Throughput-Capacity and Bit-per-Joule Performance of IEEE 802.11 Based Wireless Mesh Networks

Rima Khalaf-Bitar and Izhak Rubin

Electrical Engineering Department University of California, Los Angeles

rima, rubin@ee.ucla.edu Abstract- Energy consumption is of utmost importance in wireless mesh networks where nodes are battery operated and have limited power resources. In this paper, we develop an analytical model to estimate the throughput-capacity of a wireless mesh network and use this model to study the network’s bit-per-joule performance. We assume that each node uses an IEEE 802.11 based software controlled radio whose modulation/coding scheme and rate can be dynamically selected. A topology synthesis algorithm is used to elect nodes to act as access points as well as backbone nodes (BNs). The latter form a mesh backbone network infrastructure. We present an algorithm that is used to analytically calculate an approximation of the throughput capacity performance of the network. We use this algorithm to examine the performance efficiency of the network, expressed in terms of the throughput rate per unit energy consumed under the joint setting of nodal transmit power and data rate levels. Our results show that when traffic is uniformly distributed across the area of operations, increasing the nodal transmit power levels, as well as the data rates, tend often to enhance the bit-per-joule performance of the system. We also examine the impact of the selection of the transmit data rate and power levels used in conducting the topology synthesis of the backbone network.

I- INTRODUCTION

Wireless mesh networks (WMN) [1] are considered for

the purpose of extending wireless coverage of wireless local area networks (WLANs) that are typically operated in infrastructure mode and are operated by access points (APs). A WMN involves a wireless mesh backbone network (BNet) that serves to interconnect AP nodes. The majority of currently implemented WLANs operate with physical and MAC-layer protocols that follow the IEEE 802.11 standard [2, 3]. Rubin et al [4] have recently introduced the mobile backbone network architecture (MBN) where two classes of nodes are identified: Regular Nodes (RNs) and Backbone Capable Nodes (BCN). RNs are nodes that typically possess limited energy processing and storage resources. BCNs are nodes that have more storage and energy resources. According to the topology synthesis algorithm introduced in [5, 6], some BCNs are dynamically elected to act as BNs and form a mobile backbone (BNet). We assume that BNs provide full network coverage so that each RN or BCN is within communications range of a BN [5, 6]. Each RN/BCN is required to associate with a BN; RNs and BCNs, associated with a BN form an Access Network (ANet). We illustrate an example of an MBN topology in Fig. 1.

Figure 1: Example of an MBN topology

In comparing the backbone synthesis roles played by the

nodes in the wireless mesh network, with those of nodes in the MBN model, we note that the mesh network Access Point (AP) nodes are similar to backbone-capable nodes, while non-AP nodes act as RNs. We note that certain AP nodes are elected by the backbone synthesis algorithm to serve as BNs. For a given BN, the BCNs and RNs (AP and non-AP nodes) that are associated with this BN are identified as its client nodes. Numerous routing schemes have been introduced in the literature for hierarchical ad hoc networks such as the MBN [7, 8, 13, 14]. Our focus in this paper is on cross layer operations that provide services to such routing functions. Hence, we assume that the underlying overhead used to discover and configure routes has been subtracted, so that net capacity levels are incorporated for our cross-layer calculations.

II- ENERGY CONSUMPTION OVERVIEW

The radio module of a WLAN card can be modeled to be

in either one of four states: sleep, idle, receive and transmit [9]. We denote by Ws the energy consumption rate of a node, measured when in sleep state, and by WI the energy consumption rate of a node when in idle state. We use Wt and Wr denote the energy consumption rate of a node in transmit mode and receive mode, respectively. Ws, WI, Wt and Wr are all measured in milli-Watts. In this paper, for illustrative purposes, we demonstrate performance results that are based on measurements that pertain to the Aironet IEEE 802.11 PC4800 card. The card can operate at 4 different transmit

The 8th IFIP Annual Mediterranean Ad Hoc Networking Workshop 2009

978-1-4244-4661-2/09/$25.00 ©2009 IEEE 34

power levels (1mW, 5mW, 20mW, 50mW) and 4 different transmit data rates (1Mbps, 2Mbps, 5.5Mbps and 11Mbps). The measurements are presented in Tables 1 and 2.

