Transcript Presentation Slide

40th Annual CISS 2006 Conference on
Information Sciences and Systems
Some Optimization Trade-offs in
Wireless Network Coding
Yalin E. Sagduyu
Anthony Ephremides
University of Maryland at College Park
1
Throughput Region Optimization
i,j : average rate (packets /s)
j
i
i.
ii.
iii.
Maximum Throughput Region (TR)
Maximum Stable Throughput Region (STR)
Capacity Region
Ad Hoc Wireless Network
In general, they are all different.
2
2-User Case – Random Access
2
R
1
1 + 2 = 1
p1
p2
TR
p2
p2 (1-p1)
1
STR
2
p1 (1-p2)
–
Interacting Queues
–
Envelope over p1, p2 values: TR = STR = C
p1
1
1
(From Rao & Ephremides ’85 to Luo & Ephremides ’06)
3
General Network
(Point-to-Point or Unicast & Mostly Scheduled Access)
•
“Back-Pressure” Algorithm (Tassiulas & Ephremides ’92)
Tassiulas & Neely & Georgiadis ’06)
Max-Flow/Min-Cut
argument
• Generalized “Join Shortest Queue”
• Yields Maximum STR
(delay can be very poor)
• Arbitrary “Constraint” Sets
• Gupta & Kumar : saturated queues
 infinite delay
(completely different)
4
Challenge: Multicasting
•
Throughput Definition (per source or per destination?)
•
Network Coding Achieves Max Flow/Min-Cut limit (in “wireline” & single source)
•
Network Coding in Wireless:
– Modification of “Cut-Capacity” Definitions
– Superposed with Scheduled Access
– Time Division between different non-interfering realizations (NetCod ’05)
– MAC & Network Coding
– All for “Saturated Queues”
(1)
(2)
(3)
:
5
Stable Throughput Region
• Nothing known so far
• Potential of using “Back-Pressure” Algorithm (noted by T. Ho et al.)
• Multiple Sources
• With or without Network Coding: Find Max STR
• Simple Tandem Network
• Mostly Broadcasting
• Error-free transmissions
1
2
3
n -1
n
6
Tandem Wireless Network Model (Saturated Queues)
1
2
3
n -1
n
• Scheduled Access: Group 1: 1, 4, 7, …, Group 2: 2, 5, 8, …, Group 3: 3, 6, 9, …
Activate node group m over disjoint fractions of time tm , m  {1,2,3}.
• Random Access: Node i transmits (new or collided) packets with fixed probability pi .
• There are three separate queues at each node i :
– Qi1 stores source packets node i generates.
– Qi2 and Qi3 store relay packets incoming from right and left neighbor of node i.
crucial
point
• Plain Routing: Node i transmits one packet from queue Qi1, Qi2 or Qi3 .
• Network Coding: Node i transmits either a packet from queue Qi1 or a linear
combination of two packets, one from each of the queues Qi2 and Qi3.
7
Achievable Throughput Region under Scheduled Access
•
ir and il : total rates of packets arriving at node i from right and left neighbors.
•
 i : throughput rate from node i to destinations.
•
Throughput rates  satisfy:
 i   i r  tm(i ) ,
 i   i l  tm(i ) , i  N , for network coding
 i   i r   i l  tm(i ) , i  N , for plain routing
•
Achievable throughput
region A includes  s.t.:

3

max { i  max(  i ,  i ) }  1 for network coding
r
l
m 1 i: m ( i )  m
3
max { i   i   i }  1 for plain routing
r
l
m 1 i: m ( i )  m
For n = 3, achievable
throughput region A is:
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Stable Throughput Region under Scheduled Access
• Allow packet queues to empty.
– Packet underflow possible: node can wait to perform Network Coding
or proceed with Plain Routing.
– Consider two dynamic strategies based on instantaneous queue contents:
•
Strategy 1: Every node attempts first to transmit relay packets and transmits a source
packet only if both relay queues are empty.
 i  tm(i )
•


