource allocated to flow r be xr and letx be the vector of these allocations. Note that flow r usesxr amount of resources on each of its links and the utilityof these resources to the system is Ur(xr). The problem ofmaximizing network-wide utility is formulated as -—————————————————————————SY STEM(U,A,C):

Maximize�

r∈R

Ur(xr)

subject to Ax ≤ C.

over x ≥ 0.

(1)

—————————————————————————

602

where Ax ≤ C is the set of constraints that the sum ofthe resources of a link allocated to flows that traverse itdoes not exceed its capacity. When Ur() is differentiable andstrictly concave, and the feasible region determined by theconstraints is compact, SY STEM() is solvable using convexoptimization.

Kelly et al. [14] show that if Ur() is a logarithmic functioni.e., Ur(xr) = log(xr), the solution of SY STEM() is theset of proportionally fair allocations. Moreover, the authorsshow that SY STEM() can be solved in a distributed iterativemanner by solving the following problems -—————————————————————————USERr(Ur;λr):

Maximize Ur(wr/λr)− wr

over wr ≥ 0.(2)

NETWORK(A,C;w):Maximize

�

r∈R

wrlog(xr)

subject to Ax ≤ C.

over x ≥ 0.

(3)

—————————————————————————The USERr() subproblem is solved for each flow, with

the objective of maximizing its net utility Ur(wr/λr) − wr

where wr = λrxr is the price flow r is required to pay; andλr = dUr/dxr is the rate of price imposed by the networkfor the flow. The network solves the NETWORK() problemwith the objective of maximizing the revenue received fromthe flows, defined as

�r∈R wrlog(xr).

The distributed algorithm is derived by making use of theLagrangian multiplier (shadow cost) and driving the gradientto zero. Accordingly, the iterative update of xr at the sourceof flow r is based on the following -

d

dtxr(t) = κ[wr − xr(t)

�

j∈r

pj

� �

s:j∈s

xs(t)�] (4)

where κ is a constant step-size and pj is a cost per unit flowfunction on link j with the amount of load

�s:j∈s xs(t) on

link j as its input. The price function pj() is defined as -pj(y) = (y − (Cj − ε))+/ε2. (5)

where ε → 0 is the part of the capacity within which whenthe link usage is incurred, the price of the link increases.

Note that this distributed iterative process to update xr basedon Eqn. 4 requires collecting pj

��s:j∈s xs(t)

�for all links j

in flow r. This renders NUM an end-to-end approach.

B. NUM for Wireless NetworksIn wireless networks the capacity of a link depends on the

presence of interfering links in its vicinity. Xue et al. [23]propose to use maximal cliques in conflict graphs of linksto characterize wireless capacity constraints. While the NUMframework for wired networks assigns costs to links, Xueet al. assign costs to cliques. In addition, maximal cliquescan be used for the capacity constraint component of theSY STEM() problem. We use the notation proposed inEswaran et al. [11] - an extension of Xue et al. [23].

Like Xue et al. [23], we use the protocol model to character-ize the IEEE 802.11 wireless network in this work. Links that

are sufficiently far apart in the wireless network may transmitconcurrently. However, links within interference range are notallowed to do so. Consider a conflict graph which has nodescorresponding to all (r, j) combinations of flow r and linkj ∈ r. Edges in this graph connect nodes that may not transmitconcurrently. Let the set of maximal cliques in this graph beL. Hence, for each maximal clique l ∈ L, only one of the linksmay transmit at any time. The fraction of time the medium isbusy due to transmissions of nodes of maximal clique l is -

bl =�

(r,j)∈l

xr

cr,j(6)

where cr,j is the capacity of link j when operated in isolation.So SY STEM() can be formulated as -—————————————————————————SY STEM(U,A,C):

Maximize�

r∈R

Ur(xr)

subject to bl ≤ 1, ∀l ∈ L.

over x ≥ 0.

