1

Channel Capacity Issues For Mobile

Teams

Ameesh Pandya and Greg Pottie, UCLA Electrical Engineering Department

2

Introduction

Channel capacity models for ideal and broadband jamming environments (individual links)

Connectivity Issue: Bound on the number of nodes to have 99% of the connectivity in a fixed area for randomly (Poisson, for example) distributed nodes.

Probability of the multi-hop connection in a random mobile network

Capacity of a mobile network considering delay. Distributed network control problem

Forced change in the position of the node (UAV) to perform the requested activity (from the ground station) while maintaining highest possible QoS.

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Channel Capacity Model

Two Channel Models: Air to Air Channel.Ground to Ground Channel.

Assumptions: Isotropic antenna. Spread spectrum modulation. For Low probability of intercept (LPI), Pr/WsN0 =

0.1, where Pr is the received power and Ws is the bandwidth of spread spectrum signal.

Broadband Jammer (valid assumption).

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Air to Air Channel (No Jammer)

Channel capacity for this case is

2 Mbps is the control traffic data rate.

For 1W distance achieved at 2Mbps is 75.5 km and for 2W, 106.78 km.

High values are result of ideal channel with 100MHz bandwidth.

C WWN

P

dt= +log ( )2

021

14π

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Ground to Ground Channel (No Jammer)

Channel capacity for this case is

For 1W distance achieved at 2Mbps is 98m and for 2W, 118.22m.

Here =3.7. K = K’F where F is the

fading margin and K’ is the propagation constant.

is the path loss coeffecient.

C W KWN

Ptd

= +log ( )2 1 0

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Air to Air Channel in presence of Broadband Jammer

Capacity:

is the average jamming power at distance ‘r’ from the receiver, f is the spread factor.

For CDMA, with jamming and Nu simultaneous users, channel capacity is given by (assuming identical signal power):

For 10W jamming power distance of 50.3 Km is achieved at 10W.

C WP

dW N

rJf R

EN

t

av b

= ++

log [( )

]'2 2

0 2 20

14

15ππ

Jav'

C WP

d W NNf

E

t

ub

= ++

−log [( )

]2 2

0

14

1

21π

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Ground to Ground Channel in presence of Broadband Jammer

is the average jamming power at distance r from thereceiver, K1 is the propagation constant for jammer.

For CDMA with jamming and Nu simultaneous users, capacity is given by (assuming identical signal power):

The simulation is carried for = 4.5.

C WPtd

K

W NK

rJavf R

Eb

N

= ++

log [(

")]2 1

020 1

2 0

Jav"

C WPtd

K

W NNuf

Eb

= ++ −log [

( )]2 1

0 2 1

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Connectivity Issue

Problem Definition: Consider a closed surface of area A. Say, a

square. Randomly place n nodes in that area with some

distribution other than uniform. How large must n be to have 99% connectivity? Solution for uniformly distributed nodes could be

found by continuum percolation. Motivation:

Deals with the issue of sensor coverage. Provides guidance on minimum node density to

achieve communications connectivity.

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Approach

Nodes generated according to Poisson distribution with intensity n (# of nodes).

Region is a square with unit area. Plotting number of nodes required for 99%

connectivity as a function of radio range.

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Simulation Results

The graphs show the average of 100 iterations. Above is the plot of number of nodes connected to form

a largest cluster when the radio range is 20% of the area. For this case we require approximately 30 nodes for the

99% connectivity.

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Simulation Results

First plot displays number of nodes required for 99% connectivity with transmission range (avg for 100 runs).

The second figure shows the best fitting polynomial plot for the behavior of first figure.

The simulation result of number nodes required for the 99% connectivity, N, as a function of radio range, R is: N ~ O(e-R).

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Probability of the multi-hop Connection

(x,y) coordinates of the mobile locations ~ N(0,2)

pdf of the link distance r is: Pr{2-hop connection}=P2 [2]

Asymptotically m – hop connection probability: [2]

Upper bound on average number of hops between node pairs:

pr rr e r( ) /= −2 2

2 4 2

P e R e R P22 4 2 2 2

2< − − − = ∞/ /

Pm e m R e m R Pm< − − − − = ∞( ) / /1 2 2 4 2 2 2 4 2

E h e m R

m{ } /< −

=

∞∑

2 2 4 2

1

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Lower Bound on P2

The lower bound on P2 is function of R2/2 = 2

i.e. P2 ≥ f().

Skipping the derivation.

Numerical Integration is employed to solve the complicated integration. Hence the approximation error.

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Capacity of a Mobile Network

[3] deals with the capacity of the wireless network for a fixed channel model. The throughput per session for fixed wireless network model can at best be O(1/√n).

[4] discusses the increase in throughput for mobile ad-hoc network but assumes loose delay constraint i.e. delay is tolerable . Hence, the end result is infinite delay.

Motivation: The actual capacity of wireless ad-hoc network (with no delay constraint) will give the real picture of the QoS provision. The delay in many applications is not admissible and hence the capacity is less than the fixed wireless network.

