Ad Hoc Wireless Networks (A Communication-Theoretic Perspective) || Optimal Common Transmit Power for Ad Hoc Wireless Networks

Chapter 8

Optimal Common Transmit Powerfor Ad Hoc Wireless Networks

8.1 IntroductionIn an ad hoc wireless network, where nodes are likely to operate on limited battery life, powerconservation is an important issue. Conserving power prolongs the lifetime of a node andalso the lifetime of the network as a whole. In addition, transmitting at low power reduces theamount of excessive interference. The fundamental question which naturally arises is: ‘whatis the optimal transmit power to be used?’. This is the fundamental question that we try toanswer in this chapter. Obviously, a suitable criterion of optimality has to be introduced.

One of the goals of forming a network is to have network connectivity – that is, eachnode should be able to communicate with any of the other nodes, possibly via multiple hops.The connectivity level of an ad hoc wireless network depends on the transmit power of thenodes. If the transmit power is too small, the network might be disconnected (i.e. there maybe multiple disconnected clusters of nodes instead of a single overall connected network).However, as mentioned earlier, transmitting at excessively high power is inefficient becauseof the mutual interference in the shared radio channel and the limited battery lifetime. Thus,it is intuitively clear that the optimal transmit power is the minimum power sufficient toguarantee network connectivity [170–173].

Ideally, the transmit power of a node should be adjusted on a link-by-link basis toachieve the maximum possible power savings [35, 37–39, 172–183]. Nonetheless, due to theabsence of a central controller in a ‘pure’ ad hoc network with a flat architecture, performingpower control on a link-by-link basis is a complicated and cumbersome task. A simplersolution, which is more viable for implementation, is to have all the nodes use a commontransmit power. This is desirable, for example, in sensor networks where nodes are relativelysimple and it is difficult to modify the transmit power after deployment. In addition, theperformance difference, in terms of traffic carrying capacity, between adjusting the powerlocally and employing a common transmit power is small, especially when the numberof nodes is large [171]. The analysis of different power control strategies is consideredin [184]. Power control has also been studied in the context of code division multipleaccess (CDMA)-based packet radio networks [185–188], multicast and broadcast [189–193],medium access control (MAC) protocol design [194–197], routing and scheduling algorithmdesign [174, 175, 198–207] and topology control [173, 208–211].

Ad Hoc Wireless Networks: A Communication-Theoretic Perspective Ozan K. Tonguz and Gianluigi Ferrari© 2006 John Wiley & Sons, Ltd. ISBN: 0-470-09110-X

196 Chapter 8. Optimal Common Transmit Power for Ad Hoc Wireless Networks

In this chapter, we investigate the optimal transmit power for an ad hoc wirelessnetworking scenario where all nodes use a common transmit power. Although the optimalcommon transmit power derived in this chapter is subject to the routing and the MAC protocolconsidered, the approach can be extended to other routing and MAC protocols as well. Otherstudies which consider common transmit power exist [171, 212–214]. In [212] and [213],the minimal transmission range at which a network is connected with high probability isstudied. In [214], the authors investigate the minimal common transmit power sufficient topreserve network connectivity. These works, however, follow a graph-theoretic approachwhich only takes into account the distances between nodes. More specifically, the authorsconsider that two neighboring nodes can communicate if they are within the communicationrange of each other, and two nodes that are not neighbors can communicate if there is amulti-hop path connecting them. It is important to point out that although there may be a pathconnecting two nodes, communication between them may not be possible as the quality ofservice (QoS), in terms of tolerable bit error rate (BER) at the end of a multi-hop route, maynot be satisfied. We discuss this in more detail in section 8.3. As opposed to the conventionalgraph-theoretic approach, in this chapter, the optimal transmit power sufficient to maintainnetwork connectivity is found according to a physical layer-oriented QoS constraint given bythe maximum tolerable BER at the end of a multi-hop route with an average number of hops.

In this chapter, we evaluate the optimal transmit power both analytically (in the caseof regular topology) and via simulations (in the case of random topology). Moreover, weinvestigate the interrelation between optimal transmit power, data-rate and node spatialdensity. This chapter also investigates (i) the impact of different propagation path lossexponents (on different links of a multi-hop route) on the performance of a common transmitpower control scheme and (ii) the interrelation between transmit power, connectivity andnetwork longevity. We also validate our analysis with simulations.

The rest of this chapter is organized as follows. In section 8.2, we describe the modeland the assumptions that will be used in the derivation of the optimal transmit power. Insection 8.3, we define network connectivity. In section 8.4, we recall the basics of thecommunication-theoretic framework introduced in Chapters 2 and 3 for evaluating routeBER. The minimum transmit power sufficient to maintain network connectivity in bothregular and random topologies is analyzed in section 8.5. The performance in terms ofnode/network lifetime and effective transport capacity is evaluated in section 8.6. Numericalresults, along with their implications, are presented in section 8.7. Finally, we providediscussion of related work and conclusions in sections 8.8 and 8.9, respectively.

8.2 Model and AssumptionsIn this section, we describe the basic ad hoc wireless network communication model and thebasic assumptions considered in this chapter.

8.2.1 Network TopologyThroughout the chapter, we consider a scenario where N nodes are distributed over a surfacewith finite area A. The node spatial density is defined as the number of nodes per unit areaand is denoted as ρs � N/A. To avoid edge effects, we assume the network surface tobe the surface of a torus with length 2R on each edge, as shown in Figure 8.1. However,the analytical technique presented in this chapter can be applied to other types of surfacesas well. In a real scenario, the performance predicted by our analysis may not be extremely

8.2. Model and Assumptions 197

dlink

S1

S2

S3

D3 D2

D1

2R

2R

S1

D1

S2

D2

S3

D3

(a) (b)

Figure 8.1 Possible topologies: (a) regular and (b) random. In each case, examples of multi-hop routes are shown. (Reproduced by permission of © 2005 IEEE.)

precise for nodes on the edge of the network surface. In this case, a more precise performanceevaluation may be obtained via simulations. Nonetheless, the results presented in this chapterprovide a representative description of a realistic network behavior.

In addition to a simple scenario with a square grid network topology considered in [162],in this chapter we also consider a realistic scenario with a two-dimensional Poisson nodedistribution. In a network with a square grid topology, as shown in Figure 8.1 (a), eachnode has four nearest neighbors at a fixed distance. In contrast, the positions of nodes ina network with a two-dimensional Poisson topology are random and independent of eachother, as shown in Figure 8.1 (b). The probability mass function of the number of nodes Na

over a surface of area a in the case with two-dimensional Poisson topology is given by

Pr(Na = j) = (ρsa)j

j ! e−ρsa j = 0, 1, 2, . . . (8.1)

where, in this case, ρs corresponds to the average number of nodes per unit area or the averagenode spatial density. Considering the same value of ρs for both types of networks with regularand random topologies makes the comparison between them fair and meaningful.

