Benchmarking Working Group M. Konstantynowicz, Ed.
Internet-Draft V. Polak, Ed.
Intended status: Informational Cisco Systems
Expires: September 12, 2019 March 11, 2019
Probabilistic Loss Ratio Search for Packet Throughput (PLRsearch)
draft-vpolak-bmwg-plrsearch-01
Abstract
This document addresses challenges while applying methodologies
described in [RFC2544] to benchmarking software based NFV (Network
Function Virtualization) data planes over an extended period of time,
sometimes referred to as "soak testing". Packet throughput search
approach proposed by this document assumes that system under test is
probabilistic in nature, and not deterministic.
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Table of Contents
1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Relation To RFC2544 . . . . . . . . . . . . . . . . . . . . . 4
3. Terms And Assumptions . . . . . . . . . . . . . . . . . . . . 4
3.1. Device Under Test . . . . . . . . . . . . . . . . . . . . 4
3.2. System Under Test . . . . . . . . . . . . . . . . . . . . 4
3.3. SUT Configuration . . . . . . . . . . . . . . . . . . . . 4
3.4. SUT Setup . . . . . . . . . . . . . . . . . . . . . . . . 4
3.5. Network Traffic . . . . . . . . . . . . . . . . . . . . . 5
3.6. Packet . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.6.1. Packet Offered . . . . . . . . . . . . . . . . . . . 5
3.6.2. Packet Received . . . . . . . . . . . . . . . . . . . 5
3.6.3. Packet Lost . . . . . . . . . . . . . . . . . . . . . 5
3.6.4. Other Packets . . . . . . . . . . . . . . . . . . . . 5
3.6.5. Tasks As Packets . . . . . . . . . . . . . . . . . . 6
3.7. Traffic Profile . . . . . . . . . . . . . . . . . . . . . 6
3.8. Traffic Generator . . . . . . . . . . . . . . . . . . . . 6
3.9. Offered Load . . . . . . . . . . . . . . . . . . . . . . 6
3.10. Trial Measurement . . . . . . . . . . . . . . . . . . . . 7
3.11. Trial Duration . . . . . . . . . . . . . . . . . . . . . 7
3.12. Packet Loss . . . . . . . . . . . . . . . . . . . . . . . 7
3.12.1. Loss Count . . . . . . . . . . . . . . . . . . . . . 7
3.12.2. Loss Rate . . . . . . . . . . . . . . . . . . . . . 7
3.12.3. Loss Ratio . . . . . . . . . . . . . . . . . . . . . 8
3.13. Trial Order Independent System . . . . . . . . . . . . . 8
3.14. Trial Measurement Result Distribution . . . . . . . . . . 8
3.15. Average Loss Ratio . . . . . . . . . . . . . . . . . . . 8
3.16. Duration Independent System . . . . . . . . . . . . . . . 9
3.17. Load Regions . . . . . . . . . . . . . . . . . . . . . . 9
3.17.1. Zero Loss Region . . . . . . . . . . . . . . . . . . 9
3.17.2. Guaranteed Loss Region . . . . . . . . . . . . . . . 9
3.17.3. Non-Deterministic Region . . . . . . . . . . . . . . 9
3.17.4. Normal Region Ordering . . . . . . . . . . . . . . . 10
3.18. Deterministic System . . . . . . . . . . . . . . . . . . 10
3.19. Througphput . . . . . . . . . . . . . . . . . . . . . . . 10
3.20. Deterministic Search . . . . . . . . . . . . . . . . . . 10
3.21. Probabilistic Search . . . . . . . . . . . . . . . . . . 11
3.22. Loss Ratio Function . . . . . . . . . . . . . . . . . . . 11
3.23. Target Loss Ratio . . . . . . . . . . . . . . . . . . . . 11
3.24. Critical Load . . . . . . . . . . . . . . . . . . . . . . 11
3.25. Critical Load Estimate . . . . . . . . . . . . . . . . . 11
3.26. Fitting Function . . . . . . . . . . . . . . . . . . . . 11
3.27. Shape of Fitting Function . . . . . . . . . . . . . . . . 12
3.28. Parameter Space . . . . . . . . . . . . . . . . . . . . . 12
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4. Abstract Algorithm . . . . . . . . . . . . . . . . . . . . . 12
4.1. High level description . . . . . . . . . . . . . . . . . 12
4.2. Main Ideas . . . . . . . . . . . . . . . . . . . . . . . 12
4.3. Probabilistic Notions . . . . . . . . . . . . . . . . . . 13
4.3.1. Loss Count Only . . . . . . . . . . . . . . . . . . . 13
4.3.2. Independent Trials . . . . . . . . . . . . . . . . . 13
4.3.3. Trial Durations . . . . . . . . . . . . . . . . . . . 14
