 Research Article
 Open Access
Lower Bound of EnergyLatency Tradeoff of Opportunistic Routing in Multihop Networks
 Ruifeng Zhang^{1, 2}Email author,
 JeanMarie Gorce^{3},
 Olivier Berder^{1} and
 Olivier Sentieys^{1}
https://doi.org/10.1155/2011/265083
© Ruifeng Zhang et al. 2011
 Received: 1 May 2010
 Accepted: 13 October 2010
 Published: 18 October 2010
Abstract
Opportunistic networking aims at exploiting sporadic radio links to improve the connectivity of multihop networks and to foster data transmissions. The broadcast nature of wireless channels is an important feature that can be exploited to improve transmissions by using several potential receivers. Opportunistic relaying is thus the first brick for opportunistic networking. However, the advantage of opportunistic relaying may be degraded due to energy increase related to having multiple active receivers. This paper proposes a thorough analysis of opportunistic relaying efficiency under different realistic radio channel conditions. The study is intended to find the best tradeoff between two objectives: energy and latency minimizations, with a hard reliability constraint. We derive an optimal bound, namely, the Pareto front of the related optimization problem, which offers a good insight into the benefits of opportunistic routings compared with classical multihop routing schemes. Meanwhile, the lower bound provides a framework to optimize the parameters at the physical layer, MAC layer, and routing layer from the viewpoint of cross layer during the design or planning phase of a network.
Keywords
 Relay Selection
 Packet Error Rate
 Forwarding Node
 Delay Constraint
 Additive White Gaussian Noise Channel
1. Introduction
Opportunistic networking refers to all techniques which provide advantages from the use of spontaneous radio links in mobile adhoc networks (MANETs) [1]. In classical networking instead, the link between pairwise node is distinguished as being connected or not. This kind of classification relies on the traditional layered design of networking: the PHY and MAC layers ensure a perfect connection of some preselected links and the routing layer selects the path from these links.
However, in practice, due to the complexity of wireless environment and possible movement of nodes or surrounding objects, a network is an evolving process with unstable links and always remaining in an intermediate state, that is, partially connected, as presented in [2–5]. Moreover, these works have shown that unreliable links can be effectively exploited to improve the performance of the MAC or routing layer. More specifically, opportunism may improve the performance of multihop routing by selecting one relay according to its instantaneous linking status. Additionally, in our previous work [6], we quantified how unreliable links can improve the connectivity of Wireless Sensor Networks (WSNs) by exploiting the broadcast nature of the wireless medium, where all nodes are assumed to be simultaneously active and to try to receive a packet.
The objective of this paper consists of the evaluation of the maximal efficiency that can be achieved with such opportunistic routing schemes, namely, finding an optimal set of forwarding nodes to balance energy cost and the transmission delay. The efficiency of opportunistic communications can be evaluated from different viewpoints. For multihop networks, we identify three important performance parameters: the endtoend reliability, the endtoend delay (referred to as latency hereafter), and the energy consumption. Accordingly, to evaluate opportunistic communications fairly, we introduce a multiobjective optimization framework [10]. Due to the fundamentality of reliability, we consider it as a hard constraint in this paper. The use of acknowledged transmissions allows to fulfill this constraint, at least from a theoretical point of view. The other two constraints, that is, energy and latency, are considered as two competing objective functions that should be simultaneously minimized.