In this chapter, we only consider ad hoc wireless networks with stationary nodes.Examples of such networks are sensor networks [34] and wireless mesh networks [164].The extension to a scenario where nodes are mobile can be done following the approachproposed in Chapter 6.

8.2.2 RoutingWe assume a simple routing strategy such that a packet is relayed hop-by-hop, through asequence of nearest neighboring nodes, until it reaches the destination. In addition, we assumethat a source node discovers a route prior to data transmission [4]. Discovery of a multi-hoproute from a source to a destination is a crucial phase in a wireless networking scenariowith a flat architecture. The focus of this chapter, however, is on the characterization of thesteady-state behavior of on-going peer-to-peer (P2P) multi-hop communications. Therefore,we will assume that a route between the source and the destination exists. We discuss routingin networks with regular and random topologies in the following subsections.


Square Grid Topology

Due to the regularity of this topology, the distance to the nearest neighbor, denoted by dlink,is fixed, and a route is constituted by a sequence of hops with equal length. The distance dlinkcan be computed as follows. One can first observe that constructing a square lattice of N

nodes over a surface of a torus with area A is equivalent to fitting N small square tiles of aread2

link into a large square of area A. Hence, it must hold that Nd2link = A and, therefore, the

distance to the nearest neighbor can be written as

dlink =√

A

N= 1√

ρs. (8.2)

Two-Dimensional Poisson Topology

In the case of random topology, we still consider a routing scheme where each intermediatenode in a multi-hop route relays the packets to its nearest neighbor in the direction of thedestination. In particular, we assume that an intermediate node in the route selects the nearestnode within a sector of angle θ towards the direction of the destination as the next hop [41].An example of a multi-hop route constructed in this way is shown in Figure 8.2. In thiscase, a route can be visualized as a deviation from the straight line between the source andthe destination, referred to as the reference path. Unlike in a scenario with grid topology, ina network with two-dimensional Poisson topology, the distance from a node to its nearestneighbor is not a constant. Let W be a random variable denoting the distance to the nearestneighbor in a two-dimensional Poisson node distribution. The derivation of the cumulativedistribution function (CDF) of W , i.e. FW (w), is fairly complicated and is therefore relegatedto Appendix E. It is shown in Appendix E that, keeping the node spatial density fixed, forlarge N (i.e. as N → ∞), the CDF of the distance to the nearest neighbor in a torus is

FW(w) =

0 w < 0

1 − e−ρsπw20 ≤ w < R

1 − e−ρsχ R ≤ w <√

2R

1 w ≥ √2R

(8.3)

where χ �[4w2(π/4 − ξ) + 4R

√w2 − R2

]and ξ � cos−1(R/w). Note that the total

surface area of a torus with length 2R on each edge is equal to 4R2. Hence, the node spatialdensity in this case is ρs = N/4R2. Substituting this value back in (8.3) and computingFW (R), it follows that Pr[W ≤ R], i.e. FW (R), is almost 1 if N ≥ 10. In other words, W isalmost surely lower than R. Consequently, for N � 10 (which will be true for all scenariosconsidered in this chapter), we can neglect the case where W > R without significant impacton the accuracy of the analytical results.

Using the fact that W ≤ R with very high probability and generalizing the CDF givenin (8.3) to a scenario where a node looks for a neighboring node within a sector of angle θ ,the CDF of W can be written as

F(θ)W (w) =

1 w > R

1 − e−ρsθw2/2 0 ≤ w ≤ R

0 otherwise.

(8.4)

8.3. Connectivity 199

Source Destination

W

θ

Φ

Figure 8.2 Possible multi-hop route in a random topology. W corresponds to the distanceto the nearest neighbor, � corresponds the projection angle and θ corresponds to the angleat which a node looks for a neighbor in the direction of the destination. (Reproduced bypermission of © 2004 IEEE.)

8.2.3 Medium Access Control ProtocolIn this chapter, we consider the simple reservation-based MAC protocol introduced in Chap-ter 3 and defined as reserve-and-go (RESGO). In this protocol, a source node first reservesintermediate nodes on a route for relaying its packets to the destination – characterizationof this phase is beyond the scope of this chapter. A transmission can begin after a routeis discovered and reserved. The main idea of the protocol is that a source node or a relaynode generates an exponential random backoff time before it transmits or relays each packet.After the random backoff time expires, a node can start transmitting a packet. The randombackoff time helps reduce interference among nodes in the same route and also among nodesin different routes. Throughout this chapter, we assume that the random backoff time isexponential with mean 1/λ. In other words, given that a node has packets to send, packetsare transmitted with rate λ (dimension [pck/s]). Note that this is generally different from thetraffic generation rate: however, by assuming that a node transmits for a sufficiently longtime, it is reasonable to assume that, on average, the packet transmission rate coincides withthe packet generation rate.

8.3 ConnectivityAs discussed earlier, the optimal common transmit power is the minimum power sufficientto preserve network connectivity. In this section, we formalize the definition of networkconnectivity. Conceptually, an ad hoc wireless network is often viewed as a graph, wherevertices represent the nodes and edges represent the links connecting neighboring nodes.From a graph-theoretical perspective, a network is connected if there is a path (possibly multi-hop) connecting any node to any other node in the network. However, using this notion ofconnectivity for an ad hoc wireless network, where a communication channel is error-prone,can be misleading. Since the wireless links are susceptible to errors, the QoS in terms of routeBER deteriorates as the number of hops in a route increases. Consequently, the performancemay be unacceptable although there is a sequence of links to the destination.

In order to take the physical layer characteristics into account, in this chapter, we considernetwork connectivity from a communication-theoretic viewpoint. In particular, a network issaid to be connected if any source node can communicate with a BER lower than a prescribed


Tier 1 Tier 2 Tier 3 …

Tx Rx

Figure 8.3 Tier structure of a network with square grid topology. (Reproduced bypermission of © 2005 IEEE.)

value BERth to a destination node placed at the end of a multi-hop route with an averagenumber of hops [56]. To be conservative, in most of this chapter we consider an ideal worst-case scenario where an information bit is relayed on each link of a route toward a destinationwithout retransmissions. However, it will be shown that use of retransmission techniqueslowers the BER and thus decreases the minimum power required to maintain the networkconnectivity. We will discuss this in more detail in section 8.7.1.