1 i
tm(i )
r


1 i
tm(i )
l

for network coding,
 i  i r  i l  tm(i ) for plain routing
Strategy 2: Every node attempts first to transmit a source packet and transmits relay
packets only if the source queue is empty.
 i  max( i r , i l )  tm(i ) for network coding,
 i  i r  i l  tm(i ) for plain routing
– Strategy 2 expands the stability region STR(S) to the boundary of TR(A).
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Optimization
•
i,j = i , i  Mi
(multicasting)
 min  min {i , j }
•
minimum transmitted
throughput
• Maximize
over
,
    i | M i |
“sum”-delivered
throughput
min or 
  A or   S
AND schedule t (or p , for random access)
10
Throughput Optimization Trade-offs
• Assume saturated queues (or non-saturated queues together with strategy 2.).
• Trade-offs:  min = 0 for optimal values of  (under broadcasting i.e. Mi = N – {i}, i  N )
 arg max min  arg max 
 ,t
 ,t
 Network coding doubles  without improvement in min , as n increases.
0.5
Linear Optimization with
Linear Constraints.
0.45
Network coding for optimal min
throughput per source-destination pair
0.4
Plain routing for optimal min
0.35
Network coding for optimal 
0.3
Objectives of maximizing
min and  under broadcast
communication cannot be
achieved simultaneously.
Plain routing for optimal 
0.25
0.2
0.15
0.1
0.05
0
2
4
6
8
10
12
number of nodes n
14
16
18
20
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Throughput Optimization Trade-offs (Cont’d.)
Consider three different unicast traffic demands (with |Mi| = 1, i N):
•
0.5
0.5
0.45
0.45
0.4
0.4
throughput per source-destination pair
throughput per source-destination pair
– Best demand: destinations are the one-hop neighbors of sources.
– Least favorable demand: destinations have the largest hop-distances form sources.
– Uniform demand: destinations are uniformly and independently chosen for sources.
0.35
0.3
Network coding for uniform demand
Plain routing for uniform demand
0.25
Network coding for least favorable demand
Plain routing for least favorable demand
0.2
Network coding or plain routing for best demand
0.15
0.1
0.05
0
Network coding with optimal time allocation for uniform demand
Plain routing with optimal time allocation for uniform demand
Network coding with equal time allocation for uniform demand
0.35
Plain routing with equal time allocation for uniform demand
0.3
0.25
0.2
0.15
0.1
0.05
2
4
6
8
10
12
number of nodes n
14
16
18
20
0
2
4
6
8
10
12
14
16
18
20
number of nodes n
 Network coding can double both min and  compared to plain routing, as n
increases.
 Throughput trade-off strongly depend on communication demands.
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Joint Optimization of Throughput Measures
• Performance objectives of maximizing min and  may conflict with each other.
– Formulate the problem of maximizing  subject to min ≥ .
– Linear programming solution:

For broadcast
communication:
1
 n 1
(
2


(
3
n

10
))
for
0



, if n mod 3  0 or 1
 3
3n  5
( )  
under network coding
n

1
1

(2   (3n  8)) for 0   
, if n mod 3  2
 3
3n  4
  ( ) 
n 1
1
for 0   
under plain routing
3
3n  5
 ( )
2( n  1)
3
Network Coding
n(n  1)
n(n  1)
or
3n  4
3n  5
n 1
3
Plain Routing
1
3n
1
1
or
3n  4
3n  5

13
Additional Measures
•
Energy Efficiency :
Et (  )
transmission
Ep (  )
&
processing for network coding
(is it higher than simple queue management?)
•
Network Coding helps if 3p < t .
•
For stable operation: Et (  ) & Ep (  ) are non-linear functions of schedule t
•
Trade-off between Energy & Throughput
14
Extension to Random Access
•
Assume saturated queues (otherwise, the problem involves interacting queues).
– Source packet transmissions:
Method A: Transmit new source packets at any time slot
(no feedback - possible loss)
Method B: Transmit source packets until they are received by both neighbors (feedback + repetition)
Method C: Transmit linear combinations of source packets (feedback + open-ended)
0.14
throughput per source-destination pair
Linear optimization with
Non-linear constraints.
Network coding with method A
Network coding with method B
0.12
Network coding with method C
Plain routing with method A
0.1
Plain routing with method B
Plain routing with method C
(Logarithmic barrier method is used.)
0.08
0.06
0.04
0.02
0
4
6
8
10
12
number of nodes n
14
16
18
20
15
Future
• Cooperative Communications vs. Competitive Communications.
(ISIT 2006)
• Sharing of Resources.
• Beyond Tandem.
• What if Energy is finite?
(Volume / joule)
16