Optimal Transmission Range for Wireless Ad Hoc
Networks Based on Energy Efficiency
Jing Deng, Member, IEEE, Yunghsiang S. Han, Member, IEEE, Po-Ning Chen, Senior Member, IEEE,
and Pramod K. Varshney, Fellow, IEEE
Abstract—The transmission range that achieves the most eco-nomical use of energy in wireless ad hoc networks is studied for uniformly distributed network nodes. By assuming the existence of forwarding neighbors and the knowledge of their locations, the average per-hop packet progress for a transmission range that is universal for all nodes is derived. This progress is then used to iden-tify the optimal per-hop transmission range that gives the maximal energy efficiency. Equipped with this analytical result, the relation between the most energy-economical transmission range and the node density, as well as the path-loss exponent, is numerically in-vestigated. It is observed that when the path-loss exponent is high (such as four), the optimal transmission ranges are almost identical over the range of node densities that we studied. However, when the path-loss exponent is only two, the optimal transmission range decreases noticeably as the node density increases. Simulation re-sults also confirm the optimality of the per-hop transmission range that we found analytically.
Index Terms—Energy efficiency, optimal transmission range, wireless ad hoc networks.
I. INTRODUCTION
T
HE RESEARCH on wireless ad hoc networks has expe-rienced a rapid growth over the last few years. Unique properties of ad hoc networks, such as operation without pre-existing infrastructure, fast deployment, and self-configuration, make them suitable for communication in tactical operations, search and rescue missions, and home networking. While most studies in this area have concentrated on the design of rout-ing protocols, medium access control protocols, and security issues, we investigate the efficiency of energy consumption in wireless ad hoc networks in this work. Due to their portability and fast-deployment in potentially harsh scenarios, nodes in ad Paper approved by Y. Fang, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received December 13, 2004; re-vised April 7, 2006. This work is supported in part by the National Science Coun-cil of Taiwan, R.O.C., under Grant NSC 90-2213-E-260-007 and Grant NSC 91-2213-E-260-021, and in part by Chung-Shan Institute of Science and Tech-nology, Taiwan, under Grant XC93B95P. The work of J. Deng was supported in part by Louisiana Board of Regents under RCS Grant LEQSF (2005-08)-RD-A-43. This paper was presented in part at the IEEE Wireless Communications and Networking Conference, Atlanta, GA, March 2004.J. Deng is with the Department of Computer Science, University of New Orleans, New Orleans, LA 70148 USA (e-mail: jing@cs.uno.edu).
Y. S. Han is with the Graduate Institute of Communication Engineering, National Taipei University, Taipei, Taiwan 10617, R.O.C. (e-mail: yshan@ mail.ntpu.edu.tw).
P.-N. Chen is with the Department of Communication Engineering, Na-tional Chiao-Tung University, Hsinchu, Taiwan 300, R.O.C. (e-mail: qponing@ mail.nctu.edu.tw).
P. K. Varshney is with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 USA (e-mail: varshney@ ecs.syr.edu).
Digital Object Identifier 10.1109/TCOMM.2007.904395
hoc networks are usually powered by batteries with finite ca-pacity. It is always desirable to extend the lifetime of ad hoc network nodes without sacrificing their functionality. Thus, the study of energy-efficient mechanisms is of importance.
In wireless ad hoc networks, energy consumption at each node is mainly due to system operation, data processing, and wireless transmission and reception. While there are studies on increasing battery capacity and reducing energy consumption of system operation and data processing, energy consumption economy of radio transceivers has not received as much atten-tion. Such a study is also quite essential for an energy-efficient system design [1]. In some previous work, the radio transmis-sion range of nodes in wireless networks was optimized based on local neighborhood information so that desirable network topologies can be dynamically established with less transmis-sion interference [2]–[6]. In this paper, the radio transmistransmis-sion range is considered to be a static system parameter that is deter-mined a priori, i.e., during system design, and used throughout the lifetime of a wireless ad hoc network.
When two communicating nodes are not in range of each other in wireless ad hoc networks, they need to rely on multihop trans-missions. In such a case, packet forwarding or packet routing becomes imperative. The selected value of radio transmission range considerably affects the network topology and node en-ergy consumption. On the one hand, a large transmission range increases the distance progress of data packets toward their final destinations. This is unfortunately achieved at the expense of high energy consumption per transmission. On the other hand, a short transmission range uses less energy to forward packets to the next hop, but a large number of hops are required for packets to reach their destinations. Thus, there exists an optimum value of the radio transmission range.
There have been some publications [7]–[10] that concentrated on the optimization of radio transmission range in wireless net-works. In [7], the optimal transmission radii1that maximize the
expected packet progress in the desired direction were deter-mined for different transmission protocols in a multihop packet radio network with randomly distributed terminals. The optimal transmission radii were expressed in terms of the number of terminals in range. It was found that the optimal transmission radius for slotted ALOHA without capture capability covers on an average eight nearest neighbors in the direction of the packet’s final destination. The study concentrated on improving system throughput by limiting the transmission interference in a wire-less network with heavy traffic load. The energy consumption,
1We use transmission “range” and “radius” interchangeably in our paper.