Rate Transmit (1mW)

Transmit (5mW)

Transmit (20mW)

Transmit (50mW)

1 Mbps 1.42 W 1.5 W 1.68 W 1.78 W 2 Mbps 1.43 W 1.51 W 1.69 W 1.8 W

5.5 Mbps 1.45 W 1.53 W 1.71 W 1.82 W 11 Mbps 1.49 W 1.55 W 1.73 W 1.84 W

Table 1: Energy consumption rates upon transmission at different transmit power levels

Rate Sleep Idle Receive

1 Mbps 0.075 W 1.34 W 1.3 W 2 Mbps 0.075 W 1.34 W 1.31 W

5.5 Mbps 0.075 W 1.34 W 1.39 W 11 Mbps 0.075 W 1.34 W 1.41 W Table 2: Energy consumption rates in sleep, idle and receive states at

different data rates

III- SYSTEM MODEL

We consider a system of nRN regular nodes, and nBCN backbone capable nodes. When performing the topology synthesis algorithm, all nodes (RNs and BCNs) use the same transmit power level Pe and the same transmit data rate value Re. Upon completion of the topology synthesis algorithm, nBN nodes are elected from the set of BCNs to form a connected Backbone Network (BNet). Thus, the total number of client nodes, i.e., nodes that are members of the various Access Nets (ANets), nANet is equal to RN BCN BNn n n+ − . Since each ANet is managed by a single BN, the total number of ANets, NANet is equal to the total number of elected BNs.

In considering a mesh-backbone network, we assume

that the ANet and BNet segments employ different frequency bands. We assume the use of a hybrid FDMA/CSMA/CA MAC whereby BNet segment transmissions occur in a different frequency band than ANet segment transmissions. We classify the mesh network to consist of two segments: The BNet and the ANet segments. The BNet segment consists of the topological layout of the BNet and the spectral / capacity assets that involve transmissions initiated by a BN and directed across a link to another BN. The ANet segment consists of the topological layout of all access networks (ANets) and the spectral / capacity assets that involve transmissions initiated by a BN and directed across a link to its client ANet node, as well as all intra-ANet transmissions. Each packet has a payload size of Lb bits, a physical layer header of PHY bits, and a MAC layer header of MAC bits. The size of the acknowledgment packet is ACK bits. Time is divided into slots, and the duration of each slot is σ. There are Nf end-to-end flows present in the network. For simplicity, we assume that all flows have the same packet rate.

Each node on the BNet segment can choose to transmit at m different transmit power levels and l data rates. However, at any given point in time, the network manager regulates the operation so that all nodes on the BNet choose the same transmit power level, PBNet and operate at the same transmit data rate, RBNet, thus using the same modulation/coding scheme. Similarly, all client nodes are regulated by their corresponding managing BNs to choose, at any given time, the same transmit power level PANet and the same transmit data rate RANet. The values for PBNet, RBNet, PANet, and RANet are chosen to guarantee network connectivity and at the same time to yield an efficient high throughput network operation. On the BNet segment, each selected combination of PBNet and RBNet values gives rise to a communications range

BNetTd , a carrier sensing

rangeBNetCSd , and an interference range

BNetId . Similarly,

on the ANet segment, each combination of PANet and RANet values gives rise to a communications (or transmission) range

ANetTd , a carrier sensing range ANetCSd , and an

interference rangeANetId .

A. Spatial Re-use Factor Approximation

Because the BNet forms a multi-hop network, a BN i, does not necessarily contend with all other BNs in the network, but only with a subset of them whose number is denoted by nvBN (also denoted as nvBN(i)). This set of contending nodes consists of nodes that fall into 2 categories:

i- Nodes that are located within the carrier sense range of

transmitter BN i. These are nodes that are located in the disk that has BN i as its center and the carrier sensing range ( )

BNetCSd i as its radius. Nodes located in this

disk will not be able to transmit while BN i is in transmission mode as their carrier sensing signal will be activated.

ii. Nodes that are located within the interference range of the link layer of receiver BN j. This forms a disk that has the receiving BN j as its center and the interference range ( )

BNetI jd as its radius. Any node in this

interference disk that transmits while BN j is in reception mode may hinder j’s reception.

