6290 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, …kalman.kaist.ac.kr/assets/papers/06557449.pdf · 2019-12-08 · 6290 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO

6290 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 10, OCTOBER 2013

Parallel Opportunistic Routing in Wireless NetworksWon-Yong Shin, Member, IEEE, Sae-Young Chung, Senior Member, IEEE, and Yong H. Lee, Senior Member, IEEE

Abstract—We study benefits of opportunistic routing in a largewireless ad hoc network by examining how the power, delay, andtotal throughput scale as the number of source–destination pairsincreases up to the operatingmaximum. Our opportunistic routingis novel in a sense that it is massively parallel, i.e., it is performedby many nodes simultaneously to maximize the opportunistic gainwhile controlling the interuser interference. The scaling behaviorof conventionalmultihop transmission that does not employ oppor-tunistic routing is also examined for comparison. Our main resultsindicate that our opportunistic routing can exhibit a net improve-ment in overall power–delay tradeoff over the conventional routingby providing up to a logarithmic boost in the scaling law. Such again is possible since the receivers can tolerate more interferencedue to the increased received signal power provided by the multi-user diversity gain, which means that having more simultaneoustransmissions is possible.

Index Terms—Multihop, multiuser diversity, opportunisticrouting, source–destination pair, wireless ad hoc network.

I. INTRODUCTION

I N [1], Gupta and Kumar introduced and studied thethroughput scaling in large wireless ad hoc networks.

They showed that a total throughput scaling of[bps/Hz] can be obtained by using a multihop strategy whensource–destination (S–D) pairs are randomly distributed in aunit area.1 Multihop schemes were then further developed andanalyzed in the literature [3]–[10], while their throughput perS–D pair scales far less than . Recent studies [11], [12]have shown that we can actually achieve scaling for

Manuscript received July 14, 2009; revised July 01, 2011; accepted February02, 2013. Date of publication July 11, 2013; date of current version September11, 2013. This work was done when W.-Y. Shin was with KAIST. Thiswork was supported in part by the Korea Communications Commission,Korea (KCA-2012-08-911-04-001), and in part by the Basic Science ResearchProgram through the National Research Foundation of Korea (NRF) funded bythe Ministry of Education, Science, and Technology (2012R1A1A1044151).This paper was presented in part at the 2007 IEEE International Symposium onInformation Theory.W.-Y. Shin was with the Korea Advanced Institute of Science and Tech-

nology, Daejeon 305-701, Korea. He is now with the Division of MobileSystems Engineering, College of International Studies, Dankook University,Yongin 448-701, Korea (e-mail: [email protected]).S.-Y. Chung andY. H. Lee are with the Department of Electrical Engineering,

Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea(e-mail: [email protected]; [email protected]).Communicated by S. Ulukus, Associate Editor for Communication

Networks.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/TIT.2013.2272884

1We use the following notation: 1) means that there existconstants and such that for all . 2)

means that . 3) if .

4) if . v) ifand [2].

an arbitrarily small , i.e., an almost linear scaling of thetotal throughput, by using a hierarchical cooperation strategy,thereby achieving the best result we can hope for.Besides the studies to improve the throughput up to the linear

scaling, an important factor that we need to consider in practicalwireless networks is the presence of multipath fading. The effectof fading on the scaling laws was studied in [3], [6], [7], [13],and [14], where it was shown that achievable throughput scalinglaws do not fundamentally change if all nodes are assumed tohave their own traffic demands (i.e., there are S–D pairs) [6],[7], [13] or the effect of fading is averaged out [3], [6], [13],while it was found in [14] that the presence of fading can re-duce the achievable throughput up to . However, fading canbe beneficial by utilizing the multiuser diversity gain providedby the randomness of fading in multiuser environments, e.g.,opportunistic scheduling [15], opportunistic beamforming [16],and random beamforming [17] in broadcast channels. Scenariosexploiting the opportunistic gain were also studied in cooper-ative networks by applying an opportunistic two-hop relayingprotocol [18] and in cognitive radio networks with opportunisticscheduling [19]. In [20] and [21], strategies for improving thethroughput scaling over nonfaded environments were shown inwireless network models that do not incorporate geometric pathloss. In [22], it was shown how fading improves the throughputusing opportunistic routing when a single active S–D pair existsin a wireless ad hoc network.In this paper, we analyze the benefits of fading by utilizing

