PLR Search #

Motivation for PLRsearch #

Network providers are interested in throughput a system can sustain.

RFC 2544¹ 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 search² per RFC 2544 and 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.

We need another algorithm, which takes this indeterminism into account.

Generic Algorithm #

Detailed description of the PLRsearch algorithm is included in the IETF draft Probabilistic Loss Ratio Search for Packet Throughput³ that is in the process of being standardized in the IETF Benchmarking Methodology Working Group (BMWG).

Terms #

The rest of this page assumes the reader is familiar with the following terms defined in the IETF draft:

• Trial Order Independent System

• Duration Independent System

• Target Loss Ratio

• Critical Load

• Offered Load regions

  • Zero Loss Region
  • Non-Deterministic Region
  • Guaranteed Loss Region

• Fitting Function

  • Stretch Function
  • Erf Function

• Bayesian Inference

  • Prior distribution
  • Posterior Distribution

• Numeric Integration

  • Monte Carlo
  • Importance Sampling

FD.io CSIT Implementation Specifics #

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 aggregate counts of packets sent and packets lost), but debug output from traffic generator lists unidirectional values.

In CSIT, tests that employ PLRsearch are identified as SOAK tests, the search time is set to 30 minuts.

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.

And of course, the 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).

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.

Fitting Functions #

Current implementation uses two fitting functions, called “stretch” and “erf”. 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).

Both fitting functions have several mathematically equivalent formulas, each can lead to an arithmetic overflow or underflow in different sub-terms. 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.

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 dimensionless, a transformation is needed. Dimentionality is inherited from max_rate value.

The “mrr” parameter follows a Lomax distribution⁴ 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 distribution⁵.

Integrator #

After few measurements, the posterior distribution of fitting function arguments gets quite concentrated into a small area. The integrator is using Monte Carlo⁶ with importance sampling⁷ where the biased distribution is bivariate Gaussian⁸ distribution, with deliberately larger variance. If the generated sample falls outside (-1, 1) interval, another sample is generated.

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). The center is used directly, covariance matrix is scaled up by a heurictic constant (8.0 by default). The following additional features are applied 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 sum of the weights of data seen so far (within the iteration). 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 of the two weights.

This combination showed the best behavior, as the integrator usually follows two phases. First phase (where inherited biased distribution or single big sample are dominating) is mainly important for locating the new area the posterior distribution is concentrated at. The second phase (dominated by whole sample population) is actually relevant for the critical rate estimation.

Offered Load Selection #

First two measurements are hardcoded to happen at the middle of rate interval and at max_rate. Next two measurements follow MRR-like logic, offered load is decreased so that it would reach target loss ratio if offered load decrease lead to equal decrease of loss rate.

The rest of measurements start directly in between erf and stretch estimate average. There is one workaround implemented, aimed at reducing the number of consequent zero loss measurements (per fitting function). The workaround first stores every measurement result which loss ratio was the targed loss ratio or higher. Sorted list (called lossy loads) of such results is maintained.

When a sequence of one or more zero loss measurement results is encountered, a smallest of lossy loads is drained from the list. If the estimate average is smaller than the drained value, a weighted average of this estimate and the drained value is used as the next offered load. The weight of the estimate decreases exponentially with the length of consecutive zero loss results.

This behavior helps the algorithm with convergence speed, as it does not need so many zero loss result to get near critical region. Using the smallest (not drained yet) of lossy loads makes it sure the new offered load is unlikely to result in big loss region. Draining even if the estimate is large enough helps to discard early measurements when loss hapened at too low offered load. Current implementation adds 4 copies of lossy loads and drains 3 of them, which leads to fairly stable behavior even for somewhat inconsistent SUTs.

Caveats #

As high loss count measurements add many bits of information, they need a large amount of small loss count measurements to balance them, making the algorithm converge quite slowly. Typically, this happens when few initial measurements suggest spread way bigger then later measurements. The workaround in offered load selection helps, but more intelligent workarounds could get faster convergence still.

Some systems evidently do not follow the assumption of repeated measurements having the same average loss rate (when the offered load is the same). The idea of estimating the trend is not implemented at all, as the observed trends have varied characteristics.

Probably, using a more realistic fitting functions will give better estimates than trend analysis.

Bottom Line #

The notion of Throughput is easy to grasp, but it is harder to measure with any accuracy for non-deterministic systems.

Even though the notion of critical rate is harder to grasp than the notion of throughput, it is easier to measure using probabilistic methods.

In testing, the difference between througput measurements and critical rate measurements is usually small.

In pactice, rules of thumb such as “send at max 95% of purported throughput” are common. The correct benchmarking analysis should ask “Which notion is 95% of throughput an approximation to?” before attempting to answer “Is 95% of critical rate safe enough?”.

