## 1. Introduction

Interference effects constrain scalability performance of ad hoc networks as Gupta and Kumar (Gupta & Kumar, 2000) showed that the throughput capacity of a fixed wireless network decreases when the number of total nodes*multiuser diversity* (Knopp & Humblet, 1995), in which a source node transmits a packet to the nearest neighbor, and that relay delivers the packet to the destination when this destination becomes the closest neighbor of the relay. The scheme was shown (Grossglauser & Tse, 2001) to increase the throughput capacity of MANETs, such that it remains constant as the number of users in the network increases, taking advantage that communication among nearest nodes copes the interference due to farther nodes.

On the other hand, detailed and straightforward models for interference computation in dense ad hoc networks have not been extensively studied. Grid models have been proposed to compute interference (Gobriel et al., 2004), (Liu & Haenggi, 2005), which take advantage of the regular placement of the nodes. This orderly topology is a good starting point for static networks; however, it does not apply for MANETs. Also, some previous works have assumed a transmission or a reception range for communication among nodes without considering the effect from the entire network (Tobagi & Kleinrock, 1975), (Deng et al., 2004). This approximation can be good for low density networks, but it may imply in inaccurate results for dense networks. One problem with such approximation is the difficulty in finding an analytical description for the random topology inherent to ad hoc networks. In other cases, analytical models use graph theory (Rickenbach et al., 2005), (Qin-yun et al. 2005). While they are good for higher layer analysis, like routing, such models may not be appropriate for a more detailed communication channel study because they do not consider physical parameters like Euclidean distance, fading and path loss, for example.

This chapter analyzes an improved channel communication model, from the model proposed by Moraes et al. (Moraes et al., 2008) that permits to obtain the measured signal to noise and interference ratio (SNIR) by a receiver node, and consequently its spectral Shannon capacity (or spectral efficiency) (Cover & Thomas, 1991) at any point in the network when it communicates with a close neighbor. This model considers Euclidean distance, path loss and Rayleigh fading. The nodes are assumed to move according to a random mobility pattern and the parameter *θ* represents the fraction of sender nodes in the network. Monte-Carlo simulations (Robert & Casella, 2004) are used to validate the model. Furthermore, previous works had assumed the receiver node located at the center of the network (Lau & Leung, 1992), (Shepard, 1996), (Hajek et al., 1997). The results presented here are more general which shows that the received SNIR and spectral efficiency tend to a constant as*θ* parameter.

The remaining of this chapter is organized as follows. Section 2 introduces the network model. Section 3 presents the average number of feasible receiving neighbor nodes as a function of the network parameters. Section 4 explains the interference and spectral efficiency computation. Section 5 shows the results. Section 6 explains the autonomous technique for node state determination. Finally, Section 7 concludes the chapter summarizing the main results obtained.

## 2. Model

The modeling problem addressed here is that of a wireless ad hoc network with nodes assumed mobile. The model consists of a normalized unit circular area (or disk) containing*uniform mobility model* (Bansal & Liu, 2003). This model satisfies the following properties (Bansal & Liu, 2003): (a) the position of the nodes are independent of each other at any time

Any node can operate either as a sender or as a receiver. At a given time*θ* is defined as

A node* bits/sec* from sender node

where the summation is over all sender nodes*SNIR* level necessary for reliable communication,*L* is the processing gain of the system, and *I* is the total interference at node

where

The goal is to find an equation relating the total interference measured by a receiver node that is communicating with a neighbor node as a function of the number of total users

## 3. Feasible Receivers Near a Sender

In order to obtain the interference generated by nodes outside the neighborhood of a receiver node, we first need to find the average radius size containing a sender node and how many feasible receivers are within this range.

If the density of nodes in the disk isthen the average radius for one sender node (

Hence, the average number of receiving nodes, called

which is constant for a given *θ*. Eq. (5) is a benchmark for obtaining the average number of receiving nodes as a function of the network parameter *θ*.

Thus, the radius

## 4. Interference and Capacity Computation [1] -

In the previous section, the average radius

For a packet to be successfully received, Eq. (1) must be satisfied. Hence, consider a receiver at any location in the network for a given time

Let us assume that the sender is at distance

power

where

In order to obtain the overall expected interference at the receiver caused by all transmitting nodes in the disk, let us consider a differential element area

Because the nodes are uniformly distributed in the disk, the transmitting nodes inside the differential element of area generate, at the receiver, the following amount of interference²

The total interference is obtained by integrating Eq. (8) over the disk area and the result depends on the value of

### A. The caseα 2 ![]()

For some propagation environments (Rappaport, 2002) the path loss parameter is modeled to be always greater than two, i.e.,

*At a given time*

*, for a receiver node located at distance*

*from the center in a unit area disk network containing*

*mobile nodes uniformly distributed, where*

*, and assuming the sender located at distance*

*from this receiver, then the receiver SNIR is given by*

*where**is the standard deviation of the attenuation Gaussian random variable in decibels due to shadowing (Akl et al., 2001), and*

Define

By substituting this result in Eq. (11), we arrive at

where

is a constant for a given position

The SNIR can be obtained by using Eqs. (1), (4), (6), and (14) to arrive at

Where

which finishes the proof.

