**4. Achievable capacity limit of HPNs in a mesh network**

#### **4.1. Practical considerations**

In order to obtain the achievable capacity bound for the HPN (the dual channel dual radio) based mesh network we consider a typical static wireless mesh network. Suppose the network is assumed to consist of varying *n* number of HPNs upto 50 nodes with a fixed area of deployment region (i.e., 5 Km by 5 Km). Also to generalize our derivations and only apply specific cases later with numerical examples, we employ the approach presented by Kyasanur and Vaidya (2005) in order to investigate the impact of number of channels and interfaces on the capacity of multi-channel wireless networks. In our derivations, the term "channel" will refer to a part of frequency spectrum with some specified bandwidth and the term "radio" will mean the network interface card. Let us assume that the HPNs based mesh network has *c* channels and every node is equipped with *m* interfaces so that the relation between the number of interface cards and channels is 2 *m c* . Each interface card can only transmit and receive data on any one channel at a given time, that is half-duplex communication. Thus, the mesh network of *m* interfaces per node, and *c* channels will be noted as *m c*, -network. Suppose each channel can support a fixed data rate of *R Rmultipath* , independent of number of non-overlapping channels of the network. Then, the total data rate possible by using all *c* non-overlapping channels is *Rc* . The number of non-overlapping channels can be increased by utilizing extra frequency spectrum of the standard technologies. For example, IEEE 802.11a standard technology uses 5 GHz band and has a capability of 24+ non-overlapping channels (c = 24+) each of 20 MHz bandwidth size (W = 20 MHz). Moreover, the IEEE 802.11n standard technology implements MIMO channels with bandwidth size of 40 MHz (Cisco systems, 2011).

#### **4.2. Capacity limit for regular placement in real network**

Consider Figure 6 that shows the topology of HPNs up to a maximum of 50 nodes placed regularly in a 5km by 5km of an area. This network scenario reflects a typical wireless mesh network set-ups in rural and remote areas where inter node distance is large and the landscape affects network performance. It should be seen that the separation distance between the source and the destination HPNs are assumed to take the longest route with a mean line joining the two nodes computed to be 6505 m. The regular placement of nodes ensures that there are no any two HPNs that are placed within a radius less than 700 m. The main reason for this decision is to avoid interference between close neighbours. It will be discussed in detail how this placement criteria is ensured using the carrier sense multiple access with collision avoidance (CSMA/CA) in IEEE 802.11 standards (IEEE 802.11 standard working group, 1999). In this topology setting, the regular placement of nodes on a fixed area will be termed as an arbitrary network. That is, the location of nodes and traffic patterns can be controlled as introduced by Gupta and Kumar (2000). Controlling nodes' placement locations and the traffic patterns makes the derived capacity bounds to be viewed as *the best case* capacity bounds with results remaining applicable to any network. As introduced by Gupta and Kumar, the aggregate *end to end* network throughput over a given flow or a set of flows is measured in terms of "bit-meters/sec". That is the network is said to transport one "bitmeter/sec" when one bit has been transported across a distance of one meter in one second.

*Theorem 1:* The E2E upper bound on capacity of a statically assigned channel network of type *m c*, -arbitrary regular placement of nodes when, *<sup>c</sup> n m* , is given as *mc nR* ,

bit-meters/sec.

160 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

**4. Achievable capacity limit of HPNs in a mesh network** 

channels with bandwidth size of 40 MHz (Cisco systems, 2011).

**4.2. Capacity limit for regular placement in real network** 

Consider Figure 6 that shows the topology of HPNs up to a maximum of 50 nodes placed regularly in a 5km by 5km of an area. This network scenario reflects a typical wireless mesh network set-ups in rural and remote areas where inter node distance is large and the landscape affects network performance. It should be seen that the separation distance between the source and the destination HPNs are assumed to take the longest route with a mean line joining the two nodes computed to be 6505 m. The regular placement of nodes ensures that there are no any two HPNs that are placed within a radius less than 700 m. The main reason for this decision is to avoid interference between close neighbours. It will be discussed in detail how this placement criteria is ensured using the carrier sense multiple access with

**4.1. Practical considerations** 

In conclusion, if the number of physical paths is two then the expression of capacity over multipath phenomenon (13) simply reduces to the expression (11) of the direct and reflected paths. Clearly from expression (13), one notes that increasing the multiplicity of paths of a single wireless link and the number of antennas at each HPN in (11) or (13) will increase the capacity limit of the wireless mesh links, depicted in (4) due to MIMO technology benefits.

In order to obtain the achievable capacity bound for the HPN (the dual channel dual radio) based mesh network we consider a typical static wireless mesh network. Suppose the network is assumed to consist of varying *n* number of HPNs upto 50 nodes with a fixed area of deployment region (i.e., 5 Km by 5 Km). Also to generalize our derivations and only apply specific cases later with numerical examples, we employ the approach presented by Kyasanur and Vaidya (2005) in order to investigate the impact of number of channels and interfaces on the capacity of multi-channel wireless networks. In our derivations, the term "channel" will refer to a part of frequency spectrum with some specified bandwidth and the term "radio" will mean the network interface card. Let us assume that the HPNs based mesh network has *c* channels and every node is equipped with *m* interfaces so that the relation between the number of interface cards and channels is 2 *m c* . Each interface card can only transmit and receive data on any one channel at a given time, that is half-duplex communication. Thus, the mesh network of *m* interfaces per node, and *c* channels will be noted as *m c*, -network. Suppose each channel can support a fixed data rate of *R Rmultipath* , independent of number of non-overlapping channels of the network. Then, the total data rate possible by using all *c* non-overlapping channels is *Rc* . The number of non-overlapping channels can be increased by utilizing extra frequency spectrum of the standard technologies. For example, IEEE 802.11a standard technology uses 5 GHz band and has a capability of 24+ non-overlapping channels (c = 24+) each of 20 MHz bandwidth size (W = 20 MHz). Moreover, the IEEE 802.11n standard technology implements MIMO

*Proof:* For the best case capacity limit, let's assume that multiple interfaces of HPNs receive and transmit on interference free channels. This assumption is reasonable with the HPNs that transmit directionally but receive and ensure connectivity omnidirectionally. As the number of channels is much larger than the number of interfaces. Thus, given that each HPN has a constant radio range, the spatial reuse is considered to be proportional to the physical area of the network. Let the node density be uniform with distribution regularity equals to one (i.e., probability equals to one) throught the deployment area.

