Five chapters are included in this thesis: Chap 1 is the introduction, which reviews the funda-mentals of Wireless Sensor Network, and describes the motivation of this work. We introduce the “Maximum Information Rate Deployment”(MIRD) with detail performance analyses and related discussions in Chapter 2. The critical concern of network lifetime is included in Chap-ter 3. Here we develop the “Maximum Information Capacity Deployment”(MICD), along with the corresponding performance simulations and related discussion. The previous two chapters both are discussed in 2 − area case, we then generalize MIRD into K − area case in Chap 4. Finally, Chapter 5 is the conclusion of our work.
Chapter 2
Maximum Information Rate Deployment
In some applications of WSN, the main concern is whether the events in the network can be detected successfully. A necessary prerequisite is that possible event locations are covered by sensors. Once a region is fully covered by some sensors, the information generated in this region would be obtained. An interesting question arises : Given two regions with different areas, A1 and A2, and each region has its own information generating rate. For a limited number of sensors, N, how to deploy sensors in each region that would obtain the maximum total information rate? Here we will address this problem and the result will lead to the
“Maximum Information Rate Deployment” method. The presentation in Section [2.1] is mainly based on [1], [4], [5].
2.1 Coverage and Deployment
Many wireless sensor networks are aimed at surveillance of certain geographical regions, for example, to detect wildfires or rare animals in a habitat. Putting all communication aspects aside, such an event can only be detected if there are sensors close enough so as to sense the event. Two important questions arise:
• Given a sensor deployment, i.e., a particular placement of sensors over a certain
geo-graphical region, which points in this area are covered by sensors? Coverage is thus an important issue in sensor networks. If any point an event taking place at is not covered by sensors, the corresponding information is lost.
• Given an area to be monitored and some coverage requirements, what number of sensors is needed and where should they be placed? This question, labeled as the deployment problem, can be posed under several interesting constraints, for example, cost constraints, presence of obstacles, availability of different types of sensors, and so forth.
2.1.1 Sensing models
A sensor transforms environmental stimuli into electrical signals. The quality of the resulting signal depends on three factors. The first is the distance between the sensor and the event.
The second is the directionality of the sensor. The last factor is the possibility that the same sensor can generate different outputs for the same stimulus at different times. In our work we focus only on the first factor and assume omnidirectional sensing and no random variations. Here two sensing models are introduced :
• In the Boolean Sensing Model, all sensors have a common sensing range r and initial energy E. Events within this sensing range are detected reliably, and events outside this range are not detected at all. Accordingly, the output signal for a sensor at posi-tion p observing an event at posiposi-tion q can be expressed as:
s(p, q) =
α : kp − qk ≤ r, 0 : otherwise.
(2.1)
where k · k is the Euclidean distance between p and q and α is a constant sensor value.
• In the General Sensing Model, the sensor possesses a certain maximal sensing range r but within this range the sensor output obeys a power law instead of being uniform :
s(p, q) =
α
kp − qkβ : r0 ≤ kp − qk ≤ r, 0 : otherwise.
(2.2)
where r0 is a certain minimum distance to avoid division by zero and β is a positive real number depending on the sensing model and sensor technology. For example, the relationship between the source signal power and the sensed signal power for acoustic signals can be modeled with β = 2.
In our work, Boolean Sensing Model is applied to simplify the discussion without aug-menting the main concept. It helps to clarify the the main idea of our work.
2.1.2 Coverage measures
The term of “Coverage” has different meaning in the literature. In general, coverage measures refer to a sensor network deployed to monitor some specified region A. This region is assumed to be two dimensional. Some of the coverage measures are the following:
1. The area coverage fa specifies the percentage of A being covered. If fa = 1, we say that full area coverage is achieved, which implies there is no information loss of this region.
2. The node coverage fn describes the percentage of nodes whose sensing range can be fully covered by the sensing ranges of other nodes. When the overlapping neighbors are awake, such a node can be switched into sleep mode without reducing the area coverage.
In our work, the area coverage fa is adopted in considering the general idea of deployment.
2.1.3 Random deployment: Poisson Point Process
Some of the coverage measures have been investigated for random deployment in several references, for example, [4], [5]. The most common assumption for a random deployment is the Poisson Point Process. For example, N sensors are deployed in the region A by a Poisson Point Process with average sensor density D > 0, where D = N/A and Ai is a partition inside A. We therefore conclude that the number of sensors N(Ai) deployed in the interested region Ai has a Poisson distribution with mean D · Ai, i.e.,
P r[ N(Ai) = K ] = e(−D·Ai)· (D · Ai)K
K! , f or K = 0, 1, . . . . (2.3)
In the literature, most existing works in sensor network consider a uniform sensor density in the whole network. In our work we apply Poisson point process to match the nature of Wireless Sensor Network, which is “Randomness”. Poisson point process is popular, for example, for modeling the number of stars in space or the number of bacteria cultivated on a Petri dish. The striking feature of such a Poisson point process is that it matches the intuition most people have on “random deployments”. Now we are going to answer questions regarding certain coverage measures for sensor networks under such a random deployment.
2.1.4 Coverage of random deployment: Boolean Sensing Model
We first discuss the case of an infinite sensor network in the two-dimensional plane to avoid any boundary effects. It is straightforward to find the area coverage fa for a Poisson point process of sensor density D under the Boolean Sensing Model. Let q be a randomly chosen point in the sensor field. What we are asking for is the probability that there is at least one sensor at position p with kp − qk being smaller than the common sensing range r. Consider the situation that a number of sensors and a selected point q are given. This point is covered if there is at least one sensor presenting in the circle of radius r around q. This circle Ai has
area πr2 and the probability to find at least one sensor within it is :
fa = P r[ N(Ai) ≥ 1 ] = 1 − P r[ N(Ai) = 0 ] = 1 − e−D·πr2 . (2.4)
To satisfy a specific area coverage fa, this equation can be solved to determine the required sensor density D of the Poisson point process, given as:
D(fa) = −ln(1 − fa)
πr2 . (2.5)
As a numerical example shown below, let us assume that r = 1 m and the desired coverage is fa= 0.99. In this case, a sensor density of D ≈ 1.47 sensors per m2 is needed. To achieve an even better coverage of fa = 0.999, this number grows to D ≈ 2.2(sensors/m2) , which implies that adding one sensor to the area is more efficient to obtain more information when area coverage fa is small. We can observe this effect in Fig. 2.1.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Area Coverage
Sensor density (number of sensors/square meters) Area Coverage fa
Figure 2.1: Area Coverage fa.
2.1.5 Information-generating model: Poisson Point Process
Without loss of generality, consider a situation in which event occurs in unit area 1 m2 at random instants of time with an average rate of λE events per second. For example, an event could represent the appearance of animal or the breakdown of a component in some bridge system. Let N(t) be the number of event occurrences within the time interval [0, t], then N(t) is a nondecreasing, integer-valued, continuous-time random process known as Poisson Process. We therefore conclude that the number of event occurrences during the time inter-val [0,t] has a Poisson distribution with mean λE· t, written as:
P r[ N(t) = K ] = e(−λE·t)· (λE · t)K
K! , f or K = 0, 1, . . . (2.6)
In our work, we assume that every event requires a constant packet size of l bits to record the information for each transmission. For example, the average information capacity generated at random instants of time in an area A with an average rate of λE is λE· A · l (bits/second).