1
Fluid Models and Energy Issues

Wireless sensor networks consist of hundreds to thousands of sensor nodes with limited computational and energy resources. Sensors are densely deployed over an area of interest, where they gather and disseminate local data using multi-hop communications, i.e. using other nodes as relays. A typical network configuration includes a large collection of stationary sensors operating in an unattended mode, which need to send their data to a node which collects the networks' information, the so-called sink node.

Traditionally, network designers have used either computer simulations or analytical frameworks to predict and analyze a system's behavior. Modeling large sensor networks, however, raises several challenges due to scalability problems and high computational costs. With regard to simulations, several software tools have been extended and developed to deal with large wireless networks, see [ZEN 98, SIM 03, LEV 03] just to name a few. As for analytical modeling, to the best of our knowledge, the only work dealing with large sensor networks is presented in [DOU 04], which employs percolation techniques.

This chapter presents spatial fluid-based models for the analysis of large-scale wireless networks. The technique is said to be fluid-based because it represents the sensor nodes as a fluid entity. Sensor location is smoothed out in continuous space by introducing the concept of local sensor density, i.e. the number of sensors per area unit at a given point.

The approach is applied to describe a network scenario where nodes are static and need to send the result of their sensing activity to a sink node. Sensors may send packets to the sink in a multi-hop fashion. Although this technique requires the introduction of simplified assumptions, that are necessary to maintain the problem tractable, these models account for (1) node energy consumption, (2) node contention over the radio channel and (3) traffic routing.

By the end of the chapter, three fundamental contributions are provided with respect to existing literature:

1. 1) because of the fluid approach, very large networks can be studied while maintaining the model complexity extremely low;
2. 2) the behavior of the network can be studied as a function of the bidimensional spatial distribution of the nodes, possibly under non-homogeneous node deployment;
3. 3) the approach provides a very flexible and powerful tool, which can account for various routing strategies, sensor behaviors and network control schemes, such as congestion control mechanisms.

1.1. The fluid-based approach

The fluid approach is motivated by the observation that large-scale sensor networks can be represented by a continuous fluid entity distributed on the network area. This section describes the general framework, and the notation used to specify the model is summarized in Table 1.1.

Table 1.1. Model notation

1.1.1. Sensor density and traffic generation

Sensors are randomly placed over an area in the plane according to a Poisson point process with local intensity ?(r), hereinafter also called the sensor density, which can vary from point to point. Let us identify each point in the plane by means of its coordinates r = (x, y).

The Poisson assumption implies that the number of sensors contained in an area A is distributed according to a Poisson distribution with parameter G(A), defined as:

The mean number of sensors present in the network is denoted by N, with ?(r) dr = N. As an example, to define a system where there are (an average of) N sensors uniformly distributed over a disk of unit radius and the sink is located at the center of the disk (i.e. Sink = (0, 0)), it is correct to write:

Finally, it is fair to assume that a sensor s in position r generates traffic at rate ?s(r). By aggregating all traffic generated by sensors over an infinitesimal area centered at point r, the generation rate density is defined as ?(r), which depends on the position r. This quantity, measured in packets per second per area unit, is proportional to both the local generation rate of a sensor and the local sensor density and corresponds to the mean number of packets per second generated by an infinitesimal area. It is defined as:

1.1.2. Data routing

The next hop used by a sensor to send a packet to the sink is determined in a probabilistic way. Indeed, the exact location of the sensors is unknown, thus u(r´|r) can be defined as the probability density that a packet transmitted by a sensor in position r uses a sensor in position r´ as its next hop. Since u(r´|r) must be a valid probability density, it is correct to have:

Probability density u(r´|r) depends on the particular routing policy.

1.1.3. Local and relay traffic rates

Each sensor can be both a traffic source and a relay for other sensors. The traffic rate density ?(r) is equal to the sum of the traffic locally generated by the sensors at point r, and the traffic relayed for other nodes. By assuming that the system is stable, the total traffic rate density ?(r) can be computed by solving the following integral equation:

where ?(r) accounts for the traffic locally generated, and the integral computes the rate density of the relayed traffic using u(r´|r) introduced above. Note that the expression in [1.4] represents the traffic rate density of successfully transmitted packets. The actual traffic rate density must account also for retransmissions, as explained in the following.

1.1.4. Channel contention and data transmission

The channel contention model computes the actual traffic rate density ?*(r) at a node in r, as well as the mean packet service time at the same point, denoted by s(r). Packets that are not received correctly need to be retransmitted by the sender. The average packet retransmission probability at r is denoted by PR(r): it depends on the particular protocol adopted to access the channel and is, in general, location dependent, i.e. it can be different from point to point within the network area. By assuming that the packet transmission process is memoryless, the actual traffic rate density at point r, which also accounts for retransmitted packets, is given by

1.1.5. Mean packet delivery delay

To compute the mean time needed to deliver a packet to the sink, the mean delivery delay is introduced at point r, D(r), and is defined as the time required by a packet originated in r to reach the sink. By denoting with q(r) the mean queueing delay experienced by a packet at point r, D(r) can be expressed as

where s(r) is the mean service time previously introduced. Equation [1.6] states that the mean delivery delay at point r can be expressed as the sum of the delay experienced by a packet at point r plus the mean delivery delay associated with the next hop. The delivery delay in all different points of the network can be computed recursively starting from D(0, 0) = 0, i.e. no delay is experienced by a packet at the sink.

1.1.6. Sensor active/sleep behavior

The fluid model accounts for the active/sleep dynamics of the nodes by introducing the probability Pa that a sensor is active. Since only active sensors generate traffic, [1.2] becomes ?(r) = ?s(r)Pa?(r), where the sensor...