4.3.4. Target Loss Ratio . . . . . . . . . . . . . . . . . . 14
4.3.5. Critical Load . . . . . . . . . . . . . . . . . . . . 14
4.3.6. Load Regions . . . . . . . . . . . . . . . . . . . . 15
4.3.7. Finite Models . . . . . . . . . . . . . . . . . . . . 15
4.4. PLRsearch Building Blocks . . . . . . . . . . . . . . . . 15
4.4.1. Bayesian Inference . . . . . . . . . . . . . . . . . 15
4.4.2. Iterative Search . . . . . . . . . . . . . . . . . . 16
4.4.3. Poisson Distribution . . . . . . . . . . . . . . . . 16
4.4.4. Fitting Functions . . . . . . . . . . . . . . . . . . 16
4.4.5. Measurement Impact . . . . . . . . . . . . . . . . . 17
4.4.6. Fitting Function Coefficients Distribution . . . . . 17
4.4.7. Integration . . . . . . . . . . . . . . . . . . . . . 18
4.4.8. Optimizations . . . . . . . . . . . . . . . . . . . . 18
5. Sample Implementation Specifics: FD.io CSIT . . . . . . . . . 18
5.1. Measurement Delay . . . . . . . . . . . . . . . . . . . . 18
5.2. Rounding Errors and Underflows . . . . . . . . . . . . . 19
5.3. Fitting Functions . . . . . . . . . . . . . . . . . . . . 19
5.3.1. Stretch Function . . . . . . . . . . . . . . . . . . 20
5.3.2. Erf Function . . . . . . . . . . . . . . . . . . . . 20
5.4. Prior Distributions . . . . . . . . . . . . . . . . . . . 21
5.5. Integrator . . . . . . . . . . . . . . . . . . . . . . . 21
6. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 22
7. Security Considerations . . . . . . . . . . . . . . . . . . . 22
8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 22
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 22
9.1. Normative References . . . . . . . . . . . . . . . . . . 22
9.2. Informative References . . . . . . . . . . . . . . . . . 22
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 22
1. Motivation
Network providers are interested in throughput a system can sustain.
[RFC2544] assumes loss ratio is given by a deterministic function of
offered load. But NFV software systems are not deterministic enough.
This makes deterministic algorithms (such as Binary Search per
[RFC2544] and [draft-vpolak-mkonstan-bmwg-mlrsearch] with single
trial) to return results, which when repeated show relatively high
standard deviation, thus making it harder to tell what "the
throughput" actually is.
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We need another algorithm, which takes this indeterminism into
account.
2. Relation To RFC2544
The aim of this document is to become an extension of [RFC2544]
suitable for benchmarking networking setups such as software based
NFV systems.
3. Terms And Assumptions
3.1. Device Under Test
In software networking, "device" denotes a specific piece of software
tasked with packet processing. Such device is surrounded with other
software components (such as operating system kernel). It is not
possible to run devices without also running the other components,
and hardware resources are shared between both.
For purposes of testing, the whole set of hardware and software
components is called "system under test" (SUT). As SUT is the part
of the whole test setup performance of which can be measured by
[RFC2544] methods, this document uses SUT instead of [RFC2544] DUT.
Device under test (DUT) can be re-introduced when analysing test
results using whitebox techniques, but this document sticks to
blackbox testing.
3.2. System Under Test
System under test (SUT) is a part of the whole test setup whose
performance is to be benchmarked. The complete methodology contains
other parts, whose performance is either already established, or not
affecting the benchmarking result.
3.3. SUT Configuration
Usually, system under test allows different configurations, affecting
its performance. The rest of this document assumes a single
configuration has been chosen.
3.4. SUT Setup
Similarly to [RFC2544], it is assumed that the system under test has
been updated with all the packet forwarding information it needs,
before the trial measurements (see below) start.