In addition, note that this notion of connectivity corresponds to requiring that, on average,a communication between a source and a destination can be guaranteed with a desiredquality. However, it does not guarantee that a source can communicate with every node inthe network with this QoS. A more stringent connectivity requirement, such that a source cancommunicate with every node in the network with the desired QoS, can also be enforced.The approach proposed in this chapter can be straightforwardly extended by considering theBER at the end of a multi-hop route with the maximum possible number of hops. In the nextsubsection, we derive a simple analytical expression for the average number of hops in thecases with grid and random topologies.

8.3.1 Square Grid TopologyDue to the spatial invariance on a torus, we can assume without any loss of generality that asource node is at the center of the network (see Figure 8.3). If a destination node is selectedat random, the minimum number of hops to reach the destination can range from 1 to 2imax,where imax is the maximum tier order. In other words, it takes one hop to reach a destinationwhich is a neighbor of a source node in Tier 1, and it takes 2imax hops to reach the farthestnode from the center in Tier imax. The average number of hops can be obtained by countingthe number of hops on a route from the source to each destination node and finding theaverage value. Assuming that each destination is equally likely, in Chapter 7 it has alreadybeen shown that the average number of hops on a route can be written as

ngrid = 1

N − 1

[4

imax∑i=1

i + 4imax∑i=1

2i + 8imax∑i=1

i−1∑j=1

(i + j)

]. (8.5)

8.3. Connectivity 201

The first summation term in (8.5) corresponds to the number of hops to reach any of the fournodes in alignment with the source at the center of the network in all possible tiers; the secondsummation corresponds to the number of hops to reach nodes on the four corners of each tier;finally, the third summation corresponds to the number of hops to reach the other nodes ineach tier. With straightforward algebra, (8.5) can be simplified to

ngrid = 2

N − 1

(2i3

max + 3i2max + imax

). (8.6)

Since imax � √N/2 when the number of nodes is sufficiently large, from (8.6), one obtains:

ngrid �√

N

2+ 1√

N+ 3

2�

√N

2. (8.7)

8.3.2 Two-Dimensional Poisson TopologyIn this case, we define an average route as a route between a source and a destination separatedby an average Euclidean distance between two randomly chosen points on a torus. We nowpropose an approach for computing the average path length. Let Z be the random variabledenoting the distance between a source and a destination. It can be shown that the probabilitydensity function (PDF) of Z is [41]

fZ(z) =

πz

2R20 ≤ z < R

πz

2R2 − 2z cos−1(R/z)

R2 R ≤ z <√

2R.

(8.8)

Based on (8.8), the following average distance between a source and a destination node canbe obtained:

Z =∫ √

2R

0zfZ(z) dz = R

3

[√2 + ln(1 + √

2)]. (8.9)

Intuitively, the average number of hops in a route must be proportional to the average hoplength, i.e. the average distance between two neighbors. For instance, one can expect that aroute will consist of many hops if each hop length is short. In our analysis, we assume thateach hop deviates, with respect to the reference path, by an angle �, where � is uniformlydistributed in the interval (−θ/2, θ/2) – the angles � and θ are shown in the exampleconsidered in Figure 8.2. By projecting each hop onto the reference path, the average numberof hops between a source and a destination can be approximated as follows:53

nrand � Z

E[W cos �] (8.10)

where E[W cos �] is the average projected hop length. In general, W and � are notindependent. However, as discussed in section 8.2.1, since it is highly likely that W ≤ R,

53To be precise, in the derivation of nrand in (8.10) one should consider E[W cos �/θ ] at the denominator on theright-hand side. However, in the remainder of this chapter, we assume that θ is given and all the expected values haveto be interpreted as conditional expectations. In general, the angle θ is likely to be known. For example, in a routingprotocol where a node does not select the next relay node that is in the opposite direction toward the destination, theangle θ is π .


one can conclude that W and � are basically independent (see the proof in Appendix E).Therefore, it follows that E[W cos �] � E[W ]E[cos �]. The expected values of W andcos � can be straightforwardly obtained from the PDFs of W and �. In fact, assumingthat the network area is large (i.e. R is large), it can be shown that E[W ] � √

π/2ρsθ andE[cos �] � (2/θ) sin(θ/2) [41]. Finally, using (8.9), (8.10) and the simplified expression forE[W cos �], it can be shown that

nrand �√

N√

θ3[√

2 + ln(1 + √2)]

6√

2π sin(θ/2)(8.11)

where we have used the fact that ρs = N/A.

8.4 BER at the End of a Multi-hop RouteSince the network connectivity is defined in terms of BER quality at the end of a multi-hoproute, in this section we present an approach to evaluate the route BER.

8.4.1 Square Grid TopologyIn Chapter 3, we proposed a rigorous detection-theoretic approach for the evaluation of theBER at the end of a multi-hop route in an ad hoc wireless network with square grid topology.The reader is referred to Chapter 3 for more details on this analysis. We now simply recallthe final expressions for the route BER.

Assuming that a bit detected erroneously at the end of a link is not corrected in successivelinks, the BER at the end of a route with ngrid links, denoted as BERgrid

route, can be written as

BERgridroute = 1 − (1 − BERlink)

ngrid (8.12)

where BERlink is the average link BER, which depends on the specific MAC protocol in use.Since we assume that each intermediate node decodes the signal before forwarding to thenext node in a route, this is regarded as decode-and-forward transmission. The case whereeach intermediate node simply combines the received signal and relays the signal withoutdecoding can also be considered following the approach in [215,216]. In Chapter 3, an exactexpression for the average link BER, i.e. BERlink, is derived in the case of binary signaling.In particular, our study shows that the dominant interference signals come from the nodes inthe first tier around the receiver, and interference from nodes in other tiers is not significant.

The route BER from the exact interference analysis presented in Chapter 3 shows theexistence of a BER floor for increasing values of the node spatial density or transmit power(see Chapter 3 for more details). This is intuitively expected because after the networkbecomes dense enough, increasing the node spatial density (i.e. reducing the hop length)no longer improves the link SNR, as the interfering nodes also become closer enough to thereceiver. However, the route BER predicted by the Gaussian assumption for the interferencesignal presents a much lower BER floor. Via a careful analysis, in the case where (i) theRESGO MAC protocol is used and (ii) communications are characterized by a strong line-of-sight (LOS), the route BER floor can be approximated with the following expression:

BERRESGOroute,floor = ngridξRESGOλL

4Rb(8.13)

8.4. BER at the End of a Multi-hop Route 203

where λ is the average packet transmission rate, L is the packet length, Rb is the data-rateand ξRESGO is a constant and can be thought of as a ‘tuning factor’. By experimenting withdifferent values of ξRESGO, it can be shown that an accurate approximation is obtained byconsidering ξRESGO = 3. Motivated by these considerations, in Chapter 3 it is shown that anaccurate approximate expression for the route BER can be written as

