ThroughputDelay Tradeoff in Wireless Networks

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Throughput-Delay Trade-off in Wireless Networks
INFOCOM 2004
E. Gamal, J. Mammen, B. Prabhakar, and D.
Shah Department of EE and CS, Stanford University
Presented by Ning Li
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Outline

• Related Work
• Network Model
• Summary of Main Results
• T-D Trade-off for Static Networks
• T-D Trade-off for Mobile Networks
• Conclusions

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Related Work Fixed Networks

• 1 Gupta and Kumar, The Capacity of Wireless
  Networks.
• Static network.
• n nodes are independently and uniformly
  distributed in a disk of unit area in the plane.
• Each node has a randomly chosen destination.
• Nodes are homogeneous, each capable of
  transmitting at W bps.
• Per node throughput of can be achieved w.h.p.

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Related Work Mobile Networks

• 2 Grossglauser and Tse, Mobility Increases the
  Capacity of Ad Hoc Wireless Networks.
• Each of the n nodes moves on a unit sphere.
• The movement of each node is independent with a
  uniform stationary and ergodic distribution on
  the unit sphere.
• By allowing only one intermediate relaying node,
  the scheme can achieve an throughput capacity of
  per node, with a possibly large delay.
• Neither work considers how delay scales with the
  number of nodes.

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Network Model

• Definition of Throughput
• A throughput ?0 is said to be feasible or
  achievable if every node can send at a rate of ?
  bps to its chosen destination. Let T(n) be the
  maximum feasible throughput with high
  probability.
• Definition of Delay
• The delay of a packet in a network is the time it
  takes the packet to reach the destination after
  it leaves the source.
• The average packet delay for a network with n
  nodes, D(n), is obtained by averaging over all
  packets, all source-destination pairs, and all
  random network configurations.
• In fixed networks, it is proportional to the
  number of intermediate relay nodes.
• In mobile networks, in addition to number of hops
  it depends on the velocity of nodes.
• Packet size scales as T(n) so that D(n) captures
  the dynamics of the network/scheme and not the
  transmission delay.

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Interference Model

• n nodes are distributed in a unit torus uniformly
  at random. Each node is capable of transmitting
  at W bits per second.
• Relax Protocol Model a transmission from node Xi
  to Xj is successful if
• ?Xk - Xj? ? (1?) ?Xi - Xj? for every other node
  Xk that is transmitting over the same channel
  simultaneously.

Xi
Xj
Xk
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Main Results

• Propose a scheme that can achieve the optimal T-D
  trade-off in fixed networks
• Model the delay in mobile networks when each node
  moves according to independent Brownian motion.
• Obtain the delay for mobile networks
• Where is the average distance traveled
  in one hop
• Devise an optimal scheme for T-D trade-off in
  mobile networks.

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T-D Trade-off Fixed Networks
Scheme 1

• Divide the unit torus using a square grid into
  square cells, each of area a(n).
• A cellular time-division multi-access (TDMA)
  transmission scheme is used, in which, each cell
  becomes active, i.e., its nodes can transmit
  successfully to nodes in the cell or in
  neighboring cells, at regularly scheduled cell
  timeslots.
• Let the straight line connecting a source S to
  its destination D be denoted as an S-D line. A
  source S transmits data to its destination D by
  hops along the adjacent cells lying on its S-D
  line.
• When a cell becomes active, it transmits a single
  packet for each of the S-D lines passing through
  it. This is again performed using a TDMA scheme
  that slots each cell timeslot into packet
  time-slots.

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T-D Trade-off Scheme 1
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Theorem 1 T-D Trade-off

• Lemma 1 Each cell will have at least one node
  whp, thus guaranteeing successful transmission
  along each S-D line.
• Lemma 2 Each cell can be active for a constant
  fraction of time, independent of n.
• Lemma 3 Maximum number of S-D lines passing
  through any cell is bounded, i.e.,
  .

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Proof Delay Bound

• Average of hops per packet
• Each hop covers a distance of
• hops per S-D pair i
• Average hops per packet
• Time spent in each hop is constant.
• Each cell is activated once every constant number
  of cell timeslots
• Since packet size scales in proportion to the
  throughput T(n), each packet stays at a cell for
  a constant time.

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Optimality of Scheme 1
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T-D Trade-off Mobile Networks
Scheme 2

• Divide the unit torus into n square cells, each
  of area 1/n.
• Each packet is relayed at most once.
• Each cell becomes active once in every 1c1 cell
  time-slots.
• Transmission is always between two nodes in the
  same cell.
• Consider a cell with at least two nodes in an
  active timeslot, choose a node i in the cell at
  random. A time-slot is divided into two sub-slots
  A and B
• Sub-slot A i acts as a source and sends its own
  packet to a randomly chosen node j in the same
  cell
• Sub-slot B i acts as a relay and sends packet
  destined for randomly chosen node k in the same
  cell.

