Network Dynamics & Simulation Science Laboratory
Approximation Algorithms
for Throughput Maximization
in Wireless Networks
with Delay Constraints
Guanhong
Pei
∗
Ⱐ嘮V匮⁁湩氠䭵浡K
†
,
Srinivasan
Parthasarathy
‡
, and
Aravind
Srinivasan
§
∗
Dept. of ECE and Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA
†
Dept. of CS and Virginia Bioinformatics Institute, Virginia Tech, Blacksburg, VA
‡
IBM T.J. Watson Research Center, Hawthorne, NY
§
Dept. of CS and Institute for Advanced Computer Studies, University of Maryland, College Park, MD
IEEE INFOCOM 2011
Network Dynamics & Simulation Science Laboratory
Fundamental Problems
•
Given a set
C
of end

to

end source

destination
connections (or sessions) in an arbitrary multi

hop wireless network,
–
Maximize
total rate
of communication possible for sessions in
C
–
Minimize
end

to

end (average) delay
of communication possible
for sessions in
C
NP

Complete
Cross

layer
Optimization
Traffic Control
Transport Layer
Routing
Network Layer
Scheduling
MAC Layer
Links
to make transmissions
Paths
for each connection
Traffic rates
for each connection
Avg. time for packets to travel to destination
To tackle
these
problems
Network Dynamics & Simulation Science Laboratory
Problem Statement of Our Work
•
PROBLEM
:
Delay

Constrained
Throughput Maximization (
DCTM
)
•
GIVEN
:
an
arbitrary
multi

hop
wireless network
represented by a directed graph
, and a set
of
connections (
s

t
pairs), with a target end

to

end delay
for each connection
c
in
.
•
GOAL
:
find a
stable rate vector
with
flow paths
and a
suitable
scheduling scheme
, such that the
total rate
is maximized and
end

to

end delay
is bounded by
for each connection
c
.
Transport Layer
Mac Layer
Network Layer
O
(
n
ε
)

inapproximable
general interference
O
(1)

inapproximable
unit

disk graph interference
Network Dynamics & Simulation Science Laboratory
Summary of Results
(1)
Total throughput is at least a factor of
of
the maximum possible (with given delay constraint vector ),
where ; Each flow rate is
(2)
Avg. end

to

end delay is bounded by
Theorem: for DCTM
For computing a rate vector and flow paths under a random

access scheduling scheme w/
delay
,
throughput
&
stability
guarantees
Multi

commodity Flow Framework
Avg. end

to

end delay of each flow
Avg. end

to

end delay of entire system:
Theorem:
Delay Bounds
for a Given Set of Flows
Provable
Worst

case
Bounds
:
Path of
flow
f
Network Dynamics & Simulation Science Laboratory
Outline of the Rest
•
Related Work
•
Preliminaries
•
Approach
•
Extensions
•
Conclusion
Network Dynamics & Simulation Science Laboratory
Related Work on Delay Bounds
n
: #nodes;
m
: #links;
I
max
: max interference degree;
θ
max
: max congestion
(#
ﬂows
through a link);
C
(
N
)
: chromatic number of link interference graph
Jagabathula
,
Shah
08
Jayachandran
,
Andrews
10
Le,
Jagannathan
,
Modiano
09
Gupta,
Shroff
09
Neely
09
Kar, Luo,
Sarkar
09
Jagabathula
,
Shah
08
Jayachandran
,
Andrews
10
Gupta,
Shroff
09
Neely
09
Kar, Luo,
Sarkar
09
Le,
Jagannathan
,
Modiano
09
Network Dynamics & Simulation Science Laboratory
Outline
•
Related Work
•
Preliminaries
–
Network Model
–
Traffic Model
–
Queueing Model
–
Metrics: Throughput & Delay
•
Approach
•
Extensions
•
Conclusion
Network Dynamics & Simulation Science Laboratory
Wireless Network Model
•
The network is modeled as
a graph
–
Set of nodes:
–
Set of transmission links:
•
Wireless Interference
–
Graph

based interference model
–
Interference set for each link
•
and : interfering
⇔
no successful
tx
at the same time
–
Interference relationship
•
Binary & symmetric
Data
→
←
ACK
link
sender
receiver
Network Dynamics & Simulation Science Laboratory
Traffic Model
•
Traffic: end

to

end
–
A set
of sessions, and for each session
c
a set
of flows
–
Session
c
: a source

