ACCURACY ANALYSIS OF DATA AGGREGATION FOR NETWORK MONITORING

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1 ACCURACY AALYSIS OF DATA AGGREGATIO FOR ETORK OITORIG koletta Sora Imperal College London UK Tng He, Petros Zeros, Bong Jun Ko, Kang-on Lee IB T.J. atson Research Center Hawthorne Y, US Kn K. Leung Imperal College London UK Abstract The qualty o computng certan aggregaton unctons based on ncomplete measurements or the purpose o dstrbuted network montorng s consdered. etwork montorng plays a undamental role n network management systems by provdng tmely normaton on the network status, whch s crucal or admnstraton purposes. To reduce network overhead and or easer assmlaton, ths normaton s usually presented by calculatng a ew key aggregate metrcs. The aggregates are perodcally computed rom a large number o detaled events collected contnuously durng the course o the network operatons. Under errors nduced by network delays, the accuracy o typcal aggregaton unctons used n network management systems s evaluated both analytcally and by smulatons. The results provde a quantable trade-o between accuracy and tmelness o the normaton acqured, whch can then be used to desgn and optmze network management systems. I. ITRODUCTIO etwork montorng s a crtcal uncton or the ecent and robust operaton o modern networks, provdng tmely normaton on the status o network elements and communcaton lnks to network admnstrators. etwork montorng systems contnuously collect numerous measurements o key perormance ndcators and events such as bandwdth utlzaton and lnk status and process them nto more meanngul, aggregate metrcs such as average delay and network avalablty. The results o aggregaton are then presented to network operators or urther dagnosng potental servce problems and choosng correctve actons. Aggregate metrcs also orm the quanttatve bass around whch servce level agreements (SLAs between network provders and ther customers are usually structured [4][5], as complance wth (or devaton Research was sponsored by the U.S. Army Research Laboratory and the U.K. nstry o Deense and was accomplshed under Agreement umber 9F The vews and conclusons contaned n ths document are those o the author(s and should not be nterpreted as representng the ocal polces, ether expressed or mpled, o the U.S. Army Research Laboratory, the U.S. Government, the U.K. nstry o Deense or the U.K. Government. The U.S. and U.K. Governments are authorzed to reproduce and dstrbute reprnts or Government purposes notwthstandng any copyrght notaton hereon /08/$ IEEE rom agreed-upon levels can be readly assessed. For ths reason, accurate computaton o aggregate metrcs becomes a partcularly mportant ssue or any network montorng system and the topc o study or ths work. etwork anagement Center etwork Elements : ontorng Agent Fgure : Processng center and montorng agents o a dstrbuted network management system The accuracy o computed aggregates s aected by errors ntroduced by the measurement process as well as the transmsson o the collected data to the processng staton. The latter case occurs when montorng o the network s dstrbuted to several montorng agents and processng o the measured data takes place n a remote network management center (as n Fgure. Such deployment can cause transmsson delays or loss o sampled data, whch skew the results o the perodc calculaton o aggregate metrcs. Furthermore, dependng on the sze o the network and the network montorng archtecture, the computaton o the aggregate metrc tsel mght take place dstrbutedly, wth partal results transerred over several levels o aggregaton, a process that ntroduces urther errors n the calculatons. An nterestng case scenaro appears n the mltary context; we can envson the headquarters as the network management centre, whch needs to manage the network nodes (any mltary unt physcally equpped wth wreless communcaton capabltes. In order to montor the well beng o all the nodes and have a complete vew o the eld, enablng tmely decson makng, both network-related and eld-related (collected through sensng the area events, metrcs and updates need to be orwarded to the headquarters. The moblty o the nodes, whch mght ntroduce a delay tolerant aspect n the crulaton o ths normaton, as well as the hostle envronment result n delayed or mssng data. o 7

