October 16, 2003

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Lecture 6

• October 16, 2003

2

• Last Lecture
• slotted Aloha for simple collision channels
• Collision Channels
• If two or more packets are simultaneously
  transmitted, they will collide and they are all
  lost.
• Better signal processing at the physical layer
  permits a better packet capture for the channel.
• MPR Multi-Packet Reception Capability

3

• Example of MPR
• Beamforming - Limits the interference
  (collision) zone
• CDMA
• Multiuser detection
• Coding - dirty paper coding
• How to combine Aloha with beamforming?
• Earlier work Zander,1990 has analyzed the
  performance of a
• joint beamforming and Aloha protocol,which
  uses
• - directional antennas for transmission
• - omni-dir antennas for reception

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System Model

• Large random network
• Generated by a two-dimensional Poisson Process,
• with average number of terminals/unit area
  
• The maximum range of a node r 1
• Expected number of nodes within this range is M
• M a measure of the connectivity in the network
• Assumption
• Network in high load state.
• All nodes are backlogged and have packets to
  transmit.

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System Model cont.
Transmit a packet to a distant range - use
multiple hops
P
r
f
Final destination
O
?
? forward projector of the packet toward its
final destination
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• Slotted Aloha
• - Transmission probability in subsequent slots
  is p
• Beamforming Transmission Antenna
• - Horizontal beamwidth is

area
G is horizontal antenna gain

• Path loss propagation model
• propagation exponent is
• model
• Target SIR for correct reception is

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System Model - Performance measure

• Normalized expected forward projector
• - Expected projector in meters
  (distance units)
• Z dimensionless measure
• Z expected projector in the number of
  hopped-over terminals
• (average distance between two nearest terminals
  )
• successful transmission only node O transmits
• node P does not transmit
• no node in collision region
  transmit

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gt

For dense networks gt Collision region
for this case close to a circle with radius
gt
Maximization, with respect to M, then p
Note on collision region determined as the
region of interferers for which SIR at P
?0
9
gt
Optimum choices for M, p
- Even for moderate gains, it results
very good improvement Ex. G30,
beamwidth gt
10-fold improvement
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Multi-Packet Reception

• The concept of multi-packet reception is more
  general and
• include other particular areas as well
• CDMA, Multi-user detection etc.

• The multi-packet reception capability of a node
  can be described
• by the receiver MPR matrix.

- MPR matrix contains the reception
probabilities
P k packets are received n packets are
transmitted in the neighborhood
MPR matrix
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Multi-Packet Reception cont.
- MPR matrix include the effects of -
Noise - Type of modulation - signal processing
(beamforming,CDMA, MUD,etc.) - The MPR matrix
of conventional collision channels
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Slotted Aloha for Ad-hoc network with MPR
capability
Slotted Aloha for multi-hop networks is harder
to analyze, but there has been work to
characterize the performance for regular
networks. Note MPR matrix for node
MPR matrix for network Slotted Aloha for Regular
Network - Manhattan Network - All nodes are
located on a regular grid
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• Performance of slotted Aloha
• in Manhattan networks with MPR capability

Assumption - Every node in the network is
backlogged and the probability of
transmission is p - Traffic in the network
is uniform - Average path length
hops The max achievable end-to-end throughput
for slotted Aloha is


- Expected number of correctly received
packets given that k packets are
transmitted
- MPR matrix for the network
It is usually not trivial to decide the MPR
matrix,especially in Ad-hoc networks. (usually
the network MPR matrix node MPR matrix)
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Instability in Slotted Aloha

• Observations
• Slotted Aloha very simple scheme requires very
  little overhead and coordination.
• (only slot synchronization required)
• One problem Instability
• If the backlog is large beyond unstable
  limit it tends to become larger and larger.
• Solution for stabilizing Aloha
• - Change transmission probability ,
  according to the estimated backlog.
• the number of backlogged nodes
• - assume all arrivals are immediately
  backlogged.
• the estimated number of backlogged
  nodes

Pseudo-Bayesian Algorithm
Assumption number of nodes attempting
transmission Poisson with rate
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Instability in Slotted Aloha cont.
How to estimate ? - use the feedback
information on collision status.
For perfect estimation of
It can be shown to be stable for
Pseudo-Bayesian Algorithm
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Alternate solution for the instability - TDMA
scheduling users.
User1 User2 User3 User4 User5

• Aloha achieves lower delays when arrival rates
  are low
• TDMA very large delays with large number of
  users
• Delay for Aloha independent of the number of
  users

- TDMA part of the reservation access schemes
family. - Other conflict resolution schemes
have been proposed and analyzed in the
literature. - Another multiple access technique
which we will discuss shortly is CDMA.
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Power Control

• In general, we decide if a packet can be decoded
  correctly based on its received SIR.
• - Improving SIR improves the chances of
  correct reception.
• - One important technique is power control.
• Power Control
• Select your power level that you exactly meet
  your target SIR
• 
  battery drain
  
• - If SIR gt , use too much power
• interference with others
• - If SIR lt , packets cannot be received
  correctly

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Power Control cont.