**Figure 6.** Regular placement of HPNs in a 5 km x 5 km

The physical area of deployment, *<sup>A</sup>* , can be related to the total number of HPNs by *<sup>n</sup> <sup>A</sup>* . Consider also that, the capacity of each channel, *R* , is proportional to the physical area in accordance to the relation, *<sup>n</sup> R kA k* for some constant *<sup>k</sup>* (in bits/s/square meters). Suppose each source HPN can generate packets from higher layers protocol at a rate of bits/sec and the mean seperation distance between the source and destination HPN pairs is *L* meters (via multiple hops), then the E2E network capacity of the network is (Gupta & Kumar, 2000):

$$
\lambda \mathbf{u} \overline{L}\_t \text{ bit}-\text{meters}/\text{sec} \tag{14}
$$

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 163

*dCB d AB d AD dCD* , 1 , and , 1 , (17)

(18)

times the length of the hop around

active transmitter within a distance of 1 *r* . In IEEE 802.11a/b/g/n standards the medium access control (MAC) layer protocols execute carrier sense multiple access with collision avoidance (CSMA/CA) mechanism that ensures that this condition is satisfied. Figure 7 illustrates this type of collision avoidance mechanism. To illustrate this concept further, suppose node A is transmitting a bit to node B, while node C is simultaneously transmitting a bit to node D and both sessions are over a common frequency channel, W. Then, using the interference protocol model and the geometry sufficient for successful reception, node E

Adding the two inequalities together, and applying the triangle inequality to (17), we can

 , ,, <sup>2</sup> *dBD d AB dCD*

Therefore, in collision avoidance (CSMA/CA) principle of IEEE standards , expression (18)

each receiver. As shown by Figure 7, the total area covered by all hops must be bounded above by the total area of the deployment (domain, A). The seperation distance between receiver B and transmitter C is at least *AB AB* and that of transmitter A and receiver D

min*AB AB*

min*CD CD*

cannot transmit at the same time. Mathematically, one has

can be viewed as each hop covering a disk of radius 2

obtain the inequality in (18),

is at least *CD CD* .

**Figure 7.** Topology of HPNs and Geometry

The expression in (14) is evaluated without taking into account the lower layer number of frequency channels, interference, path loss effects and number of interface cards. In addition, in order to relate this high level network capacity with actual number of hops in a multi-hop wireless network, the overall bits transported in the network can be evaluated as follows. Suppose bit *b* , 1*b n* (bits/sec), traverses *h b* hops on the path from its source

to its destination, where the *h*th hop traverses a distance of *<sup>h</sup> b r* , then the overall bits transported in the network in every second is summed and is related to (14) as:

$$\mathsf{Adv}\,\overline{\mathsf{L}} \le \sum\_{b=1}^{\lambda n} \sum\_{h=1}^{h(b)} r\_{b}^{h} \text{ } \mathsf{bit}-\mathsf{meters}/\sec\text{ } \tag{15}$$

The inequality in (15) holds since the mean length of the line joining the source and destination, is at most equal to the distance traversed by a bit from its sources to its destination (Kyasanur & Vaidya, 2005).

Let us define to be the total number of hops traversed by all bits in a second, i.e., <sup>1</sup> *n <sup>b</sup> b* . Therefore, the number of bits transmitted by all nodes in a second (including bits forwarded) is equal to (bits/sec). Since each HPN node has *m* interfaces, and each interface transmits over a frequency channel of bandwidth *W* , with a data rate *R* possible per channel, the total bits per second that can be transmitted by all interfaces is at most 2 *Rmn* (transporting a bit across one hop requires two interfaces, one each at the transmitting and the receiving nodes). Consequently, the relation between a single channel single link rate, the number of interface cards creating single links, the number of nodes in the network and the total number of hops traversed by all bits in every second is given by,

$$\mathbf{X} \le \frac{Rmm}{2}, \text{bits/sec}\tag{16}$$

It should be noted that under the interference protocol model (Gupta & Kumar, 2000), a transmission over a hop of length *r* in a path loss link is successful only if there can be no active transmitter within a distance of 1 *r* . In IEEE 802.11a/b/g/n standards the medium access control (MAC) layer protocols execute carrier sense multiple access with collision avoidance (CSMA/CA) mechanism that ensures that this condition is satisfied. Figure 7 illustrates this type of collision avoidance mechanism. To illustrate this concept further, suppose node A is transmitting a bit to node B, while node C is simultaneously transmitting a bit to node D and both sessions are over a common frequency channel, W. Then, using the interference protocol model and the geometry sufficient for successful reception, node E cannot transmit at the same time. Mathematically, one has

$$d\left(\mathbb{C}, B\right) \ge \left(1 + \Delta\right) d\left(A, B\right) \text{ and } d\left(A, D\right) \ge \left(1 + \Delta\right) d\left(\mathbb{C}, D\right) \tag{17}$$

 Adding the two inequalities together, and applying the triangle inequality to (17), we can obtain the inequality in (18),

$$d\left(B, D\right) \geq \frac{\Lambda}{2} \left(d\left(A, B\right) + d\left(C, D\right)\right) \tag{18}$$

Therefore, in collision avoidance (CSMA/CA) principle of IEEE standards , expression (18) can be viewed as each hop covering a disk of radius 2 times the length of the hop around each receiver. As shown by Figure 7, the total area covered by all hops must be bounded above by the total area of the deployment (domain, A). The seperation distance between receiver B and transmitter C is at least *AB AB* and that of transmitter A and receiver D is at least *CD CD* .

**Figure 7.** Topology of HPNs and Geometry

162 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

to its destination, where the *h*th hop traverses a distance of *<sup>h</sup>*

ported in the network in every second is summed and is related to (14) as:

*h b <sup>n</sup> <sup>h</sup> b*

1 1

The inequality in (15) holds since the mean length of the line joining the source and destination, is at most equal to the distance traversed by a bit from its sources to its

Let us define to be the total number of hops traversed by all bits in a second, i.e.,

 . Therefore, the number of bits transmitted by all nodes in a second (including bits forwarded) is equal to (bits/sec). Since each HPN node has *m* interfaces, and each interface transmits over a frequency channel of bandwidth *W* , with a data rate *R* possible per channel, the total bits per second that can be transmitted by all interfaces is at

*Rmn* (transporting a bit across one hop requires two interfaces, one each at the trans-

, bits / sec 2

mitting and the receiving nodes). Consequently, the relation between a single channel single link rate, the number of interface cards creating single links, the number of nodes in the network and the total number of hops traversed by all bits in every second is given by,

It should be noted that under the interference protocol model (Gupta & Kumar, 2000), a transmission over a hop of length *r* in a path loss link is successful only if there can be no

*b h nL r* 

accordance to the relation, *<sup>n</sup> R kA k*

follows. Suppose bit *b* , 1*b n*

destination (Kyasanur & Vaidya, 2005).