2 Recent work on computng n-network aggregates n the eld o sensor networks [9][][3][4][9] prmarly ocuses on the trade-o between accuracy and energy consumpton or extendng the letme o the network under certan topologcal assumptons regardng the dstrbuton o computaton (e.g. tree-based, herarchcal, centralzed, etc. Results rom studes on montorng o network varables [5][6][7] explore the trade-o between communcaton overhead and accuracy o aggregaton. Ths work provdes an analyss o the error that s nduced n the calculaton o the typcal aggregaton unctons used or network montorng, when the measurement samples are delayed or mssng, ndependent o the partcular topology used or delverng the (partal results, or the communcaton costs or transmttng the underlyng measurements. As such, t provdes a characterzaton o the qualty o aggregaton rrespectve o the underlyng network montorng archtecture, whch extends the concept o Qualty o anagement (Qo ntroduced n []. ore speccally, ths work s contrbuton extends to two man drectons; analyzng the error o aggregaton unctons used or network montorng based on a stochastc model or descrbng the generaton and transmsson o measured events; and presentng smulaton results that provde nsghts on how the error s aected by both network (communcaton delay, event generaton rate and desgn parameters (aggregaton perod. The rest o ths paper s organzed as ollows: Secton dscusses the system model and the basc assumptons, whch the accuracy analyss o aggregaton unctons n Secton 3 s based on. Secton 4 presents smulaton results that very the analyss. Related work s overvewed n Secton 5 and the paper concludes n Secton 6. II. EVETS, ODELS AD QUALITY OF AGGREGATIO In order to analyze the error o aggregaton unctons n the presence o delayed or lost measurements, we need to dene a model or the generaton o events that are montored by the montorng system. e rst outlne a generc model or the network montorng uncton and event generaton and then dene the qualty o aggregaton. A. Event Generaton n etwork ontorng Systems etwork admnstrators ace a multtude o unpredctable stuatons, such as network outages, sudden trac spkes, devce and applcaton msconguraton, as well as ssues caused by thrd-party peerng networks. Inormaton about these stuatons may be manested as events, whch typcally nclude a number o characterstcs such as the tme the stuaton arose, a value that descrbes the ntensty o the stuaton, ndcaton o managed resources aected by the stuaton, etc. These events, along wth measurements on varous perormance ndcators (bandwdth utlzaton, lnk delay, dropped sessons, etc. are collected by the network montorng system, or urther processng, archvng and presentaton to the network admnstrators. In large network management deployments, collecton o the events and measurements s perormed dstrbutedly by montorng agents that report the measured samples to a centralzed network management center (Fgure. There, calculaton o user-dened metrcs takes place, whch typcally nvolves the use o aggregaton unctons such as SU, AVERAGE, COUT, and I/A. Then, the results are presented to a network admnstrator or decson makng. In the remander o ths paper, we examne the behavor o typcal aggregaton unctons used n the computaton o user-dened perormance metrcs at the network management center, as captured events mght be delayed durng ther transmsson rom the montorng agents to the network management center. ore detals on events and the model used hereater can be ound n []. B. Dscrete Event odel For the analyss o the eect o the network delay on the accuracy o the results obtaned through processng usng aggregaton unctons, we assume the ollowng event model based on stochastc processes. There are managed network elements n the network, represented by a set {,,, }, and each such network element contnuously collects normaton by means o a montorng agent. The -th event captured by a montor s represented by (x (, t (, where x ( s the ntensty o the event and t ( s an ncreasng sequence o the tme representng the tme the event occurs where t ( > t (-. The events captured at the montor occur accordng to a homogeneous Posson process wth arrval rate. A specal case o the events arrval process s that the arrval rates at each montor are homogeneous, n whch case we denote by the arrval rate at each montor,.e.,. The ntensty o an event x ( at a montor s drawn rom a random varable. e assume { },,,, are..d. random varables, or whch ] m and Var( σ. henever a montorng agent captures such an event, t sends t to the next level or aggregaton. The network delay s assumed to be a random varable, D, exponentally dstrbuted wth mean /. (A specal case s when or all,, The network management center aggregates the events that t has receved perodcally, wth a perod T. The aggregaton takes place at tme t agg kt+h or all the events that have occured wthn the nterval [(k-t, kt. Even though the addtonal tme h acltates or the delay nduced n the network, so that events occurred wthn the accordng tme nterval reach the aggregaton pont on tme, there can stll exst some events that mss ths deadlne. Ths results n some data mssng at the tme when the o 7

