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Efficient verification of performability guarantees

Published in the Fifth International Workshop on Performability Modeling of Computer and Communication Systems(PMCCS-5),September15-16,2001,Erlangen,Germany.

main contribution of this paper is a novel operational technique for determining whether a can-didate con?guration for an arbitrary system meets performability requirements.Our algorithm generates only the indispensable portion of the failure scenario space,by?rst examining the sce-narios with largest probabilities of occurrence and stopping as soon as a decision can be made. This is not done at the expense of accuracy;an exhaustive search would yield exactly the same results.We decouple the generation of failure scenarios from the computation of a performance level for each scenario(so they can be solved by independent models),and rely on heuristics to do as little as possible of both.Results show that our algorithm works very well for our case study of a planetary-scale content delivery network:on the average,it only took a few minutes of compu-tation time to determine whether each sample system can or cannot satisfy a given performability requirement.

Many alternative performability metrics can be evaluated for any given system[7,8,9],but the most common ones fail to capture the system designer’s requirements.Another contribution of this paper is a natural,multi-part way of specifying performability requirements,as a set of performance levels with associated time durations.

2Specifying performability requirements

We identi?ed some desirable properties of performability speci?cations while studying QoS guar-antees in storage systems[10,11].First,system designers care about the fraction of the time in which outages occur(i.e.,when performance deviates from baseline goals);but it is also nec-essary to quantify the extent to which performance degrades during an outage,for arbitrarily low performance may not be acceptable during most of the time.Metrics that characterize the distribution of rewards over a period of time can capture this level of information,whereas met-rics(such as expected reward)that aggregate rewards observed during both outage and baseline intervals into an average[7]cannot.Second,speci?cations should be ef?ciently and accurately veri?able against candidate con?gurations.In particular,for many aggregate measures it is neces-sary to consider all possible failure scenarios before determining whether the designer’s require-ments are met.Given the exponentially large size of the scenario space,this is clearly impractical and wasteful.

We therefore believe that designers should adopt multi-part performability speci?cations that re?ect the individual characteristics of each performance level users are able to tolerate.A per-formability speci?cation consists of pairs,where and.Each denotes a steady-state performance level,and

the corresponding denotes the fraction of the system’s lifetime during which its performance must be as good as or better(denoted as;we also use the variations,, ).For example,if a web server has a guaranteed latency of at most100ms during60%or more of the time and at most500ms during95%or more of the time,the corresponding speci?ca-tion is(100ms,0.6),(500ms,0.95).A multi-part performability speci?cation can be thought of as a bound on the cumulative distribution function(CDF)of the performance metric.Figure1shows a performability speci?cation corresponding to our example and the CDFs of latency distribution for one conforming and one non-conforming con?guration.

Let be the set of all failure scenarios the system can be in.Associated with each scenario is,its probability of occurrence,and a performance level, the performance experienced in that scenario.To verify whether a performability speci?cation is satis?ed,we need to determine whether the sum of the probabilities of all scenarios in which

performance is or better is at least.A system meets a performability requirement if:

for all(1) where the indicator function is equal to one if expression is true and zero otherwise.

3Verifying performability requirements

As stated,evaluating Equation(1)requires us to examine all failure scenarios,where can be a very large number.In this section,we present our approach for determining whether the equation holds while heuristically minimizing the amount of work.This is accomplished by (a)generating as few scenarios as possible,and(b)computing the performance levels for only the scenarios that are generated.

The architecture of the performability veri?er consists of three modules,that operate on tex-tual representations of workload and system descriptions[10].In what follows,a system con-?guration is a description of the system being studied:the types of its components,the way they are con?gured,and their interconnections.A failure scenario is a system con?guration plus additional information about the failures that have occurred and have not yet been repaired. Performance predictor:Given a workload speci?cation and a failure scenario,it returns a pre-diction of the performance that will be observed when servicing the input workload while in the input failure scenario.This module can be implemented in a variety of ways(e.g.,a simulator,or an analytical model solver).

Failure-scenario generator:This module takes as input a system con?guration,and generates failure scenarios in order of decreasing probability.The ability to generate the most likely sce-narios?rst is critical to verifying that the con?guration meets performability requirements with a minimum number of performance predictions,and thus minimizing the computational require-ments.If a system has components,,,which fail independently of each other with corresponding availabilities,,,,the occurrence probability of a failure scenario with components,,,failed is

Latency (ms)

P r o b a b i l i t y The conforming con?guration has a latency CDF

higher than the performability speci?cation line in

the entire range of latency values,while the latency

CDF of the non-conforming con?guration is below

the bound between 500–550ms.Notice that the mean

latency of the failing con?guration is,at 190ms,

lower than the bound prescribed by the performa-

bility speci?cation (235ms).A veri?cation based

on this common aggregate metric would have found

this con?guration acceptable,even though it is non-

conforming in a part of the range.

Figure 1:Sample performability speci?cation and performance CDFs.while a decision has not been made

obtain ,

compute

if

decide “system meets requirements”

if

decide “system does not meet requirements”

The failure-scenario generator provides the next (,)pair,and the performance predictor computes .The above algorithm computes progressively tighter lower and upper bounds on the latency CDF of the given con?guration;it reports a conforming con?guration when the lower bound is known to be above the performability speci?cation for the entire range,or a non-conforming con?guration when the upper bound falls below the speci?cation in any part of the range,as shown in Figure 1.