(7)

—————————————————————————Since cr,j is a constant, all the constraints are convex and a

solution exists and can be found via the Lagrangian method.Let a flow r ∈ l if ∃ link j ∈ r and j ∈ l. The update of thesource rates follows the following rule [11] -

d

dtxr(t) = k[wr − xr(t)

�

l∈L,r∈l

�

(r,j)∈l

pj

�bl

�

cr,j]. (8)

wherepj(y) = (y − (1− ε))+/ε2. (9)

IV. NUM-BASED RATE SELECTION (NUM-RS)We now present details of our NUM-RS algorithm in §IV-A.

§IV-B presents the baseline schemes to which we compareNUM-RS.A. NUM-RS

Multi-rate wireless networks impact the NUM formulationin Eqns. 6-7 via cr,j - i.e., link capacities in isolation. Whilefor a constant-rate wireless network all cr,j are assigned thesame value, multi-rate wireless networks use rate selection todetermine the rate at which to transmit.

Since link layer retransmissions are not used in our gener-alized opportunistic reception framework, the rate at whicha flow’s data is injected into the network may not be thesame as the rate at which it is received at the destination.We refer to the rate at which the source injects packets in themedium as source rate and the rate at which data is receivedat the destination, i.e., end-to-end throughput, as received rate.Given the set of transmission rates employed on the links offlow r, source rate and received rate are related by a factor βr

explained below.Consider the example shown in Fig. 2 where a flow sends

data from source S to destination D through multiple links.We define a link by its source node and immediate downstreamnode i.e., the intended next-hop target for conventional routing.For example, link j connects intermediate nodes Ra and Rb.For simplicity, let j− 1 denote the link immediately upstreamof link j and j + 1 denote the link immediately downstream.

603

... ...

er,j,k(zj)

er,j(zj) er,k(zk)

αr,1 . pr,1,(j-1)(zr,1)

αr,2 . pr,2,(j-1)(zr,2)

αr,1=1 αr,jαr,1 . (1-er,1,u(zr,1))

αr,2 . (1-er,2,u(zr,2))

S Ra Rb DRs Rt ...Link j Link k

Received : xrSent : xr/βr Forwarded : αr,(j+1) . xr,j/βr

βr

Fig. 2. Opportunistic reception in flow r: Destination D receives packetsat rate xr , Source S transmits packets at rate xr/βr , Intermediate node Rbforwards packets at rate xr · αr,(j+1)/βr

Let the received rate for flow r at destination D be xr

(cf. Fig. 2). Out of the multiple discrete transmission ratesprovided by IEEE 802.11, let transmission rate index z

(starting from 0) denote the z-th largest transmission rateavailable. Hence, let the index for transmission rate selectedon link j of flow r be zr,j ∈ [0, Z − 1] where Z is thenumber of transmission rates available. Let throughput on linkj for transmission rate index zr,j be cr,j(zr,j) and the packeterror rate be er,j(zr,j). While zr,j , er,j and cr,j are discretevariables, they can be adapted into continuous variables usinglinear interpolation.

Let j < k denote that for flow r, k is a downstream linkof j - e.g., link k in Fig. 2 is a downstream link of link j.Let er,j,k(z) denote the PER of the transmission of source oflink j at the destination of link k at transmission rate index z

where both links belong to the same flow r - as illustrated inFig. 2. Hence (1−er,j,k(z)) is the probability of opportunisticreception of Ra’s transmissions at Rt.

Nodes in opportunistic reception forward only those packetsthat have not been received by any downstream nodes. Theprobability of destination of a downstream link k ≥ j receivinga packet that is not received by any further downstream linki > k is -

pr,j,k(zr,j) = (1− er,j,k(zr,j)) ·�

i>k

er,j,i(zr,j) (10)

where it is assumed that the reception of a transmission atdifferent nodes is governed by independent error rates - asobserved in [16], [18]. While (1 − er,j,k(zr,j)) denotes theprobability of destination of link k receiving a packet fromsource of link j,

�i>k er,j,i(zr,j) is the probability of no nodes

downstream of link k’s destination receiving this packet.The fraction of the source rate that an intermediate node

has to forward is the sum of the transmissions of its upstreamnodes that have been received by the node but not by any otherdownstream nodes. Let αr,j be the fraction of the source ratethat is forwarded by the source of link j. For the first link i

of flow r, αr,i = 1. Hence αr,j is defined as -

αr,j =�

k<j

αr,k · pr,k,j−1(zr,k) (11)

where (j − 1) refers to the link immediately upstream of linkj and αr,k · pr,k,j−1(zr,k) is the fraction of the data receivedfrom the source of link k that needs to be forwarded by sourceof link j (cf. Fig. 2).