3 aggregaton occurs, ntroducng an error on these calculatons, as llustrated n Fgure. e assume that events are tme stamped, so that those receved ater the accordng tme nterval are dropped and do not nterere wth later calculatons. e denote by S,k the set o events captured by the montor n [(k-t, kt, by,k S,k the set o events that mss the deadlne kt+h, and by L,k S,k,k S,k the set o events that do not mss the deadlne and are ncluded n the aggregaton. Snce the events arrval process and the network delay n the tme nterval [(k T, kt are ndependent o k, we can omt the ndex k, by smply denotng the above sets o events by S,, and L, respectvely. e urther denote by S,, and L the correspondng sets o all events rom all montors,.e., S S,, and. U L U L U ssed Event h (k-t C. Aggregaton Functons Typcal aggregaton unctons used or network montorng can be classed nto summary unctons such as count (and hstogram, sum, average, varance (and standard devaton, and exemplary unctons such as /mn, medan, and any percentle. e reer to [0] or the dstncton between summary and exemplary aggregaton unctons. These can be combned wth addtonal preprocessng, e.g., multplyng weghts or applyng de-duplcaton (.e., lterng out duplcates to the nput values, and post-processng ncludng comparng the aggregated values aganst gven thresholds. Derent partal results and types o unctons to be used are requred or next levels o aggregaton or derent unctons. For example, calculaton o average requres the sum and count o underlyng layers. D. Qualty o Aggregaton Functons e ocus our study on accuracy as the property that prmarly characterzes the qualty o aggregaton. The accuracy o a partcular aggregaton uncton s captured by lookng at the derence between the output o the aggregaton uncton appled to the (ground-truth set o events S and that appled to the ncluded set L. Formally, kt ontorng agent (capture o events Fgure Event odel. Some events arrve at the network management center out o order or too late. h etwork anagement Center we dene an aggregaton uncton F(S as an operator that takes a set o events S as the nput and outputs a real number, then the qualty o F under our event model s measured by the absolute error: F F( S F( L. In the ollowng secton, we provde analytcal results o the expected value F ] and varance Var( F or our aggregaton unctons: CT (Count, SU (Sum, AVG (Average, and A (axmum. e wll also consder the weghted generalzaton o some o the above aggregaton unctons, denoted by appendng a prex - to the correspondng uncton, e.g., -CT. ote that the mean square error, (F(S F(L ] can be also derved rom our analyss; snce E [( F( S F( L ] Var( F + E [ F ], t suces to analyze the mean and varance o the absolute error. III. AALYSIS OF QUALITY OF AGGREGATIO Based on our event model n Secton II.B, the expected number and the varance o events occurrng at montor wthn a tme nterval T are: S ] Var( S T ( The probablty that an event, montored and sent at tme t wll not be aggregated wthn the approprate tme nterval s equal to the probablty that the montored event wll not arrve to the aggregaton ste beore the aggregaton tme t agg kt+h due to the delay ncurred by the network. As proven n [], the number o mssed events wthn a tme nterval T s a random varable ollowng the Posson dstrbuton, wth mean and varance: h ( e T ] Var( e ( Smlarly, the number o events that are ncluded n the aggregaton (receved events ollows a Posson dstrbuton o mean and varance ( h e L ] ( ( T Var L T e (3 The total expected number o all events, all mssed events, and all ncluded events n the network can be calculated by summaton over all montors usng equatons (, ( and (3 respectvely. A. Error analyss: COUT hen the aggregaton uncton s the enumeraton o events, the expected absolute error and ts varance are equal to those o the number o mssed events. Thereore, E T ( [ CT ] ] e ( CT T ( Var( e Var (4 (5 I the enumeraton s weghted, the way the weghts are 3 o 7

