Slide - Filebox - Virginia Tech

hordeprobableBiotechnology

Oct 4, 2013 (3 years and 8 months ago)

109 views

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