BERroute � max

1 −1 − Q

√√√√ 2αPt/dγ

link

Pthermal + Pgridint

ngrid

,ngridξRESGOλL

4Rb

(8.14)

where the first term in the maximum corresponds to the route BER at the end of a route withan average number of hops in the case where binary phase shift keying (BPSK) is used. Moreprecisely, in this expression Pt is the transmit power, γ is the path loss exponent (γ = 2 inmost of this book),

α � GtGrc2

(4π)2f 2c

(8.15)

Gt and Gr are the transmitter and receiver antenna gains (Gt = Gr = 1 for omnidirectionalantennas), fc = 2.4 GHz is the carrier frequency, c is the speed of light,

Pthermal = FkT0B (8.16)

F is the noise figure, k = 1.38 × 10−23 J/K is Boltzmann’s constant, T0 is the roomtemperature (T0 = 300 K), B is the transmission bandwidth and P

gridint is the average

interference power which, with the RESGO MAC protocol, can be given the followingexpression:54

Pgridint = ptranP

gridtotal = (1 − e−λL/Rb)P

gridtotal (8.17)

where

Pgridtotal = αPtIgrid (8.18)

and

Igrid � 1

dγ

link

imax∑i=1

[4

iγ+ 4

(√

2i)γ+

i−1∑j=1

8

(√

i2 + j2)γ− 1

]. (8.19)

54Actually, ptran = 1 − exp(−λL/Rb) is the probability that a node will transmit in a vulnerable interval giventhat it always has a packet to send. This is a pessimistic assumption, as there could be occasions when a node hasnothing to send (see Chapter 3 for more details). Thus, the actual probability that a node will transmit should belower. After a careful analysis, we have found that the actual probability that a relay node will transmit does nothave a simple closed-form expression. This is partly due to the fact that the arrival process of the relay node is nolonger a Poisson process. Thus, we believe that it is better to use a pessimistic assumption which allows us to derivean expression for transmission probability that provides insight into the problem. Since we are using a pessimisticassumption, the actual performance with the RESGO MAC protocol is expected to be better than what is predictedby our analysis.


8.4.2 Random TopologyIn the case of random topology, it is difficult to obtain exact analytical expressions for the linkBER and the route BER. Therefore, we resort to a simulation approach for the evaluation ofthe BER. For each communication link, transmitter–receiver pair is generated. The distancebetween the transmitter and the receiver (i.e. W ) is randomly generated from the hop lengthdistribution given in (8.4). Then, a number of interfering nodes within the circle of radius 2W

centered at the receiver are generated according to a two-dimensional Poisson distribution.Their positions are uniformly distributed. For each node, we schedule transmission of packetsaccording to the RESGO MAC protocol. For each bit received at the receiver, a threshold-based decision is made. The detected bit is declared as an error if it differs from the originalbit sent by the transmitter. The link BER is obtained by computing the ratio of the number oferroneous bits and the total number of received bits. To obtain the route BER, we generate aroute with nrand links and compute the route BER as follows:

BERrandroute = 1 −

nrand∏i=1

BERlinki (8.20)

where BERlinki is the BER of the ith link of a route. The average route BER is then obtainedby computing the arithmetic average of a large number of realizations of BERrand

route.

8.5 Optimal Common Transmit PowerIn this section, we derive the optimal common transmit power for ad hoc wireless networkswith grid and random topologies, respectively.

8.5.1 Optimal Common Transmit Power for Networks with SquareGrid Topology

With the exact BER analyzed in section 8.4.1, one can always numerically find the minimumtransmit power which would satisfy the desired BER threshold, provided that the BERthreshold is above the BER floor. However, it is more insightful to derive an analyticalexpression for the optimal transmit power. In this section, we will derive an expression forthe optimal transmit power, for a given maximum tolerable route BER, from the approximateroute BER expression given in (8.14).

In a network with grid topology, the common transmit power used by each node shouldbe large enough so that the BER at the end of a multi-hop route with an average numberof hops ngrid, given by (8.5), is lower than the maximum tolerable value, denoted as BERth.If the required BERth is higher than BERroute,floor, there exists a power which can satisfy thisrequirement. This transmit power is such that the following inequality must be satisfied:

1 − (1 − BERlinkgrid)ngrid ≤ BERth. (8.21)

Under the Gaussian assumption for the interference noise, one can rewrite (8.21) as

1 −1 − Q


link

Pthermal + E[P gridint ]

ngrid

≤ BERth. (8.22)

8.6. Performance Metrics 205

Substituting FkT0Rb for Pthermal, the expression given in (8.17) for E[P gridint ], and

rewriting (8.22) in terms of Pt, one obtains the following expression for the optimal transmitpower:

P ∗t = FkT0Rb

[2α

dγ

link�− α

(1 − e−λL/Rb

)Igrid

]−1

(dimension [W]) (8.23)

where

� �{Q−1

[1 − (1 − BERth)

1/ngrid]}2

(8.24)

and Igrid is given by (8.19). The expression given in (8.23) corresponds to the optimaltransmit power when BERroute,floor ≤ BERth for a given data-rate Rb, node spatial density(dlink � 1/

√ρs), number of nodes in the network (ngrid � √

N/2), antenna gains and carrierfrequency (α depends on them).

8.5.2 Optimal Common Transmit Power for Networks with RandomTopology

In a network with a two-dimensional Poisson node distribution, the hop length is random.Consequently, it is difficult to find a closed-form expression for the optimal transmit power.In this case, we resort to simulations. For a given data-rate and transmit power, we computethe average route BER with the approach described in section 8.4.2. The same process isrepeated for different combinations of data-rate and transmit power. At the end, we thusobtain the average route BERs for different combinations of data-rate and transmit power,and these are used to obtain contours of power–data-rate pairs in correspondence to whichthe route BER is equal to maximum tolerable BERth values.

8.6 Performance Metrics8.6.1 Node and Network LifetimeNetwork lifetime is an important performance indicator for wireless ad hoc and sensornetworks. There are several definitions of network lifetime. While network lifetime is definedas the time to the first node failure in [217, 218], it is defined in terms of the fraction ofsurviving nodes in the network in [219]. In this chapter, following [217,218], we consider thetime to the first node failure as the network lifetime (i.e. a worst-case approach).