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3.5. Network Traffic
Network traffic is a type of interaction between system under test
and the rest of the system (traffic generator), used to gather
information about the system under test performance. PLRsearch is
applicable only to areas where network traffic consists of packets.
3.6. Packet
Unit of interaction between traffic generator and the system under
test. Term "packet" is used also as an abstractions of Ethernet
frames.
3.6.1. Packet Offered
Packet can be offered, which means it is sent from traffic generator
to the system under test.
Each offered packet is assumed to become received or lost in a short
time.
3.6.2. Packet Received
Packet can be received, which means the traffic generator verifies it
has been processed. Typically, when it is succesfully sent from the
system under test to traffic generator.
It is assumed that each received packet has been caused by an offered
packet, so the number of packets received cannot be larger than the
number of packets offered.
3.6.3. Packet Lost
Packet can be lost, which means sent but not received in a timely
manner.
It is assumed that each lost packet has been caused by an offered
packet, so the number of packets lost cannot be larger than the
number of packets offered.
Usually, the number of packets lost is computed as the number of
packets offered, minus the number of packets received.
3.6.4. Other Packets
PLRsearch is not considering other packet behaviors known from
networking (duplicated, reordered, greatly delayed), assuming the
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test specification reclassifies those behaviors to fit into the first
three categories.
3.6.5. Tasks As Packets
Ethernet frames are the prime example of packets, but other units are
possible.
For example, a task processing system can fit the description.
Packet offered can stand for task submitted, packet received for task
processed successfully, and packet lost for task aborted (or not
processed successfully for some other reason).
In networking context, such a task can be a route update.
3.7. Traffic Profile
Usually, the performance of the system under test depends on a "type"
of a particular packet (for example size), and "composition" if the
network traffic consists of a mixture of different packet types.
Also, some systems under test contain multiple "ports" packets can be
offered to and received from.
All such qualities together (but not including properties of trial
measurements) are called traffic profile.
Similarly to system under test configuration, this document assumes
only one traffic profile has been chosen for a particular test.
3.8. Traffic Generator
Traffic generator is the part of the whole test setup, distinct from
the system under test, responsible both for offering packets in a
highly predictable manner (so the number of packets offered is
known), and for counting received packets in a precise enough way (to
distinguish lost packets from tolerably delayed packets).
Traffic generator must offer only packets compatible with the traffic
profile, and only count similarly compatible packets as received.
3.9. Offered Load
Offered load is an aggregate rate (measured in packets per second) of
network traffic offered to the system under test, the rate is kept
constant for the duration of trial measurement.
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3.10. Trial Measurement
Trial measurement is a process of stressing (previously setup) system
under test by offering traffic of a particular offered load, for a
particular duration.
After that, the system has a short time to become idle, while the
traffic generator decides how many packets were lost.
After that, another trial measurement (possibly with different
offered load and duration) can be immediately performed. Traffic
generator should ignore received packets caused by packets offered in
previous trial measurements.
3.11. Trial Duration
Duration for which the traffic generator was offering packets at
constant offered load.
In theory, care has to be taken to ensure the offered load and trial
duration predict integer number of packets to offer, and that the
traffic generator really sends appropriate number of packets within
precisely enough timed duration. In practice, such consideration do
not change PLRsearch result in any significant way.
3.12. Packet Loss
Packet loss is any quantity describing a result of trial measurement.
It can be loss count, loss rate or loss ratio. Packet loss is zero
(or non-zero) if either of the three quantities are zero (or non-
zero, respecively).
3.12.1. Loss Count
Number of packets lost (or delayed too much) at a trial measurement
by the system under test as determined by packet generator. Measured
in packets.
3.12.2. Loss Rate
Loss rate is computed as loss count divided by trial duration.
Measured in packets per second.
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3.12.3. Loss Ratio
Loss ratio is computed as loss count divided by number of packets
offered. Measured as a real (in practice rational) number between
zero or one (including).
3.13. Trial Order Independent System
Trial order independent system is a system under test, proven (or
just assumed) to produce trial measurement results that display trial
order independence.
That means when a pair of consequent trial measurements are
performed, the probability to observe a pair of specific results is
the same, as the probability to observe the reversed pair of results
whe performing the reversed pair of consequent measurements.
PLRsearch assumes the system under test is trial order independent.