The destination of the flow may receive transmission fromall the links in the flow. This received rate xr for flow r isrelated to the source rate by the fraction βr (cf. Fig. 2) -

βr =�

j∈r

αr,j · (1− er,j,u(zr,j)) (12)

where u is the last link in flow r, i.e., the target of link u is thesame as the destination of flow r. αr,j ·(1−er,j,u(zr,j)) denotesthe fraction of transmissions received by the destination nodedue to transmissions from link j’s source (cf. Fig. 2).

Note that we solve for transmission rates and received ratesin the NUM formulation. Given the transmission rates of allforwarder nodes, the source rate is then determined from thereceived rate.

Now, the fraction of time that a medium is busy for amaximal clique l is -

bl =�

(r,j)∈l

xr · αr,j

cr,j · βr(13)

where xr is the received rate of flow r, xr/βr is its sourcerate, and xr · αr,j/βr is the rate at which the source of linkj ∈ r transmits. Hence xr · αr,j/(cr,j · βr) is the fraction oftime that the medium is occupied due to transmission of linkj’s source.

Thus our system of equations is -—————————————————————————SY STEM(U,A,C):

Maximize�

r∈R

Ur(xr)

subject to bl ≤ 1, ∀l ∈ L.

over x ≥ 0,

zr,j ≥ 0 ∀r, j ∈ r,

zr,j ≤ argmaxy

{er,j(y) �= 1} ∀j ∈ r.

(14)

—————————————————————————where bl ≤ 1 states that the total fraction of the medium thatcan be used due to transmissions of links in maximal cliquel should be less than 1. zr,j ≤ argmaxy{er,j(y) �= 1} statesthat the transmission rate for any link j should not exceedthe maximum rate that leads to a PER less than 1 i.e., is ableto deliver some packets to the next-hop node. Note that if atransmission rate used by source of link j does not deliver anypackets to the destination of link j, it will not deliver packetsto any other downstream nodes either. Hence the constraint onzr,j does not limit opportunistic reception. Finally, to ensureproportional fairness, Ur(xr) = log(xr).

This end-to-end approach adopted by NUM-RS allows it toconsider the utilization of the medium in different parts of thenetwork through the maximal clique medium use constraint.In contrast, hop-by-hop rate selection schemes are obliviousof medium utilization and simply maximize some measure oflink throughput.

The constraint bl ≤ 1 may not be convex as bl combinesmultiple functions of several xr-s and zr,j-s. As a result,the feasible region may not be compact and a unique solu-tion to SY STEM() may not exist. We assume though thatSY STEM() can indeed be solved by convex optimizationusing transformations (like those used in [17], [19], [21]) of theconstraint bl ≤ 1. We leave the exploration of transformations

604

for future work. Vitally, results in §VI show that more than95% of the time the received rate determined by NUM-RSis higher than that when other schemes are employed - anindicator of the wide validity of our approach.

Now, consider the Lagrangian -L(x, z, v;µ) =

�

r∈R

Ur(xr)−�

l∈L

µl · (bl − 1− vl) (15)

where v is a vector of slack variables and µ is a vector ofLagrangian multipliers. Hence at a maxima of L() we have -

d

dtxr(t) = κ1 · xr(

∂Ur(xr)

∂xr−

�

l∈L,r∈l

µl

�

(r,j)∈l

αr,j

cr,j · βr)

(16)d

dtzr,j(t) = κ2 · zr,j(

�

l∈L,r∈l

µl · xr

�

(r,j)∈l

γ)) (17)

where γ is -

γ =αr,j

c2r,j · βr

∂cr,j

∂z+

αr,j

cr,j · β2r

· ∂βr

∂z− 1

cr,j · βr

∂αrj

∂z(18)

and µl is the shadow cost of congestion charged at eachmaximal clique l defined as -

µl(t) = pl(�

(r,j)∈l

xr · αr,j

cr,j · βr) (19)

where pl is from Eqn. 9. κ1 (cf. Eqn. 16) is the step-sizefor received rates and κ2 (cf. Eqn. 17) is the step size fortransmission rates.B. Baselines for Comparison

We compare NUM-RS to three schemes in which trans-mission rates are determined using hop-by-hop rate selectionalgorithms. To keep the comparisons fair, the received ratesin baseline schemes are determined using a multi-rate andopportunistic reception aware NUM - a straightforward adapta-tion of NUM-RS which is here on referred to as NUM’. Theconvergence and optimality properties of the regular NUMframework [14] apply to NUM’.