Several previous works on opportunistic routings, such as [7, 9, 11–13], provide the analysis of energy and latency performances. In [7, 11], energy and latency performances of a routing scheme called GeRaF are analyzed, and the effects of node density, traffic load, and duty cycle are evaluated. The simulations in [12] show the impacts of node density, radio channel quality and traffic rate on the energy consumption at each node, the average delay of packet and the goodput of opportunistic protocol. It is concluded that the benefit of opportunistic scheme is about 10% decrease in power and 40% reduction in delay. Whereas these analysis are based on an unrealistic disc link model [5, 6], which relies on the definition of a reception threshold level and is not well adapted to the research of opportunistic communications due to the neglect of propagation phenomena, for example, fading and shadowing. Furthermore, the energy efficiency of the protocol CAGIF [13] is studied in a fading channel, where the whole set of neighbor nodes try to receive packets from the source node, which may degrade energy performance. In order to improve the energy efficiency, an efficient selection mechanism of relay nodes is proposed in [9], rather than choosing all the neighbors as relay candidates. Simulation results of [9] in a shadowing channel indicate that the energy efficiency is greatly improved.
However, in the aforementioned studies, a fixed transmission power is considered, and the number of relay candidates is chosen according to a given routing policy, without providing any proof of optimality. Therefore, these studies are insufficient to determine whether the relative low performances of opportunistic routing are intrinsic to this kind of routing or due to the specific protocol (relay selection policy, fixed power choice, etc.).
Concerning this question, we propose in this paper to analyze the lower bound of the energylatency tradeoff for opportunistic communications under a hard endtoend reliability constraint. To obtain this bound, we consider the size of candidate cluster and the transmission power as variables of the optimization problem. Firstly, we do not focus on the relay selection mechanism here but on the two following issues: what is the best set of relay candidates and what is the performance of the optimized set of candidates? Then, according to the theoretical analysis, we propose an opportunistic routing protocol to reach the best performance of a network.
With regard to the routing policy, we assume that for a given cluster, only the candidate closest to the destination is selected to forward the packet. Such a strategy obviously relies on the assumption that each node has the full knowledge of the position of itself and the destination. Once a node has a packet to send, it appends the locations of itself and the intended relay cluster to the packet and then broadcasts it. The relay candidates which successfully receive the packet (solid nodes in Figure 1) assess their own priorities of acting as relay, based on how close they are to the destination. The best relay which is the closest to the destination forwards the packet, as shown in Figure 1. In contrast with the the aforementioned schemes, this scheme utilizes an optimized candidate cluster, instead of all active neighbors, to receive the packet for the purpose of saving energy and taking advantage of the spatial diversity.
 (i)
A general framework for evaluating the maximal efficiency of the opportunistic routing principle is provided. Energy and latency are compromised under an endtoend reliability constraint.
 (ii)
The Pareto front of energylatency tradeoff is derived in different scenarios. A closedform expression of energylatency tradeoff, when the number of relay candidates is fixed, and an algorithm to find the optimal number of relay candidates are proposed. The simulation results verify the correctness of this lower bound in a 2dimension Poisson distributed network. The numerical analysis show that opportunistic routing is inefficient in an Additive White Gaussian Noise (AWGN) channel; however, it is efficient in Rayleigh block fading and Rayleigh fast fading channels on the condition of a smallsize cluster.
 (iii)
The lower bound of energy efficiency and its corresponding maximal delay are derived.
 (iv)
An opportunistic protocol is proposed to minimize energy consumption under a delay constraint.
The remaining part of this paper is organized as follows: Section 2 describes the models and metrics used in this paper. In Section 3, the lower bound of energydelay tradeoff of onehop transmission is deduced, and the lower bound of energy efficiency is obtained for the delaytolerant applications. The result about onehop transmission is extended to the scenario of multihop transmissions in Section 4, and the gain of opportunistic communications on the energy efficiency is analyzed. In Section 5, the lower bounds proposed are applied to optimize the physical parameters of a network. In Section 6, a novel opportunistic protocol is introduced, and the simulation results using this protocol verify the theoretical analysis. Section 7 discusses the significance of these results and gives some conclusions.
2. Models and Metrics
Notations.