In practice, most system under test are not entirely trial order
independent, but it is not easy to devise an algorithm taking that
into account.
3.14. Trial Measurement Result Distribution
When a trial order independent system is subjected to repeated trial
measurements of constant offered load and duration, Law of Large
Numbers implies the observed loss count frequencies will converge to
a specific probability distribution over possible loss counts.
This probability distribution is called trial measurement result
distribution, and it depends on all properties fixed when defining
it. That includes the system under test, its chosen configuration,
the chosen traffic profile, the offered load and the trial duration.
As the system is trial order independent, trial measurement result
distribution does not depend on results of few initial trial
measurements, of any offered load or (finite) duration.
3.15. Average Loss Ratio
Probability distribution over some (finite) set of states enables
computation of probability-weighted average of any quantity evaluated
on the states (called the expected value of the quantity).
Average loss ratio is simply the expected value of loss ratio for a
given trial measurement result distribution.
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3.16. Duration Independent System
Duration independent system is a trial order independent system,
whose trial measurement result distribution is proven (or just
assumed) to display practical independence from trial duration. See
definition of trial duration for discussion on practical versus
theoretical.
The only requirement is for average loss ratio to be independent of
trial duration.
In theory, that would necessitate each trial measurement result
distribution to be a binomial distribution. In practice, more
distributions are allowed.
PLRsearch assumes the system under test is duration independent, at
least for trial durations typically chosen for trial measurements
initiated by PLRsearch.
3.17. Load Regions
For a duration independent system, trial measurement result
distribution depends only on offered load.
It is convenient to name some areas of offered load space by possible
trial results.
3.17.1. Zero Loss Region
A particular offered load value is said to belong to zero loss
region, if the probability of seeing non-zero loss trial measurement
result is exactly zero, or at least practically indistinguishable
from zero.
3.17.2. Guaranteed Loss Region
A particular offered load value is said to belong to guaranteed loss
region, if the probability of seeing zero loss trial measurement
result (for non-negligible count of packets offered) is exactly zero,
or at least practically indistinguishable from zero.
3.17.3. Non-Deterministic Region
A particular offered load value is said to belong to non-
deterministic region, if the probability of seeing zero loss trial
measurement result (for non-negligible count of packets offered)
practically distinguishable from both zero and one.
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3.17.4. Normal Region Ordering
Although theoretically the three regions can be arbitrary sets, this
document assumes they are intervals, where zero loss region contains
values smaller than non-deterministic region, which in turn contains
values smaller than guaranteed loss region.
3.18. Deterministic System
A hypothetical duration independent system with normal region
ordering, whose non-deterministic region is extremely narrow; only
present due to "practical distinguishibility" and cases when the
expected number of packets offered is not and integer.
A duration independent system which is not deterministic is called
non- deterministic system.
3.19. Througphput
Throughput is the highest offered load provably causing zero packet
loss for trial measurements of duration at least 60 seconds.
For duration independent systems with normal region ordering, the
throughput is the highest value within the zero loss region.
3.20. Deterministic Search
Any algorithm that assumes each measurement is a proof of the offered
load belonging to zero loss region (or not) is called deterministic
search.
This definition includes algorithms based on "composite measurements"
which perform multiple trial measurements, somehow re-classifying
results pointing at non-deterministic region.
Binary Search is an example of deterministic search.
Single run of a deterministic search launched against a deterministic
system is guaranteed to find the throughput with any prescribed
precision (not better than non-deterministic region width).
Multiple runs of a deterministic search launched against a non-
deterministic system can return varied results within non-
deterministic region. The exact distribution of deterministic search
results depends on the algorithm used.
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3.21. Probabilistic Search
Any algorithm which performs probabilistic computations based on
observed results of trial measurements, and which does not assume
that non-deterministic region is practically absent is called
probabilistic search.
A probabilistic search algorithm, which would assume that non-
deterministic region is practically absent, does not really need to
perform probabilistic computations, so it would become a
deterministic search.
While probabilistic search for estimating throughput is possible, it
would need a careful model for boundary between zero loss region and
non-deterministic region, and it would need a lot of measurements of
almost surely zero loss to reach good precision.
3.22. Loss Ratio Function
For any duration independent system, the average loss ratio depends
only on offered load (for a particular test setup).
Loss ratio function is the name used for the function mapping offered
load to average loss ratio.