 <sup>1</sup> *n <sup>b</sup> b* 

most

2

Kumar, 2000):

The physical area of deployment, *<sup>A</sup>* , can be related to the total number of HPNs by *<sup>n</sup> <sup>A</sup>*

Consider also that, the capacity of each channel, *R* , is proportional to the physical area in

pose each source HPN can generate packets from higher layers protocol at a rate of

bits/sec and the mean seperation distance between the source and destination HPN pairs is *L* meters (via multiple hops), then the E2E network capacity of the network is (Gupta &

The expression in (14) is evaluated without taking into account the lower layer number of frequency channels, interference, path loss effects and number of interface cards. In addition, in order to relate this high level network capacity with actual number of hops in a multi-hop wireless network, the overall bits transported in the network can be evaluated as

for some constant *<sup>k</sup>* (in bits/s/square meters). Sup-

*nL*, bit meters / sec (14)

(bits/sec), traverses *h b* hops on the path from its source

*b*

, (15)

*Rmn* (16)

, bit meters / sec

*r* , then the overall bits trans-

.

From the geometry of Figure 7, we sum over all channels (which can potentially transport *Rc* bits per second) and obtain the constraint formulated as,

$$\sum\_{b=1}^{\text{An}} \sum\_{h=1}^{h(b)} \frac{\pi \Delta^2}{4} \left( r\_b^{h} \right)^2 \le ARc\_1 \tag{19}$$

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 165

*n*

varies over space

, is given as,

*m*

distance is allowed as much possible as the size of the area can accommodate. The diagram indicates one of the possible settlement distribution patterns of the Internet users in com-

*Theorem 2:* The E2E upper bound on capacity of a statically assigned channel network of

munity based networks such as the case of Peebles valley mesh (PVM) networks.

type *m c*, -arbitrary irregular placement of nodes when, *<sup>c</sup>*

*mc nL O Rn*

*p*

bit-meters/sec.

**Figure 8.** Irregular placement of HPNs in a 5 km x 5 km

we let the node density

If we let <sup>1</sup>

then the area *<sup>A</sup>* is defined as *<sup>n</sup> <sup>A</sup>*

the expected average node density,

*n <sup>b</sup> b* 

in the network and all channels is at most

ing bits forwarded). From (22), we found out that

*Proof:* Let us consider that in irregular static networks, the node density

(i.e., an area) but stays constant at any given time since nodes are taken to be static. Suppose

nodes, *n* . Additionally, HPN nodes have *m* interfaces per node and with a data rate of *R* possible per channel. Then, the total bits per second that can be transmitted by all interfaces

> 2 *Rnmc* .

as the number of bits transmitted by all nodes in a second (includ-

to vary over space with irregularity rate (probability), 0 1 *p*

*<sup>p</sup>* . Therefore, capacity of the network will depend on

*p* of an irregular placement as well as the number of

this can be rewritten as,

$$\sum\_{b=1}^{\lambda n} \sum\_{h=1}^{h(b)} \frac{1}{\mathbf{X}} \left( r\_b^h \right)^2 \le \frac{4ARc}{\pi \Delta^2 \mathbf{X}} \tag{20}$$

Since the expression on the left hand side in (20) is convex, one obtains,

$$\left(\sum\_{b=1}^{\lambda n} \sum\_{h=1}^{h(b)} \frac{1}{\mathbf{X}} r\_b^h \right)^2 \le \sum\_{b=1}^{\lambda n} \sum\_{h=1}^{h(b)} \frac{1}{\mathbf{X}} \left(r\_b^h \right)^2 \tag{21}$$

Therefore, from (20) and (21) one gets,

$$\sum\_{b=1}^{2n} \sum\_{h=1}^{h(b)} r\_b^h \le \sqrt{\frac{4ARcX}{\pi \Delta^2}}\tag{22}$$

Substituting for from (16) in (22), and using expression (15) we have,

$$\mathcal{C}\_{\text{mech}} \le nR \sqrt{\frac{2mc}{\delta \pi \Delta^2}}, \text{ bit}-\text{meters}/\text{sec} \tag{23}$$

Therefore, the E2E asymptotically upper bound capacity limit for a scaling number nodes with node density , and static channel assignment without channel switching mechanisms in HPN network is given by

$$\begin{aligned} \mathcal{An\overline{L}} &= \mathcal{O}\left(n\mathcal{R}\sqrt{\frac{mc}{\delta}}\right), \text{ bit}-\text{meters / sec / arc /arg / range node density} \\ \mathcal{An\overline{L}} &= \mathcal{O}\left(n\mathcal{R}\sqrt{mc}\right), \text{ bit}-\text{meters / sec / constant node density} \end{aligned} \tag{24}$$

#### **4.3. Capacity limit for irregular placement in real network**

Consider Figure 8 that shows the topology of HPNs up to a maximum of 50 nodes placed irregularly in a 5km by 5km of an area. This network scenario reflects typical wireless mesh network set-ups in rural and remote areas where inter node distance is large and the landscape affects network performance. To avoid interference, it is assumed that no any two HPNs are placed within a radius less than 400 m at the edge and less than 700m toward the centre of the deployment area. However, between any two HPNs the largest separation distance is allowed as much possible as the size of the area can accommodate. The diagram indicates one of the possible settlement distribution patterns of the Internet users in community based networks such as the case of Peebles valley mesh (PVM) networks.

*Theorem 2:* The E2E upper bound on capacity of a statically assigned channel network of type *m c*, -arbitrary irregular placement of nodes when, *<sup>c</sup> n m* , is given as,

*mc nL O Rn p* bit-meters/sec.

164 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

*h b <sup>n</sup> <sup>h</sup>*

1 1

1 1

*b h*

Since the expression on the left hand side in (20) is convex, one obtains,

*b h*

*Rc* bits per second) and obtain the constraint formulated as,

this can be rewritten as,

with node density

in HPN network is given by

Therefore, from (20) and (21) one gets,

**4.3. Capacity limit for irregular placement in real network** 

 

From the geometry of Figure 7, we sum over all channels (which can potentially transport

 2 2

*b*

<sup>2</sup>

1 4 *h b <sup>n</sup> <sup>h</sup> b*

*r*

1 1 *h b h b n n h h b b*

*r r*

 

2

1 1 1 1

<sup>4</sup> *h b <sup>n</sup> <sup>h</sup> b*

2 <sup>2</sup> , bit meters / sec *mesh*

Therefore, the E2E asymptotically upper bound capacity limit for a scaling number nodes

, bit meters / sec

*mc nL nR for varying node density*

*nL nR mc for constant node density*

Consider Figure 8 that shows the topology of HPNs up to a maximum of 50 nodes placed irregularly in a 5km by 5km of an area. This network scenario reflects typical wireless mesh network set-ups in rural and remote areas where inter node distance is large and the landscape affects network performance. To avoid interference, it is assumed that no any two HPNs are placed within a radius less than 400 m at the edge and less than 700m toward the centre of the deployment area. However, between any two HPNs the largest separation