The baseline schemes are -1) ORU-RS: Rate selection approaches like RBAR [12] that

do not envisage opportunistic reception select a transmissionrate so as to maximize link throughput i.e.,

z∗r,j = argmax

z{cr,j(z) · (1− er,j(z))} (20)

where z∗r,j is the transmission rate selected, cr,j(z) is the rateat which packets are injected into the medium and (1−er,j(z))is the probability of packet reception at the destination of linkj when the transmission rate index is z.

This approach leads to opportunistic reception unaware rateselection (referred to as ORU-RS). Note that opportunisticreception is still enabled in spite of the rate selection schemebeing oblivious of it.

2) Modrate: Modrate [3], an extension of ETT [5], is op-portunistic reception aware and adopts the following approachfor rate selection -

z∗r,j = argmin

z{�

k≥j

1

cr,j(z) · pr,j,k(z)} (21)

where z∗r,j is the transmission rate selected, 1/cr,j(z) is thetime required to transmit a single packet and pr,j,k(z) is theprobability of destination of link k receiving the transmissionof source of link j such that nodes downstream of link k’sdestination do not receive it.

3) ORA-RS: Modrate does not differentiate between pack-ets that are delivered one hop away, or opportunisticallydelivered across several hops. We propose a hop-by-hoprate selection scheme that maximizes the cumulative “dis-tinct” throughput received at all next-hop nodes weighted byprogress in terms of hops i.e.,

z∗r,j = argmax

z{�

k≥j

(k − j + 1) · cr,j(z) · pr,j,k(z)} (22)

where z∗r,j is the transmission rate selected, and (k − j + 1)represents the number of links between the source of link j

and destination of link k. cr,j(z) · pr,j,k(z) is the “distinct”throughput of the source of link j’s transmission received atlink k’s destination i.e., packets not received by any otherdownstream node. We refer to this opportunistic receptionaware rate selection approach as ORA-RS.

V. OVERHEAD COMPARISON

We now present a detailed qualitative comparison ofthe overhead imposed by NUM-RS, NUM’ (i.e., baselineschemes), and regular NUM for wireless networks (like [11],[23]). Some of the overhead incurred by NUM’ and NUM-RSis inherited from NUM (cf. §V-A). Part of the overhead ofNUM-RS can be attributed to its multi-rate and opportunisticreception aware capabilities which, crucially, is common withNUM’ (cf. §V-B). Finally, §V-C shows that the additionaloverhead imposed by NUM-RS compared to NUM’ is insignif-icant.

NUM’ and NUM-RS require per-flow information (e.g.,source rates) as well as per-link information (e.g., transmissionrates). Per-flow information may be stored at the sourceor destination of the flow, while per link information isstored at the source node of a link. Per flow computation(e.g., update of received rate) and per link computation (e.g.,update of transmission rates) is similarly performed by flowsource/destination and source of a link, respectively. To keepthe overhead low, all information that needs to be conveyedto other nodes is piggybacked onto data packets.

Regular NUM for wireless networks, NUM’, and NUM-RS require a “dissemination in maximal cliques” mechanismwhere a link j needs to inform every interfering link (i.e.,links in any maximal clique of which it is a member) of somevariable ρr,j . This is accomplished in two ways. First, nodesoverhear all transmissions in the neighborhood. Second, inaddition to its own information ρr,j , a node randomly selectsa few interfering links’ information ρr,k and piggybacks thoseonto data packets as well. This provides a way to disseminateinformation up to two hops away at a low overhead.

Like NUM, NUM’ and NUM-RS also require a “cumulativedownstream propagation” mechanism. Say every link j in flowr has a variable ρr,j . Cumulative downstream propagationrequires link j to forward the value of

�k≤j ρr,k to its

immediate downstream link. This imposes an overhead of onlya single variable on each link of the flow.