This function is initially unknown.
3.23. Target Loss Ratio
Input parameter of PLRsearch. The average loss ratio the output of
PLRsearch aims to achieve.
3.24. Critical Load
Aggregate rate of network traffic, which would lead to average loss
ratio exactly matching target loss ratio (when used as the offered
load for infinite many trial measurement).
3.25. Critical Load Estimate
Any quantitative description of the possible critical load PLRsearch
is able to give after observing finite amount of trial measurements.
3.26. Fitting Function
Any function PLRsearch uses internally instead of the unknown loss
ratio function. Typically chosen from small set of formulas (shapes)
with few parameters to tweak.
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3.27. Shape of Fitting Function
Any formula with few undetermined parameters.
3.28. Parameter Space
A subset of Real Coordinate Space. A point of parameter space is a
vector of real numbers. Fitting function is defined by shape (a
formula with parameters) and point of parameter space (specifying
values for the parameters).
4. Abstract Algorithm
4.1. High level description
PLRsearch accepts some input arguments, then iteratively performs
trial measurements at varying offered loads (and durations), and
returns some estimates of critical load.
PLRsearch input arguments form three groups.
First group has a single argument: measurer. This is a callback
(function) accepting offered load and duration, and returning the
measured loss count.
Second group consists of load related arguments required for measurer
to work correctly, typically minimal and maximal load to offer.
Also, target loss ratio (if not hardcoded) is a required argument.
Third group consists of time related arguments. Typically the
duration for the first trial measurement, duration increment per
subsequent trial measurement and total time for search. Some
PLRsearch implementation may use estimation accuracy parameters as an
exit condition instead of total search time.
The returned quantities should describe the final (or best) estimate
of critical load. Implementers can chose any description that suits
their users, typically it is average and standard deviation, or lower
and upper boundary.
4.2. Main Ideas
The search tries to perform measurements at offered load close to the
critical load, because measurement results at offered loads far from
the critical load give less information on precise location of the
critical load. As virtually every trial measurement result alters
the estimate of the critical load, offered loads vary as they
approach the critical load.
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PLRsearch uses Bayesian Inference, computed using numerical
integration, which takes long time to get reliable enough results.
Therefore it takes some time before the most recent measurement
result starts affecting subsequent offered loads and critical rate
estimates.
During the search, PLRsearch spawns few processes that perform
numerical computations, the main process is calling the measurer to
perform trial measurements, without any significant delays between
them. The durations of the trial measurements are increasing
linearly, as higher number of trial measurement results take longer
to process.
4.3. Probabilistic Notions
Before internals of PLRsearch are described, we need to define
notions valid for situations when loss ratio is not entirely
determined by offered load.
Some of the notions already incorporate assumptions the PLRsearch
algorithm applies.
4.3.1. Loss Count Only
It is assumed that the traffic generator detects duplicate packets on
receive, and reports this as an error.
No latency (or other information) is taken into account.
4.3.2. Independent Trials
PLRsearch still assumes the system under test can be subjected to
trial measurements. The loss count is no longer determined
precisely, but it is assumed that for every system under test, its
configuration, traffic type and trial duration, there is a
probability distribution over possible loss counts.
This implies trial measurements are probabilistic, but the
distribution is independent of possible previous trial measurements.
Independence from previous measurements is not guaranteed in the real
world. The previous measurements may improve performance (via long-
term warmup effects), or decrease performance (due to long-term
resource leaks).
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4.3.3. Trial Durations
[RFC2544] motivates the usage of at least 60 second duration by the
idea of the system under test slowly running out of resources (such
as memory buffers).
Practical results when measuring NFV software systems show that
relative change of trial duration has negligible effects on average
loss ratio, compared to relative change in offered load.
While the standard deviation of loss ratio usually shows some effects
of trial duration, they are hard to model; so further assumtions in
PLRsearch will make it insensitive to trial duration.
4.3.4. Target Loss Ratio
Loss ratio function could be used to generalize throughput as the
biggest offered load which still leads to zero average loss ratio.
Unfortunately, most realistic loss ratio functions always predict
non- zero (even if negligible) average loss ratio.
On the other hand, users do not really require the average loss ratio
to be an exact zero. Most users are satisfied when the average loss
ratio is small enough.