Symbol  Description  Value 

 Pathloss exponent (≥2)  3 
 Amplifier proportional offset (>1)  14.0 
 SignaltoNoise Ratio  
 ACK Ratio  0.08125 
 Overhead Ratio  
 Number of bits per symbol 

 Bandwidth of channel  250 KHz 
 Minimum transmission distance  
 Onehop transmission distance  
 Carrier frequency  2.4 GHz 
 Receiver antenna gain  1 
 Transmitter antenna gain  1 
 Circuitry loss  1 
 Modulation order  
 Noise level  −150 dB m/Hz 
 Number of bits in an ACK packet  78 
 Number of bits per packet  2560 
 Number of bits in the overhead of a data packet  0 
 Number of cooperative receivers  
 Optimal number of cooperative receivers  
 Minimum number of cooperative receivers  
 Link probability  
 Optimal link probability  
 Minimum optimal transmission power  
 Optimal transmission power  
 Receiver circuitry power  59.1 mW 
 Startup power  38.7 mW 
 Transmission power  
 Transmitter circuitry power  59.1 mW 
 Bit rate  250 Kbps 
 Code rate  1 
 Symbol rate  
 Delay from queuing  
 Startup time  0 
2.1. System Model
with , where is the surface of , represents the exponential function.
We consider the case of a source node forwarding a packet to a sink/destination node . is one of 's neighbors which is closer to than .
where and are the Euclidian distance between and and that between and , respectively.
is aware of its own location and those of its neighbors and the destination and gets their link probability s, such that can select a set of forwarding candidates among its neighbors according to some kind of priority, for example, the distance to the destination node. Let denote the forwarding candidate set, which includes all the nodes involved in the local collaborative forwarding. The number of nodes in is .
2.2. Energy Consumption Models
where represents the length's ratio between ACK and data packets. We also assume that an ACK packet is much smaller than a data packet [17], that is, .
where . The related parameters are described in Table 1.
2.3. Realistic Unreliable Link Models
where denotes the transmission distance between nodes and , for the sake of simplicity. Refer to Table 1 for other parameters.
where and rely on the modulation type and order; for example, for Multiple Quadrature Amplitude Modulation (MQAM), and . For BPSK, and .
where is the link probability between and node as defined in (9).
where is the number of transmissions, and are the successful transmission probability of data packet and ACK packet, respectively, calculated by (14).
where stands for for simplification.
2.4. Mean Energy Distance Ratio Per Bit ( )
It should be noticed that this metric integrates all physical and link layers parameters, so that we can use this metric to analyze the joint PHY/MAC efficiency.
2.5. Mean Delay Distance Ratio ( )
The delay of a packet to be transmitted over one hop, , is defined as the sum of three delay components. The first component is the queuing delay during which a packet waits for being transmitted, . The second component is the transmission delay that is equal to . The third component is . Note that we neglect the propagation delay because the transmission distance between two nodes is usually short in multihop networks.
Note that includes all factors of physical and link layers also, so and are the effective metrics to measure the effect of these parameters on the energy efficiency and the delay of a network.
3. EnergyDelay Tradeoff for OneHopTransmission
In this section, we analyze the energylatency tradeoff under the reliability constraint in the scenario of onehop transmission. The optimal transmission power and the optimal number of receivers will be analyzed and the closedform expression of lower bound of energydelay tradeoff is obtained.
where ddr refers to the delay constraint. Consequently, minimizing the energy consumption under a delay constraint can be achieved by finding the three parameters for onehop transmission, where is the optimal transmission power, is the optimal number of opportunistic relay candidates, is the optimal transmission distance for each relay candidate.
This is a mixed integer nonlinear programming (MINLP) problem that can be solved using a branchandbound algorithm [19], but it is time consuming. We propose in the following an alternative which relies on an analytic solution when the size of the forwarding set is constant. A simple iterative procedure is then proposed to find the optimal size.
3.1. EnergyDelay Tradeoff for a Given Number of Receivers
First, we consider the scenario where is fixed.
Theorem 1.
Proof.
Refer to Appendix A.
where and are constant. The priority between energy and delay is balanced with the transmission power .