The actual contention area is calculated as the area

formed by the union of the above mentioned two disks. To simplify our mathematical analysis, we increase the latter area by defining the contention area AsBN(i,j) as a disk that covers the actual contention area, and as such has the smallest possible radius. Thus, as noted in Fig. 2, the contention disk provides tight covering of the actual contention area. Clearly, the diameter of the latter covering contention disk is equal to:

max( , )RBNet IBnet IBNet csBNet FBNetd d d d d= + + (1)

35

Figure 2: Diagram showing the circular areas that contain

the set of contending nodes Assuming traffic flows to be distributed so that they

geographically well cover the effective a*a area of operations, whose size is denoted by A, and that distinct successful transmissions must belong to disjoint contention areas, we use a disk covering approach to calculate the number of disks that can be packed in the area to derive an approximate expression for the BNet’s attainable spatial reuse factor (SRFBNet) level. SRFBNet is thus approximated as:

2(0.5 )BNetRBNet

ASRFdΠ

(2)

The Spatial re-use factor of the ANet, SRFANet can be

calculated in a similar fashion.

B. Average Path Length Approximation We first start by calculating the average path length of

the BNet. Since BNs are uniformly distributed, the probability density function characterizing the random variable v denoting the distance between two such nodal locations, as shown in [15], is given by:

04

4( ) ( ) Vvf v f v

a= ⋅

2 2

2 1 2 2 2 2 1 20

12 , 02 2

1( ) sin 2 cos , 2 2

0,

a av v for v a

a af v a a v a a a v fora v av v

elsewhere

π

− −

⎧ − + ≤ ≤⎪⎪⎪= + − − − − ≤ ≤⎨⎪⎪⎪⎩

(3)

Hence, the parameter BNetL which expresses the average hop length of a path connecting randomly chosen source and destination nodes is given by:

min( * , )( / )

2 ( 1)*

* ( )BNetBNet

BNet

i dF aceil a dF

BNet Vi v i dF

L i f v dv= = −

= ∑ ∫ (4)

We next calculate the average path length across an ANet. When calculating this average path length, it is important to

note that a certain fraction of the flows, denoted asφ , would require routing on the BNet. φ is given by:

( ( , ) )ANETTP d i j dφ = > , (5)

where d(i,j) represents a random variable denoting the distance between any two randomly chosen nodes i and j. Consequently,

2 (1 ) = 1ANetL φ φ φ= + − + (6) C. Identification of a Successful Packet Transmission

A transmission from node i (a BCN or an RN) to node j (a BCN or RN) is successfully received if node j is not in transmission state and if node i’s signal is received at node j at a signal to interference and noise ratio (SINR) whose value is not lower than a minimum required threshold γ. The associated SINR itself is a random variable given as

, . .

, . .(1 )

i ijI d

i j i ijId

k kjk

PGw p P

N

SINR PGw p P

N P G∈Ω

⎧⎪⎪⎪⎪= ⎨

−⎪+⎪

⎪⎪⎩

∑

(7)

where SINRij denotes the SINR experienced at node j when node i is transmitting, Pi is the transmit power of node i. N denotes the thermal noise power at the receiver; Ω denotes the set of other nodes, excluding nodes i and j, that transmit messages during the same time that the i-to-j transmission takes place; Gij denotes the channel gain between nodes i and j. In this paper, we adopt a propagation model whereby the signal attenuates in accordance with the power of distance, i.e., Gij = ηd(i,j)-α where d(i,j) denotes the distance between i and j, η is a constant, and α is the attenuation factor, 2 4α≤ ≤ . We note that our approach can be also applied to other, more complex, propagation models.

PId is the probability that, given that node i has initiated a transmission to node j, and given that node j itself does not initiate at this time a packet transmission, no other transmission at this time is sensed by node j’s receiver. Thus, setting p to denote the probability that a node initiates a transmission in an idle slot (including retransmitted packets), we have:

CSn -2IdP = (1- p) , (8)

where nCS is the number of nodes (BNs or client nodes) in the carrier sensing region of the sending node.

