We have so far expressed the power consumption of the whole network in terms of the energy consumed by each node participating in opportunistic relaying (i.e. Et, Ec, and Ed) as well as several design parameters, including transmission range D, duty-cycle period Td, and wakeup density p. In this section, we use the power consumption model thus derived to investigate the performance trade-offs between network energy consumption and message forwarding delay. We show that the ECL and the AECOL must be balanced carefully for minimizing network power consumption.
To proceed, we consider an optimization problem with a goal of minimizing the network power consumption under the constraint of the message delay requirement.
Specifically, the objective function is
p,D,Tmind
pAEd
Td + E[E[Hi,s]]Et+ π2pD2(Ec− Ed) Tm
, (3.21)
where wakeup density p, transmission range D, and duty-cycle period Td are opti-mization variables. The delay constraint is
E[E[Hi,s]]Td
E[Li,s] ≤ T1, (3.22)
where T1 is the maximal delay time for a message to advance over unit distance.
We describe in the following how the optimization problem can be solved and the properties that can be observed in an optimized RWN.
3.3.1 Jointly Solving for Design Parameters
Under the delay requirement, it can be shown that the combination of (p∗, D∗, Td∗) that minimizes the power consumption must satisfy Td∗ = E[Li,s]T1/E[E[Hi,s]]. Sub-stituting this equality into (3.21), we can express the network power consumption as
follows:
where the product of the last two terms can be considered as the average energy spent in delivering a message: the second term is the average number of epochs to deliver the message and the last term is the average energy consumption for a message per epoch.
We note that the range-independent portion of the energy consumed by a trans-mitter Et0 is often small in the proportion of the entire network power consump-tion [48, 74]. In addiconsump-tion, the range-independent component of the energy consumed by a candidate forwarder Ec0 includes the energy consumed for message detection Ed
and is often small as far as the network power consumption is concerned [75]. In light of these, in the following analysis we omit Et0 and (Ec0− Ed) in (3.23) for finding the optimal design parameters that can minimize network power consumption. We show the validity of the assumptions in Section 3.5 when these constant energy components are included.
To find the optimal values of of design parameters p∗, D∗, and Td∗, we can solve for ∂P∂pN = 0 and ∂P∂DN = 0 by using the approximation expression of PN in (3.23). It can be thus derived that the optimal values must satisfy the following two equations:
(α + 2)
D = π
We propose the procedures to solve for the optimal values of p, D, and Td as follows:
(i) The optimal value of γ is first solved in (3.24), where the path loss exponent α and the ratio between Ec1 and Et1 are given.
(ii) The optimal transmission range D is then solved in (3.25) based on the optimal value of γ and other given parameters. Since γ = πpD2, the optimal wakeup density p is also solved.
(iii) Finally, the optimal duration of the duty-cycle period Td is solved in (3.22) to meet the delivery delay requirement.
3.3.2 Key Properties in Optimized RWNs
We explore in this section several important properties of an optimized RWN where the three design parameters (p, D, Td) are set to the optimal solution for (3.21). These properties provide insights for trade-offs between network energy consumption and message forwarding delay in RWNs.
The first property regards the optimal number of nodes that should wake up within the transmission range of a relay node to participate in the relay activities.
Theorem 3.9. The optimal expected number of nodes γ awakened within the trans-mission range in a time epoch to participate in message forwarding depends only on the path loss exponent α and the response-to-transmit energy ratio Ec1/Et1. In addi-tion, γ can be lower bounded by
γ∗ > 1− α +q
Proof. The property can be obtained by substituting the inequality of e−12γ < (1 +
1
2γ +18γ2)−1 into (3.24).
Ec1/Et1
Figure 3-6: Optimal number of nodes awakened to forward messages in a time epoch within the transmission range as a function of response-to-transmit energy ratio Ec1/Et1.
Surprisingly, the optimal number of nodes to wake up within the transmission range does not depend on factors such as the message origination rate 1/Tm, the required delivery speed 1/T1, and the detection energy Ed. From (3.24), it can be observed that the optimal number of participating nodes decreases monotonically as the response-to-transmit energy ratio increases and as the path loss exponent increases. As shown in Fig. 3-6, for a typically region where 0 < Ec1 ≤ Et1, on average at least one node shall wake up in the transmission range in a time epoch. It can also be observed that the number of nodes awakened to participate in the relay activities within the transmission range shall be smaller than 10 for minimizing network power consumption in most network environments of different response-to-transmit energy ratios.
Note that γ is a function of p and D, and from (3.25) one can know D∗ ∝ (Et1/Ed)α+2−1 and p∗ ∝ (Et1/Ed)α+22 . In other words, the optimal transmission range is expected to rise as the ratio of Et1/Edreduces. A corollary to this finding is that always taking the shortest hop or persistent listening to potential incoming traffic cannot be the optimal strategy for minimizing network energy consumption in RWNs, as long as the detection energy is nonzero. Another corollary that can be directly obtained is that the optimal p can serve as a lower bound of the node density deployed in the
network for minimizing network power consumption.
The second property regards the optimal proportion of battery energy that should be spent in detecting whether a message needs to be forwarded.
Theorem 3.10. Under the optimal settings of p, D, and Td, the ratio between ECL and AECOL is α−13 .
Proof. This property can be obtained by substituting p∗, D∗, and Td∗ into (3.23).
In this theorem, the optimal ratio depends only on the path loss exponent. The higher the path loss exponent, the less proportion of energy shall be spent in trans-missions. Given that α is typically between 2 to 4, the energy used for detecting messages is surprisingly of the same order of the energy used for transmitting and re-ceiving messages. What is equally surprising is that such an optimal proportion does not depend on any of the node energy parameters Ed, Et1, Ec1 nor on the application parameters T1 and Tm.
The third property regards the impact of the message delivery delay requirement and the message origination rate on the overall power consumption.
Theorem 3.11. The power consumption under optimal setting is proportional to the
α−1
α+2-th power of the required mean delivery speed and proportional to the α+23 -th power of the mean message origination rate.
Proof. This property can be obtained by substituting p∗, D∗, and Td∗ into (4.7) such
In this theorem, a rule of thumb for the optimal energy-delay trade-off in RWNs is that one needs to invest less than an α-fold increase in total power in order to increase the mean delivery speed by a factor of 10 when the path loss exponent is α. The investment of network energy thus can be quite effective for reducing the message delivery delay in an optimized RWN.