destination pair
–
Flow
f
: a path between
–
Exogenous
Arrival processes: general
i.i.d
.
•
: # arrival packets at the source of
f
at time
t
•
First moment:
; Second moment:
–
Rate of a flow
f
:
f
1
f
2
f
3
Session
c
:
Network Dynamics & Simulation Science Laboratory
Definitions: Queueing Model
•
Each link
l
is associated with a queue
•
Definitions
–
: # packets queued on link
l
at time
t
–
: # arrival packets on link
l
at time
t
–
: # departure packets on link
l
at time
t
–
: service rate offered to link
l
at time
t
•
For simplicity, we assume uniform capacity:
•
For heterogeneous link capacities, normalizing the quantities
by link capacity will reduce the case to uniform
capacity.
Service
Departure
Arrival
Queue
Network Dynamics & Simulation Science Laboratory
f
1
f
2
f
3
Session
c
:
Queueing Dynamics
: the
i
th
link of flow
f
: size of the logical
queue of flow
f
on link
l
at
t
Queue Evolution:
2
f
1
f
3
f
4
f
5
f
6
f
f
:
Logical Queues
for Each Flow
Multiple
Input Flows
Single
Server
: Queue on
Q
Q
Q
a
l,f
1
l,f
2
l,f
3
l,f
1
a
l,f
2
a
l,f
3
d
l,f
1
d
l,f
2
d
l,f
3
Packet
Service
Rate
Depar

ture
Arrival
Network Dynamics & Simulation Science Laboratory
Long

term average backlog:
Performance of Wireless Networks
•
Metrics
–
Throughput
–
Delay
•
Throughput
–
Total throughput rate
–
Throughput region
•
Delay
–
Average per

flow end

to

end delay
–
Average network end

to

end delay
Intuitively,
By Little’s Law
Explained next
Network Dynamics & Simulation Science Laboratory
Throughput Region
•
Queue

stability of a System
iff
•
Traffic Rate Vector
–
Avg. rate for each flow
f
C
apacity region:
Λ
OPT
the set of all stable
traffic vectors
γ

scaled
region
γ
Λ
OPT
, (
0<
γ
<1
)
Max

Weight
Scheduling
Throughput region:
Λ
S
the set of all stable
traffic vectors under
S
Suboptimal
scheduling
scheme
S
NP

Complete
Network Dynamics & Simulation Science Laboratory
Outline
•
Related Work
•
Preliminaries
•
Approach
–
Solution Ideas: Step I
–
Solution Ideas: Step II
•
Extensions
•
Conclusion
Network Dynamics & Simulation Science Laboratory
Solution Ideas: Step I
•
Step I: Upper

Bounding
End

to

end Delays
Avg. delay bound
:
(per

flow)
(network)
Given
:
flow
f
with avg. rate
and a path
Scheduling
protocol
random

access scheduling (in which channel
access probability is a function of flow rates)
:
:
path length
Note
:
generally, delay bounds and
Multi

commodity
Flow Framework
Network Dynamics & Simulation Science Laboratory
Random

Access Scheduling
•
Mechanism
–
Each link
l
makes channel access attempt when
w/ prob.
–
Each flow
f
on
l
then gets serviced w/ prob.
•
Property
–
The expected service rate for
a constant
f
:
arrival
departure
service
rate
Network Dynamics & Simulation Science Laboratory
Random

Access Scheduling cont.
•
Throughput Region
Necessary
Condition
Sufficient
Condition
Λ
S
=
Λ
OPT
Efficiency
ratio
Random

access scheduling is stable if
Any
stable scheduling scheme requires
The
efficiency ratio
of the random

access scheduling scheme is
Theorem
max # links in any interference set that can
make successful transmissions simultaneously
:
Interference Degree
Network Dynamics & Simulation Science Laboratory
Challenges for Queueing Analysis
•
Arrival Processes:
Multi

hop Traffic
–
Exogenous arrival processes are
general
–
Endogenous arrival processes are not regular
•
Intricacy Caused by
Interdependency
–
Success of
tx
on one link depends on
tx
on other links
–
(where ) and change over time.
They depend on the status of other link

flow pairs
Network Dynamics & Simulation Science Laboratory
Queueing Delay
•
Bounding Delay = Bounding Queue Sizes
–
Arrival rate is
λ
; according to Little’s Law,
•
Queueing Reduction & Isolation Techniques
f
1
f
2
f
3
f
4
f
2
Reductions
≠
Queueing Approximation
Provide provable worst

