Appendix B

Proof of Theorem 1, Chapter 5

We now investigate the conditions for the effective transport capacity in the case with reserve-and-go (RESGO) medium access control (MAC) protocol to be quantitatively the same as inthe ideal (no inter-node interference, INI) case.

In the case with RESGO MAC protocol, in Chapter 3 it has been shown that the routeBER at some point reaches a floor which has the following expression:

BERRESGOroute,floor = ξRESGOcaLOSnhλL

Rb(B.1)

where ξRESGO = 3 and caLOS = 0.25. If this route BER floor is lower than the maximumtolerable BER at the end of a multi-hop route, then connectivity is lost and the effectivetransport capacity with the RESGO MAC protocol will never approach that in the ideal(no INI) scenario. Therefore, the first condition to be satisfied in order for the effectivetransport capacity with the RESGO MAC protocol to be equal to the effective transportcapacity in the ideal case is the fact that the route BER floor has to be lower than the minimumtolerable BER at the end of a multi-hop route, i.e.

ξRESGOcaLOSnhλL

Rb< BERmax

route (B.2)

which establishes the first condition (5.29) of the theorem.Provided that (B.2) is satisfied, in Chapter 3 it has been shown that the route BER can

be accurately characterized through a Gaussian assumption for the interference noise. In thiscase, a definition of the link signal-to-noise ratio (SNR) becomes meaningful. In particular,the link SNR can be rewritten as follows:

SNRRESGOlink = Pr

P RESGOnoise

(B.3)

where Pr = αρSPt and

P RESGOnoise = Pthermal + P RESGO

int = FkT0Rb + αPtρS(1 − e−λL/Rb) A(N). (B.4)

By defining

ε1 � FkT0, ε2 � λL, ε3 � αPtρS A(N) (B.5)

Ad Hoc Wireless Networks: A Communication-Theoretic Perspective Ozan K. Tonguz and Gianluigi Ferrari© 2006 John Wiley & Sons, Ltd. ISBN: 0-470-09110-X

288 Appendix B. Proof of Theorem 1, Chapter 5

the noise power with RESGO MAC protocol can be written as

P RESGOnoise = ε1Rb + (1 − e−ε2/Rb)ε3. (B.6)

At this point, in order to have connectivity it is necessary that the link SNR is larger than thethreshold value SNRmin

link, given, in the case of uncoded BPSK transmission with strong LOS,by (5.18). In the case with RESGO MAC protocol, the condition SNRRESGO

link ≥ SNRminlink,

required to support communication over a multi-hop route, can be rewritten as

P RESGOnoise ≤ αρSPt

SNRminlink

� P thnoise. (B.7)

Since (i) Pthermal is a monotonically increasing function of Rb and (ii) P RESGOint is a

monotonically decreasing function of Rb, it is possible to conclude that the data-rate forwhich (B.7) is satisfied with equality is the maximum data-rate guaranteeing that the effectivetransport capacity in a realistic network communication scenario and the RESGO MACprotocol is the same as in the ideal case. We denote this data-rate as R

RESGO,maxb . For a given

value of the traffic load λL and a given number of nodes N , in order to find RRESGO,maxb we

study the behavior of P RESGOnoise as a function of Rb. In particular, one can write

∂P RESGOnoise

∂Rb= ε1 − e−ε2/Rb

ε3ε2

R2b

. (B.8)

Imposing ∂P RESGOnoise /∂Rb = 0, one obtains

eε2/RbR2b = ε3ε2

ε1. (B.9)

Defining g(Rb) � eε2/RbR2b, since

limRb→0+ g(Rb) = lim

Rb→∞ g(Rb) = ∞ (B.10)

and g(Rb) > 0, (B.9) has at least one solution (at most two) provided that the minimum ofg(Rb) is lower than the right-hand side, given by ε3ε2/ε1, of (B.9). Since

g′(Rb) = eε2/Rb(2Rb − ε2) (B.11)

it follows that g(Rb) has an absolute minimum for RRESGO,g−minb = ε2/2. Hence, it must

hold that

g(RRESGO,g−minb ) = e2

(ε2

2

)2

<ε2ε3

ε1(B.12)

which can be rewritten as

λL

PtρS<

4α A(N)

e2FkT0. (B.13)

There are two cases of interest.