however, was not considered in the paper. Similar assumptions were made in [8], which further allowed all nodes to adjust their transmission radii independently at any time. It was found in [8] that a higher throughput could be obtained by transmitting pack-ets to the nearest neighbor in the forward direction. In [9], the authors evaluated the optimum transmission ranges in a packet radio network in the presence of signal fading and shadowing. A distributed position-based self-reconfigurable network protocol that minimizes energy consumption was proposed in [10]. It was shown in [10] that the proposed protocol can stay close to the minimum energy solution when it is applied to mobile networks. The optimization of transmission range as a system design issue was studied in [11]. The wireless network was assumed to have high node density, and to consist of nodes with rela-tively low mobility and short transmission range. As justified by the assumption of high node density, the authors further as-sumed that intermediate routing nodes are always available at the desired location whenever they are needed. Considering the nodes without power control capability, the authors argued that the optimal transmission range can be set at the system de-sign stage. Specifically, they showed that the optimal one-hop transmission progressive distance is independent of the physical network topology, the number of transmission sources, and the total transmission distance; and that it only depends on the prop-agation environment and radio transceiver device parameters.
A similar assumption was made in [12], even though the node density was only considered for the energy consumption of overhearing nodes. They investigated the problem of select-ing an energy-efficient transmission power to minimize global energy consumption for ad hoc networks. They concluded that the average neighborhood size is a useful parameter in finding the optimal balance point.
Zuniga and Krishnamachari [13] studied the optimal trans-mission radius that minimizes the settling time for flooding in large-scale sensor networks. In the paper, the settling time was evaluated at the time when all the nodes in the network have for-warded the flooded packet. Regional contention and contention delay were then analyzed.
A bit-meter-per-joule metric for energy consumption in wire-less ad hoc sensor networks was investigated in [14]. The paper presented a system-level characterization of energy consump-tion for sensor networks. The study assumed that the sensor network has a relay architecture, and all the traffic is sent from sensor nodes toward a distant base station. Also, it was assumed that the source always chooses, among all relay neighbors, the one that has the lowest bit-meter-per-joule metric to relay its data packets. In the analysis, the power efficiency metric in terms of average watt per meter for each radio transmission was first calculated, and was then extended to determine the global energy consumption. The analysis showed as to how the over-all energy consumption varies with transceiver characteristics, node density, data traffic distribution, and base-station location. In this paper, we determine the optimal transmission range that achieves the most economical use of energy under the as-sumption of uniformly distributed network nodes. Assuming the existence of forwarding neighbors and the knowledge of their locations, we first derive the average per-hop packet progress
for a transmission range that is universal for all nodes, and then use the result to determine the optimal per-hop transmission range that gives the maximal energy efficiency. The relation-ship between the most energy-economical per-hop transmission range and the node density, as well as the path-loss exponent, is then numerically investigated. We observed that the optimal transmission range varies markedly in accordance with the node densities at low path-loss exponent values (such as two) but re-mains nearly constant at high path-loss exponent values (such as four). We also found that the node density needs to be extremely high for the result of [11] to be valid, and the optimal transmis-sion radius under low to medium node densities is actually far away from the results reported in [11].
Simulations were performed to investigate the applicability of the optimal per-hop transmission range that we derived to the situation where the energy efficiency of the entire path from the originating source node to the final destination is considered. Results showed that the overall energy efficiency almost peaks at the same transmission range as the per-hop energy efficiency. In order to account for the situation that forwarding neighbors may not exist, contrary to the assumption made in our analysis, we have also simulated an extended connectivity-guaranteed transmission strategy that allows a network node to increase its initially preset transmission range until an appropriate for-warding neighbor appears. We found that the optimal radii of the connectivity-guaranteed transmission strategy, as well as the corresponding maximum energy efficiency, are almost identical to those obtained from the original strategy.
In summary, contrary to the dynamic transmission range em-ployed in [5], [8], and [10], our study determines a single static optimal energy-efficient transmission range for all nodes in the network. Compared with [11], our study does not make the as-sumption that a relay node that is closest to the destination can always be found; thus, the wireless networks that we study need not be highly dense. Compared with [14], the network that we consider does not have any base station or common receiver; also, we do not assume that the destination is far away from the source.
Our paper is organized as follows. The analysis of the single-hop energy-efficient radius is presented in Section II. Analytical and simulation results along with discussions are provided in Section III. In Section IV, we propose an extended connectivity-guaranteed transmission strategy and compare it with our orig-inal strategy. Section V concludes the paper.
II. ANALYSIS OFFIRST-HOPDISTANCE-ENERGYEFFICIENCY In this section, we analyze the distance-energy efficiency for the first hop in a wireless ad hoc network with randomly dis-tributed nodes as we consider the snapshot at the time of the
first-hop transmission, even if a multihop transmission is
sub-sequently required for the packet to reach its ultimate desti-nation. Specifically, the first-hop distance-energy efficiency is defined as the ratio of the average progress of a packet during its first transmission and the energy consumption of that trans-mission. As any intermediate relay transmission can be viewed as a new first-hop transmission for the remaining route, the
first-hop distance-energy efficiency should be consistent with the overall distance-energy efficiency of the entire route in a homogeneous environment. (This will be later substantiated by simulations in Section III.)
A. Network Model and Transmission Strategy
Suppose that a source node S is located at the center of a circle of radius x, where x is the largest possible distance between S and any destination. In other words, the source node will not send any packet to nodes outside the circle. The destination node D, to which S intends to transmit a data packet, is assumed to be uniformly distributed over the entire circle.
