In this section, we consider the network power consumption for all-to-all broadcast in the RGN. Specifically, each node in the network initiates a message to be broadcasted to all the other nodes with a given message origination period.
4.2.1 The Notion of Additional Energy Consumption over Listening
If there is no message in the network, nodes in the network wake up periodically to listen to potential incoming messages. We refer to such energy consumption as the energy consumption for listening (ECL). This function is usually a simple function of sleep mechanism.
On the other hand, the additional energy consumed over listening (AECOL) is de-fined as the energy used for gossiping on the top of the energy consumed for listening.
According to this definition, if there is no traffic in the network, the average AECOL of the network is zero. The AECOL depends heavily on the gossiping mechanism.
To minimize the energy consumption of the network, the ECL and AECOL must be balanced carefully.
4.2.2 Mean ECL and AECOL for Network-Wide Broadcast in a Linear RGN
Consider a 1-D regular RGN in which nodes are placed with unit-length apart, shown in Figure 4-1. In the network, each node has a message to broadcast to all the other nodes. Suppose that all nodes starts to gossip randomly at time epoch 1.
First, we compute the energy consumption of the whole network in a duty-cycle period. Define the random variable Wk as the AECOL of the network in epoch k.
1 2 3 4
0 5
unit length
V
Figure 4-1: One-dimensional network with unit-length spaced nodes.
In an epoch, each gossiping node consumes exactly energy Et + Ed. The AECOL of a gossiping node is Et and the AECOL of a non-gossiping node is certainly zero.
As a result, Wk depends only the number of gossiping nodes and the transmission energy. The expected value of AECOL spent in epoch k is E[Wk] = pnV Et. On the other hand, the ECL of the whole network depends only on the number of nodes in the network and the detection energy. The ECL of the whole network in an epoch is ρV Ed.
Next, we consider the AECOL for all messages to be broadcasted in the whole network. Define a random variable H as the number of time epochs it takes for all the messages to be broadcasted in the whole network. One can obtain the following theorem for AECOL of the whole network.
Theorem 4.1. The expected value of AECOL incurred by an all-to-all broadcast in the RGN is E[H]E[W ] where E[H] represents the expected number of time epochs for a single all-to-all broadcast and E[W ] stands for the expected value of AECOL during a time epoch.
Finally, given the average rate of message origination Tm, the average power con-sumption of the whole network can be represented as:
ρV Ed
Td + E[H]pnV Et
Tm .
(4.1)
4.2.3 Sensor Field Approximation for the Linear RGN
We desire expressions to gain better insights for the optimization of the design pa-rameters. Unfortunately, we can not find simple closed-form expression for E[H]. In this section, we approximate the discrete RGN as a continuous one to obtain close approximations. Specifically, a 1-D uniform random gossip sensor filed is a network
in which infinite many nodes are uniformly placed on a segment [0, V ]. Over any con-tinuous segment of length δ, there exists pδ nodes gossiping on average in a duty-cycle period.
In the 1-D random gossip sensor filed, consider a message originated from one end to be broadcasted to the other end. Define the random variable H′ as the number of time epochs for the message to be broadcasted over a distance L away. Given the operational assumption that the message is broadcasted at time epoch 1, the expected value of the H′ can be bounded by the following proposition.
Proposition 4.2. The expected value of the number of time epochs for a message to be broadcasted over a distance L away in the random gossip sensor field can be bounded as:
In (4.2), µ1 stands for the expected forwarding distance per epoch and P0 repre-sents the probability that a message does not advance to any new node in a sleep epoch. Note that in a large-size WRN in which the network size is much larger than a single hop transmission, the extra cost in epoch 1 can be reasonably omitted.
From Proposition 4.2.3, it can be observed that the expected number of time epochs for a message to be broadcasted in the whole network in the random gossip sensor field depends highly on the traveling distance of a message. Denote the maxi-mal distance between any two nodes in the whole network as Lmax. Henceforth, for the ease of analysis, we approximate E[H]≈ Lmax/µ1.
Now let us compute the expected forwarding distance in the 1-D random gossip sensor field. Define the random variable X as the distance a message gains in a time epoch. In the 1-D uniform random gossip sensor field, except the first epoch, the distance gains for a message in a time epoch is a truncated exponential distribution:
P (X ≤ x) =
Note that there is a finite probability of e−pD that a message does not advance during an epoch. From equation (4.3), the expected forwarding distance in an epoch will be:
µ1 = D− 1− e−pD
p . (4.4)
4.2.4 Multi-Dimensional RGNs
The results in the 1-D RGN can not be directly applied in a multi-dimensional RGN.
Why? One significant difference between the 1-D RGN and a multi-dimensional RGN is the presence of network boundaries. In a 2-D or a 3-D RGN, the expected number of hops for a message to be propagated within the whole network indeed depends on the shape of the network. As a result of this fact and other minor reasons, we can not find closed-form expressions of the expected forwarding distance in a higher-dimensional network.
Instead, we derive the following approximations for an n-dimensional (n-D) ran-dom gossip sensor field.
Proposition 4.3. For an n-D uniform random gossip sensor field, the expected for-warding distance in a time epoch can be approximated as:
µ1 ≈ D
1− 1− e−pnDn pnDn
.
(4.5)
As it turns out, the quantity pnDn is a key parameter in the proposition. This quantity is exactly the expected number of nodes gossiping in a time epoch within an n-dimensional cube of side D.