Appendix B. Proof of Theorem 1, Chapter 5 289

• Condition (B.13) is not satisfied. In this case, P RESGOnoise is an increasing function of Rb

and larger than P RESGOnoise |Rb=0 = ε3. There can exist a maximum data-rate such that an

average number of hops is sustained if

ε3 ≤ ε1PtρSkT0SNRmin

link

Fα(B.14)

which can be rewritten as

α2 A(N)

k2T 20 SNRmin

link

≤ 1. (B.15)

For a carrier frequency fc = 2.4 GHz, it is possible to show that, for normal values ofN , (B.15) is never satisfied.

• Condition (B.13) is satisfied. In this case, P RESGOnoise has a minimum.

Assuming condition (B.13) is satisfied, (B.9) has two solutions. Based on the first limit in(B.10), we can conclude that the first solution corresponds to a data-rate slightly larger than0. The first solution corresponds to a relative maximum of P RESGO

noise . As an example, forF = 6 dB, λL = 10 s−1, fc = 2.4 GHz, Pt = 2 × 10−7 W and ρS = 10−4 m−2, itfollows that ε2ε3/ε1 � 1.2 × 106 A(N), where A(N) > 9. One can then argue that thesecond solution of (B.9) should be much larger than 0. This is the solution of interest, sinceit corresponds to a relative minimum of P RESGO

noise . By defining this solution as RRESGO,minb , it

follows that

eε2/RRESGO,minb

(R

RESGO,minb

)2 = ε2ε3

ε1. (B.16)

Since eε2/RRESGO,minb � 1, from (B.16) one obtains

RRESGO,minb �

√ε2ε3

ε1=√

λLαPtρS A(N)

FkT0. (B.17)

The following cases can be distinguished.

• If P RESGOnoise |RRESGO,min

bsatisfies (B.7), then the maximum allowed data-rate R

RESGO,maxb

lies between RRESGO,minb and Rmax

b (the maximizing data-rate in the ideal (no INI) casegiven by (5.19)).

• On the other hand, if P RESGOnoise |RRESGO,min

bdoes not satisfy (B.7), then an average number

of hops can never be sustained, and the effective transport capacity with RESGO MACprotocol is always lower than that without INI.

Before finding a necessary condition such that P RESGOnoise (R

RESGO,minb ) satisfies (B.7), we gra-

phically show the behaviors of Pthermal, P RESGOint , and P RESGO

noise (given by the sum of the twoprevious powers). Figure B.1 shows the relative behaviors, as a function of the data-rate Rb,of the three powers considered above. Two possible values of the λL product are consideredin both cases. In particular, one can observe that for λL = 10 s−1, R

RESGO,maxb � Rmax

b .

290 Appendix B. Proof of Theorem 1, Chapter 5

0.0 5.0 104

1.0 105

1.5 105

2.0 105

Rb [b/s]

0

1 10-15

2 10-15

3 10-15

4 10-15

5 10-15

Power [W]

Pthermal

PintRESGO

PnoiseRESGO =Pthermal+Pint

RESGO

fc=2.4 GHz

L=10

F=6 dB

S=10-3

m-2

BERroute

=10-3

L=100Critical power threshold

to support average connectivity

RbRESGO,max

( L=100 s-1

)

RbRESGO,min

( L=100 s-1

)

Pt=2x10-7

W

N=1000

Gt=Gr=fl=1

RbRESGO,min

( L=10 s-1

)

max

RbRESGO,max

( L=10 s-1

)

Figure B.1 Comparison, versus the data-rate Rb, between the thermal noise power Pthermaland the interference power P RESGO

int in the case of the RESGO MAC protocol, for a spatialdensity ρS = 10−3 m−2.

Hence, for λL = 10 s−1 it is expected that the effective transport capacity with the RESGOMAC protocol is almost the same as in the no INI case. Considering a larger product λL =100 s−1, one can see that P RESGO

noise |RRESGO,minb

is almost equal to the critical noise threshold

P thnoise. In this case, one can conclude that the effective transport capacity with the RESGO

MAC protocol will very likely be lower than that without INI. In general, one can concludethat the effective transport capacity with the RESGO MAC protocol coincides with that inthe ideal (no INI) case for any data-rate such that

max

{R

RESGO,minb ,

ξRESGOcaLOSnhλL

BERmaxroute

}< Rb < R

RESGO,maxb

where the second term in the maximum at the left-hand side of the previous inequality comesfrom (B.1).