Due to the limited radio range (or equivalently, limited trans-mission energy), a packet from its originating source node to its destination node may need to be sequentially routed by a cer-tain number of intermediate nodes, which we term as routers. It is assumed that all nodes, including the source node and the intermediate nodes, employ a common transmission radius r. Consequently, direct transmission to the destination occurs only when the destination node is within distance r from the source node.
Any node within the transmission range of a node is called its
neighbor. We assume that each node knows the locations of all
its neighbors and the location of the destination node.2 Based
on this assumption, a transmission strategy can be designed as follows.
Step1) The source node S transmits a packet to the
destina-tion node D directly, if D is located within distance
r from S.
Step2) When the destination node D is outside the
trans-mission range of the source node S, the packet is sent to the neighbor that is closer in distance to the destination node D than the source node S, and that is closest to the destination D among all neighbors.
Step3) Since the source node S knows the locations of all
neighbor nodes and the destination node, it will not send out the packet when there does not exist any neighbor satisfying condition given in Step 2), and will postpone the transmission until such a neighbor appears.
As pointed out in [12], the selection of transmission radius influences energy consumption and network connectivity. It can be shown that the probability of having no forwarding neighbor is usually negligibly small. Further discussions on the connec-tivity issue will be provided in Section IV.
The probability that n nodes appear in an area of size A is given by (ρA)ne−ρA/n!, where ρ is the density parameter for
this two-dimensional Poisson point process [7].3 The appear-ance of nodes in any two nonoverlapping areas are assumed to be independent.
2This can be achieved by measuring the strength of the received signals from
the neighboring nodes and by the use of locationing service such as geographical location service (GLS) [15], [16].
3Note that we have used the definition of node density as the average number
of nodes per unit area. An alternative definition is the number of neighbors per node within its transmission radius. Since we focus on the network-wide average, these two definitions would lead to similar results.
Fig. 1. Illustration of a relaying node.
The energy consumption corresponding to each transmission can be formulated as [11]
Et(r) = k1rω+ k2
where r is the radio transmission range, ω is the path-loss ex-ponent, and k1 and k2 are the parameters determined by the
characteristic of the transceiver design and the channel. Let Er
be the energy consumption of receiving, decoding, and process-ing data packets at the receiver. It is to be noted that Erdoes not
include the energy consumption of the overhearing nodes in the neighborhood of the sender. Inclusion of such extra energy con-sumption may affect transmission range optimization. Thus, we infer that the single-transmission energy consumption is given by Et(r) + Er. Throughout this work, we do not count the
ex-tra energy consumption due to packet reex-transmissions similar to [11] and [14].
We next determine the average progress of a transmitted packet in a single hop.
B. Average Single-Transmission Progress
Denote the distance between the source node S and the des-tination node D by v. When v≤ r, direct transmission to the destination node D can be attained; hence, the distance progress of the transmitted packet to the destination node D is v. In the situation where v > r, the source node has to locate an ap-propriate neighbor for subsequent packet routing. In this case, we define the distance progress as the difference between the before-hop distance (between the sender and the destination) and the after-hop distance (between the relay node and the des-tination) [14]. The distance progress toward the destination node
D is, therefore, equal to (v− z), where z is the distance between
the first-hop router T and the destination node D (cf., Fig. 1). Denote by P the random variable corresponding to the dis-tance progress for a single transmission. Let V and Z be, respec-tively, the random variables corresponding to v and z discussed earlier. Define a new random variable H as
H =
1, if a neighbor satisfying Step 2) in the transmission strategy is available
0, otherwise.
With the given notations, the problem of finding the average single-transmission progress is equivalent to the derivation of
the expected value of P , where P = V , if V ≤ r
[cf. transmission strategy Step 1)] V − Z, if V > r ∩ H = 1
[cf. transmission strategy Step 2)] 0, if V > r∩ H = 0.
[cf. transmission strategy Step 3)] (1)
We note that if (V ≤ r) ∪ (V > r ∩ H = 1) is false, then no transmission will take place according to transmission strategy Step 3); hence, no energy is consumed, and the progress is zero. It can, therefore, be easily verified that E[P ] = E[P|(V ≤ r)
∪(V > r ∩ H = 1)].
We now proceed to derive the expectation of P given that (V ≤ r) ∪ (V > r ∩ H = 1) is true. Observe that
Pr{P > p |(V ≤ r) ∪ (V > r ∩ H = 1)} = Pr{(P > p) ∩ [(V ≤ r) ∪ (V > r ∩ H = 1)]} Pr{(V ≤ r) ∪ (V > r ∩ H = 1)} = Pr P > p∩ V ≤ r∪(P > p)∩ (V > r ∩ H = 1) Pr{(V ≤ r) ∪ (V > r ∩ H = 1)} = Pr{P > p ∩ V ≤ r} + Pr {P > p ∩ V > r ∩ H = 1} Pr{V ≤ r} + Pr {V > r ∩ H = 1}
where the last step follows from the fact that both the events in the numerator and the denominator are mutually exclusive. Substituting (1) into the previous expression yields
Pr{P > p |(V ≤ r) ∪ (V > r ∩ H = 1)}
= Pr{p < V ≤ r} + Pr {V − Z > p ∩ V > r ∩ H = 1} Pr{V ≤ r} + Pr {V > r ∩ H = 1} .
