Ness Shroff, OSU

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Expected Delay Analysis for Wireless Networks

• Ness Shroff, OSU
• Gagan Raj Gupta, Purdue

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Heterogeneous Wireless Networks

• Link interference
• Set based
• Single-hop traffic
• Exogenous source
• Dynamic Arrivals
• Assume IID in time
• Fixed size packets
• Time-varying wireless links
• Assume Markovian ON/OFF
• Different capacities
• Fixed number of packets served per unit time

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Scheduled Links
Scheduled Link
Interference Set
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Wireless Networks

• Link interference
• Set based
• Single-hop traffic
• Exogenous source
• Dynamic Arrivals
• Assume IID in time
• Fixed size packets
• Time-varying wireless links
• Assume Markovian ON/OFF
• Different capacities
• Fixed number of packets served per unit time

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Quality of Service Guarantees

• Most works on scheduling in wireless networks
  focus on stability throughput-optimality
• Stability is a first-order property
• Very-long time-scale allocation
• Implies that queue sizes dont blow up
• Do the queues become large?
• To answer this question, we need to understand
  delay performance

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Problem Statement

• Given
• Statistics of the arrival process
• Capacity of the Links
• Probability the link is ON
• Interference constraints
• Compute
• Lower and Upper bounds on the Expected Delay of
  the System
• Scheduler
• MWM (a throughput optimal scheduler)

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Challenges

• Interference constraints complicate the service
  process
• Among the links that can be scheduled (i.e.,
  state ON)
• Schedule the set of non-interfering links

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Challenges

• Arrivals are dynamic
• Schedule is dependent on the state of the system
• Complex correlations between the service,
  backlog and arrival process

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Challenges

• The schedules change dynamically
• Difficult to provide guarantees for the service
  process
• Different links may have different capacity
• May not be fully utilized if the queue is small

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Limited Work on Delay Analysis

• Development of throughput optimal schemes
• Guarantees stability for the full capacity region
  Tassiulas Ephremedes92
• Fluid Limits
• Stability results for switches Prabhakar00
• Asymptotic optimality in the heavy traffic regime
  (through workload process minimization)
  Stolyar04
• Large Deviations
• Characterize scheduling algorithms that minimize
  buffer overflow probability Lin07
• Method of Lyapunov Drifts
• Generate upper bounds on delay performance
• Order results (typically loose) Marsan01, Neely
  Modiano06, Neely08, Gupta Shroff08

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Approach

• Establish upper bound via Lyapunov drifts
• Characterize the expected decrease in the
  potential function
• Expected decrease proportional to queue length
• Captures the effect of weight of the matching on
  the drift
• Establish fundamental lower bound on delay using
  the resource constraints
• Identify bottlenecks in the system
• Analyze the queuing in these bottlenecks by
  reduction to a single queue system
• Most of the queuing takes place at these
  bottlenecks

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Upper Bound Analysis Queue Evolution

• Queue evolution
• Define residual Capacity
• Then queue evolution can be written as

Link Capacity
Scheduling
Link state
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Generalized-MWM policy

• Compute the maximum weight matching where weight
  of each link (edge) is given by

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a
c
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1-hop interference model
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Upper Bound Analysis Potential Function

• Build on the method of Lyapunov Drifts
• Devise a suitable Lyapunov (potential) function
• Bound the expected change in the potential

Weight of the matching
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Upper Bound
C

• Requires knowledge of the arrival rates
• The bound does not capture statistical
  multiplexing of packets

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Fundamental Lower Bound

• Identify constrained sets (bottlenecks) in the
  system Padhye05
• No more than one of these links can be active at
  the same time

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Lower Bound Analysis
Sum of Queue lengths QX
Server ON when at least one of the links is ON
A1
A2
A3
A4
Unit Capacity Server
S
A5
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Queuing bound for a bottleneck
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Lower Bound Analysis (cont.)
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Simulations (Grid)

• Grid Network
• Random load
• Unit Capacity at each link
• Links always ON
• Poisson Arrivals
• Secondary interference model

7x9 Grid Topology
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Simulation Results (Grid Topology)
Queue Length
Delay
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Simulations (Tree)

• Tree Network
• Links with different loads
• Different capacity
• Links on with probability 95
• Poisson Arrivals
• Secondary interference model

Tree Topology

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Simulation Results (Tree Topology)
Queue Length
Delay
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Open Problems

• Delay analysis for other scheduling policies
• LQF (Longest Queue First)
• Delay analysis for multi-hop wireless networks
• Design of a provably delay efficient scheduler

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Thanks
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Delay Approximation
Decoupling the contribution of each link
Queue Approximation
Exclusive set with the tightest constraint
Delay Approximation
For Poisson arrivals, delay is O(1/1-u) where u
is the largest real number such that for every
epsilongt0 (load/(uepsilon)) is not in C. i.e.,
for a given load, Delay does not grow with N