case bounds
R
1
R
2
Reduction 1:
decoupling paths
Construct a queueing system
R
1
, which
consists of independent tandem queueing
systems corresponding to each flow,
s.t
.
avg
queue sizes do not decrease
Reduction 2:
decoupling links
Construct a queueing system
R
2
based on
R
1
,
s.t
. we can isolate each
single queue for queueing analysis,
s.t
.
avg
queue sizes do not decrease
for a Given Set of Flows
Network Dynamics & Simulation Science Laboratory
Queueing Analysis cont.
•
For a system with a set of flows, where each flow
f
has a path and avg. rate , our random
access scheduling scheme guarantees that under
general graph

based interference model
(1)
Avg. end

to

end delay of each flow
f
is bounded by
(2)
Avg. end

to

end delay of entire system is bounded by
Theorem
Network Dynamics & Simulation Science Laboratory
Solution Ideas: Step II
•
Step II:
Bi

criteria Approx. Algorithms
to find a
stable rate vector
LP formulation
:
Randomized
rounding
Maximizing
Constraining delay
by
:
Ensured
Guaranteed
Low delay
bounds
High
throughput
Stability
Stability Condition
Multi

commodity
Flow Framework
Note
(from Step I): generally, delay bounds and
Network Dynamics & Simulation Science Laboratory
LP Formulation
•
Goal: Maximize Total Throughput
•
Constraints:
–
To find proper paths and stable rates for the set of source

destination connections under input delay constraints
–
Stability condition as the congestion constraints
–
Delay constraints
–
Flow conservation constraints
•
Path Reconstruction & Filtering
–
To screen out long paths for each connection
c
–
Loss in total rate is at most a half
Network Dynamics & Simulation Science Laboratory
•
Goal
–
To choose a set of flows to assign “large” rates without violating
the congestion constraints too much
–
Not to compromise delay bounds and the optimality of total rate
–
choose a subset of paths with “large” rate, to minimize maximum
congestion
•
Randomized Rounding Approach
Lead to the approx. factor
Randomized Rounding
F. T. Leighton, C. J. Lu, S. B.
Rao
, and A.
Srinivasan
“New algorithmic aspects of the local lemma with
applications to routing and partitioning.”
SIAM Journal of Computing
, 31:626
–
641, 2001
Complex pre

and post

processing for
the rounding step are required
Not a simple regular rounding
But a
dependent
rounding
Network Dynamics & Simulation Science Laboratory
Outline
•
Related Work
•
Preliminaries
•
Approach
•
Extensions
•
Conclusion
Network Dynamics & Simulation Science Laboratory
Extensions of Results
•
Asynchronous Systems
–
Links’ accesses to the wireless medium are not synchronized
–
Similar to 802.11
•
Channel Adaptive Systems
–
Each channel uses a unique band of frequency,
s.t
.
•
Negligible inter

channel interference
–
Links can switch among channels for transmission
•
Adds to the total capacity and capacity region of the system
•
Similar Results Hold
–
Our multi

commodity flow optimization framework applies in
both settings above
Network Dynamics & Simulation Science Laboratory
Outline
•
Related Work
•
Preliminaries
•
Approach
•
Extensions
•
Conclusion
Network Dynamics & Simulation Science Laboratory
Conclusion
•
Provide Light

weight Algorithms
–
As a multi

commodity flow optimization framework
–
For those
NP

Complete
problems with worst

case bounds for
various performance metrics in multi

hop wireless networks
•
combination of queueing analysis and optimization techniques for
multi

hop arbitrary networks
•
Provide Analytical Tools
–
Novel & Practical
–
For understanding performance of wireless networks
•
network optimization and exploration of
trade

offs
among
delay
,
throughput
,
#channels
, with varying network size, #connections
–
Instructive to network design, planning & management in
practice
Network Dynamics & Simulation Science Laboratory
Network Dynamics & Simulation Science Laboratory
Challenges
•
Challenges
:
–
Important yet hard

to

solve (
NP

Complete
) problem in a multi

hop wireless scenario even without considering any delay
guarantees.
–
Involving end

to

end queuing delay analysis & bounding, cross

layer optimization, flow rate control and routing. Only very
limited analytical results are known with recent progresses,
especially for end

to

end delay on arbitrary networks.
•
Discussed later
Network Dynamics & Simulation Science Laboratory
Multi