(2) The statistics specified in Section II-A then immediately implies
Pr{p < V ≤ r} = (r 2− p2) x2 1{p < r} and Pr{V ≤ r} = r 2 x2 (4)
where 1{·} denotes the set indicator function, and x repre-sents the largest possible distance between the source and any destination. It remains to determine Pr{V > r ∩ H = 1} and Pr{V − Z > p ∩ V > r ∩ H = 1}.
Let AS D denote the area of the overlapping region between
the circle centered at S with radius r and the circle centered at
D with radius v, i.e., the shaded region in Fig. 1. We can divide AS Dinto two regions by the circle centered at D with radius z.
The areas of these two regions are, respectively, denoted as A1
and A2as shown in Fig. 1. Then
Pr{V > r ∩ H = 1} = x 0 Pr{V > r ∩ H = 1|V = v}dPV(v) = x r
Pr{at least one neighbor exists in area AS D}dPV(v)
(5) = x r 1− e−ρ AS D(v ,r )2v x2dv = 1−r 2 x2 − 2 x2 x r ve−ρAS D(v ,r )dv (6) where the lower integration limit is r in (5) because of the condition of V > r, PV(v) = P r(V ≤ v), and AS D(v, r) = r2cos−1 r 2v + v2cos−1 1− r 2 2v2 −1 2r (2v + r)(2v− r). (7)
This completes the determination of Pr{V > r ∩ H = 1}. From Fig. 1, we have
Pr{Z ≥ z ∩ H = 1|V = v} = Pr{H = 1|V = v}, if z≤ v − r Pr{no neighbors inA1and
at least one inA2}, if v− r < z < v
0, if z≥ v.
By the independence of node appearance in nonoverlapping regions, the above expression for v− r < z < v can be rewritten as
Pr{Z ≥ z ∩ H = 1|V = v}
= Pr{no neighbors in A1} Pr{at least one in A2}
= e−ρA1(z ,v ,r )1− e−ρA2(z ,v ,r ) = e−ρA1(z ,v ,r )− e−ρAS D(v ,r ) where A1(z, v, r) = r2cos−1 r2+ v2− z2 2rv + z2cos−1 z2+ v2− r2 2vz −1 2 (r + v + z)(v + z− r)(r +v − z)(r +z −v). (8) Therefore Pr{Z < z ∩ H = 1|V = v} = Pr{H = 1|V = v} − Pr {Z ≥ z ∩ H = 1|V = v} = 0, if z≤ v − r 1− e−ρA1(z ,v ,r ), if v− r < z < v 1− e−ρAS D(v ,r ), if z≥ v (9) where we have used Pr{H = 1|V = v} = 1 − e−ρAS D(v ,r )in the above derivation.
Using (9), we obtain Pr{V − Z > p ∩ V > r ∩ H = 1} = x r Pr{Z < v − p ∩ H = 1|V = v} dPV(v) = x r Pr{Z < v − p ∩ H = 1|V = v}2v x2dv = 1− r 2 x2− 2 x2 x r ve−ρA1(v−p,p,r)dv 1{p < r}. (10)
This completes the determination of Pr{V − Z > p ∩ V >
r∩ H = 1}.
Finally, substituting (4), (6), and (10) into (2), we obtain that for p > 0 Pr{P > p |(V ≤ r) ∪ (V > r ∩ H = 1)} = x2− p2− 2x r ve−ρA1(v−p,v,r)dv x2− 2x r ve−ρ AS D(v ,r )dv 1{p < r}.
The expected value of P , given the validity of (V ≤ r) ∪ (V >
r∩ H = 1), is then equal to (21.9) in [17] EP|(V ≤ r) ∪ (V > r ∩ H = 1) = ∞ 0 Pr{P > p |(V ≤ r) ∪ (V > r ∩ H = 1)} dp = 1 x2− 2x r ve−ρ AS D(v ,r )dv · r 0 x2− p2− 2 x r ve−ρA1(v−p,v,r)dv dp =3x 2r− r3− 6r 0 x r ve−ρA1 (v−p,v,r)dv dp 3x2− 2x r ve−ρ AS D(v ,r )dv
and the single-transmission distance-energy efficiency e(r) is given by e(r) = E P|(V ≤ r) ∪ (V > r ∩ H = 1) k1rω+ k2+ Er = 3x 2r− r3− 6r 0 x r ve−ρA1(v−p,v,r)dv dp 3 (k1rω+ k2+ Er) x2− 2x r ve−ρ AS D(v ,r )dv . (11) By reformulating A1(z, v, r)=r2A¯1(z/r, v/r) and AS D(v, r)= r2A¯
S D(v/r), and defining ¯z = z/r and ¯v = v/r, where
¯ A1(¯z, ¯v)=cos−1 1 + ¯v2− ¯z2 2¯v +¯z2cos−1 ¯ z2+ ¯v2− 1 2¯v ¯z −1 2 (1 + ¯v + ¯z)(¯v + ¯z−1)(1+ ¯v− ¯z)(1+ ¯z− ¯v), (12) and ¯ AS D(¯v) = cos−1 1 2¯v2 + ¯v2cos−1 1− 1 2¯v2 −1 2 (2¯v + 1)(2¯v− 1) (13) we can simplify the single-transmission distance-energy effi-ciency e(r) to e(r) = r 3k1(rω+ k0)· g1(r) g2(r) (14) where k0= (k2+ Er)/k1 g1(r) = 2 + 6 1 0 x/r 1 ¯ v 1− e−ρr2A¯1(¯v−¯p, ¯v ) d¯v d¯p and g2(r) = 1 + 2 x/r 1 ¯ v 1− e−ρr2A¯S D(¯v ) d¯v.
