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Ph.D. advisor:
Prof. Jean-Yves Le Boudec
EPFL, Lausanne, July 17, 2003
Outline
Part I
Equation-based Rate Control
Part II
Expedited Forwarding
Part III
Input-queued Switch
In the thesis, but not in the slides:
 increase-decrease controls (Chapter 3)
 fairness of bandwidth sharing
 analysis and synthesis
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Part I
Equation-based Rate Control
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Problem
 New transmission control protocols proposed
for some packet senders in the Internet
 a design goal is to offer a better transport
for streaming sources, than offered by TCP
 In today’s Internet, TCP is the most used
 Axiom: transport protocols other than TCP,
should be TCP-friendly—another design goal
TCP-friendliness: Throughput <= TCP throughput
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Problem (cont’d)
 Equation-based rate control
 a new set of transmission control protocols
 An instance: TFRC, IETF proposed standard (Jan 2003)
 Past studies of equation-based rate controls
mostly restricted to simulations
 lack of a formal study
 understanding needed before a wide-spread deployment
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Problem (cont’d)
Equation-based rate control: basic control principles
 given: a TCP throughput formula
p = loss-event rate
 p estimated on-line
 at an instant t, send rate set as
Problem: Is equation-based rate control TCP-friendly ?
(TCP throughput formula depends also on other factors,
e.g. an event-average of the round-trip time)
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Where is the Problem ?
 The estimators are updated at some special points in time
 the send rate updated at the special instants
(sampling bias)
t = an arbitrary instant
Tn = the nth update of the estimators, a special instant
 x->f(x) is non-linear, the estimators are non-fixed values
(non-linearity)
 Other factors
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send rate
Equation-based rate control:
the basic control law
n L
... Tn  L
...
... Tn  3
n 3
n  2
n 1
Tn  2 Tn 1
Tn
...
Tn  1
Tn = instant of a loss-event
 n = a loss-event interval
 Additional control laws ignored in this slide
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We first check: is the control conservative
We say a control is conservative iff
p = loss-event rate as seen by this protocol
 Conservativeness is not the same as TCP-friendliness
 We come back to TCP-friendliness later
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When the basic control is conservative
 Assume: the send rate is a stationary ergodic process
In practice:
 the conditions are true, or almost
 the result explains overly conservativeness
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Sketch of the Proof
Palm inversion:
Throughput:
May make the control
conservative ? !
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Sketch of the Proof (Cont’d)
 1/f(1/x) is assumed to be convex, thus, it is above its tangents
 take the tangent at 1/p
 the “overshoot” bounded by a function of p and
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When 1/f(1/x) is convex
Check some typical TCP throughput
formulae:
SQRT:
PFTK-standard
almost convex
PFTK-standard:
PFTK-simplified
convex
PFTK-simplified:
SQRT
convex
b = number of packets acknowledged by an ack
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On Covariance of the Estimator and
the Next Loss-event Interval
 Recall (C1)
= a “measure” how well
predicts
It holds:
 if
is a bad predictor, that leads to conservativeness
 if the loss-event intervals are independent, then (C1) holds
with equality
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Claim
 Assume: the estimator and the next sample of the loss-event
interval are negatively or slightly positive correlated
Consider a region where the loss-event interval estimator takes
its values
 the more convex 1/f(1/x) is in this region
=> the more conservative
 the more variable the is
=> the more conservative
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Numerical example:
Is the basic control conservative ?
SQRT:
PFTK-simplified:
 loss-event intervals: i.i.d., generalized exponential density
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ns-2 and lab: Is TFRC conservative ?
ns-2
lab
PFTK-simplified
PFTK-standard
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8
4
L=8
L=2
Setup: a RED link shared by TFRC and TCP connections
 The same qualitative behavior as observed on the previous slide
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We turn to check: is TFRC TCP-friendly
First check: is
Internet, LAN to LAN,
EPFL sender
negative or slightly positive
Internet, LAN to
a cable-modem at EPFL
Lab
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Check is TFRC conservative
PFTK-standard
L=8
 setup: equal number of TCP and
TFRC connections (1,2,4,6,8,10),
for the experiments (1,2,3,4,5,6)
mostly conservative
slight deviation, anyway
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Check: is TFRC TCP-friendly
TCP-friendly ? - no, not always
although, it is mostly conservative !
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Conservativeness does not
imply TCP-friendliness !
Breakdown TCP-friendliness into:
 Does TCP conform to its formula ?
 Does TFRC see no better loss-event rate than TCP ?
 Does TFRC see no better average round-trip
times than TCP ?
 Is TFRC conservative ?
 If all conditions hold => TCP-friendliness
 If the control is non-TCP-friendly,
then at least one condition must not hold
 The breakdown is more than a set of sufficient conditions
- it tells us about the strength of individual factors
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Check the factors separately !
Does TCP conform
to its formula ?
 No
Does TFRC see no
better loss-event rate
than TCP ?
 No
Does TFRC see no
better loss-event rate
than TCP ?