When node i is transmitting to a link layer neighboring node j and given that node j’s transceiver is in reception mode, the associated conditional channel state, denoted as CS, is defined as a binary vector whose state is set to be Idle when no other transmission takes place at this slot, while it is set to be Busy, otherwise. Thus, when the system is in CS = Idle state, the packet transmission initiated by

36

node i is received successfully at node j. The probability distribution function of CS is calculated by noting the following relation:

, . .

, . . (1 )

I d

Id

Idle w p P

C SBusy w p P

⎧⎪⎪= ⎨ −⎪⎪⎩

(9)

IV. THROUGHPUT CAPACITY CALCULATION

As we mentioned in our system model earlier, BNs typically possess higher capabilities in terms of computational complexity, power levels, and transmit data rates than client nodes. Thus, depending on the network’s topology as well as on the values for BNetP , ANetP , ANetR , BNetR , either the ANets or BNet can become the system's throughput bottleneck. We now present a step-by-step process to calculate the maximum throughput attained in an MBN in either case.

- Step 1: Calculation of the Probability of Successful

Packet Transmission on the BNet. In the following, we calculate the probability, denoted as

PSSBNet, that a successful transmission will be initiated in a slot by any node residing in a single hop region AsBN(i,j). We assume that the message traffic loading the BNet is uniform and that each node on the BNet transmits at a rate of

BNetp packets/idle slot. To calculate the probability of successful transmission, we consider the slot at which the tagged transmission from BN i to BN j is initiated. We assume the data transmission over this slot to be successfully initiated if the SINR level measured at the intended receiver is higher than the specified threshold level. Recall that, as an approximation, we have assumed that at most a single such successful transmission will take place at a slot within such a region. For this purpose, using our assumption of homogeneous loading conditions to apply at the different nodes on the BNet, it is sufficient to examine the occurrence of such an event by focusing on a single BN, say BN i, that transmits a packet to a neighboring node, identified as BN j. Using our model, it is sufficient to examine events occurring within the single hop region AsBN(i,j) to determine the probability of success of such a transmission. Noting the assumed symmetry, the probability that a successful transmission will occur in a time slot when initiated by any node residing in AsBN(i,j) = AsBN, is obtained by multiplying the single pair result by the number of BNs located in AsBN, as shown below. We set CS = idle, if no node in the carrier sensing region of BN i is transmitting at the same time that the tagged transmission takes place, and we set it equal to Busy, otherwise. We thus obtain:

Pr( )BNetPss successful packet transmissionby a BN in a slot= (10)

1Pr(

)

vBNn

isuccessful packet transmission

in a slot by BN i=

=∑ (11)

Pr(

/ , )*Pr( , )vBNn successful packet transmission in a slot

i transmits j idle i transmits j idle

= (12)

(1 )

Pr( ( ) / , )vBN BNet BNetn p p

SINR i j i transmits j idleγ= −

→ ≥ (13)

,

(1 )

Pr( ( ) , / , )vBN BNet BNet

CS idle busy

n p p

SINR i j CS i transmits j idleγ=

= −

→ ≥∑ (14)

=

2

0

2

1_

(1 )*

Pr( ( ) ,

)

(1 ) Pr( ( ) ,

/ , , )

vBN BNet BNetn

Id k

nId m k

vBN

vBN

interferers

interferers

n p p

P SINR i j k

/ i transmits, j idle,CS = idle

P SINR i j

k i transmits j idle CS busy

γ

γ

−

=

−

=

= −

→ ≥

+ − → ≥

=

∑

∑

(15)

We note that when the conditional channel state CS is

equal to Idle, the only interferers that would impede BN i’s transmission to j would be those that are positioned at a distance from the transmitter that is further away than the carrier sense range. When the conditional channel state is Busy, nodes that are located within carrier sense range of BN i, the number of which is nCSBN (i)-2, when we exclude BNs i and j, as well as nodes located outside this range, the number of which is denoted as nHBN(i), can impede the tagged transmission. Consequently, we proceed to calculate the expression given by Eq. (15) as follows:

BNetPss =

( ) 1

( ) 1

( )

(1 )

(1 ) [Pr( ( )1 / ) *Pr( / )]

BNet

BNet BNet

vBN BNet

vBN BNet

csBN

n icsBN HBN

n ivBN

pi

n p pnn p SINR i jk

k outside interferers,i transmits, j idle,CS =idlek outside interferers i transmits, j idle,CS =idle

n p

γ−

−= −

+ − → ≥∑ =

+

1 2

( )(1 )(1 (1 ) )( ) 2 ( ) * [Pr( ( )1 1

/ )

/

BNet BNet

1 2

csBNn i

HBN

p pn i n icsBN SINR i jk k

k inside,k outside interferers,i transmits, j idle,CS =busy ]* Pr(k inside,k outside interferers1 2

i transmits, j idle,CS =b

γ

− − −−

→ ≥∑ ∑= =

]usy) (16)

37

We note that the expression in Eq. (16) denotes the probability that a slot contains the start of a successful transmission initiated by a node with region AS. In a single hop BNet where all nodes are within communications range, carrier sense and interference range of one another, a transmission from BN i to BN j that is successful in its first slot will be successfully completed since all BNs in the network will refrain from transmitting during i’s transmission time as their carrier sensing signal has been activated. In a multi-hop BNet however, BNs that are outside the carrier sensing range of BN i can transmit at any time during i’s transmission, thus potentially causing interference at BN j. Thus, even if the transmission from i to j started out successfully, interference caused by those nodes hidden from BN i might cause this transmission to fail. We define the vulnerable period TvBNet, measured in slots, as the period in which BN i’s transmission can be interfered with by a node located outside of its carrier sensing range. When operating in IEEE 802.11’s default mode, not using the RTS/CTS handshake, the vulnerable period consists of the time needed to transmit the data packet and is given by:

/b BNetBNet

L RTvσ

= . (17)

Any node in AS hidden from the transmitter BN i that attempts a packet transmission during the vulnerable period Tv may cause interference at the receiver j, thus hindering its successful reception. For successful reception at BN j, a packet has to be initiated successfully in its first time slot, the probability of which is given by Eq. (16), and the SINR at BN j has to be sustained above the threshold γ for the subsequent TvBNet-1 slots. If a node hidden from BN i does transmit a packet during the remainder period lasting for TvBNet-1 slots, the resulting SINR level measured at the intended receiver has to be greater than the threshold γ to ensure successful reception of the whole tagged packet.

We denote the probability that a packet transmission from BN i to BN j is successful in subsequent TvBNet-1 time slots, given that it was successful in the first time slot, and given that no node in AsBN(i,j) has initiated transmission at the same time slot as BN i, by FSBNet

P (FS used to identify

the term “future slots”). It is given by:

1)

1

( )*(

( )( )

(1 )

( ) ( ) (1 )

* ( / , )

BNetBN

BNBNBN

BNet

H vBNet

HHH k

BNet BNetk

n i TFS

n in i k

P p

n ip p

kP SINR

i transmits j receives, k hidden BN interferersγ

−

=

−

= −

+ −

>

∑ (18)

Therefore, the probability that a packet’s transmission is

successfully initiated and successfully received can be approximated by:

( )

1

( ) 1

( ) 1

(

* [ Pr( ( )

/ )*Pr( /

(1 ) *

1 )iHBN

k

BNBNet BNet FSBN BNet

BNet BNetn

ics

iBNvBN

nBNet

ncs

SINR i j

k BN outside interferers,i transmits, j idle,CS = idlek outside interferers i transmits, j idle,CS = idl

Pss

n p p Pv

n p p

γ=

−

−

→ ≥

=

−

+ −

∑

1 2

( ) 2

( )( ) 2

(1 )(1 (1 ) )

1 1

)]

)

Pr( ( )