commodity Flows w/
Delay Guarantees
•
Generally, delay bounds and
•
LP Formulation
–
Goal: to maximize total throughput
–
Constraints:
•
To find proper paths and rates for the set of source

destination
connections
•
To incorporate stability condition and delay constraints
•
Randomized Rounding
–
To obtain flows with lower

bounded throughput rates, without
compromising the total rate and delay constraints.
Network Dynamics & Simulation Science Laboratory
Max
Total Rate
S.t
.
Delay Constraints
Congestion Constraints
(to
ensure stability
)
Flow conservation at all nodes,
e
.g., at the
source nodes:
Flow

conservation
Constraints
LP Formulation
Network Dynamics & Simulation Science Laboratory
LP Formulation cont.
•
Delay Constraints Explained
–
One

hop delay
for any packet on a link is at least
1
slot
–
The end

to

end delay of session
c
’s
packets is at least
–
When we choose
cost(l)
to be
1
for each link
l
, the delay constraints
boil down to that , which is a necessary condition
for the validity of any
•
Solve the LP
•
Reconstruct the paths from the solution to the LP
•
Filtering Step:
–
To screen out the paths for each connection
c
that are longer than
–
The total rate remains at least a half of that from the original solution
to the LP
Intuitively, LHS is lower

bound of delay;
RHS is input delay constraint parameter
To make sure paths are “
short
”
Network Dynamics & Simulation Science Laboratory
Randomized Rounding
•
Goal
–
To find a throughput rate vector for the paths constructed after
solving the LP
–
Throughput rates should be “large” (i.e., lower

bounded)
–
Not compromising the optimality of the total rate and delay
constraints.
•
Steps
–
Preprocessing
–
Rounding
–
Post

processing
Network Dynamics & Simulation Science Laboratory
Step 1: Preprocessing
•
Sub

step 1: Bin

packing
–
Bin

pack paths into groups
•
Sub

step 2:
Minimax
Integer Program (MIP)
Formulation
–
Formulate a
Minimax
Integer Program
•
that
minimizes maximum congestion
and
•
that chooses one path in each group with “large” flow rate
–
Solving such an MIP is generally
NP

Complete
•
Need to employ approximate algorithms to solve this problem
•
But before applying the solution techniques, need to perform
modification and reformulation as in the following 3 sub

steps
Network Dynamics & Simulation Science Laboratory
Step 1: Preprocessing cont.
•
Sub

step 3: Path Refinement
–
“Short

cut” a path that goes through an
interference set for over a constant
K
0
times: possible under unit

disk graph
interference model
–
The maximum number of links of a
path that lie in the same interference set is under
K
0
•
Sub

step 4: Relaxation of Congestion Constraints
–
Partition the plane into
1/8
×
1/8
square
grid cells,
s.t
.:
the number of congestion constraints that involve a given path is
at most
–
Possible under unit

disk graph interference model
Network Dynamics & Simulation Science Laboratory
Step 1: Preprocessing cont.
•
Sub

step 5: MIP Reformulation
–
Formulate a relaxation of the previous MIP with
•
Refined paths from Sub

step 3, and
•
Grid

cell

based congestion constraints from Sub

step 4
•
After Preprocessing
–
Path lengths do not increase
•
Ready to Perform Rounding
–
To obtain exactly one path from each group with “large” flow rate
Network Dynamics & Simulation Science Laboratory
Step 2: Rounding
•
Randomized Rounding Approach to MIP
–
By F. T. Leighton, C. J. Lu, S. B.
Rao
, and A.
Srinivasan
•
“New algorithmic aspects of the local lemma with applications to
routing and partitioning.”
SIAM Journal of Computing
, 31:626
–
641, 2001
–
Produce a set of flows w/ a rate vector ,
s.t
.
•
The total rate is order

optimal under the delay constraints by
•
Each flow has a rate of at least
1
•
The congestion is at most
•
Need to accommodate the solution for the
original congestion constraints
Network Dynamics & Simulation Science Laboratory
Step 3: Post

processing
•
Choosing Proper Flow Rates
–
Scale down the flow rates by a reasonably small factor of
,
s.t
. the original congestion constraints of the LP
will be satisfied
–
That is how the factor of comes into the
approximation ratio
Network Dynamics & Simulation Science Laboratory
Summary of Results
•
Random access scheduling
•
If Stability condition is
satified
•
We have
Comments 0
Log in to post a comment