In this section, we present a simple analysis for computing the average lifetime of a node.We assume that the RESGO MAC protocol is used and every node has an initial finite batteryenergy denoted by Ebatt. For a given data-rate Rb, the time taken to transmit one packet isL/Rb. Therefore, the total amount of energy consumed per transmitted packet can be writtenas

Epacket = Pt × L

Rb(dimension [J]). (8.25)

Since packets are transmitted with average rate λ, the average energy depleted per second issimply λEpacket. Finally, the total time it takes to completely exhaust the initial battery energy


can be written as

τ = Ebatt

λEpacket= EbattRb

λL(dimension [s]). (8.26)

Note that due to the uniform traffic assumption (i.e. all nodes generate approximately thesame amount of traffic load), on average, all nodes exhaust their battery at the same time.In addition, this simple analysis does not take into account the energy consumed when a nodeis receiving and processing packets. In reality, the lifetime of a node will be shorter thanwhat is predicted by our analysis. Recent studies have shown that the energy consumptionratio when a device is in idle mode, receiving mode and transmitting mode is 1 : 1.2 : 1.68(see e.g., [220]). Therefore, transmitting is the most ‘expensive’ (from a battery consumptionperspective) activity. Power consumption in idle and receiving modes can be taken intoaccount by properly extending the proposed approach.

8.6.2 Effective Transport CapacityThe effective transport capacity (dimension [b m/s]) is an important wireless networkperformance metric which simultaneously captures the amount of information generated andthe distance over which it can be transported in the network (see Chapter 5 for more details).As is evident from (8.12), the BER at the end of a multi-hop route increases as the distancetraversed by an information bit increases, i.e. as the number of hops increases. Consequently,the distance that an information bit can traverse while maintaining sufficiently low BER maybe lower than the length of a multi-hop route with an average number of hops (i.e. a messagecould reach the destination, but the final BER could be so high that the destination could notdecode the received message correctly). The following QoS condition, in terms of tolerableBER at the end of a multi-hop route with nh hops, formalizes the requirement of sufficientlylow route BER:

BER(nh)route ≤ BERth (8.27)

where BER(nh)route is the BER at the end of an nh-hop route. We refer to the distance which

a bit can traverse, while satisfying a desired route BER QoS condition, as the maximumsustainable distance. The effective transport capacity of a single route can then be defined as

Csreff � λ × L × min(ds, da) (8.28)

where ds is the maximum sustainable distance and da is the distance associated with anaverage multi-hop route. Note that in (8.28) the traffic transmission rate is used, whereasone should consider a traffic generation rate which represents the average rate of informationflowing in a route. However, since we have assumed in this chapter that a node alwayshas packets to send, the average flow rate in a route, in this case, is equal to the averagetransmission rate λ. The value min(ds, da) can be interpreted as the average sustainabledistance. It represents the distance that a bit, on average, can traverse. A fundamentalunderlying assumption in (8.28) is that only the source node of a route contributes ‘effective’information (on average, a source node transmits λL bits every second). In this sense, theintermediate nodes act as relay nodes, but they do not contribute to the effective transportcapacity in terms of information bits. However, relay nodes do contribute to the effectivetransport capacity in the sense of increasing the distance traversed by each bit transmitted bythe source.

8.6. Performance Metrics 207

In this chapter, the effective transport capacity of the network is obtained by addingthe single-route effective transport capacities of all the disjoint routes in the network.55

Assuming that each route has an average number of hops, the number of disjoint routescorresponding to a scenario where each node belongs to a particular route (either as thesource, the destination or a relay) in networks with grid and random topologies can beestimated as N/ngrid and N/nrand, respectively. The extension of the proposed analysis toa scenario where multi-hop routes can cross can be done by following the approach in [58].

In the following subsection, we derive analytical expressions for ds and da in networkswith square grid and two-dimensional Poisson topologies, respectively. The correspondingeffective transport capacities in both cases can be obtained using expression (8.28).

Square Grid Topology

First, we compute the maximum sustainable distance. In the case of a square grid topology,the route BER at the end of a generic nh-hop route can be obtained by replacing ngrid in (8.14)with nh. Thus, condition (8.27) will be satisfied if

max

1 −1 − Q


link

Pthermal + E[P gridint ]

nh

,nhξRESGOλL

4Rb

≤ BERth. (8.29)

Solving inequality (8.29) for n, the maximum sustainable number of hops can be written as

ngrids = min

{ln(1 − BERth)

ln(1 − BERlinkgrid),

4Rb BERth

ξRESGOλL

}. (8.30)

Since all hops in a network with grid topology have the same length dlink, the maximumsustainable distance is simply the product of the maximum sustainable number of hops andthe hop length:

dgrids = n

grids dlink. (8.31)

Similarly, the average route distance in a grid topology, i.e. dgrida , can be obtained as the

product of the average number of hops ngrid with the hop length dlink, i.e. dgrida = ngrid dlink.

Random Topology

As in the case of a network with grid topology, in order to obtain the single-route(and the aggregate) effective transport capacity of a network with random topology, weneed to compute the maximum sustainable distance and the average route distance. In thiscase, however, the maximum sustainable distance is obtained via computer simulations.We construct a route by recursively generating links whose lengths are obtained accordingto the CDF given in (8.4). After generation of a link, the route BER is then computed andcompared with the threshold value, i.e. BERth. If the BER is lower than BERth, a new link willbe generated and added to the route. This procedure is repeated until the route BER exceeds

55The disjoint routes assumption is particularly assumed here for simplicity in calculating the network’s effectivetransport capacity. However, other results derived in this chapter hold regardless of whether routes are disjoint ornot.


Table 8.1 Major network parameter values used in the considered networking scenarios

Parameters Values

Number of nodes in the network (N) 289 nodesNode spatial density (ρs) 10−7m−2

Packet length (L) 103 bitsPacket arrival rate at each node (λ) 0.5 pck/sPath loss exponent (γ ) 2Carrier frequency (fc) 2.4 GHzRoom temperature (T0) 300 KNoise figure (F ) 6 dB

BERth. The maximum sustainable distance is obtained by adding the lengths of the linksof the multi-hop route formed. A sufficiently large number of multi-hop routes are generatedand the average sustainable distance is computed by averaging over the maximum sustainabledistances corresponding to the route realizations.

What now remains to be computed is the average route distance in a random topology,i.e. d rand

a . The average number of hops in a route between a source and a destination is nrand.To find the average distance, one can compute the expected value of the route distance in anrand-hop route, that is

d randa = E

[nrand∑i=1

Wi

](8.32)

where Wi is the hop length of the ith link. Since nrand is a deterministic quantity, (8.32) canbe simplified to

d randa = nrandE[W ]. (8.33)

8.7 Results and DiscussionNumerical results, along with their implications, are presented and discussed in this section.The values of the major network parameters are given in Table 8.1, unless specified otherwise.