opportunistic routing in multihop transmissions when there aremultiple randomly located S–D pairs in a large wireless ad hocnetwork. Our routing protocol describes how multiple nodesperform opportunistic routing simultaneously (or equivalently,in parallel) in a massive scale. To our knowledge, such an at-tempt for the network model has never been conducted in theliterature. Since the throughput scaling of a multihop protocolis far less than linear, it is natural to assume that only a subset ofS–D pairs are active at a time and active S–D pairs are chosenin a round robin fashion. In this paper, we consider a generalscenario where the number of active S–D pairs scales as a func-tion of . We are interested in improving the number of simul-taneously supportable S–D pairs, while maintaining a constantthroughput per S–D pair by using opportunistic routing.In most network applications, power and delay are also key

performance measures along with the throughput. The tradeoffamong these measures has been examined in terms of scalinglaws in some papers [8]–[10], [23]. In this paper, we analyze apower–delay–throughput tradeoff of both opportunistic routingand regular multihop routing as the number of S–D pairs in-creases up to the operating maximum, while per-node transmis-sion rate is set to a constant. We first show the existence of afundamental tradeoff between the total transmission power con-sumed by all hops per S–D pair, the average number of hops

0018-9448 © 2013 IEEE

SHIN et al.: PARALLEL OPPORTUNISTIC ROUTING IN WIRELESS NETWORKS 6291

per S–D pair, i.e., delay, and the number of active S–D pairs,which is proportional to the total throughput since we assumeper-node transmission rate is a constant. It is examined whetherpower can be reduced at the expense of increased delay for bothrouting scenarios, but a net improvement in overall power–delaytradeoff can be obtained with opportunistic routing. The im-provement comes from the multiuser diversity gain over theconventional multihop routing. This increases the average re-ceived signal power, which in turn makes it possible to havemore simultaneous transmissions since more interference is tol-erated while per-node transmission rate is maintained. Morespecifically, we show that such a multiuser diversity gain leadsto a logarithmic performance improvement.The rest of this paper is organized as follows. Section II de-

scribes our system and channel models. In Section III, our pro-tocols with and without opportunistic routing are described. InSection IV, the power–delay–throughput tradeoff for these twoprotocols is analyzed and compared. Finally, Section V summa-rizes the paper with some concluding remarks.Throughout this paper, denotes the expectation. Unless

otherwise stated, all logarithms are assumed to be to the base 2.

II. SYSTEM AND CHANNEL MODELS

We consider a 2-D wireless network that consists of nodesuniformly and independently distributed on a square of unit area(i.e., dense network [1], [8], [9], [11]). We randomly pick S–Dpairings such that each node is the destination of exactly onesource. We assume that there are randomly located S–Dpairs, which can be active simultaneously, where scalesslower than . Note that sources can generate their owndata traffic at the same time.In this paper, to utilize the opportunistic gain, we adopt the

physical channel model that can capture opportunism by mod-eling a realistic fading. The received signal at node

at a given time instance is then given by

where is the signal transmitted by node , is the cir-cularly symmetric complex additive-white Gaussian noise withzero mean and variance , and is the set of si-multaneously transmitting nodes. The channel gain is givenby

(1)

where is the complex fading process between nodes and, which is assumed to be Rayleigh with and in-dependent for different ’s and ’s. Moreover, we assume theblock fading model, where is constant during one packettransmission and changes to a new independent value for thenext transmission. and denote the distance betweennodes and and the path-loss exponent, respectively. We as-sume that channel state information (CSI) is available at all thereceivers, but not at the transmitters.

Our model for successful reception of a transmission overone hop basically follows the physical model in [1]. The de-tailed argument is described as follows. Since there is no CSIat the transmitter, we assume that each source node transmitsdata to its destination at a fixed target rate independentof . A similar assumption was also made in some earlier work[1], [3]–[14]. As in the earlier studies [15]–[19] dealing withopportunism under the block fading model, we suppose that apacket is decoded successfully if the received signal-to-interfer-ence-and-noise ratio (SINR) exceeds a predetermined threshold

, which is independent of , i.e.,. Then, the total throughput of the network

would be given by if no transmission fails i.e., thereis no outage.