A. Overhead Common with NUM for Wireless NetworksEach of the following overheads is incurred equally by

NUM, NUM’, and NUM-RS.

605

• The maximal clique approach for NUM in wirelessnetworks, explored by Xue et al. [23] and Eswaranet al. [11], includes distributed discovery of maximalcliques. Each node exchanges information with nodes upto two hops away to construct local conflict graphs. Thisoverhead is incurred by all the three schemes.

• In NUM-RS, each link needs to compute the congestioncost (cf. Eqn. 19). This requires a link j ∈ r to dissemi-nate xr ·αr,j/(cr,j ·βr) in all maximal cliques of which itis a member. This overhead is incurred by NUM’ as well.Moreover, it is also analogous to exchanging xr/cr,j inNUM.

• In NUM-RS, once every link is aware ofxr · αr,j/(cr,j · βr) of every link j that it interfereswith, it performs a cumulative downstream propagationof the congestion cost (cf. Eqn. 19) to enable update ofxr (cf. Eqn. 16). Similar actions are required by NUMand NUM’.

B. Overhead Common with NUM’The following overhead is not incurred by NUM for wireless

networks, but is incurred by both NUM’ and NUM-RS periteration due to their opportunistic reception awareness.

• Once xr is updated by the destination node, xr/βr ispropagated upstream by each link for NUM-RS andNUM’. While this information is being propagated up-stream, NUM-RS updates transmission rates of links (cf.Eqn. 17) at no extra communication cost.

C. Overhead Increment over NUM’Each of the following overheads applies once to NUM’

and per-iteration to NUM-RS due to the fact that NUM-RSupdates transmission rates in addition to received rates in eachiteration.

• Link j ∈ r needs to compute αr,j . This requires acumulative downstream propagation up to link j ofαr,k · pr,k,j−1(zr,k) of upstream links k ∈ r. Duringroute discovery, a source node of a link can keeptrack of the link quality (e.g., in terms of SNR) toevery downstream node. This enables link k to computepr,k,j−1(zr,k). Hence in a h hop flow, cumulative down-stream propagations of length 1, 2, 3, ..., h are required.This leads to an average per link communication overheadof (h+ 1)/2 variables per iteration.

• The destination node of flow r needs to be aware of βr

(cf. Eqn. 12). This is calculated by cumulative down-stream propagation of αr,j · (1− er,j,u(zr,j)) where u isthe last hop in the flow.

Effectively, our approach of jointly solving for transmissionrate and received rate in the NUM-RS formulation imposesonly a small overhead increment on NUM’.

VI. RESULTSIn this section we present an evaluation of NUM-RS. We

compare its performance in terms of received rate againstORU-RS, Modrate, and ORA-RS. Note that each of theseschemes is evaluated in conjunction with opportunistic recep-tion. We first discuss the results of a 5-hop scenario to illustrate

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THROUGHPUT [PPS] FOR IEEE 802.11A TRANSMISSION RATES FOR1500BYTE PACKETS

Mbps 6 9 12 18 24 36 48 54PPS 416 586 736 990 1200 1512 1744 1834

the convergence of the distributed iterative baseline and NUM-RS schemes (cf. §VI-A). We then present results for a singleflow comprised of different number of hops in §VI-B. §VI-Cdiscusses results for multiple such competing flows deployedin proximity.

We use NS-3 [2] as the simulation platform. The wirelessMAC protocol used is IEEE 802.11a, which provides 8 trans-mission rates ranging from 6Mbps to 54Mbps. Table I showsthe throughput of these transmission rates at very high SNRsfor 1500 Byte packets in packets-per-second (referred to aspps here on). Fig. 4 shows the packet error rate vs. SNR forIEEE 802.11a. Note that only the data packet is transmitted atthe given rate; control packets may not be transmitted at thesame rate as data packets.