4 assgned depends on the speccs o the applcaton. In the ollowng we assume that the probablty o the event beng mssed s uncorrelated wth the weght value that would be assgned to ths event. Assumng that the weght gven to event rom the montor s a random varable wth mean w and varance σ, ndependent o the event but not necessarly dentcally dstrbuted across montors, the expected absolute error s gven by CT S L, (6 and we assume the weghts to be non-negatve, we get CT, (7 Accordng to [], and assumng that events across montors are uncorrelated we can calculate these moments as ollows: CT Var( ] CT e T ( ] h ( e e T ( ] σ ( σ + w ] w, + ( ] Var( B. Error analyss: SU The SU aggregaton uncton s dened as the sum o the values (.e., ntensty o the events,.e., SU ( S. The absolute error o the SU S aggregaton uncton s then gven by S L. (0 SU Let us assume that s are nonnegatve, as t s usually the case or the quanttes measured n network montorng (e.g. delay, packet loss rate etc. In ths case, we have SU, and we can calculate the expected value and the varance o the absolute sum error, treatng t as a sample sum o random sample sze, as the number o mssed events s a dscrete random varable []. Snce the number o mssed events s ndependent o the ntenstes o those events, we have : and E ( SU T [ ] ] ] m e (8 (9 ( Var( SU ] σ ( σ + m + ( ] Var( ( T e ( The absolute error or the calculaton o the weghted sum where weghts are..d. or events o the same montor, and ndependent (but not dentcally dstrbuted or derent montors can be calculated by: SU (, (3 e can get the expected value and the varance o the absolute error or weghted sum aggregaton: and Var[ SU SU ] ] ( ] ( m w + Cov(, e ] ] Var( + ( ] Var( ( T ( Var( + ( m w Cov(, h ( e T e +, (4 (5 where a dependence o the values o the weghts on the underlyng values, such exsts, can be accommodated. The varance o the product, can be estmated as n [3]. ote that the above results can also be used or the case that event values are ndependent but not dentcally dstrbuted across montors (by vewng value. ( as the new event C. Error analyss: AVERAGE The AVERAGE aggregaton uncton s dened as the sample average o a set o events values (.e., ntensty,.e. AVERAGE( S. S S Then the absolute error o the average value s gven by AVG. (6 S S L L Snce S L, and L φ, we can rewrte the above ormula by dvdng the set o events nto mssed events ( and receved events (L: AVG S L S L. (7 S L L The mert o (7 s that gven and L, the two summatons are about dsont sets o events, whch have 4 o 7

5 ndependent values. The computaton o the expected value o (7 s complcated by the absolute operator, and thus we try to provde bounds nstead. By trangle nequalty, we can obtan a trval lower bound o zero and an upper bound: AVG] E + E S S L L (8 Pr mss m + Pr mss m m { } { } T h ( e e T assumng homogeneous transmsson delay. Smlar technques can be used to obtan bounds on the error varance. Speccally, t can been shown that a lower bound o the error varance s gven by p e Var( AVG 0, σ 4m p, (9 p T T h ( e where p e, the probablty that an event msses T the deadlne, and an upper bound s gven by e Var( AVG σ. (0 T D. Error analyss: A and I e present results only or A uncton snce I uncton can be analyzed n an dentcal manner. e assume that the probablty densty uncton o the ntensty values s known and gven by (thereore ther cumulatve dstrbuton uncton, F, s known as well. e can use order statstcs to nd the dstrbuton o the mum value among a set o realzatons o, gven the sze o the set, K: K ( K K ( F ( x ( x. ( The true mum value wll be the mum value among the set o all the events that have occurred n the accordng tme nterval. e can obtan the probablty densty uncton o the true mum value, condtonng on the number o total events, K, as ollows: true ( F ( ( x p ( K e ( x ( T. (! ( F ( x T ( x e Smlarly, the probablty densty uncton o the estmated mum value, can be obtaned as the densty uncton o the mum value o the set o ncluded values, condtonng on the number o the receved events, L : est ( p ( L (3 T h ( e ( T e ( x T h ( e exp[ ( F ( x( T e ] e can use ( and (3 to calculate the respectve expected values, m true, mest.the expected value o the error n the mum value estmaton wll then be: A ] mtrue mest. (4 For example, event values are unormly dstrbuted n an nterval [ 0, a], then t can be shown that a mtrue a ( e, T h T a[ exp( ( T e ( e / ] mest a, h T ( T e ( e / whch gve a closed-orm soluton to (4. Snce n general absolute error and squared error do not yeld closed-orm solutons, we turn to a smpler error measure: the ndcator error, whose mean gves the probablty o gettng a wrong estmate. In partcular, all event ntensty values are..d., then the arrval tmestamp o event wth the mum value s unormly dstrbuted wthn each perod. Thereore, the probablty o mssng ths event s the same as the margnal probablty o mssng any other event,.e. h T Pe e ( e / T. (5 In general, the absolute and squared error analyss or exemplary aggregaton unctons, such as mn/ and p-percentle, do not gve closed-orm solutons. However, the result n (5 holds or any exemplary aggregaton uncton that s solely based on event values, e.g., percentle. IV. SIULATIOS To very the analyss, we smulate the proposed aggregaton unctons under varous