III. ROUTING PROTOCOLS

In this section, we describe our routing protocols withand without opportunistic routing. We simply use a multihopstrategy in both cases using the nodes other than S–D pairs asrelays. Hence, we do not assume the use of any sophisticatedmultiuser detection schemes at the receiver.2

Next let us introduce the scaling parameters and .The average number of hops per S–D pair is interpreted as theaverage delay and is denoted as . The parameter de-notes the average total transmit power used by all hops for anS–D pair. Assuming that the transmit power is the same for eachhop, we see that is equal to times the transmit powerper hop. In addition, for both routing schemes, let us supposethat

(2)

As a consequence of this choice of power , all the transmitpower is scaled in such a way that the average total interferencepower from the set , consisting of simultane-ously transmitting nodes, is given by , which will be an-alyzed in a later section. Note that this power scaling strategydoes not affect the tradeoff among the orders of power ,delay , and total throughput (see Section IV-A formore detailed description).

A. Opportunistic Routing

Opportunistic routing was originally introduced in [24] and[25] and was further developed in various network scenarios[26]–[29]. When a packet is sent by a transmitting node, it maybe possible that there are multiple receivers successfully de-coding the packet. Among the relay nodes that successfully de-code the transmitted packet for the current hop, the one that isclosest to the destination becomes the transmitter for the nexthop. Since the packet can travel farther at each hop using this op-portunistic routing, the average number of hops can be reduced.Note that the existing protocol in [24]–[29] was designed simplyfor the case where there exists a single S–D pair, and thus, it did

2If scales between and for an arbitrarily small ,which is the operating regimes in our work, then multihop protocols are suffi-cient to satisfy the order optimality in dense networks (the detailed proof is notshown in this paper).


Fig. 1. S–D paths passing through the shaded cell.

not incorporate interference between links, which is a criticalproblem in wireless networks.We modify this routing to apply it to our network, composed

of multiple nodes performing opportunistic routing simultane-ously in a massive scale. Then, we need to carefully design arouting protocol while solving the interference problem causedby simultaneously transmitting nodes. The per-hop distance ofthis opportunistic transmission is random. However, we canmake sure that there are multiple successfully receiving nodesin a given square cell with high probability (w.h.p.) if we con-trol the size of the cell and the distance between the transmitterand the cell. Then, one of the successfully receiving nodes canbe the transmitter for the next hop. Short signaling messages[24], [25] need to be exchanged between some candidate relaynodes and the corresponding transmitting node in order todecide who will be the transmitter for the next hop.3 Thesemessages are transmitted using a different time slot from thatfor data packets to avoid any interference. More specifically,it is assumed that the two different messages are transmittedat even and odd time slots, respectively, which causes only afactor 2 loss in performance, thus resulting in no degradationin terms of scaling laws.4

As shown in Fig. 1, we divide the whole area intosquare cells with per-cell area . Note that

holds since the average distance be-tween an S–D pair is given by . We assume XY routing,i.e., the route for an S–D pair consists of a horizontal and avertical paths. Suppose that routing is performed first hori-zontally and then vertically for each S–D pair, as illustratedin Fig. 1 ( and denote a source and the correspondingdestination node, respectively, for ). Then, for each hopin the S–D path, some relay nodes that successfully decodetheir packets are selected opportunistically for transmissionin the next hop (the relaying node selection strategy will bedescribed later in detail). That is, the route for each S–D pairis not predetermined. Nodes operate according to the 25-timedivision multiple access (TDMA) scheme. This means thatthe total time is divided into 25 time slots and nodes in eachcell transmit 1/25th of the time, while all transmitters in a

3Alternatively, a timer-based strategy can be used for selecting the transmitterfor the next hop [30].4Since our aim is to study the performance in the limit of infinite packet length

under the block fading model, if the packet length scales fast enough in , thenwe may conclude that these signaling messages have a negligible overhead.