Node locations in all simulations are generated randomlywhile keeping the deployment connected. The source anddestination of each flow are randomly chosen. Like Srcr inRoofnet [5], the route from a source to destination is deter-mined using the ETT [5] metric. As observed in Afanasyevet al. [3], ETT calculation expects a certain transmission rateto be selected for any link. But like Afanasyev et al., we mayselect a different transmission rate for the source of a link dueto the rate selection scheme employed.A. Example

We now present results for a 5-hop flow from node n1 to n6

via n2, n3, n4, and n5 (in order) to illustrate the convergenceproperties of the distributed iterative NUM’ and NUM-RS.

For all relevant links, the packet error rate for differenttransmission rates is shown in Table II. Links like n1 → n3

that do not deliver any data at any rate, i.e., er,j(z) = 1 forall z are omitted. The only transmission rate that results inopportunistic reception is transmission on n2 → n4 at 6Mbpswhich results in packets being dropped with probability 0.06.

For the baseline schemes, NUM’ is used to determineonly the received rates. The transmission rate selected by thebaseline schemes (and NUM-RS) are shown in Table III.

The convergence of the received rate for the flow using thebaseline schemes is shown in Fig. 3(a) (all curves overlap).The convergence properties of the regular NUM frameworkapply directly to the baseline schemes. As a result, the conver-gence of the distributed iterative approach to the centralized

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EXAMPLE: PACKET ERROR RATES (PER) FOR TRANSMISSION RATES

PER ↓ 6 9 12 18 24 36 48 54

n1 → n2 0 0.08 0.51 1 1 1 1 1n2 → n3 0 0 0 0 0.93 1 1 1n2 → n4 0.06 1 1 1 1 1 1 1n3 → n4 0 0 0 0 0.16 1 1 1n4 → n5 0.14 1 1 1 1 1 1 1n5 → n6 0 0 0 0.02 1 1 1 1

solution is nearly perfect. For this example, all the baselineschemes select the same set of transmission rates and hencethey determine the same received rate - 123pps.

NUM-RS determines both received rates and transmissionrates using the NUM framework. The convergence of receivedrates and transmission rates for the 5 links for NUM-RS isshown in Figs. 3(b) and 3(c), respectively. Note that Fig. 3(c)plots transmission rate index (cf. §IV-A) vs. iteration. Trans-mission rate index of 0 corresponds to 6Mbps, 1 correspondsto 9Mbps, and so on. Vitally, while none of the baselineschemes are able to provide opportunistic reception due tothe transmission rates they select (cf. Table III), NUM-RS’choice of transmission rates do enable opportunistic receptionof node n2’s transmissions at node n4.

For NUM-RS, the received rate variable converges to133pps and is verified by solving SY STEM() (cf. Eqn. 14)in Matlab. We refer to the ratio of the total received rates of allflows, i.e.

�r xr, for NUM-RS to that for the baseline scheme

as gain. Hence for this example, the gain of NUM-RS overall the baseline schemes is 1.08×.

In the distributed iterative process the received rates areinitialized to a near zero value for all schemes. To speed upthe convergence of the received rate xr, initially received ratestep size (i.e., κ for baseline schemes or κ1 for NUM-RS) isset to 2.0. When an iteration decrements the received rate xr

for the first time, this step size is reduced to 0.02 to avoidunnecessary oscillations. For NUM-RS, the transmission ratesare initialized to values corresponding to those selected byORA-RS and the transmission rate step size (i.e., κ2) is setto 0.02. Fig. 3 shows that for the baseline schemes as wellas NUM-RS, the distributed iterative process converges to theoptimal value in less than 500 iterations. Note that the numberof iterations required to converge depends on the selection ofstep-sizes (i.e., κ, κ1, κ2). Larger step sizes may lead to fasterconvergence at the risk of oscillation.

The results from this example lead to two conclusions. First,in spite of the fact that the constraint on maximal cliques’

TABLE IIIEXAMPLE: TRANSMISSION RATES [MBPS]

Scheme ↓ Node → n1 n2 n3 n4 n5

ORU-RS, Modrate, ORA-RS 9 18 24 6 18NUM-RS 6 6 6 6 6

medium use in Eqn. 14 may not be provably convex, theNUM-RS framework can still be employed to improve theperformance of opportunistic reception in terms of receivedrate. Second, the delay in convergence for iterative NUM-RSis nearly the same as that for NUM’.