8.7.1 Optimal Transmit Power and Data RateIn Figure 8.4, the optimal common transmit power is shown as a function of the data-rate.The optimal power–data-rate curves are shown for various values of BERth. For each caseof BERth, both the optimal transmit power computed numerically with the exact interferenceanalysis and the optimal transmit power computed with the approximation given in (8.23)are compared. It can be observed that the optimal transmit power increases as the data-rateincreases. This can be explained as follows. Although transmitting packets at a higher data-rate reduces the vulnerable time (and hence gives smaller interference), increasing the data-rate (i.e. bandwidth) also increases the thermal noise. Therefore, the minimum transmit powerrequired to sustain the network connectivity has to increase accordingly. We also observe a

8.7. Results and Discussion 209

0 5 10 15 2010

0

101

102

103

Rb [Mb/s]

Pt*

[mW]

ApproximationExact

BERth

= 10−2

BERth

= 10−3 BERth

= 5x10−3

Figure 8.4 Optimal common transmit power in a network with square grid topology. Onlyinterference from nodes in the first tier around the receiver is considered.

close match between the optimal transmit power predicted with the approximation and thatobtained numerically from the exact interference analysis.

In addition, it can be observed from Figure 8.4 that there is a critical data-rate, beyondwhich the desired BERth cannot be satisfied for any transmit power. The critical data-rateoccurs at the point where the BERfloor for that particular data-rate becomes higher than thedesired BERth. Consequently, for an uncoded transmission and without retransmission, notransmit power can achieve the desired BER threshold.

The optimal transmit power is also minimized at the data-rate near the critical point.This suggests that the data-rate also plays an important role in the design of wireless ad hocand sensor networks; that is, for a given node spatial density, if the data-rate is carefullychosen, the transmit power can be minimized, prolonging the network’s lifetime.

A scenario with random topology is shown in Figure 8.5 for the case where BERth =10−3. Similar to the case of a grid topology, the optimal transmit power increases as the data-rate increases. However, in the case of random topology, as expected, a much higher transmitpower is required to sustain connectivity. This is due to the fact that the hop length is random.In other words, a long hop is likely to be present in a multi-hop route, and this significantlyreduces the route BER [41]. In addition, in the case of random topology, higher data-rates arerequired to support connectivity.

The optimal transmit power shown in Figure 8.4 is computed based on the assumptionthat there is no packet retransmission on each link of a route. In some applications, packetretransmission may also be used. In Figure 8.6, we compare the optimal transmit powerin three different scenarios: (i) no retransmission on each link; (ii) one retransmission isallowed on each link; and (iii) five retransmissions are allowed on each link. The valuefor BERth considered in this case is 10−3. It can be observed that allowing retransmissioncan decrease the minimum transmit power required to keep the network connected. This isbecause retransmission improves the BER, and thus a lower transmit power is required toget a packet to the destination with the desired quality. It can also be observed that thecritical data-rate shifts leftward as the number of retransmission increases. Thus, by allowingretransmission, the network connectivity can be supported at lower data-rates.


6 8 10 12 14300

400

500

600

700

800

Rb [Mb/s]

P*t

[mW]

Figure 8.5 Optimal common transmit power in a network with random topology and BERth= 10−3.

0 2 4 6 8 1010

0

101

102

103

Rb [Mb/s]

Pt*

[mW]

ApproximationExact

r = 0

r = 1

r = 5

Figure 8.6 Optimal common transmit power in a network with square grid topology andBERth = 10−3. Different numbers of retransmissions are compared.

8.7.2 Optimal Transmit Power and Node Spatial Density

Figure 8.7 illustrates the optimal transmit power as a function of the node spatial density.In this scenario, the data-rate is fixed at Rb = 4 Mb/s, and the considered BERth are 10−2

and 10−3, respectively. Both the optimal transmit power computed numerically with theexact interference analysis and the optimal transmit power computed with the approximationgiven in (8.23) are compared. It can be observed that the optimal transmit power decreaseslinearly, on a log scale, as the node spatial density increases. This can be explained as follows.When the network is sparse (i.e. low node spatial density), the minimum transmit power has tobe increased in order to preserve connectivity. However, as the network becomes denser, the


10−6

10−5

10−2

10−1

100

101

102

ρs [m−2]

Pt*

[mW]

ApproximationExact

BERth

= 10−2

BERth

= 10−3

Figure 8.7 Optimal common transmit power in a network with grid topology as a functionof the node spatial density.

nodes are closer to their neighbors, and therefore the power required to sustain connectivitydecreases.

Note that the BERroute,floor for the scenario considered in Figure 8.7 is fixed becausethe data-rate Rb, the number of nodes N , the traffic transmission rate λ, and the packetlength L are fixed. As long as the desired BERth is greater than this BERroute,floor, thereis always a minimum power to sustain the connectivity, at any node spatial density. On theother hand, in the case where BERth is less than the BERfloor, no transmit power can satisfythe desired connectivity criteria at any node spatial density, unless forward error correctionor retransmission techniques are used.

8.7.3 Effects of Strong Propagation Path LossIn a realistic communication environment, some links may experience higher path loss thanothers, due to impairments such as shadowing [161]. In order to study how path loss affectsthe optimal common transmit power, next we consider scenarios where some links in a multi-hop route are characterized by higher path loss than the others. We study the impact of thesehigh path loss links on the optimal transmit power by varying the number of these links in amulti-hop route with an average number of hops.

Figure 8.8 illustrates the optimal common transmit power as a function of the data-rate ina scenario where the RESGO MAC protocol is used and the network topology is square grid.Four scenarios are considered: (i) all links in an average multi-hop route are characterizedby a path loss exponent γ = 2; (ii) m = 1 link is characterized by a path loss exponentγ = 3, whereas the other links in the route are still characterized by γ = 2; (iii) m = 3links are characterized by a path loss exponent γ = 3, whereas the other links in the routeare still characterized by γ = 2; and (iv) all links in the route are characterized by a path lossexponent γ = 3. In the scenarios where the path loss exponents for different links are not thesame, attenuation of the signals originating from interfering nodes will also vary. If the pathloss for each interfering node, relative to the receiver, were known, then the exact interferencepower could be computed. However, this would only correspond to a specific realization of


2 4 6 8 1010

−1

100

101

102

103

Rb [Mb/s]

Pt*

[mW] all links with γ = 2 m = 1 m = 3all links with γ = 3

Figure 8.8 Optimal common transmit power in a network with square grid topology, forvarious configurations of the path loss exponents for the links.

the network, where one needs to determine the propagation loss on each individual link fromany node to any other node.

In order to estimate the interference power, we assume, for simplicity, that the interferencepower from interfering nodes attenuate with a common path loss exponent γc, given by thefollowing heuristic expression:

γc = ngrid − m

ngrid2 + m

ngrid3. (8.34)

In other words, expression (8.34) for the common path loss exponent corresponds to aweighted average between 2 and 3. The intuition behind (8.34) is that if the number of linksper route with γ = 2 is large, then the path loss exponent of each interfering node to thereceiver should be close to 2; on the other hand, if the number of links with γ = 3 is large,then the path loss exponent of each interfering node to the receiver should be close to 3.We emphasize that the expression in (8.34) is only a heuristic. However, the results obtainedwith this analysis provide, at least trendwise, meaningful insights.