Fig. 2. Grouping of interfering cells in the 25-TDMA scheme. The first layerrepresents the outer eight shaded cells.

cell transmit simultaneously.5 Fig. 2 shows an example ofsimultaneously transmitting cells depicted as shaded cells.Our routing protocol consists of two transmission modes, i.e.,

Modes 1 and 2, where Mode 2 is used for the last two hops tothe destination and Mode 1 is used for all other hops (refer toFig. 1 for the brief operation of two modes).6

Mode 1: We use an example in Fig. 3 to describe this mode.Transmitting nodes in Cell A send packets simultaneously,where one of those can be either source or relay node . Arelay node that successfully decodes the packet and is two (CellB) or three (Cell C) cells apart from the transmitter horizontally(or vertically), for example or in Fig. 3, is arbitrarilychosen for the next hop. If there is no such node, then anoutage occurs, i.e., none of the nodes satisfies in thecells. We do not assume any retransmission scheme in our casesince we will make the outage probability negligibly small. Ifthere are more than one candidate relay, then we choose oneamong them arbitrarily. Note that the multiuser diversity gainis roughly equal to the logarithm of the number of nodes inCells B and C, which will be rigorously analyzed in the nextsection. We perform Mode 1 until the last two hops to thedestination, and then switch to Mode 2. The reason why wehop either two or three cells at a time is because 1) hopping toan immediate neighbor cell can create huge interference to areceiving node near the boundary of the two adjacent cells7 and2) always hopping by two cells is not good since it partitionsthe cells into two groups, even and odd, and a packet can neverbe exchanged between the two groups.Mode 2: For the last two hops to the destination, Mode 2 is

used. If we use Mode 1 for the last hop, we cannot get any op-portunistic gain since the destination is predetermined. Hence,

5Under our opportunistic routing protocol, 25-TDMA scheme is used 1) toguarantee that there are no transmitting and receiving nodes near the boundaryof two adjacent cells and 2) to avoid a partitioning problem, which will be dis-cussed later in this section.6Even for the case where only one hop is needed between an S–D pair, we

can artificially introduce an additional hop so that there are at least two hops forevery S–D pair.7By hopping by one cell, the distance between a receiving node and an inter-

fering node can be arbitrarily small.


Fig. 3. Opportunistic routing protocol in Mode 1.

Fig. 4. Opportunistic routing protocol in Mode 2.

we use the following two-step procedure for Mode 2. We usethe example in Fig. 4 to explain this mode.• Step 1: In this step, a node in Cell D or E (e.g., orin Fig. 4) transmits its packet, whose signal reaches Cell F.This is similar to what happens in Mode 1 except that weare seeing this from Cell F’s perspective. Assuming thatthere exist nodes in Cell F, we arbitrarily partition Cell Finto subcells of equal size, i.e., there are roughlynodes in each subcell. One node is then opportunisticallychosen among the nodes that received the packet correctlyin each subcell. Therefore, nodes are chosen in Cell Fas potential relays for the packet.

• Step 2: In Step 2, which corresponds to the last hop, thefinal destination in Cell G or H (e.g., or in Fig. 4)sends a probing packet, i.e., short signaling message, tosee which one of the selected relay nodes in each cellwill be the transmitter for the next hop whose channel linkguarantees a successful packet transmission. Finally, thepacket from the selected relay node in cell F is transmittedto the final destination.

Although there are only candidate nodes in each cell inMode 2, whereas there were nodes in Mode 1, this does notaffect the scaling law since the multiuser diversity gain is loga-rithmic in and .

B. Nonopportunistic Routing

In this case, a plain multihop transmission [1], [8] is per-formed with a predetermined path for each S–D pair consistingof a source, a destination, and a set of relaying nodes. There-fore, there is no opportunistic gain. The whole area is also di-vided into cells with per-cell area and one trans-mitter in a cell is arbitrarily chosen while transmitting at a fixeddata rate independent of . We assume the shortest pathrouting and the 9-TDMA scheme as in [1] and [8]. However,even if interference is carefully controlled, a transmission mayfail due to fading, causing outages. In this paper, we simplyassume that for the event that an outage occurs (i.e.,

) for a certain hop, such an event is not counted asoutage, which will give an upper bound on the performance.