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Capacity of a Mobile Network

The major difference in the capacity for the fixed wireless network and mobile Ad-hoc network is the energy used for updating the routing table for the later case.

One way of updating routing table is flooding. Capacity may tend to zero if the energy

usage for updating routing table goes very high.

Hence, the throughput for the case of Ad-hoc wireless network with no loose delay constraint is of O(1/√n) – signaling for updating routing table.

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Distributed Network Control Problem

Problem Definition

Consider the network connected in a multi-hop fashion as shown in the figure above.

Suppose a request comes from ground station to node N1 to move from its current location to Y for some specific function.

This can cause a change in the QoS assurance for the network. So, objective is to minimize the Euclidean distance between N1 and Y subject to QoS constraints.

N1

Y

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Distributed Network Control Problem

Formulating as an Optimization problem. Let the current position of Node Ni (node in

consideration) be defined as Qi(t) = {Xi(t), Yi(t)}.

Let the requested position of Node Ni be Qj = {Xj, Yj}.

Objective Function:min. |Qi(t) – Qj|

The cost function is minimized subject to QoS

constraints.

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Distributed Control Algorithm

Prerequisites for QoS Constraints [1]: Power Control

Power received from transmitter j, at receiver i is given by GijFijPj. The nonnegative number Gij is the path gain in absence of fading from the jth transmitter to the ith receiver. Fij is the Rayleigh fading between each transmitter j and receiver i.

Signal to Interference ratio for user i :

Outage Probability: Oi = Pr (SIRi ≤ SIRth) where SIRth is the given threshold.

The outage probability can be expressed as

Outage probability over a path S:

SIRPG

PG ni

i ii

j ij ij i

N=+

≠∑

OSIR G P

G P

ith ik k

ii i

k i=−

+≠∏1

1

1

O OpathS is S

=− −∈∏1 1( )

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Prerequisites for QoS Constraints (Contd.)

Constellation Size M used by a hop can be closely approximated for M-QAM modulation:

In the above expression for M, K is given by The data rate of the ith hop Link Capacity: Cj packets/sec., J links. K classes of traffic, for each QoS of class k, the bandwidth

required is bk Hz. Delay guarantee in a service level agreement (SLA) is

dk,UB sec. Minimum probability of delivering the packet across the

unreliable network required in SLA : pk,LB. # of packets dynamically admitted in the kth class of traffic

: nk. Probability that link will be maintained during

transmission: pj.

M K SIR=+ ×1K

BER=

−155.

(ln )R T K SIRi i= +( / ) log ( . )1 12

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Problem Formulation

min. ( )

. .

, ,

,

_ _,

max,

Qi t Qjs tRi Ri LB i

SIRthGikPk

GiiPi

k iProuti

i

SIRthGikPk

GiiPi

k sProut path ss s

S

Pi P

bknk

Cjk K j

j

−

≥ ∀

−+≠

∏ ≤ ∀

−+≠

∏ ≤∈∏

∀

≤≤

∈∑ ∀

1 1

1

1 1

1

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Constraint Sets Description

CS 1) Data rates demanded by existing users.CS 2) Outage probability limitations demanded by users using

single hop.CS 3) Outage probability limitations for users using a multi-hop

path.CS 4) Set of all feasible powers Pi.CS 5) Link Capacity Constraint.CS 6) Delay Guarantee constraint.CS 7) Delivery probability constraint.CS 8) Guaranteed data rate to each class of traffic.CS 9) Service Level Agreement (SLA) constraint that give a class

of traffic the sole right to traverse a link j*.CS 10) Specifies end to end delay guarantee and a delay

requirement for a particular traffic class k* on a link j*.CS 11) Upper bound constraint on pj.CS 12) Positivity Constraints.

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Hurdles and Solution

All the above mentioned constraints assures the QoS.

Too many constraints to solve an optimization

problem. Prioritizing (i.e. weighted) QoS.Deriving Cost function for QoS and maximizing it.

Since all the constraints are monomials, Geometric programming might help.

Transforming into Convex optimization problem as in [1].

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Future Work

Designing “robust” distribution control algorithm for the nodes to acknowledge the special requested activities.

Deriving the capacity of mobile wireless network.

Critical probability for the network connectivity.

Designing MAC layer clustering algorithm which adapts with minimum delay to the change in the network configuration. Robust and fault-tolerant.

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References

[1] Mung Chiang, et. al., “Resource Allocation for QoS Pro-visioning in Wireless Ad Hoc Network”, GLOBECOM 2001.

[2] Leonard E. Miller, “Probability of a Two-Hop Connection in a Random Mobile Network”, Conference on Information Sciences and Systems, The John Hopkins University, 2001.

[3] Piyush Gupta and P. R. Kumar, “The Capacity of Wireless Networks”, IEEE Transactions on Information Theory, 46(2):388-404, March 2000.

[4] Matthias Grossglauser and David Tse, “Mobility Increases the Capacity of Ad-hoc Wireless Networks”, IEEE INFOCOM, 2001.