The number of maximal cliques in this 5-hop flow is 2- l1 = {n1 → n2, n2 → n3, n3 → n4, n4 → n5} andl2 = {n2 → n3, n3 → n4, n4 → n5, n5 → n6}. For thesecliques, the fraction of the time that the medium is occupiedi.e., bl (cf. Eqn. 13), sheds light on the internals of these rateselection schemes. For all the baseline schemes, bl1 = 0.99and bl2 = 0.80. In contrast, for NUM-RS, bl1 = 0.99 andbl2 = 0.90. Hence, for maximal clique l2, NUM-RS betterutilizes the medium. This is an outcome of the incorporationof rate selection with the bl constraint (cf. Eqn. 14) - ashortcoming of the baseline schemes.

Since we have shown that the simulated version of theprotocol and the results obtained by Matlab coincide, andbecause the simulations are very time-consuming, for theremainder of the paper we present Matlab results.B. Single Flow

As the opportunity to leverage opportunistic reception isdependent on the number of hops in a flow, we now presentresults for a single flow with the number of hops rangingfrom 2 to 8. One hop flows do not offer any scope foropportunistic reception and flows longer than 8 hops may notoccur frequently in multi-hop wireless networks.

We present the CDF of the gain over each baseline schemein terms of received rate in Fig. 5. Overall, as the number ofhops increases, the fraction of flows that incur a gain increases.This is because longer flows offer more transmission rates totune, increase the impact of opportunistic reception, and arecomprised of more maximal cliques. Moreover, the gain ofNUM-RS over each of the baseline schemes is nearly the same.Compared to Modrate, for 50%, 40%, 30%, 20%, and 10% of8-hop flows, NUM-RS brings at least 15%, 19%, 26%, 36%,and 54% gain, respectively.

For some cases, NUM-RS brings gains of up to 80%. Thisshows that while hop-by-hop rate selection schemes might notlag behind considerably in most cases, in some cases, using theend-to-end rate selection approach yields substantial returns.

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(c) NUM-RS vs. ORA-RSFig. 6. Single Flow - Avg. Weighted Transmission Rate RatioTABLE IV

bl OF MAXIMAL CLIQUES FOR A 8-HOP FLOW

Scheme ↓ Clique → l1 l2 l3 l4ORU-RS, Modrate 0.60 0.99 0.93 0.90ORA-RS 0.57 0.99 0.92 0.77NUM-RS 0.90 0.99 0.93 0.99

For one particular 8-hop flow, the received rate for NUM-RS, ORU-RS, Modrate, and ORA-RS is approximately 146,89, 89, and 85pps, respectively. Hence NUM-RS brings a gainof 1.6× over Modrate for this example. We present bl, thefraction of the medium used by the transmissions of links inthe 4 maximal cliques that comprise this flow, in Table IV.These maximal cliques are ordered in increasing distance fromthe source of the flow. The largest discrepancy between bl-s forNUM-RS and Modrate (and other baseline schemes) occurs formaximal cliques l1 and l4. bl1 for NUM-RS is 50% higher thanthat for the other schemes. This shows that even though thebaseline schemes may have same bl for some of the maximalcliques in the network, the fact that they fall short for othermaximal cliques impacts overall performance negatively.

For each scenario, we plot the CDF of the ratio of averageweighted transmission rate i.e.,

�j∈r αr,j · cr,j(zr,j), used by

NUM-RS to that used by the baseline scheme in Fig. 6.This represents the average transmission rate of all packets.While the gains of NUM-RS over all baseline schemes weresimilar (cf. Fig. 5), it is not so with the weighted transmissionrate ratio (cf. Fig. 6). In fact, among the baseline schemes,ORU-RS selects the highest weighted transmission rates andORA-RS selects the lowest weighted transmission rates - asobserved by comparing Figs. 6(a), 6(b), and 6(c). This is asexpected because ORA-RS may select lower transmission ratesto encourage opportunistic reception while ORU-RS does nottake opportunistic reception into account.

For less than 5% of the scenarios NUM-RS is outperformedby the baseline schemes (cf. Fig. 5). This means that in spite

of the medium use constraint in Eqn. 14 being not provablyconvex, more than 95% of the time the local or global optimalbrings a significant improvement over the baseline schemes.Overall, these results show that the hop-by-hop approach torate selection may not lead to the best performance and theend-to-end approach of NUM-RS performs best.