, bit meters / sec

*r*

*b h b h*

1 1

*b h*

Substituting for from (16) in (22), and using expression (15) we have,

*mc C nR*

, <sup>4</sup>

*r ARc*

2

2

(21)

(22)

(23)

(24)

2

, and static channel assignment without channel switching mechanisms

*ARc*

*ARc*

(19)

(20)

**Figure 8.** Irregular placement of HPNs in a 5 km x 5 km

*Proof:* Let us consider that in irregular static networks, the node density varies over space (i.e., an area) but stays constant at any given time since nodes are taken to be static. Suppose we let the node density to vary over space with irregularity rate (probability), 0 1 *p* then the area *<sup>A</sup>* is defined as *<sup>n</sup> <sup>A</sup> <sup>p</sup>* . Therefore, capacity of the network will depend on the expected average node density, *p* of an irregular placement as well as the number of nodes, *n* . Additionally, HPN nodes have *m* interfaces per node and with a data rate of *R* possible per channel. Then, the total bits per second that can be transmitted by all interfaces in the network and all channels is at most 2 *Rnmc* .

If we let <sup>1</sup> *n <sup>b</sup> b* as the number of bits transmitted by all nodes in a second (including bits forwarded). From (22), we found out that

$$\sum\_{b=1}^{hn} \sum\_{h=1}^{h(b)} r\_b^h \le \sqrt{\frac{4ARRRmmc}{2\pi\Delta^2}}\tag{25}$$

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 167

(bits/sec), traverses *h b* hops on the path from its source to its

, bit meters / sec

*hh h bb b rr r* 

(28)

*Rmn* (29)

( having intra-

*nL* , bit-meters/sec.

assumption is reasonable since HPNs within a cluster use a shorter transmission range compared to that range that is being used by nodes while communicating between clusters. The application layer generates the E2E capacity according to Gupta and Kumar model. This

capacity depends on the number of nodes and can be simplified as

**Figure 9.** Regularly clustered placement of HPNs in a 5 km x 5 km

rate since *R* drops with distance. Consequently, we have,

destination, where the *h* -th hop traverses a distance of 1 2

*h b <sup>n</sup> <sup>h</sup> b*

1 1

*b h nL r* 

cluster and inter-cluster hop distances), then one obtains by summing over all bits in the

Let us define to be the total number of hops traversed by all bits in a second, i.e.,

 . Therefore, the number of bits transmitted by all nodes in a second (including bits forwarded) is equal to (bits/sec). It is known that each HPN node has *m* interfaces, and each interface transmits over a frequency channel of bandwidth *W* , with a data rate *R* per channel, the total bits per second that can be transmitted by all interfaces is at

*Rmn* (transporting a bit across one hop requires two interfaces, one each at the trans-

,bits / sec <sup>2</sup>

mitting and the receiving nodes). But in clustered networks where bits traverse the intra cluster hops and inter cluster hops with R1 and R2 rates respectively. *R* takes the minimum

Suppose bit *b* , 1*b n*

 <sup>1</sup> *n <sup>b</sup> b* 

network:

most

2

Alternatively, it has been established that

$$\sum\_{b=1}^{hm} \sum\_{h=1}^{h(b)} r\_b^h = \mathcal{A}n\overline{L} \tag{26}$$

We have

$$\begin{aligned} \mathcal{A}n\overline{L} &\le R\sqrt{\frac{2Anmc}{\pi\mathfrak{A}^2}} = Rn\sqrt{\frac{2mc}{\delta p\,\pi\mathfrak{A}^2}}\\ \mathcal{A}n\overline{L} &= O\left(Rn\sqrt{\frac{mc}{\delta p}}\right), \text{bit -- meters / seconds} \end{aligned} \tag{27}$$

#### **4.4. Capacity limit for clustered placement in real network**

Suppose that *n* nodes are arbitrarily located (a cluster fashion) on a square of a fixed area with LOS ensured between any two neighbouring nodes shown in Figure 9. Note that the deployment area is fixed to 5 km by 5 km. To avoid interference no two HPNs can be placed at a radius less than 400 m near the edge and less than 700 m toward the centre of the deployment area. Thus, within a cluster a minimum separation distance of 700 m is considered, while any largest separation distance possible is considered between clusters. Figure 9 shows the regularly clustered topology indicating how far as possible the separation distance between the source and destination HPNs. The diagram depicts typical rural community networks such as Peebles valley mesh (PVM) networks. The community mesh network is considered to adopt such a distribution pattern and the goal would be to find the achievable capacity over wireless mesh networks.

*Theorem 3:* The E2E upper bound on capacity of statically a signed channel network of type *m c*, -arbitrary clustered placement of nodes when *<sup>c</sup> n m* is given as

$$
\lambda n \overline{L} = \mathcal{O} \left( R \sqrt{\frac{mmc}{1} \left( \frac{n\_1}{\delta\_1} + \frac{n\_2}{\delta\_2} \right)} \right) \text{ in bit-meters/sec, where } R \text{ is the min (R\iota\_1 R\iota\_2), m are number of } \delta\_1 \text{ and } \delta\_2 \text{ is the radius of the c\iota\_1.} \text{ The second part is the sum of the c\iota\_2.}
$$

nodes in a regular cluster and n2 are number of clusters in the network.

*Proof:* We assume *a clustered placement* of the mesh network as *a special case of the regular HPNs placement*. However, in this case the node densities are respectively, <sup>1</sup> 1 1 *n A* as the density of nodes within a cluster consisting of 1 *n* nodes occupying *A*1 geographical area and <sup>2</sup> 2 2 *n A* as the density of clusters consisting of 2 *n* clusters occupying *A*<sup>2</sup> of an area. This assumption is reasonable since HPNs within a cluster use a shorter transmission range compared to that range that is being used by nodes while communicating between clusters. The application layer generates the E2E capacity according to Gupta and Kumar model. This capacity depends on the number of nodes and can be simplified as *nL* , bit-meters/sec.