C. Optimum Transmission Radius in High-Density Networks
In high-density networks, i.e., when ρ is considerably large, we can approximate 1− e−ρr2A¯1(¯v−¯p, ¯v )≈ 1 and 1 − e−ρr2A¯S D(¯v )≈ 1 and obtain g1(r) g2(r)≈ 2 + 6011x/r¯vd¯vd¯p 1 + 21x/r¯vd¯v = 3− r2 x2. Therefore e(r) = r 3k1(rω+ k0)· g1(r) g2(r) ≈ 3x2r− r3 3k1x2(rω+ k0)
which gives maximum value at some positive r satisfying (ω− 3)rω +2− 3(ω − 1)x2rω− 3k0r2+ 3k0x2= 0. (15)
For the special case of ω = 2, (15) reduces to
r4+ (3x2+ 3k0)r2− 3k0x2= 0.
Thus, the optimal r for ω = 2 is equal to −(3x2+ 3k 0) + (3x2+ 3k 0)2+ 12k0x2 2 .
We depict the optimal transmission range for high-density networks in Fig. 2 for x ranging from 40 to 200 m and
k0= 222.56 m2.4It can be observed from this figure that the
optimal transmission range goes from 13.74 to 14.86 m while
x is changed from 40 to 200 m. This suggests that under high
node density, x is not a dominant factor in the determination of the optimal transmission range when the path-loss exponent
ω is 2.
For an environment with ω = 3, (15) becomes 2x2r3+ k0r2− k0x2= 0.
The real solution r for the above equation is equal to
− k0 6x2+ k2 0 6x2 −k3 0+ 54k0x6+ 6k0x3 81x6− 3k2 0 1/3 + −k3 0+ 54k0x6+ 6k0x3 81x6− 3k2 0 1/3 6x2 .
Again, the optimal transmission range for high density networks at ω = 3 is depicted in Fig. 3 for x ranging from 40 to 200 m
Fig. 2. Optimal transmission range of high-density networks (path-loss expo-nent ω = 2).
Fig. 3. Optimal transmission range of high-density networks (path-loss expo-nent ω = 3).
and k0= 222.56 m3. From this figure, we conclude that the
optimal transmission range goes from 4.78686 to 4.80901 m while x is changed from 40 to 200 m. This again suggests that, for a moderately large x, optimal transmission range is quite insensitive to the values of network radius x.
III. ANALYTICAL ANDSIMULATIONRESULTS
Analytically evaluated distance-energy efficiency and simu-lation results for its verification are summarized in this section. Quantities k1 and k2+ Er are assumed to be 6.6319× 10−5
and 1.476× 10−2, respectively, unless specified otherwise.5
5The parameters chosen are the same as in [11] for the purpose of comparison.
Systems with different types of hardware will generally lead to different set of optimum transmission ranges, as shown in Fig. 10.
Fig. 4. Distance-energy efficiency (ω = 2).
A. Analytical Results
Fig. 4 compares the analytically obtained first-hop distance-energy efficiencies calculated by (14) for different node densi-ties. The network coverage area is assumed to be a circle with radius 100 m (i.e., x = 100). The path-loss exponent ω is as-sumed to be 2. Node density varies from 0.005 to 0.04, which corresponds to 157–1256 nodes on an average in a circle with radius 100 m.
It can be observed from Fig. 4 that the first-hop distance-energy efficiency improves initially for small r, and then de-grades after r exceeds a certain value. Fig. 4 also shows that the distance-energy efficiency in a network with higher node density is higher. The explanation of this result is that the prob-ability of finding relay nodes closer to the final destination is higher when there are more nodes in the network. Thus, each hop makes more progress toward the final destination, thereby improving the distance-energy efficiency.
Additionally, we observe from Fig. 4 that the optimal trans-mission range (r∗) changes for different node densities. When the node density is 0.005, the optimal transmission range is around 25 m. It reduces to 17 m when ρ reaches 0.04. Such a decrease in r∗with an increasing ρ is due to the increase in relative first-hop progress with respect to the radio transmission range; therefore, a smaller transmission range achieves better energy efficiency when ρ is larger. It is to be noted that this observation regarding r∗agrees with that found in [11] under a strong assumption that a source node can always find a neigh-bor at the required location to forward its data packet, which is valid only in networks with very high node density. Our anal-ysis, however, shows that the same conclusion is reached for networks with low to medium node density.
Fig. 5 compares the analytical first-hop energy-distance ef-ficiencies for different node densities for a larger path-loss ex-ponent ω = 4. It shows that, when ω = 4, the optimum trans-mission range remains around 3 m for all node densities ρ lying between 0.005 and 0.04. Therefore, the node density has little
Fig. 5. Distance-energy efficiency (ω = 4).
Fig. 6. Distance-energy efficiency for different network radius x (ω = 2).
effect on the optimal transmission radius when signals encounter serious attenuation. Notably, due to a more rapid signal attenu-ation, this optimal transmission radius is markedly smaller than 17–25 m obtained at ω = 2. This result confirms that the optimal transmission range can be set at the system design stage.