Generally, it can be observed from Figure 8.8 that the power required to sustain networkconnectivity increases as the number of links with γ = 3 increases. Moreover, the transmitpower has to be increased substantially even if only one link in a multi-hop route experienceshigher path loss than the others. This has a significant implication. It suggests that the use ofa common transmit power will work well in situations where each link experiences relativelythe same level of propagation loss. If it is highly likely that at least one link per route isgoing to experience unusually high path loss, then it is not efficient to use a common transmitpower because all nodes have to significantly raise their transmit power to recover from thedegradation caused by a few high path loss links. A possible solution to this problem is tomodify the routing protocol in such a way that, at the moment of route creation, link quality isselected as one of the routing criteria [221, 222]. With this capability, a routing protocol maybe able to select a multi-hop route which bypasses high path loss links. This solution will beexplored in more detail in Chapter 9. Another potential solution is obviously considering localpower control [37, 174, 175]. These possible extensions are currently under investigation.


103

102

0

100

200

300

Pt [W]

N

103

102

0

2

4

6x 10

6

Pt [W]

[s]

(a)

(b)

Figure 8.9 (a) Increase in the connectivity level and (b) lifetime of a node as functions ofthe transmit power in a network with grid topology.

8.7.4 Connectivity Robustness to Node Spatial Density Changes

The optimal transmit power analyzed earlier is the minimum transmit power which issufficient to maintain network connectivity at a particular node spatial density. In somesituations, the node spatial density of a network can change over time, for example, whennodes join/leave the network or when the batteries of some nodes are exhausted. A networkcan lose connectivity immediately if the node spatial density is lower than the initial valuefor which the power is optimally chosen. In order to provide robustness against a possiblenode spatial density reduction, nodes may transmit at a power level higher than the requiredminimum level. In this section, we evaluate the connectivity robustness as the networkbecomes sparser. We define the initial density at which the network starts forming asρs0 � N0/A, where N0 is the initial number of nodes in the network. Keeping the networkarea constant, the node spatial density decreases if the number of nodes becomes lower thanN0.56 We denote the decrease in the number of nodes as N � N0 − Nf, where Nf is thefinal number of nodes after some nodes disappear from the network – some nodes could eithermove out of the network or their batteries could be exhausted.

Figure 8.9 (a) shows a plot of N as a function of the transmit power. In this scenario,the initial number of nodes is N0 = 289 and the network surface area is A = 108 m2.The minimum transmit power required to sustain the network connectivity for this initialnode spatial density is P ∗

t0 = 1.6 mW. In this figure, for each transmit power, the node spatialdensity and the number of nodes Nf at which the network can still maintain connectivity iscalculated.

This allows one to obtain N , which, in this figure, can be interpreted as the connectivityrobustness against a change in terms of node spatial density. For example, N = 100indicates that a network can still maintain connectivity even if 100 nodes disappear.

56The proposed analysis can be straightforwardly extended to a scenario where the number of nodes is fixed andthe network area increases.


0 50 100 150 200 250

106

107

τ [s]

∆ N

Figure 8.10 Lifetime of a node as a function of connectivity.

It can be observed from Figure 8.9 (a) that as the transmit power increases, N

increases.57 This is intuitively expected because for higher transmit power, the network cantolerate lower node spatial densities, i.e. it can work even if the network becomes sparser.However, as the transmit power level increases, the rate of gain, in terms of connectivityrobustness, asymptotically decreases.

While connectivity robustness can be increased with higher transmit power, this comes atthe expense of the lifetime of the nodes and the network. Figure 8.9 (b) illustrates the behaviorof the lifetime of a node as a function of the transmit power. In particular, the lifetime ofa node is given by (8.26). As expected, the node lifetime decreases as the transmit powerincreases to support communication in a sparser network.

As shown in Figure 8.9 (a) and (b), it is clear that there is a tradeoff between connectivityincrease and the longevity of the nodes. This tradeoff is clearly shown in Figure 8.10, wherenetwork lifetime is shown as a function of the connectivity robustness. It can be observed thatthe slope of the curve becomes steeper as N increases. This suggests that trading networklifetime for connectivity is beneficial when the level of connectivity is small; however, thebenefit decreases at high values of N . In other words, to have high protection against nodespatial density changes, significant network lifetime has to be sacrificed.

The increased robustness against node spatial density reduction is also exhibited by thebehavior of the effective transport capacity. Figure 8.11 shows the behavior of the effectivetransport capacity, in a scenario with a square grid topology, as a function of the number ofnodes. The results are obtained assuming that the data-rate is Rb = 4 Mb/s and the desiredlevel of BER is BERth = 10−3. By keeping the area constant, an increase of the number ofnodes corresponds to an increase of the node spatial density. Three levels of transmit powersare considered: (i) P ∗

t0 = 1.6 mW, (ii) 2 × P ∗t0 and (iii) 5 × P ∗

t0 .Considering the effective transport capacity curve, one is mostly interested in the region

where the network is connected. This is the region where an information bit can be transported

57The decrease in the number of nodes ( N ) is computed as if the nodes are still in a square grid topology after N nodes disappear from the network. In reality, nodes will probably disappear from the network in a randomfashion, and the topology is likely to be random after some nodes disappear. Thus, the power required to maintainnetwork connectivity is expected to be much higher than what is shown in Figure 8.9. However, the results shown inFigure 8.9 provide a meaningful insight in terms of the trend involved.


100 200 300 400 500 6000

5

10

15x 10

7

N

CTe

[b−m/s]

Pt0

*

2 x Pt0

*

5 x Pt0

*

Figure 8.11 Effective transport capacity in a network with grid topology.

over an average route while satisfying the QoS constraint on the maximum tolerable BER, i.e.BERth. The solid curve in Figure 8.11 indicates the effective transport capacity in the regionwhere the network has connectivity. The connectivity region for each considered transmitpower starts from the point where each dashed curve reaches the solid curve and ends atthe knee of the curve (e.g. around N = 425). It can be seen from Figure 8.11 that whenthe transmit power is increased from P ∗

t0 to 2 × P ∗t0 , the connectivity region of the network

increases. More specifically, the network gains connectivity at N = 289 in the case wherethe transmit power is P ∗

t0 whereas it gains the connectivity at a much lower number of nodes(e.g. around N = 160) in the case where the transmit power is 2 × P ∗

t0 . Hence, transmittingat a power higher than the minimum required level can increase the connectivity range, in thesense that communications can be supported also in sparser networks. In other words,

• if the network is already connected, increasing the transmit power does not increasethe effective transport capacity;

• if the network is not connected, increasing the transmit power might make it connected,increasing dramatically the effective transport capacity.

