MSR, Cambridge, August 5, 2003
Long-Run Behavior of Equation-Based Rate Control& Rate-Latency of Some Input-Queued Switches
2
Outline
Part ILong-run Behavior of
Equation-based Rate ControlPart II
Rate-Latency of Some Input-queued Switches
The talk takes from:
M.V., Ph.D. thesis, July 2003
3
Part ILong-Run Behavior of
Equation-Based Rate Control
4
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
5
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
6
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)
7
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
8
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
9
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
10
When the basic control is conservative assume: the send rate be a stationary ergodic process
In practice: the conditions are true, or almost the result explains overly conservativeness
11
Sketch of the Proof
Palm inversion:
Throughput: May make the control conservative ? !
12
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
13
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:
14
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
15
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
16
Numerical example: Is the basic control conservative ?
SQRT:
PFTK-simplified:
loss-event intervals: i.i.d., generalized exponential density
17
ns-2 and lab: Is TFRC conservative ?
PFTK-simplified
Setup: a RED link shared by TFRC and TCP connections
L=2
48
16
the same qualitative behavior as observed on the previous slide
PFTK-standard
L=8
ns-2 lab
18
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
19
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
20
Check: is TFRC TCP-friendly
TCP-friendly ? - no, not always although, it is mostly conservative !
21
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 RTT than TCP ? Is TFRC conservative ?
22
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
23
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 protocol adjustment
conservativeness against TCP-friendliness TCP-friendliness is difficult to verify conservativeness
amenable to a formal verification not TCP centric
24
Part IIRate-Latency of
Some Input-queued Switches
The work done in part while an intern with Dept. of Mathematics of Networks and Systems, Bell Laboratories, Murray Hill, NJ, Summer 2001
25
Problem
at any time slot, connectivity restricted to permutation matrices
switch scheduling problem: schedule crossbar connectivity with guarantees on the rate and latency
26
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 number:
27
Rate-Latency Service Curve
*
28
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 token types 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)
29
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
30
Numerical Example
worst-case deterministic w.p. 99/100
31
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:
32
Random-distortion Periodic token processes as with Chang et al, but place each token uniformly at random on the
periods
By derandomization:
33
A Numerical Example
Chang et al
Random-distortionperiodic
Random-phase periodic
rate-demand matrices drawn in a random manner
34
Concluding Remarks for Part II
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 may lead to construction of practical algorithms