From Eq. (9), taking the limit as

From Eq. (10),

Therefore, from Eqs. (19) and (20), for*SNIR* tends to a constant as

From Lemma 1, the spectral efficiency (*C* ) is straightly obtained and is given (in units of bits/s/Hz) by (Cover & Thomas, 1991)

Accordingly, from Eqs. (9), (20) and (21), we conclude that the limiting spectral efficiency goes to a constant as

### B. The caseα = 2 ![]()

For the free space propagation environment (Rappaport, 2002), the path loss parameter is modeled to be equal to two, i.e.,

*At a given time t, for a receiver node located at distance r from the center in a unit area disk network containing*

*mobile nodes uniformly distributed, where*

*, and assuming the sender located at distance*

*from this receiver, then the receiver SNIR is given by*

Consequently, the spectral efficiency is obtained (in units of bits/s/Hz) by (Cover & Thomas, 1991)

From Eq. (22), it is straightforward that

## 5. Results

In this section, the analytical results elaborated in Section 4 are compared with Monte-Carlo simulations (Robert & Casella, 2004).

Figure 2 shows the spectral efficiency as function of

Figure 3 illustrates the spectral efficiency behavior for different values of

Figure 4 confirms that the limiting spectral efficiency does not depend on

Figure 5 shows the spectral efficiency as function of

Figure 6 shows curves for spectral efficiency as function of *n* when

## 6. A Technique to Attain a Desired Value for The θ Parameter

Another challenge associated with this study is how a node can efficiently set its state (sender or receiver) in order to the network attain a given

The technique suggested here consists of a simplified part of a MAC layer protocol. Similar to the Traffic Adative Medium Access (TRAMA) protocol (Rajendran et al., 2003), our scheme has the requirement to be synchronized with cyclical periods of contention followed by transmission in which some nodes are capable of taking control of close neighbors, as found in IEEE 802.15.4 - ZigBee (ZigBee Alliance, 2009). This technique, restricted only to the contention period, is intended to be autonomous and able to distribute the states of the nodes according to the

Considering the node distribution as described in Section 2 and that each node has its unique identification (ID), the node with the lowest ID controls the network and it is called the coordinator node of the network.

The communication among nodes follows cycles which are divided in contention and transmission phases. The contention period is divided in following three phases, respectively. The announcement is the period in which each node sends its packet identification number. The dissemination is the phase when the coordinator node sends its identification to all nodes of the domain. Finally, there is the distribution phase where the node coordinator sends a random sequence indicating the status (sender or receiver) that each node in the network must assume during the following transmission period according to the

Figure 7 presents the results of a simulation implemented in JAVA (Java, 2009), using the shuffle method of Class Collections (JavaClassCollections, 2009) for the random distribution of the states (sender and receiver), which are displayed as fraction of times that three nodes randomly chosen, over 100 cycles, were senders. It is observed that the three randomly chosen nodes tend to converge their sender fraction of times to

The technique suggested here does not consider other medium access issues like channel admission control, collision resolution, node failure, etc., which is subject of future work.

## 7. Conclusions

We have analyzed interference effects and spectral Shannon capacity (or spectral efficiency) for mobile ad hoc networks using a communication channel model, which considers Euclidean distance, path loss, fading and a random mobility model. We found that, for a receiver node communicating with a close neighbor where the path loss parameter *α* is greater than two, the resultant signal to noise and interference ratio (SNIR) and consequently the spectral efficiency tend to a constant as the number of nodes n goes to infinity, regardless of the position of the receiver node in the network. Therefore, for the studied model, communication is feasible for near neighbors when the number of interferers scales. Furthermore, for the receiver nodes located at the boundary of the circular network, we show that they suffer less interference than those located inside attaining higher capacity. Also, for the case where *α* is equal to two, the capacity was shown to go to zero as *n* increases; however, the decay is very slowly making local communication still possible for a finite *n*. Model and Monte-Carlo simulation results present good agreement and validate the interference and Shannon capacity investigation performed.

It was also proposed a technique for autonomous and distributed allocation of states (sender or receiver) of nodes based on the parameter

## 8. Acknowledgements

This work was supported in part by PIBIC/POLI, by Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE) and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.

## Notes

- This represents the worst case scenario, because the other neighbors are located either closer or at the same distance r0 to the sender, so they measure either a stronger or the same SNIR value.
- Because the nodes are uniformly distributed in the disk and n grows to infinity, we approximate the sum in Eq. (1) by an integral.