**Figure 9.** Regularly clustered placement of HPNs in a 5 km x 5 km

166 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

*h b*

*h b*

2 2

*r nL* 

2 2

Suppose that *n* nodes are arbitrarily located (a cluster fashion) on a square of a fixed area with LOS ensured between any two neighbouring nodes shown in Figure 9. Note that the deployment area is fixed to 5 km by 5 km. To avoid interference no two HPNs can be placed at a radius less than 400 m near the edge and less than 700 m toward the centre of the deployment area. Thus, within a cluster a minimum separation distance of 700 m is considered, while any largest separation distance possible is considered between clusters. Figure 9 shows the regularly clustered topology indicating how far as possible the separation distance between the source and destination HPNs. The diagram depicts typical rural community networks such as Peebles valley mesh (PVM) networks. The community mesh network is considered to adopt such a distribution pattern and the goal would be to find the achieva-

*Theorem 3:* The E2E upper bound on capacity of statically a signed channel network of type

*Proof:* We assume *a clustered placement* of the mesh network as *a special case of the regular* 

density of nodes within a cluster consisting of 1 *n* nodes occupying *A*1 geographical area

as the density of clusters consisting of 2 *n* clusters occupying *A*<sup>2</sup> of an area. This

*HPNs placement*. However, in this case the node densities are respectively, <sup>1</sup>

*m c*, -arbitrary clustered placement of nodes when *<sup>c</sup>*

nodes in a regular cluster and n2 are number of clusters in the network.

,bit meters / seconds

*p*

 

*h b hn*

1 1

*Anmc mc nL R Rn*

*p*

 

*mc nL O Rn*

**4.4. Capacity limit for clustered placement in real network** 

ble capacity over wireless mesh networks.

1 2 1 2 1 *nmc n n*

 

  *b h*

*r* 

4 2

2

(25)

(26)

(27)

*n*

is given as

1

1 *n A*

as the

*m*

in bit-meters/sec, where *R* is the min (*R1*, *R2*), n1 are number of

*ARRnmc*

*h b hn*

1 1

*b h*

Alternatively, it has been established that

We have

*nL R*

and <sup>2</sup> 2

2 *n A*

Suppose bit *b* , 1*b n* (bits/sec), traverses *h b* hops on the path from its source to its destination, where the *h* -th hop traverses a distance of 1 2 *hh h bb b rr r* ( having intracluster and inter-cluster hop distances), then one obtains by summing over all bits in the network:

$$
\lambda \mathfrak{M} \overline{L} \le \sum\_{b=1}^{\lambda \mathfrak{M}} \sum\_{h=1}^{h(b)} r\_b^h \text{, bit-meters/sec} \tag{28}
$$

Let us define to be the total number of hops traversed by all bits in a second, i.e., <sup>1</sup> *n <sup>b</sup> b* . Therefore, the number of bits transmitted by all nodes in a second (including bits forwarded) is equal to (bits/sec). It is known that each HPN node has *m* interfaces, and each interface transmits over a frequency channel of bandwidth *W* , with a data rate *R* per channel, the total bits per second that can be transmitted by all interfaces is at most 2 *Rmn* (transporting a bit across one hop requires two interfaces, one each at the transmitting and the receiving nodes). But in clustered networks where bits traverse the intra cluster hops and inter cluster hops with R1 and R2 rates respectively. *R* takes the minimum rate since *R* drops with distance. Consequently, we have,

$$\text{XX} \le \frac{Rmm}{2}, \text{bits/sec}\tag{29}$$

Therefore, using similar arguments and steps provided in the *Proof of Theorem 1*, the interference constraint protocol of the clustered mesh network will still hold. The derived E2E capacity limit is then upper bound as

$$
\Delta n \overline{L} \le R \sqrt{\frac{2nmc}{\pi \Delta^2} \left(\frac{n\_1}{\delta\_1} + \frac{n\_2}{\delta\_2}\right)} / \text{ bit}-\text{meters} / \text{sec} \tag{30}
$$

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 169

**5 5**

**50 50** 

transmission power is kept the same for both cases, an increase in antenna gain due to more focussed beams increases capacity substantially. Thus, the HPNs have capacity gains over

*Parameter Data sheet BB4all TM architecture* 

*Maximum single link range (metres), d* **~6505 ~6505**

*Modulation scheme* **OFDM OFDM**

*Number of spatial streams, L* **2 2**

*Combined antenna gain (dBi), Kantenna* **9 28** 

*Channel width (MHz) of IEEE 802.11 a, W* **20 20**

*Reference distance (metres),* <sup>0</sup> *d* **5 5**

*AWGN (mWatts), N*<sup>0</sup> **1e-10 1e-10**

*Achievable data rate (Mbps) for all streams, R* **60.792 183.30**

ly. Thus, the HPNs have capacity gains over the standard IEEE 802.11n devices.

Table 2 lists parameters needed to evaluate the achievable data rate for all wireless streams of a single link IEEE 802.11n radios. A comparison is made between the achievable data rate computed using data from the IEEE 802.11n air interface data-sheet and the data from the BB4allTM architecture. It is observed that while specifications of other parameters are kept the same in both cases, the datasheet combined antenna gain is taken to be 7 dBi and that of BB4allTM architecture to be 28 dBi. With these antenna gains, the achievable data rate is 291.33 Mbps in standard architecture compared to 570 Mbps for the HPNs. This numerical result is explained as follows. When transmission power and the size of the MIMO are constant, an increase in antenna gain increases capacity substantial-

the standard IEEE 802.11a devices.

*RF Industrial, Science & Medical (ISM) band* 

*Maximum output power (mWatts) of IEEE* 

**Table 1.** IEEE 802.11a air interface single link capacity

*(GHz), f*

*802.11a radio, P*

Hence, the asymptotic end to end upper bound capacity limit for a scaling number of nodes with node density 1 and cluster density <sup>2</sup> , and statically assigned channels in HPN network is given by

$$
\Delta n \overline{L} = \mathcal{O}\left(R\sqrt{\frac{nmc}{1}\left(\frac{n\_1}{\mathcal{S}\_1} + \frac{n\_2}{\mathcal{S}\_2}\right)}\right) / \text{ bit} - \text{meters / sec}\tag{31}
$$

#### **5. Numerical examples using the Peebles valley mesh**

Tables 1 and 2 shows useful data that can be used to determine the achievable capacity limit over long links with direct LOS of about 6.505 km from one end of the network to the other. The data mimics the physical scenario of PVM as compared to the data sheet values of IEEE 802.11a and IEEE 802.11n air interfaces, respectively. The computed capacity values assume CSMA/CA and the protocol model whereby a transmission over one link is successful only if there is no active transmitter within a distance of 1 *d* . That is, the distance *d* is the range between a transmitter and receiver, and signifies a fraction of one hop distance needed to ensure collision-free transmission. The assumption of protocol model is reasonable in sparsely placed nodes in rural set-up whereby interference effects can be neglected without loss of generality. Furthermore, let the size of data carriers in OFDM scheme be, 48 *Mc* with each HPN having an 8dBi omni-directional antenna and 20 dBi being the directional antenna (i.e., the combined antenna gain is 630.96 (6.3096 times 100) and in a hilly and foliage area is taken approximately to be three between the frequency channels 5.15 GHz and 5.85 GHz (Durgin et al., 1998). Then, from the capacity limit expression in Section 3, practical data rates can be obtained.