IV. POWER–DELAY–THROUGHPUT TRADEOFF

Our goal in this section is to analyze the power–delay–throughput tradeoffs with and without opportunisticrouting. Provided that per-node transmission rate isgiven by a constant independent of , we will show later thatthere exists a tradeoff among scaling parameters , ,and for the two routing protocols that we take into ac-count. By assuming the per-node rate of , the tradeoff amongthe four parameters , , , and is thusessentially reduced to the tradeoff among the three parameters

, , and such that any one of them can bechanged freely, which in turn determines the other two. Notethat with a constant rate , the parameter is proportionalto the total throughput since if thereis no outage. Note that different protocols will lead to differentpower–delay–throughput tradeoffs.If more power is available, then per-hop distance can be ex-

tended. Since the path-loss exponent is greater than 2, therequired power increases at least quadratically in the per-hopdistance. On the other hand, the total power consumption ofmultihop transmission is linear in the number of hops per S–Dpair. Therefore, it seems advantageous to transmit to the nearestneighbor nodes if we want to minimize the total power. How-ever, this comes at the cost of increased delay due to more hops.In the following sections, we first show that there exists a funda-mental tradeoff between the total transmission power consump-tion per S–D pair, the average delay per S–D pair, and the totalthroughput, and then show that there is a net improvement in theoverall power–delay tradeoff when our opportunistic routing isutilized in the network.

A. Opportunistic Routing

The relationship among the three parameters , ,and is derived under the opportunistic routing protocoldescribed above. More specifically, we are interested in howmany S–D pairs, denoted by , can be active simultane-ously while maintaining a constant transmission rate per S–Dpair. In the following, we mainly focus on Mode 1 since Mode2 can be similarly analyzed with a slight modification. First,


let denote the SINR value seen by receiverfor the th hop of the th S–D pair, where and

. Here, denotes theset of hops for the th S–D path, where is a positive param-eter that scales as . Then, we have

(3)

where and denote the received signal powerat node from the desired transmitter for the thhop of the th S–D pair and the total interference power atnode from all interfering nodes, respectively. Specifi-cally, they are given by

(4)

and

(5)

respectively. Here, is the set of simultaneouslytransmitting nodes. Before establishing our tradeoff results, westart from the following lemma, which shows lower and upperbounds on the number of nodes in each cell available as potentialrelays.Lemma 1: Let denote the number of nodes in cell

. If , then isbetween , i.e., ,w.h.p. for a constant independent of .The proof of this lemma is given in [11]. In a similar fashion,

the number of nodes inside each subcell defined inMode 2 is be-tween w.h.p. Note thatthe upper and lower bounds on are irrelative to the cellindex . We now turn our attention to quantifying the amountof interference in our schemes in the following two lemmas.Lemma 2: If and

for a sufficiently small , then thenumber of S–D paths simultaneously passing through each cellis given by w.h.p.

Proof: This proof technique is similar to that of [8], but amore general result is provided for the case where the size ofeach cell (or equivalently, the average delay ) can be con-trolled systematically and scales as a function of . Let

denote an indicator function whose value is one if the pathof the th S–D pair passes through a fixed cell and is zero oth-erwise, where and .The total number of paths passing through the cell is given by

, which is the sum of independent andidentically distributed (i.i.d.) Bernoulli random variables withprobability

where the expectation is taken over the matching of S–D pairs aswell as the node placement. This is because S–D pairs are

randomly located with uniform distribution on the unit square.Hence, for any constant , we get the following:

from the Chernoff bound [31]. By computing the followingexpectation

where and are some positive constants independent of ,we have

Similarly, by the Chernoff bound [31], it follows that

thereby yielding

Due to the fact that there are cells in the network, byapplying the union bound over cells, it follows thatthe number of S–D paths passing through each cell is between

with proba-bility of at least

for constant independent of . This tends to one asgoes to infinity, i.e.,

, where is a constant satisfying . Thiscompletes the proof of this lemma.Using the result of Lemma 2, we upper-bound the total in-

terference as a function of three parameters , , andin the following lemma.

Lemma 3: Suppose ,, and , where is a sufficiently

small constant. When the 25-TDMA scheme is used, the totalinterference power at receiving node from simultaneouslytransmitting nodes is given by

with probability of at least

(6)

for constant independent of . Equation (6) tends to oneas increases and the expectation of is given by

(7)


