Queuing analysis for coded networks with feedback J. Sundararajan, D. Shah, M. M

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Queuing analysis for coded networks with
feedbackJ. Sundararajan, D. Shah, M. Médard, M.
Mitzenmacher, J. Barros

• Consequences.
• Queue size now grows linearly with 1/(1- ?)
• Reduces the amount of storage needed at
  intermediate nodes for performing re-encoding
• Analysis also applies when only some nodes do
  re-encoding
• ACK of degrees of freedom allows traditional
  queuing results to be applied easily in scenarios
  with network coding

Packets can be dropped from queue only upon
confirmation of decoding This means the queue
sizes will be unnecessarily long In particular,
as load factor ? approaches capacity, queue grows
quadratically as a function of 1/(1- ?)

• MAIN ACHIEVEMENT
• Propose novel ACK mechanism that allows nodes to
  manage queue occupancy effectively
• Characterize expected queue size at each node

HOW IT WORKS Acknowledge seen packets

• Key insight.
• With drop-when-decoded, the busy period of the
  virtual queue contributes to the physical queue
  size calculation
• Responding to ACK of the degrees of freedom
  ensures only queuing delay of virtual queues
  contributes to physical queue size

Almost as if there is link-by-link feedback
Extend queue management protocol to more general
(wireless) scenarios Multipath routing with
coding Multicast traffic pattern

• ASSUMPTIONS AND LIMITATIONS
• Perfect and delay-free feedback used in analysis,
  though not critical for the approach
• Field size assumed to be very large

The proposed approach to queue management will
play a key role in interfacing TCP with network
coding, especially when intermediate nodes
re-encode
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Problem setup

• Tandem network of erasure links
• Bernoulli arrival process of rate ?
• Perfect delay-free end-to-end feedback
  (End-to-end nature is motivated by TCP ACKs)
• Want to study the expected size of the queues at
  all the nodes

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Questions addressed

• With link-by-link feedback (benchmark)
• Every link performs simple ARQ no coding
• Every queue behaves like a Geom/Geom/1 queue
• Growth of the queue size as load factor ??1 is
  linear in 1/(1-?)
• With end-to-end feedback
• Need to use intermediate node re-encoding to get
  to capacity
• Degree-of-freedom queue (also called virtual
  queue) still behaves like a Geom/Geom/1 queue
• Can we ensure O(1/(1-?)) growth of physical
  queues in this setting?

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Questions addressed

• Baseline approach ACK when decoded
• Physical queue size is related to busy period of
  virtual queues
• This gives O(1/(1-?)2) growth of queues
• Also, this approach causes the delay for decoding
  at the receiver to enter the round-trip time
• This has adverse effects in congestion control
  TCP windows will close unnecessarily
• Need to ACK every degree of freedom
• Then physical queue size will be related to the
  waiting time for successful transmission
• Then we can achieve O(1/(1-?)) growth of queues
• TCP window will also progress smoothly, since
  every incoming packet will generate an ACK
  without waiting for decoding
• How to do this in a way that is simple to
  implement?

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Seeing a packet
Seen
Unseen
Decoded
Coefficient vectors of received linear
combinations, after Gaussian elimination
p1 p2 p3 p4 p5 p6 p7 p8
1 0 0
0 1 0 0
0 1 - - - -
- - - 1 - -
- - - - - 1
- - - - - - -
Witness for p4
Number of seen packets Rank of matrix
Dim of knowledge space
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A new kind of ACK
Seen
Unseen
Decoded
Coefficient vectors of received linear
combinations, after Gaussian elimination
p1 p2 p3 p4 p5 p6 p7 p8
1 0 0
0 1 0 0
0 1 - - - -
- - - 1 - -
- - - - - 1
- - - - - - -
Witness for p4

• Acknowledge degrees of freedom
• ACK a packet upon seeing it
• Allows ACK of every innovative linear
  combination, even if it does not reveal a packet
  immediately

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The queue update rule

• Store every incoming innovative linear
  combination
• Perform row reduction of the stored coefficient
  matrix and update the packets correspondingly
• Essentially, queue stores witnesses of seen
  packets
• Drop the witness of a packet if you know receiver
  has seen the packet
• Implicit ACK Although only sender gets
  receivers ACK, other nodes can infer receivers
  state from the senders coding window, which is
  embedded in the header

Innovative means the packet is linearly
independent of previously received linear
combinations
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The analysis

• Use Littles law to find the expected queue size
  using expected time spent in queue
• Arrival Packet arrives into queue of node k when
  the node first sees the packet
• Departure Packet departs when node k finds out
  that the receiver has seen the packet
• This duration can be broken into two parts
• T1 Time until receiver sees packet
• T2 Time till node k learns of receivers ACK

Lemma Let SA and SB be the set of packets seen
by two nodes A and B respectively. Assume SA\SB
is non-empty. Suppose A sends a random linear
combination of its witnesses of packets in SA and
B receives it successfully. The probability that
this transmission causes B to see the oldest
packet in SA\SB is (1 - 1/q), where q is the
field size.
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The analysis (contd.)

• Lemma implies that the virtual queues behave like
  a FIFO Geom/Geom/1 queue
• Hence, the time between node i seeing a packet
  and node i1 seeing the packet is the waiting
  time in a Geom/Geom/1 queue, with expectation
• Hence, time till receiver sees packet is
• Additional time till receivers ACK propagates to
  node k is
• Hence, using Littles law, the expected queue
  size is

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Conclusions

• Proposed a new ACK mechanism that acknowledges
  every degree of freedom
• Analyzed expected queue length for single path
  with re-encoding at one or more intermediate
  nodes, and end-to-end feedback
• Queue size now grows linearly with 1/(1- ?)
• Need to extend the protocol and analysis to the
  case of multiple paths and multiple receivers