/

* [

2

BNet

1

H

BNet BNet

BN

iBN

iiBN

1

ncsvBN

k k

e

n pnn

k outside BN interferers,i transmits, j idle,CS = busy ]* Pr(k BN inside

p p

cs SINR i j

k inside BN interferers,

γ

−

−

− − −

= = → ≥

+

∑ ∑

]2,k BN outside interferers

| i transmits, j idle,CS = busy) (19) Step 2: Calculation of the BNet segment’s throughput-

capacity in isolation

In order to calculate the end-to-end throughput capacity of the BNet, we first calculate the throughput-capacity of a single hop region on the BNet. As explained in Section III, this region consists of the union of the disk whose radius is the carrier sensing range of a BN and another disk whose radius is the interference range of a BN. The throughput of a single hop region on the BNet, when the BNet is considered in isolation, is given by:

/*

BNet

BNet BNetBNet region

cycle

Ts PssST

= (20)

where TsBNet is the average time it takes for a BN to complete a successful packet transmission to another BN ,

BNetcycleT is the average time elapsed between two

transmission attempts across the medium which is also taken as the basic regeneration cycle in our analysis. It accounts for the possibility of a successful transmission as well as for the probability of a collision and involves retransmissions, overhead and ACK transmissions. Step 3: Calculation of the ANet segment’s throughput-

capacity in isolation

In this step, we calculate the throughput of the ANet segment in isolation. This consists of the throughput contributed by the flows distributed between BNs to their client nodes, and by the flows directed between client nodes themselves. To calculate the ANET’s throughput, we first calculate the throughput of a single hop region on the ANet segment, which is formed by a disk whose radius is approximated by the carrier sensing range of the transmitting node and the interference range of the receiving node. We write:

38

*( | )*

maxANet

ANetANet

ANet ANetANet link layer p p ANet

cycle

Pss TsS SRFT == . (21)

where pANet denotes the rate (in packets/ idle slot) at which a client node transmits to another client node. ANetPss denotes the probability that a successful transmission between two client nodes occurs in a slot. ANetTs denotes the time it takes to complete a successful packet transmission, including the transport of the headers and the time to receive an acknowledgment.

ANetcycleT is the average duration between

two successful packet transmissions between two client nodes, as calculated in [12]. The end-to-end ANets throughput is given by:

/ANet ANet ANetend to end link layer

S S L− −

= (22)

Step 4: Identification of the system bottleneck

We are now ready to identify which of these two segments is the system bottleneck, as the throughput-capacity calculation method of the network will be different in either case. If the ANet segment is the system bottleneck, then the throughput of the network is dictated by the ANets’ throughput capacity as calculated in Eq. (22). If the BNet segment is the system bottleneck, then in order to fully utilize the BNet’s capacity, we equate the flow rate generated by the ANets to be routed on the BNet to the BNet’s capacity such that:

*2

ANet link layerANet to BNetend to end BNet end to end

SS S

φ− − − − − −= = , (23)

where the division by 2 is necessary to account for packets traversing 2 hops, one from the transmitting client node to the managing BN and another one from the BN to the destination client node. We then calculate the throughput rate involving the remaining ANet only (client-to-client) flows:

(1 )*ANet only end to end ANet link layerS Sφ− − − = − (24) Thus, the maximum attainable throughput of the network is given by:

MBN ANet

ANet to BNetend to end ANet only end to end

S SS S− − − − − − −

== +

(25)

V. BIT PER JOULE PERFORMANCE

We now proceed to calculate the energy consumed by

client nodes (BCNs or RNs). Let λp denote the rate at which packets are generated, expressed in units of packets/sec, by each of the fN end-to-end flows. The fraction of time that a node spends in transmitting packets is given by:

(1 )*( / ( ) / )tx p t b ANet bANetr L R PHY MAC Rρ λ= + + + , (26)

where tr is the average number of retransmissions experienced by a client node as calculated in [12].