Next, we show how to obtain . For a given set of nodes having fixed SNRs, exploiting (18) leads to the following theorem.
Theorem 2.
For a given set of forwarding nodes whose corresponding is , respectively, is minimized with respect to if and only if the are ordered in an increasing order such that the higher priority is given to the node with the smaller .
Proof.
Refer to Appendix B.
where is the derivation of with respect to .
It should be noticed that the whole lower bound of the energydelay tradeoff is obtained for the same SNR value at the receivers. Furthermore, note that this SNR constraint can be achieved for different couples of transmission power and effective distance parameters. In other words, to satisfy the optimal SNR constraint, the internode distance and the power transmission should be selected jointly according to a desired tradeoff constraint between delay and energy.
Equation (30) implies that the optimal SNR strongly relies on the function . Thus, we should consider different channels to obtained the closedform expression of its lower bound. First, we focus on AWGN, Rayleigh fast fading and Rayleigh block fading channels, then, a general solution of obtaining is given for all other scenarios.
Substituting (11), (12), and (13) into (30), respectively, yields in the different kinds of channels.
3.1.1. AWGN
where is the branch of the Lambert W function satisfying [20].
3.1.2. Rayleigh Block Fading Channel
3.1.3. Rayleigh Fast Fading Channel
Substituting , and into (29), respectively, yields in the three kinds of channels.
3.1.4. Other Scenarios
Contrarily to the abovementioned cases, in many practical situations, the closedform expression of or . Then, a numeric approach can be used.
The first problem may hold when the expression of link probability is not known. In this case, the value of has to be approached with an iterative approximation algorithm.
A second limit may be encountered for estimating a closedform expression of even if the link probability is known. For example, when a coding scheme is employed, the link probability is more complex, and a closedform expression of is untractable. In this case, a sequential quadratic programming (SQP) algorithm (see for instance [21]) can be adopted to solve the optimization problem of minimizing . Then, approximated values of and are obtained.
3.2. Optimal Number of Receivers
In the previous subsection, the lower bound of energy delay tradeoff is obtained for a fixed number of receivers. In this subsection, we analyze how to select the optimal number of receivers for a given ddr.
Theorem 3.
is convex with respect to .
Proof.
Refer to Appendix C.
Theorem 3 proves that a global minimum exists. In addition, since the optimal power is known explicitly with (28) under a delay constraint ddr, the minimum value of for a given delay constraint ddr can be obtained by searching the optimal number of receivers. The following algorithm addresses this optimization.
3.3. Lower Bound of EnergyDelay Tradeoff
3.4. Minimum Energy Consumption
In the previous section, we found the lowest point existed in each curve of lower bound of energydelay tradeoff in three kinds of channels. For delaytolerant applications, the minimum energy consumption point is very important, that is, the lowest point on the curve of lower bound of energydelay tradeoff. In the following subsection, we will derive this point.
In this subsection, as to the lowest point, we derive the lower bound of energy efficiency and corresponding energyoptimal transmission power and distance without the delay constraints.
3.4.1. Optimal Transmission Power
which is the optimal transmission power minimizing .
Algorithm 1: Search the optimal number of receivers .
, , , ddr
while do
Calculate
if then
,
return
else
,
end if
end while
Meanwhile, (39) also shows that the characteristics of the amplifier have a strong impact on . When the efficiency of the amplifier is high, that is, , reaches its maximum value. As well, it is clear that when the environment of transmission deteriorates, namely, increases, decreases correspondingly. However, is independent of .
3.4.2. Lower Bound of and Its Corresponding Delay
where is the distance between a source node and a destination node.
3.4.3. Minimum Mean Transmission Distance
This distance shows the minimal distance between a source node and a destination node; otherwise, too small hop distance results in more energy consumption or too many hops, namely, too much delay.