One of PLRsearch inputs is called target loss ratio. It is the loss
ratio users would accept as negligible.
(TODO: Link to why we think 1e-7 is acceptable loss ratio.)
4.3.5. Critical Load
Critical load (sometimes called critical rate) is the offered load
which leads to average loss ratio to become exactly equal to the
target loss ratio.
In principle, there could be such loss ratio functions which allow
more than one offered load to achieve target loss ratio. To avoid
that, PLRsearch assumes only increasing loss ratio functions are
possible.
Similarly, some loss ratio functions may never return the target loss
ratio. PLRsearch assumes loss ratio function is continuous, that the
average loss ratio approaches zero as offered load approaches zero,
and that the average loss ratio approaches one as offered load
approaches infinity.
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Under these assumptions, each loss ratio function has unique critical
load. PLRsearch attempts to locate the critical load.
4.3.6. Load Regions
Critical region is the interval of offered load close to critical
load, where single measurement is not likely to distinguish whether
the critical load is higher or lower than the current offered load.
In typical case of small target loss ratio, rates below critical
region form "zero loss region", and rates above form "high loss
region".
4.3.7. Finite Models
Of course, finite amount of trial measurements, each of finite
duration does not give enough information to pinpoint the critical
load exactly. Therefore the output of PLRsearch is just an estimate
with some precision.
Aside of the usual substitution of infinitely precise real numbers by
finitely precise floating point numbers, there are two other
instances within PLRsearch where an objects of high information are
replaced by objects of low information.
One is the probability distribution of loss count, which is replaced
by average loss ratio. The other is the loss ratio function, which
is replaced by a few parameters, to be described later.
4.4. PLRsearch Building Blocks
Here we define notions used by PLRsearch which are not applicable to
other search methods, nor probabilistic systems under test, in
general.
4.4.1. Bayesian Inference
Having reduced the model space significantly, the task of estimating
the critical load becomes simple enough so that Bayesian inference
can be used (instead of neural networks, or other Artifical
Intelligence methods).
In this case, the few parameters describing the loss ration function
become the model space. Given a prior over the model space, and
trial duration results, a posterior distribution can be computed,
together with quantities describing the critical load estimate.
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4.4.2. Iterative Search
The idea PLRsearch is to iterate trial measurements, using Bayesian
inference to compute both the current estimate of the critical load
and the next offered load to measure at.
The required numerical computations are done in parallel with the
trial measurements.
This means the result of measurement "n" comes as an (additional)
input to the computation running in parallel with measurement "n+1",
and the outputs of the computation are used for determining the
offered load for measurement "n+2".
Other schemes are possible, aimed to increase the number of
measurements (by decreasing their duration), which would have even
higher number of measurements run before a result of a measurement
affects offered load.
4.4.3. Poisson Distribution
For given offered load, number of packets lost during trial
measurement is assumed to come from Poisson distribution, and the
(unknown) Poisson parameter is expressed as average loss ratio.
Side note: Binomial Distribution is a better fit compared to Poisson
distribution (acknowledging that the number of packets lost cannot be
higher than the number of packets offered), but the difference tends
to be relevant only in high loss region. Using Poisson distribution
lowers the impact of measurements in high loss region, thus helping
the algorithm to focus on critical region better.
4.4.4. Fitting Functions
There are great many increasing functions (as candidates for the loss
ratio function).
To make the space of possible functions more tractable, some other
simplifying assumptions are needed. As the algorithm will be
examining (also) loads very close to the critical load, linear
approximation to the loss rate function around the critical load is
important. But as the search algorithm needs to evaluate the
function also far away from the critical region, the approximate
function has to be reasonably behaved for every positive offered
load, specifically it cannot predict non- positive packet loss ratio.
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Within this document, "fitting function" is the name for such a
reasonably behaved function, which approximates the loss ratio
function well in the critical region.
4.4.5. Measurement Impact
Results from trials far from the critical region are likely to affect
the critical rate estimate negatively, as the fitting function does
not need to be a good approximation there. This is true mainly for
high loss region, as in zero loss region even badly behaved fitting
function predicts loss count to be "almost zero", so seeing a
measurement confirming the loss has been zero indeed has small
impact.
Discarding some results, or "suppressing" their impact with ad-hoc
methods (other than using Poisson distribution instead of binomial)
is not used, as such methods tend to make the overall search
unstable. We rely on most of measurements being done (eventually)
within the critical region, and overweighting far-off measurements
(eventually) for well- behaved fitting functions.