Fig. 6 compares the analytical results in (14) for different network radii. The path-loss exponent ω is assumed to be 2 in Fig. 6. It clearly illustrates that the optimal transmission range decreases as ρ grows. The optimal transmission range
r∗ increases slightly as the network radius x increases. Fig. 7 shows this trend more clearly. We can also see from Fig. 7 that for the same amount of increase in x, the increment in optimal transmission radius is larger when node density ρ is smaller. The analytical results from Figs. 6 and 7 clearly show that the optimal transmission radius is no longer 15 m for low node density. The optimal transmission radius is close to 15 m only when the node density is extremely high, e.g., ρ = 1.0. Therefore,
Fig. 7. Optimal transmission range for different node densities (ω = 2).
Fig. 8. Optimum transmission range for different ω.
the results presented in [11] are applicable only for very high node density networks. Detailed discussion on simulation results will be presented in Section III-B.
In Fig. 8, we compare the optimal transmission range that maximizes the distance-energy efficiency e(r) for different node densities and path-loss exponents. When ω = 2, a decrease in optimum transmission range is observed with an increase in node density. A direct interpretation is that as ρ decreases, it is less likely to find a neighbor node to relay the data pack-ets efficiently toward the final destinations; thus, the optimum transmission range increases as ρ decreases. This interpretation, however, is not applicable to other values of ω. For instance, when ω = 4, the optimum transmission range remains relatively flat for different node densities. This can be explained with the prohibitively high energy consumption on increasing the trans-mission range when path-loss exponent is high. When path-loss exponent ω lies between 2 and 4, the trend of r∗as a function
Fig. 9. Distance-energy efficiency for ρ = 0.02 and ω = 2.
of ρ becomes less predictable, which is due to the combined effects of the two factors discussed earlier.
We can view the curves in Fig. 8 in the context of the number of neighbors m seen by the sender. For a fixed transmitted power, both the node density and the path-loss exponent have similar effects on m. In fact, m increases linearly with node density; it increases exponentially as path-loss exponent decreases. When the path-loss exponent is small (such as 2) and node density increases, the optimum transmission range is shifted to a smaller value as more neighbors are seen. As ω increases, the benefit of increasing r to increase m diminishes.
B. Simulation Results
Simulations (programs written in C language) have been per-formed to verify our analytical results. In our simulations, the network nodes are distributed in a circular region according to a two-dimensional Poisson distribution. The circle is centered at (0, 0), with radius x ranging from 50 to 150 m. The source node is fixed at (0, 0), while destination nodes are randomly chosen in the circle. The source node transmits the packets to the selected destination node in accordance with our transmission strategy. We measured the average first-hop distance-energy efficiency of each pair of source and destination. All the results presented are the average of 500 runs, each of which selects 100 destinations randomly.
In Fig. 9, we compare the numerical results on first-hop distance-energy efficiency with the simulation results under the conditions of ρ = 0.02 and ω = 2. The simulation results of first-hop distance-energy efficiency match with our numerical results quite well. As shown in this figure, the optimal transmis-sion range that maximizes first-hop distance-energy efficiency is about 18 m. As indicated in Fig. 9, the value of x only mildly affects the first-hop distance-energy efficiency in the ranges of
r that we have simulated.
In addition, we simulated the overall distance-energy effi-ciency (not just the first-hop, but the entire path from the orig-inating source node to the final destination). As illustrated in
Fig. 10. Distance-energy efficiency for ρ = 0.02 and ω = 2; k1in this figure
is 20 times the value in all other figures (i.e., k1= 1.3264× 10−3).
Fig. 9, the first-hop distance-energy efficiency and the overall distance-energy efficiency are approximately the same when r is small. Their difference is more noticeable when r becomes larger. The first-hop distance-energy efficiency is slightly larger than the overall distance-energy efficiency. This is anticipated as the last hop is usually not as efficient as all other hops.6In gen-eral, a larger x results in a little higher overall distance-energy efficiency. Such a slight difference could be due to the increase in the number of hops for a larger x. From our simulations, the average number of hops for the packets to reach the final destination is larger for a larger x; so the influence of the small last-hop progress is less significant.
We present several other simulation results in Figs. 10 and 11. We use Fig. 10 to demonstrate the effect of energy consumption characteristics on the distance-energy efficiency and optimal transmission range. In Fig. 10, the value of k1is chosen as 20×
6.6319× 10−5= 1.3264× 10−3. Such a large k1 represents
transceiver hardwares that use relatively larger portion of energy for packet transmission. As shown in Fig. 10, the peak of the distance-energy efficiency shifts to lower range of r when k1is
larger. The optimal transmission range becomes roughly 4 for such a k1value.
In Fig. 11, we show the simulation results and numerical results for ω = 4. With a higher path-loss exponent, the en-ergy consumption of each hop increases quickly as r increases. Therefore, the benefit of increasing the transmission range di-minishes quickly. Based on Fig. 11, the optimal transmission range is around 3. The similarity in shape in Figs. 10 and 11 implies that an increase of either the transmission energy con-sumption characteristics k1or path-loss exponent ω has similar
effects on the distance-energy efficiency and the optimal trans-mission range.
