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Figure 3.24 Illustration of the SSEP model.

3.5.3 Single Static Sink Edge Placement Model

We consider a static model of WSNETs and analyze its data capacity—and energy utilization. In the model, we consider a WSNET with a single stationary sink located on an edge of the rectangular network area. Also, the sink is at least rs distance away from the corners of the network area. This model is referred to as the single static sink edge placement (SSEP) model as illustrated in Figure 3.24. We refer to this kind of a sensor network as an SSEP network.

Due to randomly distributed sensors, the data capacity—of an SSEP network is limited by the total initial energy in the neighbors of the sink. Each bit of sensed data in needs to be forwarded to the sink on a multihop path. Since we assume that the sensed phenomenon is uniformly distributed over time in the network area, the average hop length of all shortest multihop paths can be computed. Using this average hop length, the average energy needed to send one bit of data to the sink can be computed. Therefore, the total energy needed to transmit bits of sensed data to the sink is easily computed. From this utilized energy and the total initial energy, we can obtain the energy utilization ratio in the SSEP model.

3.5.4 SSEP Model Description

In the SSEP model shown in Figure 3.24, we assume that the sink node is located at coordinate (xc, 0) and its location never changes. The reason for using this sink placement model is that in many real situations, the sink cannot move into the sensed area due to some adverse conditions in that area, such as the area being close to a volcano, for example. A network in the SSEP model is determined by six parameters: the area width W , the area height H , the average degree g of sensors, the transmission radius rs , and the location of the sink (xc, yc), where yc = 0.

3.5.5 Critical Region and Network Data Capacity

Definition 3.5.1 The total network data capacity of a WSNET is defined as the total amount of sensed data received by the sink. (It does not include other data such as routing overhead.)

The first critical region, denoted by V1, of the sink is defined as the shaded half circle centered at the sink and with a radius rs in Figure 3.24. It may be noted that V1 contains

an average of g/2 (half of the degree) sensors. The sink can only receive data from sensors in V1. Thus, the lifetime of the network is over if all sensors in V1 deplete their energy. Since the average total initial energy available in V1 is Ps g/2, where g = λπrs2, and transmission of one bit of data from a sensor in V1 to the sink consumes ps amount of

energy, the average value of the total network data capacity , denoted by , is given as follows:

In an ideal situation where there is no routing overhead, no collision at the MAC level, and the energy consumed in sensing and computation are negligible compared to data transmissions, the average data capacity can be close to .

3.5.6 Energy Efficiency of a WSNET in the SSEP Model

To calculate the energy utilization of an SSEP network, we need to know the average amount of energy required to transmit one-bit of sensed data along a forwarding path to the sink. This is done by finding the transmission energy efficiency as follows:

Definition 3.5.2 The transmission energy efficiency is defined as the reciprocal of the average amount of energy required to transmit one bit of sensed data along a multihop path from the original sensor to the sink.

A larger value of indicates that less energy is required for data transmission. Since the sensed data should be forwarded to the sink via a multihop path, the smaller the average number of hop counts, the higher the energy efficiency. Therefore, the potential highest energy efficiency can be achieved if sensors use the shortest-hop paths.

To compute , we need to find the average hop count for all shortest-hop paths. However, direct computation of the average hop count is not straightforward for a randomly deployed WSNET. Instead, we use a statistical method as follows. For any given sensor P with many neighbors, P always chooses a neighbor node, which is nearest to the sink, as the next forwarding node. The selected node then follows the same rule to choose the next node, and so on, until the data packet reaches the sink. In general, each forwarding operation reduces the remaining distance from a sensor to the sink by the maximum length, and this length is called the maximum one-hop progress toward the sink. Intuitively, a forwarding path established by this rule is generally the shortest-hop path from P to the sink because the network is dense. The hop count of a shortest-hop path from P to the sink is approximately equal to the distance from P to the sink divided by the average maximum one-hop progress. Therefore, the average hop count of all shortest-hop paths is approximately equal to the average distance to the sink from all sensors divided by the average maximum one-hop progress. Next, we will derive the expressions for the average distance and the average maximum one-hop progress.

Due to the randomly occurring phenomenon in the sensed area, each shortest-hop path has equal probability of being chosen as a forwarding path. For a large and dense network, the problem is equivalent to finding the average distance from all points in the network area to the sink. Assume that the sink is located at (xc, yc). For a given node with coordinate

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Figure 3.25 Partition of the W × H area.

d, as follows:

00

It is difficult to solve (3.44) directly. Instead, we consider the double integral in (3.44) in three-dimensional space. Hence, the double integral is a volume enclosed by the function z = (x − xc)2 + (y − yc)2 and x-y plane with 0 ≤ x ≤ W and 0 ≤ y ≤ H . The entire W−H area can be partitioned into eight triangles as shown in Figure 3.25. For each triangle, we can compute the volume V (x, y) above the triangle as follows, where x and y denote the lengths of the two nondiagonal edges.

The total volume of the double integral is the sum of all volumes of the eight partitioned triangles. Therefore, the solution of (3.44) is the total volume divided by W × H as follows:

d(W, H, xc, yc) = 1 [V (xc, yc) + V (yc, xc) + V (W − xc, yc) + V (yc, W − xc)

where d(W, H, xc, yc) denotes the average distance in the configurations of the area with width W , height H, and the sink location (xc, yc). Since yc = 0 in the SSEP model, the first four addition terms in (3.45) are 0.

The basic idea behind the average maximum one-hop progress has been illustrated in Figure 3.26. In Figure 3.26, P is a sensor node and Q is the sink. Any node on the arc, which is centered at Q and has a radius R = d − z, makes the same forwarding progress z = d − R from P to Q in one hop. When P forwards data to the sink, P always chooses the neighbor node, which makes the maximum progress z, as the next forwarding node. The selected node then follows the same rule to choose the next node, and so on, until the data packet reaches Q. A forwarding path established by this rule is generally the shortest-hop