Speaking about new trials, each next trial will be done at offered
load equal to the current average of the critical load. Alternative
methods for selecting offered load might be used, in an attempt to
speed up convergence. For example by employing the aforementioned
unstable ad-hoc methods.
4.4.6. Fitting Function Coefficients Distribution
To accomodate systems with different behaviours, the fitting function
is expected to have few numeric parameters affecting its shape
(mainly affecting the linear approximation in the critical region).
The general search algorithm can use whatever increasing fitting
function, some specific functions can described later.
It is up to implementer to chose a fitting function and prior
distribution of its parameters. The rest of this document assumes
each parameter is independently and uniformly distributed over a
common interval. Implementers are to add non-linear transformations
into their fitting functions if their prior is different.
Exit condition for the search is either the standard deviation of the
critical load estimate becoming small enough (or similar), or overal
search time becoming long enough.
The algorithm should report both average and standard deviation for
its critical load posterior. If the reported averages follow a trend
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(without reaching equilibrium), average and standard deviation should
refer to the equilibrium estimates based on the trend, not to
immediate posterior values.
4.4.7. Integration
The posterior distributions for fitting function parameters will not
be integrable in general.
The search algorithm utilises the fact that trial measurement takes
some time, so this time can be used for numeric integration (using
suitable method, such as Monte Carlo) to achieve sufficient
precision.
4.4.8. Optimizations
After enough trials, the posterior distribution will be concentrated
in a narrow area of the parameter space. The integration method
should take advantage of that.
Even in the concentrated area, the likelihood can be quite small, so
the integration algorithm should avoid underflow errors by some
means, for example by tracking the logarithm of the likelihood.
5. Sample Implementation Specifics: FD.io CSIT
The search receives min_rate and max_rate values, to avoid
measurements at offered loads not supporeted by the traffic
generator.
The implemented tests cases use bidirectional traffic. The algorithm
stores each rate as bidirectional rate (internally, the algorithm is
agnostic to flows and directions, it only cares about overall counts
of packets sent and packets lost), but debug output from traffic
generator lists unidirectional values.
5.1. Measurement Delay
In a sample implemenation in FD.io CSIT project, there is roughly 0.5
second delay between trials due to restrictons imposed by packet
traffic generator in use (T-Rex).
As measurements results come in, posterior distribution computation
takes more time (per sample), although there is a considerable
constant part (mostly for inverting the fitting functions).
Also, the integrator needs a fair amount of samples to reach the
region the posterior distribution is concentrated at.
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And of course, speed of the integrator depends on computing power of
the CPU the algorithm is able to use.
All those timing related effects are addressed by arithmetically
increasing trial durations with configurable coefficients (currently
5.1 seconds for the first trial, each subsequent trial being 0.1
second longer).
5.2. Rounding Errors and Underflows
In order to avoid them, the current implementation tracks natural
logarithm (instead of the original quantity) for any quantity which
is never negative. Logarithm of zero is minus infinity (not
supported by Python), so special value "None" is used instead.
Specific functions for frequent operations (such as "logarithm of sum
of exponentials") are defined to handle None correctly.
5.3. Fitting Functions
Current implementation uses two fitting functions. In general, their
estimates for critical rate differ, which adds a simple source of
systematic error, on top of randomness error reported by integrator.
Otherwise the reported stdev of critical rate estimate is
unrealistically low.
Both functions are not only increasing, but also convex (meaning the
rate of increase is also increasing).
As Primitive Function to any positive function is an increasing
function, and Primitive Function to any increasing function is convex
function; both fitting functions were constructed as double Primitive
Function to a positive function (even though the intermediate
increasing function is easier to describe).
As not any function is integrable, some more realistic functions
(especially with respect to behavior at very small offered loads) are
not easily available.
Both fitting functions have a "central point" and a "spread", varied
by simply shifting and scaling (in x-axis, the offered load
direction) the function to be doubly integrated. Scaling in y-axis
(the loss rate direction) is fixed by the requirement of transfer
rate staying nearly constant in very high offered loads.
In both fitting functions (as they are a double Primitive Function to
a symmetric function), the "central point" turns out to be equal to
the aforementioned limiting transfer rate, so the fitting function
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parameter is named "mrr", the same quantity our Maximum Receive Rate
tests are designed to measure.