4Case study:Wide-area content provider

We study a system in which a number of geographically-dispersed servers store and deliver ob-

jects to a number of user locations,also geographically dispersed.Each object served(e.g.,a

web page)consists of several sub-objects(e.g.,advertising banners and the page’s text);each

sub-object is replicated on multiple different servers.We assume that network latencies between servers and users are known in advance.To retrieve an object,a user sends one request per sub-object(all in parallel)to the location that has minimum latency to the user,among the locations

that have not crashed and contain a replica of that sub-object.The latency of the object’s retrieval

is the maximum of the latencies of retrieving the sub-objects.For simplicity,we consider only a single object and a single user location.We also assume a failure model where only servers can

fail and they do so independently and in a fail-stop manner.

The content provider’s network is speci?ed by(1)a number of servers,each with given avail-ability and latency to the user location,(2)an object,constituted of a set of sub-objects,each

with a given number of replicas and their locations,and(3)a performability requirement for the latency of retrieving the object.We then apply our performability veri?er to determine which

con?gurations meet the requirements,and report the number of failure scenarios examined to complete each veri?cation.We applied our algorithm to three such networks,using availability

and latency values obtained from real-world planetary-scale networks as follows:

1.Sites from Netcraft:This is a50-server network.We assigned availabilities to the servers

based on the uptimes of the50most-available ISP sites,obtained from https://www.doczj.com/doc/0d17662505.html,.

We used a downtime of4hours/uptime to estimate their availabilities(the minimum avail-

ability was99.89%).

2.Sites from Uptimes:A27-server network with availability values obtained from https://www.doczj.com/doc/0d17662505.html,.

We eliminated sites which reported either0%or100%availabilities unless the polling pe-

riod was over500days.Availabilities for these sites range from17.2%to100%.

3.Self-reporting sites:We used a network of32-randomly chosen internet sites that report

their own availability statistics.The minimum reported availability in this sample was

93.27%.

We assigned latency values to the servers based on average round-trip latencies of a randomly se-lected100routers spread over5continents,as reported by https://www.doczj.com/doc/0d17662505.html,. We assumed that every object consists of5sub-objects.A random,uniformly-distributed number

of replicas of each sub-object are stored at randomly-chosen servers.We verify each such random

con?guration against three performability speci?cations corresponding to different service lev-

els,from most to least stringent:premium((100ms,0.96),(300ms,0.97),(400ms,0.98),(500ms,

0.99)),standard((200ms,0.9),(300ms,0.95),(500ms,0.99)),and economy((200ms,0.1),(300ms,

0.5),(500ms,0.8)).

Our results indicate that the algorithm proposed in this paper indeed works well in practice. Table1presents the average number of scenarios examined by our algorithm for the case study. Running times on a single processor of an HP9000-N4000server were under1minute in all cases

for the netcraft and the selfreport datasets.The uptimes dataset took an average of10minutes

and a maximum of35minutes.The average time to examine a single scenario was approximately

5ms for our simple performance predictor;for more complex predictors,we would expect longer

run times.We were able to determine whether a given performability speci?cation is satis?ed

Service levels Standard

5101551015 netcraft37373711

1546372751917

uptimes122634687426019332944108632 Table1:Number of scenarios examined by our algorithm to validate each performability spec-

i?cation on each of three sample content-delivery networks,for three mean replica counts.We report averages from?ve different executions.

by examining as little as one scenario;the maximum number of scenarios that our algorithm

had to examine was433,498,for the uptimes dataset.In contrast,the number of scenarios in

the full Markov reward models(up to scenarios for the netcraft dataset)implies that exhaustive methods are not viable for this case study.

The heuristic embedded in the failure-scenario generator implicitly assumes that component availabilities are high,hence scenarios with fewer failures are more likely than scenarios with more failures.This is re?ected in our results:the performability guarantees for con?gurations in which most of the sites have high availabilities can be veri?ed by examining a relatively small number of high-probability scenarios.This is most evident for the case of the netcraft dataset, where our algorithm was able to verify that the economy requirements can be met by examining only one single scenario.This is due to the fact that the failure-free scenario in netcraft has an occurrence probability greater than94.6%and its performance was suf?cient to meet the require-ments of the economy speci?cation.In contrast,the uptimes dataset,which contains several sites with low availabilities,required examining a large number of scenarios as each scenario added only a small increment of probability.

The case study also demonstrated that performance speci?cations with either stringent re-quirements(e.g.,the premium speci?cation)or weak requirements(e.g.,the economy speci?ca-tion)are easier to verify than those with medium-strength requirements.This is because con?g-urations can often be shown to fail the stringent requirements very quickly,and,conversely,to pass the weak requirements quickly.Verifying medium-strength requirements often requires the algorithm to examine a larger number of scenarios.

5Conclusions

Many traditional performability metrics measure the system’s average performance during a time interval.We showed that this falls short of the needs of business-and mission-critical systems,for

it potentially hides time periods during which performance degrades to unacceptably low levels. Realistic metrics need to consider the time-dependent distribution of various performance levels.

We presented a multi-part style of specifying performability requirements,and a novel algo-rithm that determines whether a candidate system con?guration meets the requirements.Our al-gorithm heuristically minimizes the number of failure scenarios evaluated,thus solving the prob-

lem very rapidly and without sacri?cing any accuracy.Our case study shows that this is indeed

the case—we examined several network con?gurations using data gathered on real planetary-scale systems.All runs?nished in35minutes or less of computation time and had very low

memory requirements,as only one failure scenario is examined at a time and then discarded. Our technique scales well to large real-world systems,and compares favorably to other existing model-solving approaches.

Acknowledgements:John Wilkes proposed the multi-part speci?cation of performability re-quirements and Ralph Becker-Szendy participated in the genesis of the algorithm presented here. We thank Erik Riedel and John Wilkes for comments on an earlier draft of this paper. References

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