Algorithmic Analysis #

Motivation #

While the estimation computation is based on hard probability science; the offered load selection part of PLRsearch logic is pure heuristics, motivated by what would a human do based on measurement and computation results.

The quality of any heuristic is not affected by soundness of its motivation, just by its ability to achieve the intended goals. In case of offered load selection, the goal is to help the search to converge to the long duration estimates sooner.

But even those long duration estimates could still be of poor quality. Even though the estimate computation is Bayesian (so it is the best it could be within the applied assumptions), it can still of poor quality when compared to what a human would estimate.

One possible source of poor quality is the randomnes inherently present in Monte Carlo numeric integration, but that can be supressed by tweaking the time related input parameters.

The most likely source of poor quality then are the assumptions. Most importantly, the number and the shape of fitting functions; but also others, such as trial order independence and duration independence.

The result can have poor quality in basically two ways. One way is related to location. Both upper and lower bounds can be overestimates or underestimates, meaning the entire estimated interval between lower bound and upper bound lays above or below (respectively) of human-estimated interval. The other way is related to the estimation interval width. The interval can be too wide or too narrow, compared to human estimation.

An estimate from a particular fitting function can be classified as an overestimate (or underestimate) just by looking at time evolution (without human examining measurement results). Overestimates decrease by time, underestimates increase by time (assuming the system performance stays constant).

Quality of the width of the estimation interval needs human evaluation, and is unrelated to both rate of narrowing (both good and bad estimate intervals get narrower at approximately the same relative rate) and relatative width (depends heavily on the system being tested).

Graphical Examples #

The following pictures show the upper (red) and lower (blue) bound, as well as average of Stretch (pink) and Erf (light green) estimate, and offered load chosen (grey), as computed by PLRsearch, after each trial measurement within the 30 minute duration of a test run.

Both graphs are focusing on later estimates. Estimates computed from few initial measurements are wildly off the y-axis range shown.

The following analysis will rely on frequency of zero loss measurements and magnitude of loss ratio if nonzero.

The offered load selection strategy used implies zero loss measurements can be gleaned from the graph by looking at offered load points. When the points move up farther from lower estimate, it means the previous measurement had zero loss. After non-zero loss, the offered load starts again right between (the previous values of) the estimate curves.

The very big loss ratio results are visible as noticeable jumps of both estimates downwards. Medium and small loss ratios are much harder to distinguish just by looking at the estimate curves, the analysis is based on raw loss ratio measurement results.

The following descriptions should explain why the graphs seem to signal low quality estimate at first sight, but a more detailed look reveals the quality is good (considering the measurement results).

L2 patch #

Both fitting functions give similar estimates, the graph shows “stochasticity” of measurements (estimates increase and decrease within small time regions), and an overall trend of decreasing estimates.

On the first look, the final interval looks fairly narrow, especially compared to the region the estimates have travelled during the search. But the look at the frequency of zero loss results shows this is not a case of overestimation. Measurements at around the same offered load have higher probability of zero loss earlier (when performed farther from upper bound), but smaller probability later (when performed closer to upper bound). That means it is the performance of the system under test that decreases (slightly) over time.

With that in mind, the apparent narrowness of the interval is not a sign of low quality, just a consequence of PLRsearch assuming the performance stays constant.

Vhost #

This test case shows what looks like a quite broad estimation interval, compared to other test cases with similarly looking zero loss frequencies. Notable features are infrequent high-loss measurement results causing big drops of estimates, and lack of long-term convergence.

Any convergence in medium-sized intervals (during zero loss results) is reverted by the big loss results, as they happen quite far from the critical load estimates, and the two fitting functions extrapolate differently.

In other words, human only seeing estimates from one fitting function would expect narrower end interval, but human seeing the measured loss ratios agrees that the interval should be wider than that.

Summary #

The two graphs show the behavior of PLRsearch algorithm applied to soak test when some of PLRsearch assumptions do not hold:

• L2 patch measurement results violate the assumption of performance not changing over time.
• Vhost measurement results violate the assumption of Poisson distribution matching the loss counts.

The reported upper and lower bounds can have distance larger or smaller than a first look by a human would expect, but a more closer look reveals the quality is good, considering the circumstances.

The usefullness of the critical load estimate is of questionable value when the assumptions are violated.

Some improvements can be made via more specific workarounds, for example long term limit of L2 patch performance could be estmated by some heuristic.

Other improvements can be achieved only by asking users whether loss patterns matter. Is it better to have single digit losses distributed fairly evenly over time (as Poisson distribution would suggest), or is it better to have short periods of medium losses mixed with long periods of zero losses (as happens in Vhost test) with the same overall loss ratio?