The proof of this lemma is presented in Appendix A. Notethat depends on the path loss exponent .Now to simply find a lower bound on the throughput, suppose

that the threshold value is set to 1.8 Let us focus on the thS–D pair, where . Note that a packet fromthe th source passes through the th S–D pair’s routing paththat consists of the set of hops. Ac-cordingly, if the condition is not guaranteed forat least one among hops of the path, then the data trans-mission for the th S–D pair will fail, causing outages. To an-alyze the achievable throughput, it is thus important to examinethe probability that the source’s packet is successfully deliveredto the final destination node while satisfying for allhops . To be concrete, let denote the event that thereis no outage for the th S–D pair, i.e., the event that the condi-tion holds for at least one receiver perhop for all hops of the th S–D pair. Here, it followsthat , where andrepresent the cell indices that are both two and three cells apartfrom the desired transmitter along the routing path, respectively.This comes from the fact that two cells are taken into account forselecting one receiving node. Since the Gaussian is the worst ad-ditive noise [32], [33], treating all the interference as Gaussiannoise lower-bounds the capacity. Hence, by assuming full CSIat the receiver side, the total throughput is given by

(8)

Here, the first inequality comes from the fact that per-node trans-mission rate is given by . The secondinequality holds by applying the union bound over all hopsfor each S–D pair, where the set of hops is specified by

forMode 1 and the last two hops to thedestination (i.e., ) forMode 2. Notethat Lemma 1 is used to compute the minimum number of nodes

8If is optimized, then the achievable rates can be slightly improved. How-ever, for analytical convenience, we just assume .

in each cell (or in each subcell). In order to further compute theright-hand side of (8), we need to know the distribution of theSINR, which is difficult to obtain for a general class of channelmodels consisting of both geometric and fading effects. Instead,in [34], asymptotic upper and lower bounds on the cumulativedistribution function (CDF) of SINR were characterized.In this paper, as mentioned earlier, we assume the transmit

power in (2), which makes the analysis of scaling laws muchsimpler. Then, using (7) in Lemma 3 and (2), it follows thatthe average total interference power at receiving

node ( and ) becomes, which is the best situation we can hope for to maintain

the fixed transmission rate for each hop. This is becauseif is not , then we can scale down all transmit

powers proportionally such that withoutloss of optimality in scaling. This is because the receivedsignal power from the desired transmitter should be

to maintain a fixed rate per S–D pair andhaving such higher power from both the signal and the interfer-ence is unnecessary. On the other hand, if ,

then it follows that . We canthus scale up all transmit powers proportionally such that

in order to increase the SINR value,resulting in an improved per-node transmission rate. As aconsequence, it is possible to find the CDF of the SINR in(8) when our opportunistic routing is utilized. Let de-note the event that holds for receiving nodein the network. By using (1), (3), (4), and the condition

in Lemma 3, we then have

(9)

where , , , and denote some positive constants inde-pendent of , and is a sufficiently small constant. Here,the second equality comes from the fact that per-hop distance isgiven by . The third equality holds since the squaredchannel gain follows the chi-square distribution


with two degrees of freedom. The third inequality comes from(2) and (6). The last equality holds since it follows that

under the condition . Note that the

upper bound on the probability is iden-tical for all hops since it does not depend on . Now weare ready to derive the scaling laws for , , andin terms of by using (2), (8), and (9).Theorem 1: Suppose that

for , transmission rate per S–D pair is a positive con-stant, and , where is a suf-ficiently small constant. If and

, then the opportunistic routing achieves the power

(10)

the delay

(11)

and the total throughput w.h.p., whereis an arbitrarily small constant.

Proof: By substituting (9) into (8), the total throughputcan be lower-bounded by

where . To guaranteew.h.p. with no outage for transmissions, we

thus need the following equality:

for an arbitrarily small . Then, it follows that

which yields

if

if .

(12)

After some calculation, using (2) and (12), we obtain

and

(13)

From (13) and the condition , itfollows that

(14)

and

(15)

for an arbitrarily small , and hence, we have

and

under the constraints (14) and (15), finally resulting in (10)and (11). Let , where and are shown inLemmas 2 and 3, respectively. If we choose a constant

, independent of , satisfying

then it is seen that the conditionalways holds from (11). We also have due to

. This completes the proof of this theorem.Note that the logarithmic terms in (10) and (11) are obtained

due to the multiuser diversity gain of the opportunistic routing.The operating regimes correspond to the case where the number

of simultaneously active S–D pairs scales betweenand . Furthermore, we see that monotonically de-creases with respect to while scales almost linearly.We finally remark that using (10) and (11) yields the relationship

between the two scaling parameters and .