6Based on a uniform selection of traffic destinations, the expected value of the
last-hop progress is approximately r/2, which is smaller than the average-hop progress when an adequate number of nodes are present.
Fig. 11. Distance-energy efficiency for ρ = 0.02 and ω = 4.
IV. EXTENDEDTRANSMISSIONSTRATEGY
Although the probability of no forwarding neighbors is fairly small when ρπr2 is moderately large,7 its occurrence can still
disconnect the link between the source and the destination nodes. In order to account for the relatively rare situation that forwarding neighbors may not exist, we have simulated an ex-tended transmission strategy, which allows the network nodes to increase their initially preset transmission range until an appro-priate forwarding neighbor is found. The extended transmission strategy is the same as our original transmission strategy in Section II-A except Step 3):
Step 3’) When a node cannot find a forwarding node using the preset transmission radius based on Step 2), it increases its transmission radius until such a for-warding node appears.
It is interesting to evaluate the distance-energy efficiency of the extended transmission strategy for the first-hop as well as for the entire path. The simulation results are summarized in Fig. 12. The node density ρ is 0.02, and the path-loss expo-nent ω is 2. It can be observed that the extended transmission strategy and the original transmission strategy perform almost the same in terms of energy-distance efficiency when the trans-mission radius is large. This is an anticipated result since the probability of no-forwarding neighbors is negligibly small for a large transmission radius. When the transmission radius is small, however, the distance-energy efficiency of the extended transmission strategy becomes markedly better than that of the original transmission strategy because the actual transmission range is increased more frequently as no forwarding neighbor exists. Nevertheless, the optimal transmission ranges, as well as the resulting maximum distance-energy efficiencies, are almost identical for both original and extended transmission strate-gies. We, therefore, conclude that the energy-economic radius
7It can be shown that this probability is upper bounded by e−(23−
√ 3 2π)N (ρ ), where N (ρ) = ρπr2 is the average number of nodes within transmission
range r.
Fig. 12. Distance-energy efficiency for ρ = 0.02 and ω = 2. The bot-tom subfigure depicts the probability of no-relay neighbor in the range of transmission radius. This probability is calculated as EV[P0(v, r)] =
x r e−ρAS D
(v , r )(2v/x2)dv [18]. The optimal transmission radii, respectively,
for first-hop, first-hop (mod), overall, overall (mod) are 16, 16, 18, and 18, which result in the energy-efficiencies of 381, 376, 353, and 352, respectively.
derived in this paper, when used in a static manner, is a reason-able approach for the design of energy-efficient wireless ad hoc networks.
V. CONCLUDINGREMARK
The radio transmission range as a system parameter affects the energy consumption economy of wireless ad hoc networks. On the one hand, a large transmission range increases the expected progress of a data packet toward its final destination at the expense of a higher energy consumption per transmission. On the other hand, a short transmission range consumes less per-transmission energy, but requires a larger number of hops for a data packet to reach its destination.
Based on the underlying device energy consumption model and a two-dimensional Poisson node distribution, we have proposed an analytical model to investigate the optimal value of the radio transmission range. The optimal transmission range for a location-aware transmission strategy is then determined. Our analysis shows that the optimum transmission radius is influenced more by the node density than the network coverage area. It is observed that when the path-loss exponent is four, the optimal transmission ranges are almost identical over the range of node densities that we studied. However, the optimal transmission range decreases noticeably as the node density increases when the path-loss exponent is only two. Our results can be used to determine suitable radio transmission power for wireless ad hoc networks or wireless sensor networks in the predeployment phase.
Compared with other methods that also assume network nodes having adjustable transmission power (and, thus, trans-mission range) [5], our technique will not lead to unidirectional links and requires little maintenance once the common opti-mal transmission power is identified. This ensures the practi-cality of our technique. The examination of the connectivity-guaranteed transmission strategy that allows a sender to extend
its transmission range to force the appearance of a forwarding node, further confirms the applicability of our analysis.
In determining the distance-energy efficiency, this paper as-sumed that the transmission power Et(r) is an increasing
func-tion of transmission range r, and is given by k1rω+ k2 [11].
However, in some systems, the same transmission power may result in different effective transmission ranges due to the use of different code punctuation or modulation schemes. An example is the implementation of IEEE 802.11a, where data rates of 1 and 54 Mbps result in quite different effective transmission ranges even with the same transmission power. Therefore, a node can effectively increase its transmission range by reducing the data rate without changing its transmission power, contrary to fixing the data rate and adapting the transmission range by dynami-cally adjusting the transmission power. It would be interesting to consider such a data-rate adaptive possibility to conserve en-ergy. Furthermore, the interference among multiple traffic flows may affect the transmission range optimization by introducing packet collisions and retransmissions. We leave it as our future work.
ACKNOWLEDGMENT
The work of Deng and Han was partially performed during their visit to the CASE Center and the Department of Electrical Engineering and Computer Science, Syracuse University, Syra-cuse, NJ, USA. We would also like to express special thanks to Prof. Y. Fang and all the anonymous reviewers for their valuable comments.
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Jing Deng (S’98–M’02) received the B.E. and M.E.
degrees in electronic engineering from Tsinghua Uni-versity, Beijing, China, in 1994 and 1997, respec-tively, and the Ph.D. degree in electrical and computer engineering from Cornell University, Ithaca, NY, in 2002.