Reducing the node spatial density and/or the transmit power, and increasing the noisefigure have a deleterious effect on the performance with the RESGO MAC protocol – dueto the definition of Rmax

b in the ideal (no INI) case, it is clear that a reduction of Pt or ρS

proportionally reduces Rmaxb . In Figure B.2, where the node spatial density is reduced to 10−4

m−2, it is easy to see that P RESGOnoise |RRESGO,min

bis above P max

noise for λL = 100 s−1. Hence, in this

case the effective transport capacity with the RESGO MAC protocol is significantly lowerthan the effective transport capacity of the no INI case (i.e. the ideal case) for any valueof Rb.

Appendix B. Proof of Theorem 1, Chapter 5 291

0 1 104

2 104

3 104

Rb

0

1 10-16

2 10-16

3 10-16

4 10-16

5 10-16

6 10-16

7 10-16

8 10-16

9 10-16

1 10-15

Power

[W]

Pthermal

PintRESGO

PnoiseRESGO =Pthermal+Pint

RESGO

fc=2.4 GHz

L=10 s-1

F=6 dB

S=10-4

m-2

BERroute

max=10

-3

L=100 s-1

Critical power thresholdto support average connectivity

RbRESGO,min

( L=100 s-1

)

N=1000

Gt=Gr=fl=1

Figure B.2 Comparison, versus data-rate Rb, between the thermal noise power Pthermal andthe interference power P RESGO

int in the case of the RESGO MAC protocol, for a spatial densityρS = 10−4 m−2.

At this point, we impose that P RESGOnoise |RRESGO,min

bis lower than P th

noise, in order to guarantee

that RRESGO,maxb exists. The minimum noise power with the RESGO MAC protocol can be

written as

P RESGOnoise |RRESGO,min

b= √

ε1ε2ε3 +[

1 − exp

(−√

ε1ε2

ε3

)]ε3. (B.18)

Expression (B.18) is difficult to compare with P thnoise. Hence, we make the following

observation. It can be observed from Figures B.1 and B.2 that RRESGO,minb approximately

corresponds to the data-rate such that Pthermal � P RESGOnoise . In this case, we can write

P RESGOnoise |RRESGO,min

b� 2Pthermal|RRESGO,min

b= 2

√ε1ε2ε3. (B.19)

Hence, in order for RRESGO,maxb to exist, the following condition must be satisfied:

P RESGOnoise |RRESGO,min

b

P thnoise

≤ 1. (B.20)

From the results in Figures B.1 and B.2, ensuring that RRESGO,maxb is close to Rmax

b , seems todictate that the ratio P RESGO

noise |RRESGO,minb

/P thnoise is lower than 1

3 . Generalizing, we can impose

292 Appendix B. Proof of Theorem 1, Chapter 5

the following rule of thumb:

P RESGOnoise |RRESGO,min

b

P thnoise

= ξ, ξ ≤ 1

3(B.21)

where ξ is the ratio of the minimum total noise power in the case of the RESGO MACprotocol to the threshold value of the total noise power required to guarantee full connectivity.If ξ is equal or lower than 1

3 , then RRESGO,maxb should be approximately equal to Rmax

b .From (B.21), it is possible to derive

λL

PtρS= ξ2α

4FkT0 A(N)(SNRmin

link

)2 , ξ ≤ 1

3. (B.22)

Comparing the right-hand side of the equality in (B.22) with the right-hand side of (B.13), itis possible to verify that

ξ2α

4FkT0 A(N)(SNRmin

link

)2� 1

e2

4α A(N)

kT0. (B.23)

Hence, if condition (B.22) is satisfied, condition (B.13) is automatically satisfied.To conclude, the following two conditions, if satisfied, guarantee that an ad hoc wireless

network with RESGO MAC protocol show little loss, in terms of average transport capacity,with respect to an ideal (no INI) network communication scenario:

λL

ρSPt= ξ2α

4FkT0 A(N)(SNRmin

link

)2, ζ ≤ 1

3(B.24)

Rb = αρSPt

FkT0SNRminlink

(B.25)

which establish (5.30) and (5.31). From conditions (B.24) and (B.25), one can conclude thatif the node spatial density ρS and/or the transmit power Pt increase, by selecting a data-rateas in (5.31) the corresponding average transport capacity approaches that of the ideal case(i.e. the case with INI), provided that the product λL satisfies (B.24). In particular, for fixedPt and ρS, it is possible to find a maximum value for the product λL. Hence, if the packetgeneration rate λ increases, the packet size L has to decrease proportionally.