Channel Capacity Issues
For Mobile Teams
Ameesh Pandya and Greg Pottie,
UCLA Electrical Engineering
Department
1
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.
2
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).
3
Air to Air Channel (No Jammer)

Channel capacity for this
case is
1
Pt
C  W log 2 (1 
)
2
WN 0 4d

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.


4
Ground to Ground Channel (No Jammer)





Channel capacity for this
case is
Pt
K
C  W log (1
)
2
WN d 
0
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.
5
Air to Air Channel in presence of Broadband
Jammer
Capacity: P
C  W log 2 [1  t 2
4d



1
]
'
5 J E
W ( N 0  2 2av b )
r f R N 0
'
J av 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):
Pt
1
]
2
N

1
4d W ( N  2 u
Eb )
0
f
For 10W jamming power
distance of 50.3 Km is achieved
at 10W.
C  W log 2 [1 

6
Ground to Ground Channel in presence of
Broadband Jammer
Pt
C  W log [1  
2
d


K
]
20K J " E
W ( N   1 av b )
0 r
f 2 R N0
"
Jav
is the average jamming
power at distance r from the
receiver, K1 is the propagation
constant for jammer.
For CDMA with jamming and Nu
simultaneous users, capacity is
given by (assuming identical
signal power):
Pt
K
C  W log [1  
]
2
d W ( N  2 Nu  1 E )
0
b
f

The simulation is carried for
 = 4.5.
7
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.
8
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.

9
Simulation Results



The graphs show the average of 100 iterations.
Above is the plot of number of nodes connected to form a l
argest cluster when the radio range is 20% of the area.
For this case we require approximately 30 nodes for the
99% connectivity.
10
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).
11
Probability of the multi-hop Connection




(x,y) coordinates of the mobile locations ~ N(0,s2)
r er2 /4s 2
p
(
r
)

pdf of the link distance r is:
r
2s 2
Pr{2-hop connection}=P2 [2]
P  e R2 /4s 2  e R2 /s 2  P
2
2
Asymptotically m – hop connection probability: [2]
2 2 2
Pm  e(m1) R /4s  em2R2/4s 2  Pm

Upper bound on average number of hops between
node pairs:
 m2 R 2 / 4s 2
E {h}   e
m 1
12
Lower Bound on P2



The lower bound on P2 is
function of R2/s2 = g2 i.e.
P2 ≥ f(g).
Skipping the derivation.
Numerical Integration is
employed to solve the
complicated integration.
Hence the approximation
error.
13
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.
14
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.
15
Distributed Network Control Problem

Problem Definition
Y
N1
 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.
16
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.
17
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.
i ii
Signal to Interference ratio for user i : SIRi  PG
N

j i
Pj Gij  ni
Outage Probability: Oi = Pr (SIRi ≤ SIRth) where SIRth is the
given threshold.
1
The outage probability can be expressed as Oi 1 ki SIR G P
th ik k
1
Outage probability over a path S: OpathS 1  (1 Oi )
Gii Pi
s S
18
Prerequisites for QoS Constraints (Contd.)









Constellation Size M used by a hop can be closely approximated for
M 1 K  SIR
M-QAM modulation:
15
.
K

In the above expression for M, K is given by
ln(5BER)
th
The data rate of the i hop Ri (1/ T)log2 (1 K.SIRi )
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.
19
Problem Formulation
min. Q (t )  Q
i
j
 ni
iK j
 ( C )  dk ,UB ,k
s.t.
jJ
j
k
Ri  R
, i
i, LB
 p j  pk , LB ,k
jJ
1
1 
 Pr ,i
k
SIR
G
P
outi
b n  R , k
k i 1 th ik k
k k
k
Gii Pi
b n C
k* k*
j*
1
1  
 Pr
, S n
SIR
G
P
ss k s
out _ path_s
k*  d *
1 th ik k
Gii Pi
C
k, j
*
j
Pi  Pmax
pj  p
j,UB
b
n

C
,

j
 k k
j
b ,C j , p j ,d
,p
0
k K j
k
k ,UB k , LB
20
Constraint Sets Description
CS
CS
CS
CS
CS
CS
CS
CS
CS
1) Data rates demanded by existing users.
2) Outage probability limitations demanded by users using single hop.
3) Outage probability limitations for users using a multi-hop path.
4) Set of all feasible powers Pi.
5) Link Capacity Constraint.
6) Delay Guarantee constraint.
7) Delivery probability constraint.
8) Guaranteed data rate to each class of traffic.
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.
21
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].
22
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.
23
References




[1] Mung Chiang, et. al., “Resource Allocation for QoS Provisioning 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 The
ory, 46(2):388-404, March 2000.
[4] Matthias Grossglauser and David Tse, “Mobility
Increases the Capacity of Ad-hoc Wireless Networks”, IEEE
INFOCOM, 2001.
24