8.7.5 Practical Determination of the Optimal Transmit PowerIn this chapter, we have presented an analytical approach to determine the optimal commontransmit power for network connectivity. Next, we briefly discuss how this optimal transmitpower can be determined practically. There are at least two possible strategies based on thecapability of the devices. First, in a wireless network where a device can operate with onlya single power level, the optimal transmit power can be determined at the design phase. Forexample, one can estimate the number of nodes and the size of the coverage area that awireless sensor network will be deployed, and then configure a device such that it operates atthe optimal transmit power. Alternatively, if the transmit power of a device is preconfiguredat the manufacturing stage, in the network planning phase, one can determine the minimumnode spatial density required for this transmit power. This enables a network designer todetermine the minimum number of nodes to be deployed in the coverage area.


Second, if each device has an option of changing its transmit power, a distributed powercontrol algorithm can be used. By adapting the COMPOW algorithm [171], the minimumcommon transmit power for which the route BER satisfies the desired QoS can be determined.Assuming that each device has several discrete power levels, this can be done as follows.Starting from the maximum power level, each node creates a routing table corresponding tothis power level. This allows each node to identify all the possible destinations that it couldreach. For each destination, a node sends a probing packet to check the route BER at thedestination. The same process is repeated for the other available power levels. The minimumpower which yields the same routing table as if the maximum power were used and satisfiesthe required route BER is the optimal power level.

A careful reader may note that our approach does not take an opportunistic longer-hoprouting approach although it may become available when the transmit power becomes larger.However, we emphasize that although the transmit power is enough to reach a node furtherthan the nearest neighbor, it may still not be enough to reach the destination with the desiredlevel of BER. Certainly, there are tradeoffs between choosing the nearest neighbor andchoosing the furthest node within the transmission range. The latter is similar to selectinga route with the lowest number of hops to the destination. For more detail on the tradeoffsbetween these two routing strategies, we refer the interested reader to [221, 222].

8.8 Related WorkThere are a number of distributed power control algorithms reported in the literature, andit is important to emphasize that they are designed to achieve different objectives. In [170],Foschini and Miljanic proposed a distributed power control algorithm for a cellular networksuch that the minimum signal-to-interference ratio requirement per user is met. An algorithmproposed by Bambos and Kandukuri [223] attempted to balance the power and delay tradeoff,by using local information such as the last transmitted power level, the last interference leveland the backlog at the transmitter, to determine the appropriate power to transmit in the nexttime slot. In [224], Bharghavan et al. proposed an algorithm which aims at improving channelefficiency. In [225], Elbatt et al. proposed a power management scheme for improving theend-to-end network throughput. In [217], Chang and Tassiulas proposed a routing algorithmwhich aims at balancing the energy consumption among the nodes. However, none of thesedistributed power control algorithms were designed with the objective of ensuring networkconnectivity.

For our purpose, a distributed power control algorithm should adjust the power suchthat the minimum power is used to achieve network connectivity. Distributed powercontrol algorithms that are designed with network connectivity and network longevity astheir objectives also exist. Some examples of these algorithms are [173], [37] and [219].One example of distributed power control algorithms for determining the minimum commontransmit power that keeps the network connected is the COMPOW algorithm [171], whichhas been discussed at the end of the previous section. To the best of our knowledge, none ofthe previous studies has considered the network connectivity from a communication-theoreticviewpoint, where the desired QoS in terms of BER at the end of a multi-hop route is enforced,as presented in this chapter (and, more generally, in this book).

A similar analysis of BER at the end of a multi-hop route, as presented in this chapter,also exists in the literature. In [215], Boyer et al. provide a detailed analysis of the route BERin both the decoded relaying multi-hop channel case and the amplified relaying multi-hop

8.9. Conclusions 217

channel case. In our analysis, we only consider the decoded relaying channel case; that is weassume that each intermediate node decodes a received bit before forwarding to the next nodein the route. However, our BER analysis also include multiple access interference whereasthis is not considered in [215].

8.9 ConclusionsIn this chapter, we have investigated the optimal common transmit power for wirelesssensor networks. In particular, the optimal common transmit power has been defined asthe minimum transmit power sufficient to preserve network connectivity. As opposed tofollowing the conventional graph-theoretic approach, we consider connectivity from a morerealistic viewpoint by taking into account the characteristics of a wireless communicationchannel. Our study shows that, for a given data-rate and a given maximum tolerable BERat the end of a multi-hop route, there exists an optimal transmit power. Moreover, for agiven value of the maximum tolerable route BER, there exists a global optimal data-ratefor which the optimal common transmit power is the minimum possible. This suggeststhat the data-rate, if chosen carefully, can guarantee significant savings in terms of transmitpower, prolonging the battery life of devices and network lifetime. This is useful in designingwireless sensor networks. Since the node spatial density of a sensor network is typicallyknown a priori (i.e. the number of sensors and the coverage area are known), a networkdesigner can determine the optimal power and the optimal data-rate to operate. Conversely,in the case where the transmit power and the data-rate of a sensor node is pre-configured, anetwork designer can use the approach presented in this chapter to determine the minimumnode spatial density required to keep the network connected.

However, caution should be exercised when using the common power control schemeproposed in this chapter, as its performance is significantly affected by the variation inpropagation path loss. In other words, the common power control scheme performs wellin the case where each link in the network experiences similar propagation path loss.When variation in propagation path loss occurs, the transmit power needs to be increasedsubstantially even though there is only one link per route which experiences high path loss.A possible solution to this problem is to add intelligence into the routing protocol. Moreprecisely, a routing protocol that takes link quality as one of its route selection criteria couldbe an attractive option. However, if it is not possible to avoid a high propagation loss link bymeans of routing, local power control may be required.

In addition, there are tradeoffs between robustness to change in network connectivity andnetwork longevity. Considering a transmit power higher than the optimal value helps preserveconnectivity of networks in the presence of node spatial density changes. However, thiscomes at the expense of node/network lifetime. Moreover, in a scenario where the network isconnected, our results show that increasing the transmit power does not increase the effectivetransport capacity at all.

Finally, the analytical framework presented in this chapter can be further extended tostudy other interesting scenarios. For example, one may extend this work to estimate theoptimal transmit power for other MAC and routing protocols. The channel model can alsobe modified to take into account multi-path fading, following the approach presented inChapters 2 and 3.

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Ad Hoc Wireless Networks (A Communication-Theoretic Perspective) || Optimal Common Transmit Power for Ad Hoc Wireless Networks