B. Nonopportunistic Routing

In this section, the scaling result of nonopportunistic routingis shown for comparison. As addressed before, the total inter-ference power at receiving node needs to be , and itthus follows that due to Lemma


3 and (2). In this case, we investigate how the delay andthe power scale when there are simultaneously ac-tive S–D pairs, while maintaining a constant , as inSection IV-A. The power–delay–throughput tradeoff is derivedin the following theorem.Theorem 2: Suppose that

for and transmission rate per S–D pair is a positiveconstant. If and foran arbitrarily small , then the nonopportunistic routingachieves the power

(16)

the delay

(17)

and the total throughput .The proof of this lemma almost follows the same line as that

of Theorem 1. Note that there is no logarithmic term in the twoequations shown above. We also remark that using (16) and (17)results in the relationship

between and .

C. Performance Comparison

Now we show that the opportunistic routing exhibits anet improvement in overall power–delay tradeoff over theconventional nonopportunistic routing. Figs. 5 and 6 showhow the power and the delay scale with respectto the number of simultaneously active S–D pairs, cor-responding to the total throughput . and denotethe scaling curves with and without opportunistic routing,respectively. We only take into account the range ofbetween and for an arbitrarily small , whichis the operating regimes in our work, due to various constraintsthat we assume in the model. Hence, the multiuser diversitygain may not be guaranteed if scales faster thanfor a vanishingly small (e.g., it is shown in [7] that when

, the benefit of fading cannot be exploitedin terms of scaling laws). We observe that decreaseswhile increases as we have more active S–D pairs inboth schemes. This is because we assume a fixed transmissionrate independent of , which implies that for receiver

( and ), andneed to be and , respectively, as mentioned

earlier. Then, to maintain the interference level atas increases, more hops per S–D pair are needed, i.e.,per-hop distance needs to be reduced. Hence, from the aboveargument, we may conclude that the power is reduced at theexpense of the increased delay, and therefore, there is a funda-mental tradeoff between the two scaling parameters and

. Furthermore, it is seen that utilizing the opportunisticrouting increases the power compared to the nonopportunisticrouting case, but it can reduce the delay significantly. Thus,it is not clear whether our opportunistic routing is beneficialor not from Figs. 5 and 6. However, if we plot the power

Fig. 5. Power scaling with respect to .

Fig. 6. Delay scaling with respect to .

Fig. 7. Power–delay tradeoff.

versus the delay as in Fig. 7, then it can be clearlyseen that opportunistic routing scheme ( ) exhibits a betteroverall power–delay tradeoff than that of nonopportunisticrouting scheme ( ), while providing a logarithmic boost inthe scaling law. For example, if the delay is given by

, then the power is reduced times by usingthe opportunistic routing. In this case, it is further seen fromFig. 6 that the number of simultaneously supportableS–D pairs is improved by , i.e., logarithmic boost on thetotal throughput . This gain comes from the fact that thereceived signal power increases due to the multiuser diversitygain based on the use of opportunistic routing, which allowsmore simultaneous transmissions since more interference canbe tolerated.

V. CONCLUSION

The scaling behavior of large ad hoc networks using oppor-tunistic routing protocol in the presence of fading has been char-


acterized. Specifically, it was shown how the power, delay, andtotal throughput scale as the number of S–D pairs increases,while maintaining a constant per-node transmission rate. Weproved that for the range of simultaneously active S–D pairs be-tween and for an arbitrarily small , our par-allel opportunistic routing exhibits a net improvement in overallpower–delay tradeoff over the conventional scheme employingnonopportunistic routing, while providing up to a boost inthe scaling law due to the multiuser diversity gain.

APPENDIX

A) Proof of Lemma 3: There are interfering cells inthe th layer of 25-TDMA (refer to Fig. 2). Let denotethe total interference power at a fixed receiving node fromsimultaneously transmitting nodes in the th layer, where

for some constant independent of .Note that the distance between a receiving node and an inter-fering node in the th layer is between

. Suppose that the Euclidean distance among thelinks above is given by , thereby providing a

lower bound on . By Lemma 2, the number of simul-

taneous transmitters in each cell is given byw.h.p. Thus, from (1) and (5), the expectation is lower-bounded by

for any nodes and , where and are some positive con-stants independent of . Similarly by takingfor the Euclidean distance between a receiver and simultane-ously transmitting nodes in the th layer, we obtain

for constant independent of , which results in

Moreover, let denote the total interference power fromother transmitting nodes in the cell including a desired trans-mitter (see the shaded cell located in the center in Fig. 2). Thenas above, we have , and

hence, it follows that sinceis bounded by a certain constant for .