3.4.4. EnergyDelay Tradeoff in Different Channels
According to the analysis in previous subsections, we obtain and under a delay constraint ddr, then we have the lower bound of energydelay tradeoff for onehop transmission with opportunistic communication on the basis of (34). Because ( , , ) tightly depends on the function of link probability, , we analyze the lower bound of energydelay tradeoff in three different channels mentioned in this section.
Figure 4 shows the lower bound of energydelay tradeoff and the corresponding optimal transmission power and the optimal number of receivers. It should be noticed that the lowest point exists in each curve; this is to say, there is the most energy saving point without the delay constraints for each channel, and the corresponding mean delay is the maximum mean delay of a pair of nodes. In Section 3.4, we have analyzed the most energy saving point in detail. The left side of the lowest point shows the energy consumption increases with the decrease of the delay constraint which coincides with our intuition. However, in the right of the lowest point, the energy consumption increases with the increase of the delay because both the transmission power and the number of receives are too small which results in very small hop distance, that is, the increase of the hop number. Certainly, this work state should be avoided in practice.
Note that the optimal number of receivers corresponding to the lowest point in each curve is 2 in Rayleigh block fading and fast fading channels and is 1 in an AWGN channel. The result implies that too many nodes will lead to the waste of energy; this is to say, we should avoid acting all neighbor nodes as the relay candidates. In addition, the optimal number of receivers raises with the decrease of delay limit. As for the corresponding optimal transmission power, it is not monotonically decreased as we saw in traditional P2P communications. Conclusively, it is clear that the transmission power and the number of relay candidates should be adjusted correctly according to a delay constraint in order to avoid too much energy consumption. Algorithm 1 and (28) provide the approach to calculate the optimal transmission power and a distributed algorithm to select the optimal relay candidates will be introduced in Section 6.
Though the lower bound on the energydelay tradeoff is derived in linear networks, it will be shown by simulations in the following Section 6 that this bound is proper for 2dimensional Poisson distributed networks.
4. EnergyDelay Tradeoff of MultihopTransmissions
In this section, we extend the result of the onehop transmission case developed in Section 3 to the scenarios of multihop transmission. Meanwhile, the effect of physical parameters on lower bound of energydelay tradeoff and the energy efficiency gain of opportunistic communication are analyzed.
4.1. Lower Bound of EnergyDelay Tradeoff
where and are the endtoend energy consumption and delay between a source node and a destination node.
In order to obtain the lower bound of energylatency tradeoff of multihop transmission, the theorems about equivalent distance transmission are introduced as follows.
Theorem 4.
Proof.
Refer to Appendix D.
Theorem 5.
if and only if .
Proof.
Refer to Appendix E.
Based on Theorems 4 and 5, we conclude that regarding a pair of source and destination nodes with a given number of hops, the single scenario, which minimizes both mean energy consumption and mean transmission delay, is that each hop with uniform distance along a linear path. As a result, the optimization about energy and delay for a single hop will bring the optimization of the same performance for the multihop transmission. Hence, the results about the lower bound of energydelay tradeoff in Section 3 can be used directly in multihop transmissions.
 (1)
The model can be adapted locally. The pathloss function can be adapted in different areas. For instance, in the case of a very dense wireless sensors deployment, a subset of nodes can decide to remain active only under local estimation of the pathloss properties.
 (2)
A statistical shadowing can be added to take into account the discrepancy between local reality and the theoretical model. This point out of the scope of the paper. While, this approach is proposed by another team in a paper extending our contribution [22].
4.2. The Gain of Opportunistic Communication
Figure 5 provides an example in three kinds of channel and the physical parameters are shown in Table 1. In this example, the gain of opportunistic communications decrease from 25% to 0% with the increase of the delay constraint. The gain becomes 0 when the delay constraint is greater than 0.11 ms/m which implies that the opportunistic communication has changed to the traditional P2P communication which coincides with the result in Figure 4, where the optimal number of receiver becomes 1 for the corresponding delay constraint. In other words, when the delay constraint is greater than a threshold, the traditional P2P communication is more energy efficient than the opportunistic communication in an AWGN channel.