For each successful packet transmission, the node will receive an acknowledgment. Thus, the ratio of time that the node spends in receiving ACKs is given by:

*( / )rx p ANetANetACK Rρ λ= (27)

Since we are assuming that each awake client node generates and receives a single flow, then each such node will spend a fraction of the time 'rx ANet

ρ receiving packets. This ratio is

given by: ' *( / ( ) / )rx p b ANet bANet

L R PHY MAC Rρ λ= + + (28)

Furthermore, each awake node in the ANet will transmit an ACK for each packet that it successfully received. The fraction of the time it spends doing so is given by:

' *( / )tx p ANetANetACK Rρ λ= . (29)

Finally, the ratio of time that a node spends in the idle state is given by:

1 ( ') ( ')I tx tx rx rxANet ANet ANet ANet ANetρ ρ ρ ρ ρ= − + − + . (30)

We now proceed to calculate the energy consumption for backbone nodes. In order to do so, we must first characterize the packet flow rate,

BNpλ , expressed in units of packets per

sec, experienced at BNs. The offered packet rate to the BNet is given by:

* *p p fBNet offered

Nλ φ λ= . (31)

The packet flow rate traversing each BN, on average,

assuming all backbone nodes to be evenly loaded, is given by:

*( 1)

BNet offered BNetpBN

BN

p L

n

λλ

+= (32)

Now the fraction of time that a BN spends in transmitting packets to other BNs is given by:

*(1 )[ *( /

( ) / )]BN BNBN BN BNtx p t B b BNet

b

r L R

PHY MAC R

ρ λ κ−−

= +

+ + , (33)

where BN BNtr −

is the average number of retransmissions experienced by a BN when transmitting to its BN neighbors, as calculated in [12]. Consequently, the ratio of time a BN spends transmitting packets to its client nodes is given by:

*(1 )[(1 )

*( / ( ) / )]BN ANetBNBN ANettx p t B

b ANet b

r

L R PHY MAC R

ρ λ κ−−

= + −

+ + , (34)

where BN ANettr −

is the average number of retransmissions experienced by a BN when transmitting to its BN neighbors, as calculated in [12]. Bκ is the fraction of time a BN

39

communicates with other BNs as opposed to communicating with its client nodes, and is given by:

1BNet

BBNet

LL

κ =+

(35)

The fraction of time that a BN spends in receiving ACK packets for packets it has transmitted to a neighboring BN is given by:

* *( / ) BN BN BNrx B BNetp ACK Rρ λ κ

−= (36)

Consequently, the ratio of time a BN spends in receiving ACK packets for packets it has transmitted to its neighboring non-BN nodes is given by:

(1 )*( / ) BN ANetrx B ANetACK Rρ κ

−= − (37)

The fraction of time that a BN spends in receiving packets from other BNs is given by:

' *[ *( /

( ) / )]BN BN BNrx p B b BNet

b

L R

PHY MAC R

ρ λ κ−

=

+ + (38)

Similarly, the ratio of time a BN spends in receiving packets from its ANet clients is given by:

' *(1 )*( /

( ) / )]BNBN ANetrx p B b ANet

b

L R

PHY MAC R

ρ λ κ−

= −

+ + (39)

The fraction of time a BN spends in transmitting ACKs to packets it has received from either the BNet or the ANet is given by:

' * *( / )BN BNtx pBN B BNetACK Rρ λ κ

−= . (40)

' *(1 )*( / )BN ANettx pBN B ANetACK Rρ λ κ

−= − (41)

Thus, the fraction of time that a BN is idle is given by: 1 (

' ')

( ' ')

BN BN BN BN ANet

BN BN BN ANet

BN BN BN ANet BN BN BN ANet

I tx tx

tx tx

rx rx rx rx

ρ ρ ρρ ρρ ρ ρ ρ

− −

− −

− − − −

= − +

+ +

− + + + (42)

The total energy consumption in the network, including energy consumed by nodes that reside in the sleep state, denoted as total MBNW , is given by:

*[( * ( ')*

( ')*

( ')*

( ')* )]

*( * ( ')*

BN BNet BN BN BN BN BNet

BN ANet BN ANet ANet

BN BN BN BN BNet

BN ANet BN ANet ANet

ANet ANet

total MBN s s

BN I I tx tx t

tx tx t

rx rx r

rx rx r

ANet I I tx tx tAnet ANet AN

W n Wn W W

W

W

W

n W W

ρ ρ ρρ ρρ ρρ ρ

ρ ρ ρ

− −

− −

− −

− −

=+ + +

+ +

+ +

+ +

+ + +

( ')* )et

rx rx rANet ANet ANetWρ ρ+ +

(43)

where BNetIW ,

BNettW andBNetrW represent the energy

consumption rates of a BN on the BNet segment using a

transmit power BNetP when in idle, transmit or receive modes, respectively, and where