C. Multiple FlowsWe now present results for multiple flows such that nodes

that comprise the flows form a connected conflict graph interms of interference. The intricate interaction of multipleflows in wireless networks has fundamental implications on anetwork’s overall performance. But for the purpose of NUM-based rate allocation, multiple flows only result in maximalcliques with links from different flows. As such, thoughNUM-RS and the baseline schemes take interference due tomultiple flows into account, they are agnostic of the sourceof interference itself. Hence conclusions drawn from previousresults for a single flow apply directly to multiple flows.

As the number of flows within interference range increases,load in terms of traffic increases. Fig. 7 shows the CDF ofgains of NUM-RS over Modrate as the number of flows in thedeployment range from 1 to 3. Each flow spans 2 to 8 hops.As the number of flows increases, the gains decrease. This isbecause only a subset of the flows may bring a substantial gainwhich is thereafter amortized over a larger denominator. Onceagain, NUM-RS outperforms Modrate as well as other baselineschemes (graphs for ORU-RS and ORA-RS not shown dueto lack of space). Finally, NUM-RS leads to considerablydifferent average weighted transmission rates, whether loweror higher, for a large fraction of the cases (graphs not shown).

VII. RELATED WORKOpportunistic reception has been leveraged in a broad

swathe of works like Laneman et al. [15] under the umbrella

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Fig. 7. Multiple Flows - Received Rate Gain - NUM-RS vs. Modrateof cooperative diversity. These schemes enable all nodes thatreceive a packet to relay it to the next-hop. But this bringsforth the issue of duplicate transmissions which is addressedby ExOR [6]. ExOR’s synchronization requirements lead to therestriction that only one forwarder node can transmit at anyinstant. This robs wireless networks of a vital feature - spatialreuse of the medium. This issue is remedied by MORE [8]which combines network coding [4] and opportunistic recep-tion.

Lee et al. [17] also propose combining NUM with rateselection. The authors recognize the non-convexity of theresulting constraints and transform them into convex versions.This transformation is based on the assumption that PER onthe component links of a flow are small. But links in wirelessnetworks often have high error rates - as evidenced in Fig. 4.Additionally, Lee et al.’s scheme is not designed for oppor-tunistic reception. The constraint involving bl in our problemformulation cannot employ Lee et al.’s transformations as bl

is a combination of functions of several variables. Like Leeet al. though, we will explore convex transformations of theNUM-RS problem formulation in future work.

Zeng et al. [24] analyze the end-to-end throughput ofopportunistic reception for multi-hop wireless networks. Theauthors propose a hop-by-hop rate selection scheme similarto our baseline scheme ORA-RS (§IV-B3). This rate selec-tion approach uses geographic progress towards destinationas weights (instead of hop count as used by ORA-RS). Incontrast, NUM-RS is an end-to-end rate selection scheme.

Laufer et al. [16] demonstrate that using all possible nodesas forwarders for opportunistic reception may be detrimental.The authors propose a joint solution to determine forwardernode candidates as well as the transmission rate i.e., routing iscombined with rate selection. This approach is based on sub-stituting Dijkstra’s algorithm in link state routing algorithms.Hence effectively, Laufer et al. propose a global rate selectionscheme. In contrast, NUM-RS is an end-to-end scheme.

VIII. CONCLUSIONSWidely used rate selection schemes for multi-rate wireless

networks use a hop-by-hop approach. In this work, we proposeNUM-RS - a low-overhead, end-to-end rate selection schemefor a multi-hop wireless network that leverages opportunisticreception and utilizes the NUM framework. Our results showthat solving for transmission rates in addition to received rateswithin the NUM framework leads to substantial gains over

three baseline schemes - ORU-RS, Modrate [3], and ORA-RS. Typically, for a 8-hop flow 20% and 50% of the scenariosshow gains of at least 36% and 15%, respectively. In certaininstances, up to 80% gains are observed.

ACKNOWLEDGMENTThis work was supported by the US Army Research Office

under the Multi-University Research Initiative (MURI) grantW911NF-07-1-0318.

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