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T(n)?(1)

• Random relaying spreads the traffic of each S-D
  pair uniformly.
• In steady state, each node has a packet for every
  other S-D pair, and the traffic between each S-D
  pair is spread uniformly across all other nodes.
• Each cell is active constant fraction of the
  time, that is, ?(1) throughput is possible per
  cell.
• ?(n) (i.e. constant fraction) cells have at least
  2 nodes.
• Then overall network throughput is ?(n).
• As each packet is relayed once, net throughput
  remains ?(n).
• Due to symmetry, this is equally divided among n
  S-D pairs.
• That is, ?(1) throughput per pair is achievable.

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Model Delay in Mobile Networks

• The unit torus can be seen as a grid.
• Each node performs independent Brownian motion,
  which induces a symmetric independent random walk
  on the grid.
• Velocity v(n) is assumed to scale down with n.
  The time-step of the random walk is
  time-slots.

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Relay Queue
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Relay Queue Model

• Average delay is determined by delay at the relay
  nodes.
• To determine average delay, consider an S-D pair
  and any relay node R.
• The delay at a relay node is the queueing delay.
• An arrival can occur when S and R are in the same
  cell and a departure can occur when R and D are
  in the same cell.
• Thus inter-arrival and inter-departure times are
  identically distributed and are independent.
• This results in a GI/GI/1-FCFS queue at each
  relay
• The average queueing delay is
• where S is the inter-meeting time.

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Relay Queue Analysis

• Let (X(t),Y(t)) be the distance between two nodes
  performing random walk on the grid.
• We are interested in the inter-occurrence time of
  the event X(t),Y(t)(0,0).
• Average steps between two meetings
• Average delay

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Scheme 3(a)

• Divide the torus into square cells, each of area
  a(n).
• Each cell becomes active once every 1c1
  time-slots.
• A source S sends its packet directly to its
  destination D if it is in any of the neighboring
  cells. Otherwise, it randomly chooses a relay
  node R in an adjacent cell on the S-D line at the
  time of transmission.
• When the cell containing the relay node R is
  active, R transmits the packet directly to D, if
  D is in a neighboring cell. Otherwise, it relays
  the packet again to a randomly chosen node in a
  neighboring cell on the straight line connecting
  it to D. This process continues until the packet
  reaches the destination.

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Scheme 3(a) Trade-Off

• v(n) has to satisfy
• In each relay step the distance to the
  destination node will be decreased at least
• The packet eventually reaches its destination if
  
• of S-D pairs per cell

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Scheme 3(b)

• Divide the unit torus into square cells, each of
  area a(n). We further lay out an additional grid
  formed by square sub-cells of size . Thus each
  square cell of area a(n) contains a(n)/b(n)
  sub-cells.
• Each cell becomes active once every (1c1)
  time-slots. A cell time-slot is divided into
  ?(n?a(n)) packet time-slots.
• An active packet time-slot is divided into two
  sub-slots A and B.
• In sub-slot A, each node sends a packet to its
  destination node if it is present in the same
  cell. Otherwise, it sends its packet to a
  randomly chosen node in the same cell, which acts
  as a relay. The packet is sent using hops along
  sub-cells as in Scheme 3(a).
• In sub-slot B, each node picks another node at
  random from the same cell and sends a packet that
  is destined to it. Again, the packet is sent
  using hops along sub-cells as in Scheme 3(a).
• The scheme requires the packet size to scale as

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Scheme 3(b) T-D Trade-off
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Scheme 3(b) Sketch of Proof

• Throughput in any packet time-slot in a given
  cell,
• The S-R/R-D pairs is randomly chosen according to
  Scheme 3(b)
• Packets are communicated according to Scheme
  3(a)
• There are d(n)a(n)/b(n) sub-cells and
  mn?a(n)o(n?a(n)) nodes
• The throughput between S-R/R-D pair is
• Delaythe delay has two components,
• Hop-delay proportional to the number of hops
  along sub-cells from a source to the mobile relay
  and from the mobile relay to the destination.
  Average hops is
• Mobile-delay the time it takes the mobile-delay
  node to reach the cell containing the destination
  and to deliver the packet to it. We can use a
  similar approach as in Scheme 2. The mobile delay
  is . Since , the
  mobile-delay dominates the hop-delay.