modeling

Energy Analysis

Charlie Zhong

August 19, 2002

Outline

 Energy analysis

 Application of fix point theorem



 Models and numerical results



 Simulation results

Research Goal

Network Layer



# neighbors Traffic density Link level reliability



Data Link Layer



Modulation Radio Data Rate # of channels available





Physical Layer





We want to find a combination of algorithms with lower

energy under these constraints.

Parameters

Nn Tn Ln









MAC

Power control

(Ns)

Channel

r model

Pkt_COL

Error Control

(k,M)



N Nack Pkt_BER



Nch h R

The Mega Formula

Average energy per bit per node:



( LN  LOH )

EM  ( PT  PI )  TN   EN   EC  PI  ti

E R

TN  LN



EM is average energy spent on maintenance per cycle; PT/PI is

TX/RX power; TN is average number of packets per cycle from

network; R is radio data rate; LN/LOH is the length of info/OH bits;

N is number of transmissions; EC is average computation energy per

cycle; ti is the node idle time. e.g. receiver duty cycle when no

packet has arrived.

Average Transmit Power for Data

Transfer

 Power control sets radiated power level

P

 Efficiency P  Rad

h

T









PT

E[ PD ]   TN  [( LN  LOHD _ DATA )  E[ N ]  LOHD _ ACK  E[ N ack ]

R

 LOHD _ SETUP  E[ N setup ]  LOHD _ READY  E[ N ready ]  LOHD _ END ]

Error Control System View

Probability that data fails Probability that ACK fails



Pkt_d Pkt_a



Channel independent Probability that either data

between packets Or ACK fails

CRC+ARQ

pkt _ s  1  (1  pkt _ d )  (1  pkt _ a )









Number of transmissions: N Number of ACKs: Nack



1  pkt _ s

M



E[ N ]  E[ N ack ]  E[ N ]  (1  pkt _ d )

1  pkt _ s

MAC System View

Number of interferers: Nint Traffic rate: lo

Or radius: r







Packet size: L Radio data rate: R

MAC





Collision rate: pkt_COL

Collision Rate in Aloha

L

2 lo  Nint 

pkt _ COL  1  e R









More accurately,



 5 5  N int



  li Li  (  li ) L 

 R

pkt _ COL  1  e  i 1 i 1 









e.g. data rate:

l1  TN  E[ N ]

Collision Rate in CSMA

Number of hidden terminals: Nh

 3 2

Ne  2    r 2 

3 r  D



 8 



Nh    r 2  D  Ne







r: radius

D: node density



 5 5  Nh



  li Li  (  li ) L 

 R 1 ( N e 1)

pkt _ COL  1  e  i 1 i 1 

 (1  )

Ns

Outline

 Energy analysis

 Application of fix point theorem



 Models and numerical results



 Simulation results

A Fixed Point Problem

Simplified view:

f1

M 1

pkt 1  pkt E[N]

E[ N ] 

1  pkt



f2

 C E [ N ]

pkt  1  e

C>0



pkt  f 2 ( E[ N ])  f 2 ( f1 ( pkt ))  f ( pkt )

Ordered Set

pkt belongs to [0,1]:

 This is a real valued closed set



 It is a fully ordered set (algebraic ordering) with

bottom 0 and top 1.

 It is also a complete ordered set (CPO) since every

chain Y in it has lowest upper bound V(Y).

– Every non-decreasing sequence {xn}in [0,1] is

bounded, so it has limit x in [0,1]. Additionally, x is its

lowest upper bound.

Function

M

f ( x)  1  exp(C   x ) i



i 1



 f is monotonic

– If x1 lim f ( xn )  f (lim xn )

n n

Fixed Point Theorem

 We need one for algebraic ordered sets:

If X is a CPO with bottom ^, and f: X->X is

continuous,

Then f has a least fixed point x and we can

find x constructively by finding the lowest

upper bound of the chain:

{^, f(^), f(f(^), …..}

Intuitive Way to Look at It

1





f(f(f(0)))

Starting from bottom,

monotonically

converging to f(f(0))

the least fixed point

f(0)=1-exp(-C) >= 0



For C > 0

0

Ways to Find Fixed Point (1/2)

 Iteration in MATLAB

– Simple, fast

– Scalable to more complicated models

 Simulink model

– Pros: intuitive

– Cons: slow, internal bugs in close loop, not

scalable to more complicated models, poor plot

functionality

Ways to Find Fixed Point (2/2)

