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Guanhong Pei ∗ , V. S. Anil Kumar † , Srinivasan Parthasarathy ‡ , and Aravind Srinivasan §

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### Approximation Algorithms for Throughput Maximization in Wireless Networks with Delay Constraints

Guanhong Pei∗, V. S. Anil Kumar†, SrinivasanParthasarathy‡, and AravindSrinivasan§

∗ 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

Fundamental Problems

- Given a set C of end-to-end source-destination connections (or sessions) in an arbitrary multi-hop wireless network,
- Maximizetotal rate of communication possible for sessions in C
- Minimizeend-to-end (average) delay of communication possible for sessions in C

NP-Complete

Avg. time for packets to travel to destination

To tackle these problems

Traffic ratesfor each connection

Pathsfor each connection

Links to make transmissions

Problem Statement of Our Work

O(1)-inapproximable

unit-disk graph interference

- Problem:Delay-Constrained Throughput Maximization (DCTM)
- Given: an arbitrarymulti-hopwireless 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 vectorwith flow paths and a suitable scheduling scheme, such that the total rate is maximized and end-to-end delay is bounded byfor each connection c.

O(nε)-inapproximable

general interference

- Transport Layer

Network Layer

Mac Layer

Summary of Results

For computing a rate vector and flow paths under a random-access scheduling scheme w/ delay, throughput & stability guarantees

Multi-commodity Flow Framework

Total throughput is at least a factor of of the maximum possible (with given delay constraint vector ), where ; Each flow rate is

Avg. end-to-end delay is bounded by

Theorem: for DCTM

: Path of flow f

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

Outline of the Rest

- Related Work
- Preliminaries
- Approach
- Extensions
- Conclusion

Related Work on Delay Bounds

Jagabathula, Shah 08

Jayachandran, Andrews 10

Le, Jagannathan, Modiano09

Gupta, Shroff09

Neely 09

Kar, Luo, Sarkar09

Jagabathula, Shah 08

Jayachandran, Andrews 10

Le, Jagannathan, Modiano09

Gupta, Shroff09

Neely 09

Kar, Luo, Sarkar09

n: #nodes; m: #links; Imax: max interference degree; θmax: max congestion

(#ﬂows through a link); C(N): chromatic number of link interference graph

Outline

- Related Work
- Preliminaries
- Network Model
- Traffic Model
- Queueing Model
- Metrics: Throughput & Delay
- Approach
- Extensions
- Conclusion

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 →

receiver

link

sender

← ACK

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:

Session c:

f1

f2

f3

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.

Queue

Arrival

Departure

Service

Queueing Dynamics

f :

1f

4f

5f

3f

6f

Depar-ture

Arrival

Packet

a

l,f1

Q

Service Rate

d

l,f1

l,f1

a

l,f2

Q

d

l,f2

a

l,f2

l,f3

Q

d

l,f3

l,f3

: Queue on

Multiple Input Flows

Logical Queues for Each Flow

Single Server

Queue Evolution:

: the ith link of flow f

Session c:

f1

f2

f3

: size of the logical queue of flow f on link l at t

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,

Explained next

By Little’s Law

Long-term average backlog:

Throughput Region

- Queue-stability of a System iff
- Traffic Rate Vector
- Avg. rate for each flow f

NP-Complete

Max-Weight Scheduling

Capacity region: ΛOPTthe set of all stable traffic vectors

Throughput region: ΛSthe set of all stable traffic vectors under S

Suboptimal scheduling scheme S

γ-scaled region γΛOPT, (0<γ<1)

Outline

- Related Work
- Preliminaries
- Approach
- Solution Ideas: Step I
- Solution Ideas: Step II
- Extensions
- Conclusion

Multi-commodity Flow Framework

Solution Ideas: Step I- Step I: Upper-Bounding End-to-end Delays

: path length

Given: flow fwith avg. rate and a path

Schedulingprotocol

random-access scheduling (in which channel access probability is a function of flow rates)

:

(per-flow)

Avg. delay bound:

(network)

Note: generally, delay bounds and

Random-Access Scheduling

f :

- Mechanism
- Each link l makes channel access attempt whenw/ prob.
- Each flow f on l then gets serviced w/ prob.
- Property
- The expected service rate for

a constant

servicerate

departure

arrival

Random-Access Scheduling cont.

- Throughput Region

The efficiency ratio of the random-access scheduling scheme is

Theorem

Random-access scheduling is stable if

Sufficient Condition

ΛS =ΛOPT

Any stable scheduling scheme requires

Necessary Condition

Efficiency ratio

max # links in any interference set that can make successful transmissions simultaneously

: Interference Degree

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

Queueing Delay

Reductions ≠Queueing ApproximationProvide provable worst-case bounds

for a Given Set of Flows

- Bounding Delay = Bounding Queue Sizes
- Arrival rate is λ; according to Little’s Law,
- Queueing Reduction & Isolation Techniques

- Reduction 1: decoupling pathsConstruct a queueing system R1, which consists of independent tandem queueing systems corresponding to each flow, s.t. avg queue sizes do not decrease

- Reduction 2: decoupling linksConstruct a queueing system R2 based on R1, s.t. we can isolate each single queue for queueing analysis, s.t. avg queue sizes do not decrease

f1

f2

f2

f3

f4

R2

R1

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

Avg. end-to-end delay of each flow f is bounded by

Avg. end-to-end delay of entire system is bounded by

Theorem

Multi-commodity Flow Framework

Solution Ideas: Step II- Step II: Bi-criteria Approx. Algorithms to find a stable rate vector

Highthroughput

Maximizing

LP formulation:

Constraining delayby

Stability Condition

Low delay bounds

Ensured

Randomized rounding

:

Stability

Guaranteed

Note(from Step I): generally, delay bounds and

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

Randomized Rounding

- 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

Not a simple regular rounding

But a dependent 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

Lead to the approx. factor

Outline

- Related Work
- Preliminaries
- Approach
- Extensions
- Conclusion

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

Outline

- Related Work
- Preliminaries
- Approach
- Extensions
- Conclusion

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

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

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.

LP Formulation cont.

Intuitively, LHS is lower-bound of delay;

RHS is input delay constraint parameter

- 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

To make sure paths are “short”

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

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

Step 1: Preprocessing cont.

- Sub-step 3: Path Refinement
- “Short-cut” a path that goes through an interference set for over a constant K0times: possible under unit-disk graphinterference model
- The maximum number of links of a path that lie in the same interference set is under K0
- Sub-step 4: Relaxation of Congestion Constraints
- Partition the plane into 1/8 × 1/8 squaregrid cells, s.t.:the number of congestion constraints that involve a given path is at most
- Possible under unit-disk graph interference model

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

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

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

Summary of Results

- Random access scheduling
- If Stability condition is satified
- We have

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