#### **5.1. Single link achievable capacity**

Table 1, lists parameters needed to evaluate the achievable data rate for all wireless streams of a single link IEEE 802.11a radios. A comparison is made between the achievable data rate computed using data from the IEEE 802.11a air interface data-sheet and the data rate of the IEEE 802.11a air interface constructed from BB4allTM architecture. It is observed that while specifications of other parameters are kept the same in both cases, the combined antenna gain is taken to be 9 dBi from the data-sheet and that of BB4allTM architecture to be 28 dBi. With these antenna gains, the achievable data rate in standard architecture is 60.792 Mbps compared to 183.30 Mbps for the HPNs. This numerical result is explained as follows. When transmission power is kept the same for both cases, an increase in antenna gain due to more focussed beams increases capacity substantially. Thus, the HPNs have capacity gains over the standard IEEE 802.11a devices.


**Table 1.** IEEE 802.11a air interface single link capacity

168 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

pacity limit is then upper bound as

with node density 1

network is given by

and  *nL R*

*nL R*

expression in Section 3, practical data rates can be obtained.

**5.1. Single link achievable capacity** 

Therefore, using similar arguments and steps provided in the *Proof of Theorem 1*, the interference constraint protocol of the clustered mesh network will still hold. The derived E2E ca-

1 2

 

<sup>2</sup> , bit meters / sec *nmc n n*

, bit meters / sec 1

(30)

(31)

, and statically assigned channels in HPN

1 2

Hence, the asymptotic end to end upper bound capacity limit for a scaling number of nodes

1 2 1 2

Tables 1 and 2 shows useful data that can be used to determine the achievable capacity limit over long links with direct LOS of about 6.505 km from one end of the network to the other. The data mimics the physical scenario of PVM as compared to the data sheet values of IEEE 802.11a and IEEE 802.11n air interfaces, respectively. The computed capacity values assume CSMA/CA and the protocol model whereby a transmission over one link is successful only if there is no active transmitter within a distance of 1 *d* . That is, the distance *d* is the range between a transmitter and receiver, and signifies a fraction of one hop distance needed to ensure collision-free transmission. The assumption of protocol model is reasonable in sparsely placed nodes in rural set-up whereby interference effects can be neglected without loss of generality. Furthermore, let the size of data carriers in OFDM scheme be, 48 *Mc* with each HPN having an 8dBi omni-directional antenna and 20 dBi being the directional antenna (i.e., the combined antenna gain is 630.96 (6.3096 times 100)

in a hilly and foliage area is taken approximately to be three between the frequency

channels 5.15 GHz and 5.85 GHz (Durgin et al., 1998). Then, from the capacity limit

Table 1, lists parameters needed to evaluate the achievable data rate for all wireless streams of a single link IEEE 802.11a radios. A comparison is made between the achievable data rate computed using data from the IEEE 802.11a air interface data-sheet and the data rate of the IEEE 802.11a air interface constructed from BB4allTM architecture. It is observed that while specifications of other parameters are kept the same in both cases, the combined antenna gain is taken to be 9 dBi from the data-sheet and that of BB4allTM architecture to be 28 dBi. With these antenna gains, the achievable data rate in standard architecture is 60.792 Mbps compared to 183.30 Mbps for the HPNs. This numerical result is explained as follows. When

*nmc n n*

 

 

 

2

and cluster density <sup>2</sup>

**5. Numerical examples using the Peebles valley mesh** 

Table 2 lists parameters needed to evaluate the achievable data rate for all wireless streams of a single link IEEE 802.11n radios. A comparison is made between the achievable data rate computed using data from the IEEE 802.11n air interface data-sheet and the data from the BB4allTM architecture. It is observed that while specifications of other parameters are kept the same in both cases, the datasheet combined antenna gain is taken to be 7 dBi and that of BB4allTM architecture to be 28 dBi. With these antenna gains, the achievable data rate is 291.33 Mbps in standard architecture compared to 570 Mbps for the HPNs. This numerical result is explained as follows. When transmission power and the size of the MIMO are constant, an increase in antenna gain increases capacity substantially. Thus, the HPNs have capacity gains over the standard IEEE 802.11n devices.

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 171

*E2E achievable capacity (Mbps)* 

**0.4202**

**0.5374**

(E2E) capacity limit with practical examples of network deployments. In particular, consider

a. Regular pattern when *n* 10 and when *n* 50 , the node density is distributed with

b. Irregular pattern when *n* 10 and when *n* 50 . Assume that the average distance of source-destination pair is 6505 m. The value enables the computation of achievable capacity over direct LOS path (i.e., without multi-hops) between the source and destination nodes. Nodes are assummed to be placed irregularly with a rate (probability) *p* , taken arbitrally as 0.9. Note that 0 1 *p* . The choice of *p* depicts the severeness of the irregular placements of HPNs, with smaller values of *p* depicts more irregular environment and larger value shows that the placement of nodes in an area is carefully

c. Regularly clustered pattern above when *n* 10 and 5 clusters each of 2 nodes, as well

*No. of HPNs Achievable link capacity (Mbps)* 

**50 R(700 m) = 376.22 0.9322**

**50 R(700 m) = 376.22 0.9827**

**R2(4200 m)= 221.13 R = min (R1, R2)** 

**R2(1400 m)= 316.22 R = min (R1, R2)** 

*Regular at p = 100%* **10 R(2100 m) = 281.12 0.5192**

*Irregular at p = 90%* **10 R(2100 m) = 281.12 0.5473**

the following cases:

planned.

*HPNs placement in a 5 km x 5 km area* 

uniform probability of one.

as when *n* 50 with 5 clusters each of 10 nodes.