Both fitting functions return logarithm of loss rate, to avoid
rounding errors and underflows. Parameters and offered load are not
given as logarithms, as they are not expected to be extreme, and the
formulas are simpler that way.
Both fitting functions have several mathematically equivalent
formulas, each can lead to an overflow or underflow in different
places. Overflows can be eliminated by using different exact
formulas for different argument ranges. Underflows can be avoided by
using approximate formulas in affected argument ranges, such ranges
have their own formulas to compute. At the end, both fitting
function implementations contain multiple "if" branches,
discontinuities are a possibility at range boundaries.
Offered load for next trial measurement is the average of critical
rate estimate. During each measurement, two estimates are computed,
even though only one (in alternating order) is used for next offered
load.
5.3.1. Stretch Function
The original function (before applying logarithm) is Primitive
Function to Logistic Function. The name "stretch" is used for
related a function in context of neural networks with sigmoid
activation function.
Formula for stretch function: loss rate (r) computed from offered
load (b), mrr parameter (m) and spread parameter (a):
r = a (Log(E^(b/a) + E^(m/a)) - Log(1 + E^(m/a)))
5.3.2. Erf Function
The original function is double Primitive Function to Gaussian
Function. The name "erf" comes from error function, the first
primitive to Gaussian.
Formula for erf function: loss rate (r) computed from offered load
(b), mrr parameter (m) and spread parameter (a):
r = (b + (a (E^(-((b - m)^2/a^2)) - E^(-(m^2/a^2))))/Sqrt(Pi) + (b -
m) Erf((b - m)/a) - m Erf(m/a))/2
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5.4. Prior Distributions
The numeric integrator expects all the parameters to be distributed
(independently and) uniformly on an interval (-1, 1).
As both "mrr" and "spread" parameters are positive and not not
dimensionless, a transformation is needed. Dimentionality is
inherited from max_rate value.
The "mrr" parameter follows a Lomax Distribution with alpha equal to
one, but shifted so that mrr is always greater than 1 packet per
second.
The "stretch" parameter is generated simply as the "mrr" value raised
to a random power between zero and one; thus it follows a Reciprocal
Distribution.
5.5. Integrator
After few measurements, the posterior distribution of fitting
function arguments gets quite concentrated into a small area. The
integrator is using Monte Carlo with Importance Sampling where the
biased distribution is Bivariate Gaussian distribution, with
deliberately larger variance. If the generated sample falls outside
(-1, 1) interval, another sample is generated.
The the center and the covariance matrix for the biased distribution
is based on the first and second moments of samples seen so far
(within the computation), with the following additional features
designed to avoid hyper-focused distributions.
Each computation starts with the biased distribution inherited from
the previous computation (zero point and unit covariance matrix is
used in the first computation), but the overal weight of the data is
set to the weight of the first sample of the computation. Also, the
center is set to the first sample point. When additional samples
come, their weight (including the importance correction) is compared
to the weight of data seen so far (within the computation). If the
new sample is more than one e-fold more impactful, both weight values
(for data so far and for the new sample) are set to (geometric)
average if the two weights. Finally, the actual sample generator
uses covariance matrix scaled up by a configurable factor (8.0 by
default).
This combination showed the best behavior, as the integrator usually
follows two phases. First phase (where inherited biased distribution
or single big sasmples are dominating) is mainly important for
locating the new area the posterior distribution is concentrated at.
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The second phase (dominated by whole sample population) is actually
relevant for the critical rate estimation.
6. IANA Considerations
..
7. Security Considerations
..
8. Acknowledgements
..
9. References
9.1. Normative References
[RFC2544] Bradner, S. and J. McQuaid, "Benchmarking Methodology for
Network Interconnect Devices", RFC 2544,
DOI 10.17487/RFC2544, March 1999,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
9.2. Informative References
[draft-vpolak-mkonstan-bmwg-mlrsearch]
"Multiple Loss Ratio Search for Packet Throughput
(MLRsearch)", November 2018, .
Authors' Addresses
Maciek Konstantynowicz (editor)
Cisco Systems
Email: mkonstan@cisco.com
Vratko Polak (editor)
Cisco Systems
Email: vrpolak@cisco.com
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