EPFL, Lausanne, July 17, 2003
Ph.D. advisor: Prof. Jean-Yves Le Boudec
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Outline
Part IEquation-based Rate
ControlPart II
Expedited ForwardingPart 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 IEquation-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)
given: a TCP throughput formulap = loss-event rate
p estimated on-line
at an instant t, send rate set as
Problem: Is equation-based rate control TCP-friendly ?
Equation-based rate control: basic control principles
(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 instantTn = 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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Ln 3n 2n1n
Equation-based rate control: the basic control law
...
nT1nT 3nT LnT
Additional control laws ignored in this slide
2nT ...... ...
send rate
1nT
nT = instant of a loss-event
= a loss-event intervaln
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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)
the “overshoot” bounded by a function of p and
1/f(1/x) is assumed to be convex, thus, it is above its tangents take the tangent at 1/p
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SQRT
PFTK-standard
PFTK-simplified
convex
convex
almost convex
When 1/f(1/x) is convex
b = number of packets acknowledged by an ack
SQRT:
PFTK-standard:
PFTK-simplified:
Check some typical TCP throughput formulae:
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On Covariance of the Estimator and the Next Loss-event Interval
Recall (C1)
It holds:
if is a bad predictor, that leads to conservativeness
if the loss-event intervals are independent, then (C1) holds with equality
= a “measure” how well predicts
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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 ?
PFTK-simplified
Setup: a RED link shared by TFRC and TCP connections
L=2
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The same qualitative behavior as observed on the previous slide
PFTK-standard
L=8
ns-2 lab
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First check: is negative or slightly positive
Internet, LAN to LAN, EPFL sender
Internet, LAN to a cable-modem at EPFL
Lab
We turn to check: is TFRC TCP-friendly
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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:
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
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 ?
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Check the factors separately !
when a few connections compete, none of the conditions hold
Does TCP conform to its formula ?
Does TFRC see no better loss-event rate
than TCP ?
Does TFRC see no better loss-event rate
than TCP ?
No No No
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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 IIExpedited 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
Note that an EF flow is allowed to be any stochastic process as long as it obeys to the given set of the assumptions
A1, A2, …, AI stochastically independent
Ai is constrained with an arrival curve
Ai is such that
There exists a finite s.t.
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One Result: a Bound on Probability of the Buffer Overflow
Then, for Q(t) (= number of bits in the node at an instant t),
Assume: all I fix:
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A Method to Derive BoundsStep 1: containment into a union of the “arrival overflow events”
(by def. of a service curve and )
Step 2: use the union probability boundStep 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
Known result: for , a bound on the e2e delay-jitter is
Given:
( = set of EF flows that traverse the node n)
(= maximum number of hops an EF flow can traverse)
Problem: obtain a bound on the e2e delay-jitter
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A dimensioning rule
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)
Given, in addition:
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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:
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 nbn(t)= rn(t-e)+
Combine the last two to retrieve the asserted d’
must be > 0, for the bound to be < 1
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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 IIIInput-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)
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
2) Schedule: schedule the permutation matrices with objective to offer a ”smooth” schedule
Consider: decomposition-based schedulers
a permutation matrix
a positive real:
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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
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 ?
Known result (Chang et al, 2000)
(= subset of permutation matrices
that schedule input/output port pair ij)
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Random Permutation a rate k is an integer multiple of 1/L L = frame-length
compare with the worst-case deterministic latency
Scheduler: schedule the permutation matrices in a frame,
according to a random permutation of the tokens repeat the frame over time
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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
compare with Chang et al
By derandomization:
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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-distortionperiodic
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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