Throughputdelay Tradeoff in Wireless Networks

1
Throughput-delay Trade-off in Wireless Networks

• James Mammen and Devavrat Shah
• Joint Work With
• A. El Gamal and B. Prabhakar

SNRC 7th April, 2003
2
Outline

• Previous work
• Overview of the main results
• T-D trade-off for fixed networks
• Modeling delay in mobile networks
• T-D trade-off for mobile networks
• Conclusions

3
Wireless Networks

• Characteristics of wireless networks
• n wireless nodes capable of transmitting to and
  receiving from each other
• Topology not completely known or liable to change
• Transmitting nodes create interference for other
  nodes
• Successful transmission requires low interference
• Questions
• How does the throughput of the network scale with
  n ?
• How does the associated delay scale with n ?
• What is the T-D trade-off ?

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Random Network Model

• Introduced by Gupta Kumar for
  studying throughput scaling
  in a
  fixed network IT 00
• n nodes distributed in a unit disk uniformly and
  randomly
• Each node can transmit at W bits per second
• Each node has a randomly chosen destination
• Each node is simultaneously a source, S, a
  potential destination, D, and a relay for other
  S-D pairs
• Slotted transmission

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Models for Successful Transmission

• Model needs to capture interference effects
• Transmission from node i to node j is successful
  if
• Protocol Model
• where k is any other node transmitting
  simultaneously
• Physical Model

k
r
j
i
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Throughput of a Random Network

• Throughput t(n) is feasible if there exists a
  routing and scheduling scheme such that every S-D
  pair can communicate t(n) bits per second on an
  average.
• Throughput capacity T(n) is the maximum feasible
  throughput with high probability (whp)
• Gupta Kumar
• under protocol
  model

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Mobile Wireless Networks

• Grossglauser Tse Infocom 01
• For a mobile network with independently moving
  nodes having a stationary uniform distribution
• is feasible under the physical model
• The throughput results for fixed and mobile
  networks were obtained using different methods
• Delay not addressed
• Is it possible to trade-off some throughput for
  lower delay (T-D trade-off)?

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New Results

• We present a scheme for T-D trade-off in a fixed
  network and show its optimality
• We determine the delay in a mobile network by
  modeling the motion of nodes as a symmetric
  random walk
• We present a scheme for T-D trade-off in a mobile
  network
• We also re-derive previous throughput results for
  fixed and mobile networks within the same
  framework

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T-D Trade-off in a Fixed Network

• Theorem 1 The following T-D trade-off can be
  achieved and is optimal

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T-D Trade-off for a Mobile Network

• Theorem 2 Assuming the
  following T-D trade-off is achievable

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Wireless Network Model

• Unit square with n nodes distributed
  uniformly at random
• Each node capable of transmitting
  at W bits per second
• Relaxed protocol model
• Transmission from node i to node j is successful
  if for any other simultaneously transmitting node
  k
• T(n) maximum feasible throughput with high
  probability

k
d
j
i
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Definition of Delay

• Delay is the time taken by a packet to reach its
  destination starting from the time it leaves its
  source
• ignore queueing delay at the source as it is
  independent of n and our interest is in the
  network delay
• D(n) average delay is used as a measure of
  delay
• At each hop, the delay has two components
  - Transmission delay and
  Queueing delay
• Queueing delay delay in getting to the head of
  the queue servicing delay
• Queueing delay dominates the transmission delay

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T-D Trade-off in a Fixed Network

• Cellular scheme with square cells of are a(n)
• Routing is along adjacent cells
  falling on the straight
  line
  connecting the source and the
  destination (S-D line)
• a(n) is a parameter which
  determines T(n) and D(n)
• Theorem 3

Area a(n)
D
S
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Outline of the Proof

• When a node in a cell transmits to a node in a
  neighboring cell, there are at most c interfering
  cells where c is independent of n
• Thus each cell has effective bit-rate W/(1c)
• Each cell contains at least one node as long as
  . Hence routing
  is always possible
• The number of S-D lines passing through any cell
  is
  whp
• Therefore is
  feasible whp
• Average distance between source and destination
  is constant and hence
  hops

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Scheme for

• Modification of the scheme used by Grossglauser
  and Tse
• Each packet is relayed at most once

• The unit square is divided into square cells of
  area 1/n
• In each cell, a randomly chosen node transmits to
  another randomly chosen node in the same cell
• The randomly chosen node pair could be an S-D,
  S-relay or relay-D pair

Cells of area 1/n
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Routing and Scheduling

• Operates in 2 phases
• Phase 1 - odd slots S to random relay(or D)
• Phase 2 - even slots Random relay(or S) to D

• Relaying causes uniform spreading of the traffic
• In the steady state each node has packets for
  every other node
• A relay delivers a packet to a destination when
  both are in the same cell

S
R
D
Theorem 3 by choosing
cells of area 1/n
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Delay in a Mobile Network
Model for any packet from its source to its
destination

• Average delay is determined by the delay at the
  relay nodes
• as there are n-2 relay nodes

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Modeling Delay in a Mobile Network

• To determine the average delay it is sufficient
  to consider any S-D pair and any one relay node
• The delay at a relay node is the queueing delay

• We model the movement of nodes as a symmetric
  random walk on a torus where a node decides to go
  to any of the 4 neighboring cells with
  probability 1/4

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Modeling Delay (2)

• An arrival can occur when the source and the
  relay are in the same cell and a departure can
  occur when the relay and the destination are in
  the same cell
• Thus inter-arrival and inter-departure times are
  identically distributed and are independent
• This can be modeled as a GI/GI/1 FCFS queue
• And hence the queueing delay is proportional to
  where S is the service time

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Modeling Delay (3)

• The time scale of the random walk depends on n
  and V(n)
• Assuming velocity V(n), the time a node stays in
  a cell is
• V(n) is assumed to be a decreasing function of n
• Using recent results Dembo et al. for the cover
  time of a random walk on a torus, we obtain
• Lemma 4.1
• Theorem 4 For the scheme in theorem 3, average
  delay
• Assuming , we get

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T-D Trade-off in a Mobile Network
sub-cells of area
cells of area a(n)

• Divide the unit square into square cells of size
  a(n) and sub-cells of size b(n)
• Within each cell, route using hops along adjacent
  cells, otherwise same as before
• A destination node may not be in the same
  sub-cell by the time a packet reached that cell
  by hops across sub-cells

D
R
S
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T-D Trade-off Scheme

• Use a follow/chase scheme if the destination is
  not in the cell by the time a packet arrives then
  the packet is forwarded through hops to the new
  destination cell
• If hop-speed is greater than the node velocity
  then follow/chase scheme can be used and the
  decrease in throughput is only by a constant
  factor
• Delay has two components
• - the delay due to hops is due to the
  hops across adjacent cells taken by a packet in
  moving from a source/relay to its destination
• - the mobility delay is the time taken
  for the relay/source and destination to be chosen
  as a Tx-Rx pair when in the same cell

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T-D Tradeoff in a Mobile Network

• By choosing and
  varying a(n) we get the upper portion of the T-D
  curve where dominates
• By fixing and varying b(n), we
  obtain the lower portion of the T-D curve where
  dominates
• Mobility allows higher throughput at the cost of
  significantly higher delay

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Conclusions

• We presented an optimal T-D trade-off for a fixed
  network
• We determined the average delay in a mobile
  network
• We presented a T-D trade-off scheme for a mobile
  network
• Re-derived previous throughput results for fixed
  and mobile networks within the same framework