 No
 when a few connections compete, none of the conditions hold
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Concluding Remarks for Part I
 under the conditions we identified,
equation-based rate control is conservative
 when loss-event rate is large, it is overly conservative
 different TCP throughput formulae may yield different bias
 breakdown TCP-friendliness problem into sub-problems,
check the sub-problems separately !
 the breakdown would reveal a cause of an observed
non-TCP-friendliness
 an unknown cause may lead a protocol designer to an
improper adjustment of a protocol
 TCP-friendliness is difficult to verify
 we propose the concept of conservativeness
 conservativeness is amenable to a formal verification
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Part II
Expedited Forwarding
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Problem
 Expedited Forwarding (EF): a service of differentiated services Internet
- network of nodes
- each node offers service to the aggregate EF traffic, not per-EF-flow
 EF per-hop-behavior: PSRG, Packet Scale Rate Guarantee with a rate r
and a latency e
 EF flows: individually shaped at the network ingress
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Problem
 Obtain performance bounds to dimension EF networks
Assumption: EF flows stochastically independent at ingress
Step 1: Find probabilistic bounds on backlog, delay, and loss
for a single PSRG node, with stochastically independent EF
arrival processes, each constrained with an arrival curve
Step 2: Apply the results to a network of PSRG nodes
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Packet Scale Rate Guarantee
with a rate r and a latency e
Relations among different node abstractions:
 a property that holds for one of the node abstractions,
holds for a PSRG node
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Assumptions
 A1, A2, …, AI stochastically independent
 Ai is constrained with an arrival curve
 Ai is such that
 There exists a finite
 Note that an EF flow is allowed to be any stochastic process
as long as it obeys to the given set of the assumptions
s.t.
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One Result: a Bound
on Probability of the Buffer Overflow
 Assume:
 fix:
all I
Then, for Q(t) (= number of bits in the node at an instant t),
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A Method to Derive Bounds
Step 1: containment into a union of the “arrival overflow events”
(by def. of a service
curve and )
Step 2: use the union probability bound
Step 3: apply Hoeffding’s inequalities
key observation:
is a sum of I random variables
- independent, with bounded support, bounded means
- fits the assumptions by Hoeffding (1963)
Note: realizing that we can apply Hoeffding’s inequalities,
enabled us to obtain new performance bounds
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Numerical example
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Our Other Bounds that apply to a PSRG node
 Bounds on probability of the buffer overflow
 for identical and non-identical arrival curve constraints
 in terms of some global knowledge about the arrival
curves (for leaky-bucket shapers)
 Bounds on probability of the buffer overflow
as seen by bit and packet arrivals
 Bounds on complementary cdf of a packet delay
 Bounds on the arrival bit loss rate
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Dimensioning an EF network
 Given:
(= maximum number of hops an
EF flow can traverse)
(
= set of EF flows that traverse the node n)
 Problem: obtain a bound on the e2e delay-jitter
 Known result: for
, a bound on the e2e delay-jitter is
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A dimensioning rule
 Given, in addition:
Dimensioning rule: fix the buffer lengths such that qn=d’rn, all n
 The e2e delay-jitter is bounded by h(d’+e)
(delay-from-backlog property of PSRG nodes)
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Sketch of the Proof
 Majorize by the fresh traffic:
bits of an EF flow i seen at the node n in (s,t]
bits of an EF flow i seen at the network ingress
(fresh traffic)
= (h-1)(d+e), a bound on the delay-jitter to any node in the network
 Use one of our single-node bounds:
must be > 0, for the bound to be < 1
horizontal deviation between an arrival curve of the
aggregate EF arrival process to a node n,
an(t)=rn(at+b+a(h-1)(d+e))
and a service curve offered by the node n
bn(t)= rn(t-e)+
Combine the last two to retrieve the asserted d’
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Numerical Example
 Example networks
rn =
all n
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Concluding Remarks for Part II
 We obtained probabilistic bounds on
performance of a PSRG (r,e) node
 Our bounds hold in probability
 the bounds would be more optimistic,
than worst-case deterministic bounds
 Our bounds are exact
 Network of nodes: we showed probabilistic bounds for
a network of PSRG nodes
 The bounds are still with a bound on the EF load,
likewise to some known worst-case deterministic bounds
 With an additional global parameter, we obtained a bound
on the e2e delay-jitter that is more optimistic than a
known worst-case deterministic bound
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Part III
Input-queued Switch
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Problem
 at any time slot, connectivity restricted to permutation matrices
Switch scheduling problem: schedule crossbar connectivity
with guarantees on the rate and latency
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Problem (Cont’d)
Consider: decomposition-based schedulers
Given: M, a I x I doubly sub-stochastic rate-demand matrix
1) Decomposition: decompose M=[mij] into a sequence of
permutation matrices, s.t. for an input/output port pair ij,
intensity of the offered slots is at least mij
– Birkoff/von Neumann: a doubly stochastic matrix M
can be decomposed as
a permutation matrix
a positive real:
2) Schedule: schedule the permutation matrices
with objective to offer a ”smooth” schedule
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Rate-Latency Service Curve
*
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Scheduling Permutation Matrices
 unique token assigned to a permutation matrix
 scheduler by Chang et al can be seen as
Known result (Chang et al, 2000)
(= subset of permutation matrices
that schedule input/output port pair ij)
 superposition of point processes on a line marked by the tokens
 schedule permutation matrices as their tokens appear
Scheduler by Chang et al is for deterministic periodic individual token processes
Problem: can we have schedules with better bounds on the latency ?
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Random Permutation
 a rate k is an integer multiple of 1/L
 L = frame-length
Scheduler:
 schedule the permutation matrices in a frame,
according to a random permutation of the tokens
 repeat the frame over time
 compare with the worst-case deterministic latency
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Numerical Example
worst-case deterministic
w.p. 0.99
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Random-phase Periodic
 token processes as with Chang et al, but for a token process chose a
random phase, independently of other token processes
By derandomization:
 compare with Chang et al
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Random-distortion Periodic
 token processes as with Chang et al, but place each token uniformly
at random on the periods
By derandomization:
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A Numerical Example
Chang et al
Random-distortion
periodic
Random-phase
periodic
 rate-demand matrices drawn in a random manner
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Concluding Remarks for Part III
 We showed new bounds on the latency for a
decomposition-based input-queued switch scheduling
 The bounds are in many cases better than
previously-known bound by Chang et al
 To our knowledge, the approach is novel
 conjunction of the superposition of the token processes
and probabilistic techniques may lead to new bounds
 construction of practical algorithms
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