In Rayleigh block fading and Rayleigh fast fading channels, the gain of opportunistic communications is always greater than 0, which reveals the opportunistic communication outperforms the traditional P2P communication in these two kinds of channel, where opportunistic routings benefit from the effect of diversity and can improve the energy efficiency, while the gain decreases with the increase of the delay constraint.
According to these results, it can be concluded that opportunistic communications are more energy efficient for Rayleigh block fading channels than for AWGN channels.
5. Effect of Physical Layer Parameters
The closedform expression on energydelay tradeoff provides the framework to evaluate the energydelay performance of a network according to its parameters of physical layer, MAC layer, and routing layer. This framework can be used in the following applications.
(1) Performance Evaluation
During the design or planning phase of a network, these results of performance evaluation provide the basis of the choice of sensor node and the choice of routing and MAC protocols.
(2) Benchmark of Performance
Regarding the design of a protocol, the best performance of a network can be obtained using this framework which can act as the benchmark of performance in order to measure the performance of a protocol and to adjust the parameters of the protocol.
(3) Parameter Optimization
Likewise in the design phase of a network, we can optimize the parameters such as transmission power according to the request of performance of a network on the basis of the framework.
In this subsection, on the basis of (34), the effects of coding scheme and the type of modulation on the lower bound of energydelay tradeoff are studied, as the examples of the applications of parameter optimization.
(a) Coding
(b) Modulation
Besides the above three parameters, the other parameters such as circuitry power, strength of fading, transmit rate or the integration of several parameters can be analyzed also according to the different applications because this framework includes every parameter in physical layer. On the basis of these analysis results, we can adjust the parameters to obtain the best performance of a network.
6. Simulations
The purpose of this section is to verify the theoretical analysis of the lower bound on the energydelay tradeoff and the energy efficiency in a 2dimensional Poisson distributed network by simulations although these theoretical results are obtained in a linear network using approximation approach. First, we introduce a novel opportunistic protocol on the basis of the theoretical analysis.
6.1. Opportunistic Protocol
The analysis in Section 3 reveals that the transmission power and the number of receivers should be configured as the corresponding optimal values in order to approach the lower bound of energydelay tradeoff. While. Algorithm 1 proposed in Section 3 requires the acknowledgment of global network parameters, which can not be directly applied to distributed networks. Consequently, an algorithm in distributed manner is introduced based on the algorithm proposed in [9] and the analysis in Section 3.
The containing property in Lemma 3.4 proposed in [9] shows that a straightforward way to find an optimal set containing nodes is to add a new node into the optimal node set containing nodes. Furthermore, when a local minimum is found, it is the global minimum according to Theorem 3. Based on this idea, a distributed algorithm for finding the optimal receiver set at each hop in order to minimize the energy consumption and satisfy the delay constraint ddr is proposed in Algorithm 2.
Algorithm 2: Search the optimal set of receivers.
, , ddr
while do
for each node do
;
Sorting the nodes in
according to their effective transmission distance;
if then
;
end if
end for
if then
;
else
return
end if
end while
In Algorithm 2, is the set of neighbor nodes of a source node, is the set of nodes selected to receive the packet from the source node.
 (1)
Search the forward candidates according to Algorithm 2.
 (2)
Assign a priority to each node according to its effective transmission distance.
 (3)
Transmit the data packet including the information of relay candidates ID and theirs corresponding priorities.
 (4)
Nodes in the set of relay candidates try to receiver packet.
 (5)
A node which receives the packet correctly calculates the backoff time according to the priority and waits for the ACK packet from the nodes with the priority higher than that of itself.
 (6)
If a node does not receive any ACK packet, it broadcasts its ACK packet and then is ready to transmit the received packet to next hop or destination. If a node receives an ACK packet, the received data packet is dropped from its queue.