 Solve equations

– MATLAB solve():

• Slow, no symbolic coefficient, output order not specified by

user

– Mathematica Solve():

• Pros: fast, symbolic coefficient, output order as specified

• Cons: can not clear previous value, need to figure out how to

use vector and plot

 Find intersection of f(x) and x

Outline

 Energy analysis

 Application of fix point theorem



 Models and numerical results



 Simulation results

Model 1

 A little more complicated than the previous

model used for illustration

 Considers external input of BER



 But ignores ACK, session setup messages

for simplicity

 Finds only the value of E[N]



 Single channel MAC (Aloha)



 Supports scalar only

Simulink Model

Verified by MATLAB Iterations





N









1000 iterations Packet error rate

Model 2

 Considers ACK now

 Still ignores session setup messages



 Supports vector



 Provides the average transmit power for a

range of traffic density

Simulink Model

A Break Here

 It is becoming much more difficult to build

simulink model

 Bugs in Simulink are leading to incorrect results

 Fix point does exist for this model and this has

been verified by iteratively applying f in

MATLAB for 1000 times

 MATLAB iteration will be used from now on

Model 3

 Considers everything now

 Same has been done to CSMA MAC



 Still single channel



 Parameters used:

Name Value Comments Name Value Comments

p 1E-3 BER L_data 148 bits Data packet length

R 10k bps Radio data rate L_ack 32 bits ACK packet length

m 6 Max # of transmissions L_setup 32 bits Setup packet length

PT 4 mW Transmit power L_ready 32 bits Ready packet length

r 10 m Radius L_end 32 bits End packet length

D 0.01/m2 Node density Ns 10 Backoff window size

NN 6 # of neighbors Ne 2 # of exposed terminals

Nh 3 # of hidden terminals

Better Accuracy

Model 4

 2 channel MAC

– Session setup messages on one channel

– Data and ACK on the 2nd channel

Comparison of MAC

Less Traffic ?

Packet Loss Rate (1/2)





pkt _ loss  pkt _ ss  (1  pkt _ ss )  pkt _ s

M M M

Packet Loss Rate (2/2)

Channel Utilization (1/2)

Defined as the ratio of aggregate data rate and radio data

rate







( N N  1)

  TN  [( LN  LOHD _ DATA )  E[ N ]  LOHD _ ACK  E[ N ack ]

R

 LOHD _ SETUP  E[ N setup ]  LOHD _ READY  E[ N ready ]  LOHD _ END ]

Channel Utilization (2/2)

Energy Per Useful Bit (1/2)



E[ PD ]

Eb 

TN  LN  (1  pkt _ loss )



where

PT

E[ PD ]   TN  [( LN  LOHD _ DATA )  E[ N ]  LOHD _ ACK  E[ N ack ]

R

 LOHD _ SETUP  E[ N setup ]  LOHD _ READY  E[ N ready ]  LOHD _ END ]

Energy Per Useful Bit (2/2)

Radio Data Rate (1/4)

Radio Data Rate (2/4)

Radio Data Rate (3/4)

Radio Data Rate (4/4)

Number of Transmissions (1/4)

Number of Transmissions (2/4)

Number of Transmissions (3/4)

Number of Transmissions (4/4)

Outline

 Energy analysis

 Application of fix point theorem



 Models and numerical results



 Simulation results

Purpose of Simulation

 See the effect of inaccurate modeling in the

following areas:

– Retransmission traffic not Poisson distributed

– Channel not independent between packets

– Interaction between retransmissions and

collision rate

– Timing issues not considered in the modeling

so far

Simulation Setup









24 nodes

1 hour

Average Transmit Power



Timeout:

Data 50ms

Control msgs 20ms









Node 0 or 1

Packet Loss Rate









Node 0 or 1

Statistics









Bursty behavior resulted from receiver being off

Appendix

Error Control Design

 ARQ+CRC

 Maximum number transmission: M



 Positive acknowledgement

Packet Error Rate

 Assume channel impairment is independent

of collisions



pkt  pkt _ BER  pkt _ COL  pkt _ BER  pkt _ COL







Note: increased retransmissions will increase collision

rate

Channel Models

 Independent channel model

– For same average BER, this model results in

higher packet error rate than bursty channel

model

 Gilbert-Elliott channel model

pkt _ BER  1  (1  p) L

Assumptions for MAC Analysis

 Single channel

 Retransmissions are also Poisson distributed