Queuing analysis for coded networks with feedback
J. Sundararajan, D. Shah, M. Médard, M. Mitzenmacher, J. Barros
Packets can be dropped from
queue only upon confirmation
of decoding
• This means the queue sizes will be
unnecessarily long
MAIN ACHIEVEMENT:
 Propose novel ACK mechanism that allows
nodes to manage queue occupancy effectively
• In particular, as load factor ρ
approaches capacity, queue grows
quadratically as a function of 1/(1- ρ)
2
1
k
N
 Characterize expected queue size at each node
HOW IT WORKS:
Consequences.
 Queue size now grows linearly
with 1/(1- ρ)
IMPACT
STATUS QUO
ACHIEVEMENT DESCRIPTION
Unseen
Decoded
p1 p2 p3 p4 p5 p6 p7 p8
1
1
1
1
1
Key insight.
λ x (Time for receiver’s
ACK to propagate from
source to node k)
λ
x
(Time between node (i-1) seeing pkt and node i seeing pkt)
Almost as if there is link-by-link feedback…
ASSUMPTIONS AND LIMITATIONS:
 Perfect and delay-free feedback used in analysis,
though not critical for the approach
 Field size assumed to be very large
NEXT-PHASE GOALS
NEW INSIGHTS
Coefficient vectors of
received linear
combinations, after
Gaussian elimination
Number of seen packets = Rank of matrix
 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
0 0 0
0 0 0
-------------------
 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
Acknowledge “seen” packets
Seen
 Reduces the amount of storage
needed at intermediate nodes
for performing re-encoding
Rx
Tx
Rx
Extend queue management
protocol to more general
(wireless) scenarios
• Multipath routing with coding
• Multicast traffic pattern
The proposed approach to queue management will play a key role in interfacing TCP with
network coding, especially when intermediate nodes re-encode
Problem setup
1
2
k
N
• Tandem network of erasure links
• Bernoulli arrival process of rate λ
• 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
Questions addressed
1
2
k
N
• 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 ρ→1 is linear in 1/(1-ρ)
• 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-ρ)) growth of physical queues in this
setting?
Questions addressed
• Baseline approach: ACK when decoded
– Physical queue size is related to busy period of virtual queues
– This gives O(1/(1-ρ)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-ρ)) 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?
‘Seeing’ a packet
Seen
Unseen
Decoded
Coefficient vectors of
received linear combinations,
after Gaussian elimination
p1 p2 p3 p4 p5 p6 p7 p8
1
1
1
1
1
Number of seen packets
=
0 0 0
0 0 0
-------------------
Rank of matrix
Witness for
=
p4
Dim of knowledge space
A new kind of ACK
Seen
Unseen
Decoded
p1 p2 p3 p4 p5 p6 p7 p8
1
1
1
1
1
0 0 0
0 0 0
-------------------
Coefficient vectors of
received linear combinations,
after Gaussian elimination
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
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 receiver’s ACK,
other nodes can infer receiver’s state from the sender’s
coding window, which is embedded in the header
†Innovative
means the packet is linearly independent of previously received linear combinations
The analysis
• Use Little’s 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 receiver’s 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.
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 i+1
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 receiver’s ACK propagates to node k is
• Hence, using Little’s law, the expected queue size is:
Conclusions
• Proposed a new ACK mechanism that acknowledges
every degree of freedom
• Analyzed expected queue length for single path with reencoding at one or more intermediate nodes, and end-toend feedback
• Queue size now grows linearly with 1/(1- ρ)
• Need to extend the protocol and analysis to the case of
multiple paths and multiple receivers