ANetIW ,ANettW and

ANetrW denote the energy consumption

rates of a client node on the ANet segment or a BN transmitting to a client node using a transmit power ANetP when in idle, transmit or receive modes, respectively. The bit-per-joule performance of an MBN (BPJMBN) is then obtained by dividing the throughput rate

MBNS calculated in Section IV by the total energy consumed in the network, yielding:

MBNMBN

total MBN

SBPJW

= . (44)

VI. PERFORMANCE EVALUATIONS

We consider 150 nodes that are uniformly distributed in a 2000m* 2000m terrain. The BN election algorithm is carried out using a 5mW transmit data rate and a 2Mbps data rate. The client nodes transmit their flows using a data rate of 2Mbps and a power level of 5mW. The ANet and BNet segments operate on separate frequency bands. 40 flows are present in the network, and these flows originate and terminate at 40 client nodes in the network. The election algorithm yields 21 BNs, whose main role is to route flows between client nodes that are not within transmission range of one another. The BNs can transmit at either 20 mW or 50 mW, and can choose to transmit at data rates of either 1, 2, 5.5 or 11 Mbps. As can be seen from Fig. 3, the highest bit-per-joule performance level is attained when the BNs are operating at the highest possible data rate and transmit power levels. We also note that our analytical evaluations yield results that are very close to those produced by executing simulation runs. We now turn our attention to study the effect of the setting of the power and data rate levels used during the BN election process, on the bit-per-joule performance of the network. We have conducted an analysis of a network of 150 nodes that are distributed in a 2000 m *2000 m area. 40 flows are present in the network and the path-loss exponent α is equal to 2. Bit-per-joule performance results are presented in Fig. 4. As can be seen from the figure, increasing the power level and the data rate leads to an increase in the bit-per-joule performance of the network. Clearly, increasing the power level at which the election algorithm is being done causes the number of BNs that are forced to be awake to relay packets to decrease, and thus reduces the level of energy consumption. Increasing the data rate enhances the bit-per-joule performance of the network as it tends to increase the throughput-capacity of the network. The same behavior is seen in Fig. 5, which presents results for a network of 750 nodes in a 750 m*750 m terrain; the path-loss exponent is set to be equal to 2.5. It is interesting to note that operations at BN election power levels of 20 mW and at 50 mW yield similar bit-per-joule performance behavior. This result is explained by noting that operations at BN election power levels of 20 mW and at 50 mW yield similar SRF and average path length values.

40

Figure 3: Bit-per-joule Performance of an MBN; client nodes operate at 2Mbps and 5 mW and BNs transmit at the data rates and power

levels shown

Figure 4: Bit-per-joule Performance of an MBN; BN election carried

out at the power levels and data rates shown, α=2

Figure 5: Bit-per-joule Performance of an MBN; BN election carried

out at the power levels and data rates shown, α=2.5 VII. CONCLUSIONS

In this paper, we have presented an analytical model for

the calculation of the throughput-capacity and bit-per-joule performance of mesh-backbone IEEE 802.11 based wireless

networks, when employed under a multitude of transmit power levels, forwarding ranges and transmit data rates. For the models investigated, our main conclusion here is that transmitting at higher power levels tends to produce a better bit-per-joule performance behavior than that attained when operating at lower transmit power levels. This is due in large part to the increase in the internal traffic rate traversing the network that is induced under lower transmit power operation, subsequently leading to an increase in the number of nodes that are forced to be kept awake (acting as access points and as backbone nodes) to relay that traffic. . Generally, we have observed that the bit-per-joule performance of mesh-backbone networks is enhanced when the transmit power (Pe) and data rate (Re) levels at which the election algorithm is executed are increased. Furthermore, as the levels of both Pe and Re are increased, we have shown the throughput-capacity of the network to generally also increase, thus further contributing to the enhancement of the network's bit-per-joule performance behavior.

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