 (7)
The source node waits for the ACK packet from one of forwarding candidates. If an ACK packet is received, the source node removes the packet from the buffer; otherwise, it is ready to retransmit the data packet.
6.2. Simulation Setup
In the simulations, the lower bounds of energydelay tradeoff and are evaluated in an area of surface using the simulator Wsnet [23]. The nodes are uniquely deployed according to Poisson distribution.
All the other simulation parameters concerning a node are listed in Table 1. The distance between the source and the destination nodes is 1000 m. The source node transmits only one data packet of 320 bytes to the destination with BPSK modulation. The size of ACK packet is 26 bytes. For every hop, the transmitter will retransmit the data packet until the data packet is received by the next relay node; this is to say, there is no limit for the number of retransmissions in order to ensure the reliability. The opportunistic protocol proposed in Section 6.1 is employed to simulations. A simulation will be repeated for 1000 times in each different configuration.
 (i)
After the initial phase, the network is geographical aware; that is, each node knows the position of itself, the sink node and all the neighbor nodes in the network.
 (ii)
Each node in the network has the same fixed transmission power.
6.3. Results and Analysis
In order to analyze the variety of energydelay tradeoff with the increase of node density, the simulations are run in three cases: 200, 400, and 800 nodes in the simulation area. The transmission power of each node is configured as the optimal transmission power derived from (28).
7. Conclusions
In this paper, an unreliable link model is firstly integrated into our energy model using the specific metrics for energy efficiency and delay: , . By minimizing for AWGN, Rayleigh block fading and Rayleigh fast fading channels with and without delay constraint, we have shown that the channel state impacts the optimal number of receivers in a cluster. Meanwhile, the corresponding optimal transmission power and the optimal transmission range are obtained. The energydelay tradeoff for onehop and multihop transmissions are analyzed and compared with the tradeoff given by traditional P2P communications. The main conclusion is that opportunistic communications exploiting spatial diversity are beneficial for Rayleigh fast fading and the Rayleigh block fading channels.
Appendices
A. The Lower Bound of EnergyDelay Tradeoff
Proof.
where .
Notice that according to this assumption, we can deduce that when .
Therefore, this assumption leads to a contradiction. Finally, we can conclude that (24) and (23) is the lower bound of energydelay tradeoff when the minimum value of is available.
B. Minimum G Function
Refer to the Theorem 2. For a given nodes whose corresponding is , the minimum value of can only be obtained by giving the higher relay priority to the node whose is smaller.
Proof.
Let us introduce a new function . It can easily be seen that maximizing the value of leads to minimizing . Next, we prove this theorem by maximizing .
When , the maximum value of can be gotten directly. Then, we assume it holds for where the relationship between is and its corresponding priority is . Therefore, the theorem will be proved if we can show that it holds for .
Without loss of generality, we may assume is the SNR of the node and . Here, there are three scenarios for assigning the priority of the node, : , and .
Because when , we can obtain that .
When , using the similar method, the same result is obtained.
Thus, we conclude that the theorem holds.
C. Proof of Convexity
Refer to Theorem 3. is a convex function with respect to .
Proof.
where and .
Obviously, (C.1) is great than 0 if and . Regarding , it is easily concluded that . Next, we proof .
so, is a monotonic decreasing function. Thus, for all .
Finally, it is evident to see that is a convex function.
D. Minimum Energy Consumption inMultihop Transmissions
Proof.
where is the Lagrange multiplier.
E. Minimum Mean Delay in MultihopTransmissions
Proof.
where and are the first and second derivative of (14) with respect to . Because and in case of in many practical scenarios, (E.1) is great than 0. Thus, is a monotonic increasing function with respect to .
Declarations
Acknowledgments
This work was funded by the French ANR project BANET, "Body Area Networks and Technologies", and by the French Ministry of Industry through the European ITEA2 Geodes Project, and by INRIA.
Authors’ Affiliations
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