*Clustered* **10 R1(700 m) = 376.22**

 **50 R1(700 m) = 376.22**

**Table 3.** IEEE 802.11a of HPNs of BB4all TM architecture


**Table 2.** IEEE 802.11n air interface single link capacity

## **5.2. End to end achievable capacity under different HPN placements**

Tables 3 and 4 show the E2E numerical values of capacity, right from the ethernet at one end of the network to ethernet at the other end of the network. Consider a wireless mesh network made up of IEEE 802.11a and IEEE 802.11n (Cisco systems, 2011) HPNs. Suppose typical information available are: the radio interfaces *m*2 , the orthogonal channel *c*2 , the deployment area *Amm* 5000 5000 and the bandwidth *W Mhz* 20 and carrier frequency of 5.85 GHz. Assume that Carrier sense multiple access with collision avoidance (CSMA/CA) protocol is employed in order to identify pairs nodes that can simultaneously transmit (Kodialam and Nandagopal, 2005). In this protocol, neighbours of both an intended transmitter and receiver have to refrain from both transmission and reception in order to avoid collisions. Practically, we can let =10% of one hop distance to be sufficient enough to prevent neighbouring nodes from transmitting on the same subchannel at the same time. One hop distance is approximately 2100 m. This study also assummed an optimized link state routing (OLSR) protocol that proactively maintains fresh lists of destinations and their routes (Johnson, 2007). These routing tables are periodically distributed in the network. The protocol ensures that a route to a particular destination is immediately available. Couto et al (2005) proposed an expected transmission count (ETX) metric to calculate the expected number of retransmissions that are required for a packet to travel to and from a destination. ETX metric is adapted in this study as a default routing metric to determine the amount of successful packets at any receiver node from a transmitting neighbour within a window period. ETX metric is also viewed as a high-throughput path metric for multi-hop wireless mesh network (Couti et al., 2005). Using such information, we can illustrate the end to end (E2E) capacity limit with practical examples of network deployments. In particular, consider the following cases:

170 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

*Maximum single link range (metres), d* **~6505 ~6505** *Modulation scheme* **OFDM OFDM**

*Number of spatial streams (2xMIMO), L* **2 2** *Combined antenna gain (dBi), Kantenna* **7 28**  *Channel width (MHz) of IEEE 802.11 a, W* **40 40**

*Reference distance (metres),* <sup>0</sup> *d* **5 5**

*AWGN (mWatts), N*<sup>0</sup> **1e-10 1e-10** *Achievable data rate (Mbps) for all streams, R* **291.33 570**

**5.2. End to end achievable capacity under different HPN placements** 

Tables 3 and 4 show the E2E numerical values of capacity, right from the ethernet at one end of the network to ethernet at the other end of the network. Consider a wireless mesh network made up of IEEE 802.11a and IEEE 802.11n (Cisco systems, 2011) HPNs. Suppose typical information available are: the radio interfaces *m*2 , the orthogonal channel *c*2 , the deployment area *Amm* 5000 5000 and the bandwidth *W Mhz* 20 and carrier frequency of 5.85 GHz. Assume that Carrier sense multiple access with collision avoidance (CSMA/CA) protocol is employed in order to identify pairs nodes that can simultaneously transmit (Kodialam and Nandagopal, 2005). In this protocol, neighbours of both an intended transmitter and receiver have to refrain from both transmission and reception in order to avoid collisions. Practically, we can let =10% of one hop distance to be sufficient enough to prevent neighbouring nodes from transmitting on the same subchannel at the same time. One hop distance is approximately 2100 m. This study also assummed an optimized link state routing (OLSR) protocol that proactively maintains fresh lists of destinations and their routes (Johnson, 2007). These routing tables are periodically distributed in the network. The protocol ensures that a route to a particular destination is immediately available. Couto et al (2005) proposed an expected transmission count (ETX) metric to calculate the expected number of retransmissions that are required for a packet to travel to and from a destination. ETX metric is adapted in this study as a default routing metric to determine the amount of successful packets at any receiver node from a transmitting neighbour within a window period. ETX metric is also viewed as a high-throughput path metric for multi-hop wireless mesh network (Couti et al., 2005). Using such information, we can illustrate the end to end

*RF Industrial, Science & Medical (ISM) band* 

*Maximum output power (mWatts) of IEEE* 

**Table 2.** IEEE 802.11n air interface single link capacity

*(GHz), f*

*802.11a radio, P*

*Parameter Data sheet BB4allTM architecture* 

**5 5**

**100 100**





Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 173

decreases proportionately and if the area

gating over shorter hops. The numerical results plotted in the tables 3 and 4 also showed that HPNs distributed with irregularity rate (probability) of 90% provides the highest E2E achievable capacity limit compared to the three node placement scenarios. This means that when HPNs are distributed with irregularity rate of 90%, the probability of finding some nodes in some areas will likely reduce by 10%. However, with the same number of nodes and fixed area of deployment, the inter hop distances where nodes occur will be much smaller by 10% than in regular placements. But shorter hops imply higher capacity if and only if there is no interference as we have noted with single links. Moreover, according to Li et al. (2001), increasing or keeping constant the number of nodes placed in a fixed area automatically increases or keeps constant the average node density. The average node density is inversely proportional to the E2E capacity according to *Theorem 2.* Thus, a lower average density in an irregular node placement for the same number of nodes will yield a higher E2E capacity if and only if the area of deployment is fixed or decreased. Using similar argument, when values of

of deployment is fixed or reduced then for the same number of nodes, the capacity will increase. Interestingly, Tables 3 and 4 showed that in regularly clustered placements, the E2E capacity limit values are least compared to other placement scenarios. The explanation is motivated by viewing that there is long distances between clusters and shorter distances between HPNs within a cluster. While, the former situation exacerbates achievable capacity, the latter improves capacity. The contributing factor within a cluster is then the inter-cluster distances

At n = 50 the achievable E2E capacity for clustered placement is almost two times less than one related to regular or irregular patterns in the case of IEEE 802.11a air interface network, but in the case of IEEE 802.11n air interface network it becomes more or less comparable. In Section 3, characterization of influence of multipath, multiple antennas, and hop distance on the link capacity revealed that multipath and distance predominantly affect capacity in IEEE 802.11a air interface, while multipath and the number of antennas predominantly influence the achievable capacity in IEEE 802.11n air interface. Because clustered placements irrespective of the number of antennas per HPN provide longer hop distances between one cluster and other, one expects much worse E2E capacity value in a clustered IEEE 802.11a air inter-

It was also noted that network throughput dropped significantly from source HPN to the destination HPN or gateway. In particular, the drop was by about 99% across 3 long distance hops and by about 99% across 3 long distance hops considering regularly deployed HPNs from Tables 3 and 4, respectively. The general explanation is that, the channel gain drops with increase in propagation distance, and there are also overhead losses associated with medium access control (MAC) and the multi-hop routing such that the number of packets sent is not equal to the number of packets received successfully. Despite this observation, HPNs derived from IEEE 802.11n radios have a better E2E capacity achievable mainly due to the MIMO tech-

In arbitral network, with a combined antenna gain of 9dBi, hop distance of 700 m, bandwidth of 20 MHz, transmitted power output of 100 mWatts and 1e-10 Watts, the

nologies that are capable of combating multi-path fading (Franceschetti et al., 2009).

*p* is decreased (i.e., 0.8, 0.7, 0.6, etc), the average

that degrades the overall capacity that can be achieved.

face compared to regular and irregular placements.