He is currently Assistant Professor with the De-partment of Computer Science, University of New Orleans, New Orleans, LA. From 2002 to 2004, he visited the CASE Center and the Department of Elec-trical Engineering and Computer Science at Syracuse University, Syracuse, NY, as a Research Assistant Professor. He was a Teaching Assistant from 1998 to 1999 and a Research Assistant from 1999 to 2002 with the School of Electrical and Computer Engineering, Cornell University. He is an Associate Editor of the International Journal of Mobile Communications, Net-works, and Computing. His research interests include mobile ad hoc netNet-works, wireless sensor networks, wireless network security, energy efficient wireless networks, and information assurance.
Dr. Deng was the Co-Chair of the IEEE International Workshop on Ad Hoc and Ubiquitous Computing, Taiwan, and the Sponsorship Chair of the 1st Inter-national Conference on Multimedia Services Access Networks, Orlando, FL. He was a member of the technical program committees of many IEEE confer-ences including MASS’05, MASS’06, GLOBECOM’06, and WCNC’07. He is a member of the IEEE Computer Society, the IEEE Communications Society, and the Association for Computing Machinery.
Yunghsiang S. Han (S’90–M’93) was born in Taipei,
Taiwan, R.O.C., in 1962. He received the B.S. and M.S. degrees in electrical engineering from the Na-tional Tsing Hua University, Hsinchu, Taiwan, in 1984 and 1986, respectively, and the Ph.D. degree from the School of Computer and Information Sci-ence, Syracuse University, Syracuse, NY, in 1993.
From 1993 to 1997, he was an Associate Profes-sor with the Department of Electronic Engineering, Hua Fang College of Humanities and Technology, Taipei, Taiwan. From 1997 to 2004, he was with the Department of Computer Science and Information Engineering, National Chi Nan University, Nantou, Taiwan, where he became a Full Professor in 1998. From June to October 2001, he was a Visiting Scholar with the Department of Electrical Engineering, University of Hawaii, Manoa, and from September 2002 to January 2004, he was the SUPRIA Visiting Research Scholar with the Department of Electrical Engineering and Computer Science and CASE Center, Syracuse University, NY. He is currently with the Graduate Institute of Commu-nication Engineering, National Taipei University, Taipei, Taiwan. His research interests include wireless networks, security, and error-control coding.
Po-Ning Chen (S’93–M’95–SM’01) was born in
Taipei, Taiwan, R.O.C., in 1963. He received the B.Sc. and M.Sc. degrees in electrical engineering from the National Tsing-Hua University, Taiwan, in 1985 and 1987, respectively, and the Ph.D. degree in electrical engineering from the University of Mary-land, College Park, in 1994.
From 1985 to 1987, he was with the Image Pro-cessing Laboratory, National Tsing-Hua University, where he worked on the recognition of Chinese char-acters. After a two-year military service, he joined Star Tech. Inc. in 1989, where he developed the first prototype of finger-print recognition systems in the company. He joined Wan Ta Technology, Inc. as a Vice General Manager in 1994, conducting several projects on point-of-sale systems. He was a Member of the Research Staff with the Advanced Technol-ogy Center, Computer and Communication Laboratory, Industrial TechnolTechnol-ogy Research Institute, Taiwan, where he led a project on Java-based network man-agements. He joined the Department of Communications Engineering, National Chiao-Tung University, Taiwan, as an Associate Professor in 1996, where he became a Full Professor in 2001. His research interests include information and coding theory, large deviations theory, distributed detection, and sensor networks.
Dr. Chen was the recipient of the 2000 Young Scholar Paper Award from Academia Sinica, Taiwan.
Pramod K. Varshney (S’72–M’77–SM’82–F’00)
was born in Allahabad, India, in 1952. He received the B.S. degree in electrical engineering and com-puter science (with honors), and the M.S. and Ph.D. degrees in electrical engineering from the Univer-sity of Illinois at Urbana-Champaign, Champaign, in 1972, 1974, and 1976, respectively.
From 1972 to 1976, he was a Teaching and Re-search Assistant with the University of Illinois at Urbana-Champaign. Since 1976, he has been with Syracuse University, Syracuse, NY, where he is cur-rently a Professor of electrical engineering and computer science and the Re-search Director of the New York State Center for Advanced Technology in Computer Applications and Software Engineering. He is also an Adjunct Pro-fessor of Radiology with the Upstate Medical University, Syracuse, NY. He has served as a consultant to several major companies. His current research interests include distributed sensor networks and data fusion, detection and estimation theory, wireless communications, image processing, radar signal processing, and remote sensing. He is the author of several publications, including Distributed Detection and Data Fusion (Springer-Verlag, 1997). He is on the Editorial Board of Information Fusion.
Dr. Varshney was a James Scholar, a Bronze Tablet Senior, and a Fellow at the University of Illinois. He is a member of the Tau Beta Pi and is the recipient of the 1981 ASEE Dow Outstanding Young Faculty Award. He was the Guest Editor of the special issue on data fusion of the PROCEEDINGS OF THEIEEE in January 1997. He received the Third Millennium Medal from the IEEE and Chancellor’s Citation for exceptional academic achievement at Syracuse Uni-versity in 2000. He is a Distinguished Lecturer for the Aerospace and Electronic Systems Society of the IEEE. He was the President of the International Society of Information Fusion in 2001.
