3.4 Generalizations and Extensions for n-Dimensional Networks
3.4.1 Optimize-able Parameters are p, D, and T d
Under the delay requirement, it can be shown that the combination of (p∗, D∗, Td∗) that minimizing the power consumption must satisfy Td∗ = E[Li,s]T1/E[E[Hi,s]]. Substitut-ing this equality into (3.46), one can express the average total energy consumption
rate as:
One can find that the average energy spent to deliver a message is the product of the last two terms in equation (3.48). The second term is the average number of hops to deliver the message and the last term is the average energy consumption for a message to advance one hop.
It can be shown that the optimal setting satisfies the following two equations:
(α + n) (e−pnDn+ 1)(pnDn)2+ 2(e−pnDn − 1)pnDn E In (3.49), note that the pD-product stands for the expected number of nodes awaken in a time epoch within the transmission range for message relay per dimension. As it turns out, it is a critical quantity for the optimized random wakeup relay network.
From (3.49), we observe the following important property.
Proposition 3.14. The optimal expected number of nodes awaken in a time epoch to participate in message forwarding in the transmission range depends only on the path loss exponent, the network dimensionality, and the response-to-transmission energy ratio.
It is quite surprising to note that this optimal quantity does not depend on factors
Es1/Et1
Figure 3-7: The optimal number of nodes awaken to forward messages in a time epoch within the transmission range per dimension as a function of response-to-transmission energy ratio Ec1/Et1.
such as the message origination rate, the required delivery speed, and the message detection energy.
From (3.49), it can be derived that the optimal number of participating nodes decreases monotonically as the response-to-transmission energy ratio increases. Also, the optimal number of participating nodes decreases as the path loss exponent in-creases. The optimal number of participating nodes within the transmission range increases as the network dimensionality increases.
The optimal pD-product for the overall power consumption with the required delivery speed satisfied is shown in Figure 3-7. For a typically regime where 0 <
Ec1 ≤ Et1, on average at least one node shall wake up in the transmission range in a time epoch per dimension. Under the optimal settings in this typical regime, a message advances some distance toward the sink node in a time epoch with high probability.
Substituting the optimal pD-product into (3.50), one can determine the values of p∗ and D∗. We note specifically that D∗ ∝ (Et1/Ed)α+n−1 and p∗ ∝ (Et1/Ed)α+n1 . This result is quite intuitive; the range is expected to rise as the ratio of transmission energy
per unit length to detection energy reduces. 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 minimum energy routing in WRNs, as long as the message detection energy is nonzero.
Let us summarize how the optimal values of p, D, and Td are derived:
• The pD-product shall first be matched to the response-to-transmission energy ratio.
• The range D (and hence p) shall then be chosen to reflect the ratio of unit-length transmission energy to detection energy.
• Finally, Td is chosen to meet the delivery speed requirement.
Armed with the optimal setting of the three design parameters, we can investigate the properties of an optimized network. The first property we seek to understand is the right proportion of battery energy to be spent on (mostly fruitlessly) detecting that a neighbor node is requesting a message to be forwarded. The answers for the first question is quite clear!
Proposition 3.15. Under the optimal settings of p, D, and Td, the ratio between ECL and AECOL is α−1n+1.
In the proposition, the optimal ratio depends only on the path loss exponent and the network dimensionality. The higher the path loss exponent, the less proportion of energy shall be spent on transmission. Given that α is typically between 2 to 4, the total detection power is at the same order of total transmission power. This optimal proportion can be quite a lot higher than conventional thinking. What may be more surprising is that such optimal proportion does not depend on any of the node energy parameters Ed, Et1, Ec1 nor on the application parameters T1 and Tm.
The second property we seek to understand is how do factors such as the message delivery speed requirement and the message origination rate impact the overall power consumption? Note that the overall power consumption under the optimal setting
can be found by substituting p∗, D∗, and Td∗ into equation (3.48). The minimal power consumption of the whole network is
f (α, n, V, Ec1, Et1, Ed)E[Li,s]α+nn+1 1 T1
α−1α+n 1 Tm
n+1α+n
.
(3.51)
In (3.51), the function f (α, n, V, Ec1, Et1, Ed) depends only on the environmental pa-rameters and the node papa-rameters. From (3.51), we can observe the following prop-erty.
Proposition 3.16. The power consumption under optimal setting is exactly propor-tional to the α+nα−1-th power of required mean delivery speed and is exactly proportional to the α+nn+1-th power of mean message origination rate.
From the proposition, a rule of thumb for the optimal power-delay trade-off in a 1-D network is that one needs only to invest a 2-, 3-, or 4-fold in total power in order to increase the mean delivery speed by a factor of 10 given that the path loss exponent is 2, 3, or 4, respectively. Another observation is that, interestingly, the higher the network dimensionality is, the less sensitive the network power is to the velocity requirement.