**Table 4.** IEEE 802.11n of HPNs of BB4allTM architecture

Table 5 illustrates the achievable E2E capacity results of the BB4allTM architecture compared to closely related work on dual radio dual channel analytical results by Kyasanur and Vaidya (2005).


**Table 5.** E2E achievable capacity gain of BB4allTM architecture

#### **5.3. Discussions on E2E achievable capacity**

It should be noted from both Tables 3 and 4 that in a fixed area of 5 km by 5 km, the E2E achievable capacity evaluated shows that there is lower capacity when number of HPNs is ten than when the number is 50 in all node placement scenarios. The main reason is that a series of long links created between any two immediate nodes degrades the achievable E2E capacity. This was proven by single link capacity models in Section 3. For instance, at ten HPNs in the fixed sized network, the hop distances are much larger than the case for 50 HPNs. In each hop, the propagating signal faces path loss effects due to terrain irregularity, foliage and wireless medium conductivity. The implication is that signal traversing longer hop distances are faced with higher attenuation and lower E2E capacity than signal propagating over shorter hops. The numerical results plotted in the tables 3 and 4 also showed that HPNs distributed with irregularity rate (probability) of 90% provides the highest E2E achievable capacity limit compared to the three node placement scenarios. This means that when HPNs are distributed with irregularity rate of 90%, the probability of finding some nodes in some areas will likely reduce by 10%. However, with the same number of nodes and fixed area of deployment, the inter hop distances where nodes occur will be much smaller by 10% than in regular placements. But shorter hops imply higher capacity if and only if there is no interference as we have noted with single links. Moreover, according to Li et al. (2001), increasing or keeping constant the number of nodes placed in a fixed area automatically increases or keeps constant the average node density. The average node density is inversely proportional to the E2E capacity according to *Theorem 2.* Thus, a lower average density in an irregular node placement for the same number of nodes will yield a higher E2E capacity if and only if the area of deployment is fixed or decreased. Using similar argument, when values of *p* is decreased (i.e., 0.8, 0.7, 0.6, etc), the average decreases proportionately and if the area of deployment is fixed or reduced then for the same number of nodes, the capacity will increase. Interestingly, Tables 3 and 4 showed that in regularly clustered placements, the E2E capacity limit values are least compared to other placement scenarios. The explanation is motivated by viewing that there is long distances between clusters and shorter distances between HPNs within a cluster. While, the former situation exacerbates achievable capacity, the latter improves capacity. The contributing factor within a cluster is then the inter-cluster distances that degrades the overall capacity that can be achieved.

172 Wireless Mesh Networks – Efficient Link Scheduling, Channel Assignment and Network Planning Strategies

*Regular at p = 100%* **10 R(2100 m) = 722.24 1.3339**

*Irregular at p = 90%* **10 R(2100 m) = 722.24 1.4061**

*Clustered* **10 R1(700 m) = 912.44**

 **50 R1(700 m) = 912.44**

*Consists of IEEE 802.11a HPNs: regularly placed* 

**Table 5.** E2E achievable capacity gain of BB4allTM architecture

**5.3. Discussions on E2E achievable capacity** 

**Table 4.** IEEE 802.11n of HPNs of BB4allTM architecture

*No. of HPNs Achievable link capacity (Mbps)* 

**50 R(700 m) = 912.44 2.2609**

**50 R(700 m) = 912.44 2.3832**

**R2(4200 m)= 602.24 R = min (R1, R2)** 

**R2(1400 m)= 792.44 R = min (R1, R2)** 

Table 5 illustrates the achievable E2E capacity results of the BB4allTM architecture compared to closely related work on dual radio dual channel analytical results by Kyasanur

0.9322 2.2609 0.01

It should be noted from both Tables 3 and 4 that in a fixed area of 5 km by 5 km, the E2E achievable capacity evaluated shows that there is lower capacity when number of HPNs is ten than when the number is 50 in all node placement scenarios. The main reason is that a series of long links created between any two immediate nodes degrades the achievable E2E capacity. This was proven by single link capacity models in Section 3. For instance, at ten HPNs in the fixed sized network, the hop distances are much larger than the case for 50 HPNs. In each hop, the propagating signal faces path loss effects due to terrain irregularity, foliage and wireless medium conductivity. The implication is that signal traversing longer hop distances are faced with higher attenuation and lower E2E capacity than signal propa-

*Consists of IEEE 802.11n HPNs: regularly placed* 

*E2E achievable capacity (Mbps)* 

**1.1443**

**2.0201**

*Arbitrary network of dual radio dual* 

*channel (Kyasanur and* 

*Vaidya, 2005)* 

*HPNs placement in a 5 km x 5 km area* 

and Vaidya (2005).

Dual-radio dualchannel mesh network

*Upper bound capacity value (of 50 nodes) in Mbps in a 5 km x 5 km* At n = 50 the achievable E2E capacity for clustered placement is almost two times less than one related to regular or irregular patterns in the case of IEEE 802.11a air interface network, but in the case of IEEE 802.11n air interface network it becomes more or less comparable. In Section 3, characterization of influence of multipath, multiple antennas, and hop distance on the link capacity revealed that multipath and distance predominantly affect capacity in IEEE 802.11a air interface, while multipath and the number of antennas predominantly influence the achievable capacity in IEEE 802.11n air interface. Because clustered placements irrespective of the number of antennas per HPN provide longer hop distances between one cluster and other, one expects much worse E2E capacity value in a clustered IEEE 802.11a air interface compared to regular and irregular placements.

It was also noted that network throughput dropped significantly from source HPN to the destination HPN or gateway. In particular, the drop was by about 99% across 3 long distance hops and by about 99% across 3 long distance hops considering regularly deployed HPNs from Tables 3 and 4, respectively. The general explanation is that, the channel gain drops with increase in propagation distance, and there are also overhead losses associated with medium access control (MAC) and the multi-hop routing such that the number of packets sent is not equal to the number of packets received successfully. Despite this observation, HPNs derived from IEEE 802.11n radios have a better E2E capacity achievable mainly due to the MIMO technologies that are capable of combating multi-path fading (Franceschetti et al., 2009).

In arbitral network, with a combined antenna gain of 9dBi, hop distance of 700 m, bandwidth of 20 MHz, transmitted power output of 100 mWatts and 1e-10 Watts, the conventional analytical results of Kyasanur and Vaidya (2005) was compared with the HPNs of the BB4allTM architecture. Data from Table 5 shows that HPNs of the latter with special radios and antenna arrangements is more superior to the HPNs with standard antenna gains. While all cases considered dual radio dual channel specifications, the HPNs of the BB4allTM architecture have higher throughput antenna configurations than the work proposed by Kyasanur and Vaidya (2005).

Achievable Capacity Limit of High Performance Nodes for Wireless Mesh Networks 175

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**7. References** 

C02.