• Assume that Q transmitters use the same channel
  
• They have power
• The expression for SIR at receiver is

the power at the transmitter

- link gain
- noise power at receiver i
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Power Control cont.

• Transmitter i is supported if
• - target SIR
• gt
  

power to select if all other powers are kept fixed

• Denote
• In a system has to be hold for all

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Power Control cont.

• Example For 2 users

• both users can be supported
• only user1 can be supported
• only user2 can be supported
• none can be supported

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Power Control - Feasibility

• How many users you can support to maximize
  capacity,while maintaining SIR requirement?
• Feasibility conditions
• For Q users

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Power Control Feasibility cont.

• Def. The target SIR is said to
  achievable, if there exists a non-negative power
  vector so that holds for all .
• The target SIR is achievable if the dominant
  (largest) eigenvalue of matrix H, ( ) is
  less or equal to one.

Is achievable only when noise is zero
Power control feasibility condition

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Power Control SIR Balancing

• Another approach to power control
• - Maximize the minimum SIR in all links SIR
  Balancing Problem
• Solution
• - Power control is achieved by making every
  transmitters received
• SIR balanced (equalized), while keeping
  the balanced SIR as high
• as possible.
• - For the noiseless case , find the
  maximum achievable
• Denote

The solution (with equality) for the power vector
is the eigenvector corresponding to the
eigenvalue .
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Power Control SIR Balancing cont.

• Potential problems with SIR balancing
• - Implement SIR balancing resulting less
  than desired
• target SIR for all users.
• -gt All links drop below the threshold and
  become useless.
• Solution
• - Shut off some users,at least
  temporarily,and improve the
• connections of the remaining users.
• A question
• How to maximize the number of users in the
  system?
• - not trivial -gt several removal algorithm s

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Power Control Stepwise Removal Algorithm

• Stepwise Removal Algorithm (SRA)
• Step1 Determine
• if , use
• else
• Step2 Remove terminal k
• After removal,
  matrix
• With go to Step 1.
• It can be shown that in the optimal power
  allocation that maximizes the number of users,
  some links are completely shut off and the other
  users have SIR balanced links.

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Distributed Power Control (DPC)

• One problem with above discussion
• - need to know the link gain matrix (gains for
  all users)
• - impractical
• -gt need distributed power control
• Distributed Power Control
• - A distributed iterative power control
  algorithm based on local measurements.

- received SIR at iteration n
- transmission power of transmitter i at
iteration n
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Distributed Power Control (DPC) cont.

• DPC requires only local measurements on SIR at
  receivers
• DPC is the same as the Jacobs relaxation method
  in numerical linear algorithm
• For these distributed iterative algorithm, the
  speed of convergence is very important, because
• - The assumption is that the system is static
  (link gains do not change) until the power
  control algorithm converges.

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Unconstrained Second Order Power Control (USOPC)

• Unconstrained Second Order Power Control (USOPC)
• It can be shown that USOPC converges much faster
  than DPC.

- non-increasing sequence
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Standard Interference Function

• To support the design of various power control
  algorithm,a general framework for proving the
  convergence of iterative distributed power
  control algorithms was proposed.
• -Standard Interference Function
• A power control algorithm iterative
• I Interference function define the power
  vector of the next iteration
• Def. Assuming positive receiver noise, an
  interference function I is called standard if
  it satisfies all non-negative power vectors
• 1) Positivity
• 2) Monotonicity
• 3) Scalability

Component wise
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Standard Interference Function cont.

• Proposition If the system is feasible, the
  sequence of power vectors from the standard
  interference function will converge to the
  minimum power solution vector , starting with
  any non-negative power vector, and the rate of
  convergence is geometric.
• Observation on convergence
• - Starting with all zero power vector, the
  sequence
• is monotonically increasing.
• - If , then the
  sequence
• is monotonically decreasing.
• - Staring with any initial power vector
  , the sequence
• converges geometrically to the
  fixed point .

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Standard Interference Function cont.

• Observation
• - When approaching capacity,
• power level increases. power
  warfare effect
• DPC belongs to the class of standard
  interference function
• USOPC may not meet the first condition
  (Positivity)
• The standard interference function is a
  sufficient condition but not necessary.

Monitor the capacity Admission control If
power warfare, reject the new users