Now we focus on computing the probability

by using the Cher-noff bound, where is a constant independent of . From

the fact that for all transmitting nodes and inthe same layer and receiving node , we have

(18)

for constant independent of . Here, is the set of si-multaneously interfering nodes in the th layer. Since the Cher-noff bound for the sum of i.i.d. chi-square random variables

with two degrees of freedom is given by [35], for a certainconstant , (18) can be upper-boundedby

which tends to zero as . We remark that the

event for all is a

sufficient condition for the event . Thus, bythe union bound over all layers (including the cell with a desiredtransmitter), we have the following inequality:

where the first inequality holds since there exist layers,for some independent of . Finally, using the unionbound over nodes in the network yields that the total inter-ference power at receiving node is given by( ) with probability of at least


which tends to one as for a certainconstant

This completes the proof of this lemma.

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Won-Yong Shin (S’02–M’08) received the B.S. degree in electrical engineeringfrom Yonsei University, Seoul, Korea, in 2002. He received the M.S. and thePh.D. degrees in electrical engineering and computer science from Korea Ad-vanced Institute of Science and Technology (KAIST), Daejeon, Korea, in 2004and 2008, respectively.From February 2008 to April 2008, he was a Visiting Scholar in the School of

Engineering and Applied Sciences, Harvard University, Cambridge, MA. FromSeptember 2008 to April 2009, he was with the Brain Korea Institute and CHiPSat KAIST as a Postdoctoral Fellow. From August 2008 to April 2009, he waswith the Lumicomm, Inc., Daejeon, Korea, as a Visiting Researcher. In May2009, he joined Harvard University as a Postdoctoral Fellow and was promotedto a Research Associate in October 2011. Since March 2012, he has been withthe Division of Mobile Systems Engineering, College of International Studies,Dankook University, Yongin, Korea, where he is currently an Assistant Pro-fessor. His research interests are in the areas of information theory, communi-cations, signal processing, and their applications to multiuser networking issues.

Sae-Young Chung (S’89–M’00–SM’07) received the B.S. (summa cum laude)and M.S. degrees in electrical engineering from Seoul National University,Seoul, South Korea, in 1990 and 1992, respectively and the Ph.D. degree inelectrical engineering and computer science from the Massachusetts Instituteof Technology, Cambridge, MA, USA, in 2000.From September 2000 to December 2004, he was with Airvana, Inc.,

Chelmsford, MA, USA. Since January 2005, he has been with the Depart-ment of Electrical Engineering, Korea Advanced Institute of Science andTechnology (KAIST), Daejeon, South Korea, where he is currently a KAISTChair Professor. He has served as an Editor of the IEEE TRANSACTIONS ONCOMMUNICATIONS since 2009. He is the Technical Program Co-Chair of the2014 IEEE International Symposium on Information Theory. His researchinterests include network information theory, coding theory, and wirelesscommunications.


Yong H. Lee (S’81–M’84–SM’98) was born in Seoul, Korea, on July 12, 1955.He received the B.S. and M.S. degrees in electrical engineering from Seoul Na-tional University, Seoul, Korea, in 1978 and 1980, respectively, and the Ph.D.degree in electrical engineering from the University of Pennsylvania, Philadel-phia, PA, in 1984.From 1984 to 1988, he was an Assistant Professor with the Department of

Electrical and Computer Engineering, State University of New York, Buffalo,NY. Since 1989, he has been with the Department of Electrical Engineeringat Korea Advanced Institute of Science and Technology (KAIST), Daejeon,

Korea, where he is currently a Professor and the Provost of KAIST. His researchactivities are in the area of communication signal processing, which includesinterference management, resource allocation, synchronization, estimation, anddetection for code-division multiple access (CDMA), time-division multiple ac-cess (TDMA), orthogonal frequency division multiplexing (OFDM), and mul-tiple-input multiple-output (MIMO) systems. He is also interested in designingand implementing transceivers.

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