settngs. Speccally, we evaluate the qualty o aggregaton usng the normalzed mean absolute error, dened as the rato o mean absolute error and expected true value, and plot the error versus network parameters (e.g., communcaton delay and rate o event streams as well as desgn parameters (e.g., aggregaton perod and watng tme. oreover, to evaluate the nluence o event ntensty, we smulate two ntenstes---unorm and lognormal---wth the same mean and varance. The normalzed squared error shows the same trend and s thereore omtted due to space lmt. e rst plot the normalzed error as a uncton o the mean communcaton delay (/, as shown n Fgure 3. From the Fgure, we observe that the error s monotoncally ncreasng wth communcaton delay, as expected. 5 o 7

6 oreover, count and sum have dentcal error (because ater normalzaton, the errors or count and sum are both equal to the probablty or an event to mss the deadlne, whch s much larger than the errors or average and unctons. ormalzed mean absolute error ax Lognorm Average Lognorm Average Un Count, sum ax Un ean delay (/ Fgure 3 ormalzed mean absolute error vs. ean delay ( tot 0, Τ0, h, 0000 teratons Furthermore, the event ntensty does not aect the error except or uncton, where the error or lognormal dstrbuton s much larger than that or unorm dstrbuton. Ths s because all the other unctons only rely on the rst moment o the ntensty, whereas reles on outlers and s thereore senstve to how spread out the ntensty s. These observatons also hold or the other graphs (Fgures 4 and 5. The smulatons are consstent wth the analyss whenever closed-orm solutons are avalable. ext, n Fgure 4, we plot the error versus mean latency to urther evaluate the trade-o between tmelness and accuracy. Tmelness s expressed n terms o mean latency, whch s dened as the expected value o the tme between the generaton o an event and the tme when t s ncluded n the aggregaton, gven by lt/+h. Snce events are tme stamped and dropped they mss the deadlne, latency s only dened or the ncluded events. Accordngly, we can adust the mean latency by ether varyng T or varyng h. As expected, all the errors decay as latency ncreases. Comparng the two scenaros, however, we see that there exsts a threshold latency, below whch ncreasng T gves better tradeo than ncreasng h or the same latency value, and the opposte holds otherwse. Ths s because the latency ncreases twce slower wth T, but the error decays much aster wth h. From a desgn perspectve, ths means that gven overall latency constrant, there s an optmal par o aggregaton perod and watng tme that mnmzes the error. Lastly, n Fgure 5, we present the eect o the total event generaton rate ( tot on accuracy. As expected, tot has the least mpact on the accuracy compared wth the other parameters. The errors or count and sum are not unctons o tot as shown beore, and those or average and only decay slghtly. To summarze the smulaton results, count and sum have the lowest accuracy, and the relatve accuracy s (almost ndependent o propertes o event streams ncludng event ntensty and rate (except or uncton, whch s hghly dependent on the ntensty. For desgn purpose, we can acheve the desrable accuracy-tmelness tradeo by properly choosng aggregaton perod and watng tme. ormalzed mean absolute error Varyng h Varyng T ax lognorm Average lognorm Average un ax Un Count, Sum ean latency Fgure 4 ormalzed mean absolute error vs. ean latency: varyng T (sold lne or h (dashed lne (tot0, /, 0000 teratons. Sold lne: T vares rom 8.4 to 8, h; dashed lne: h vares rom 0. to 5, Τ0 ormalzed mean absolute error ax Lognorm Average Lognorm ax Un Count, Sum Average Un Total events' generaton rate Fgure 5 ormalzed mean absolute error vs. Event generaton rate tot (h, T0, /, 0000 teratons V. RELATED ORK Data aggregaton has been a topc o ntense study n the area o sensor networks, due to ts potental or reducng the number o transmssons requred to eectvely montor varous physcal phenomena, by provdng only aggregate statstcs on the sampled data. Ths, n turn, results n sgncant energy savngs, whch extends the letme o the resource-constrant sensor nodes. [6][8][0] study aggregaton wth respect to the specc underlyng sensor network topology (e.g. tree-based, multple path, or hybrd that s used n order to transer the partal aggregates. In contrast, our work s ndependent o the network topology and consders only the tmng o the event generaton 6 o 7

7 process and the dstrbuton o values o the events. It also provdes a theoretcal analyss o the error n the computaton o aggregaton unctons and complements expermental evaluatons such as [9] and [5]. Qualty o Data (QoD and Qualty o Inormaton (QoI wth respect to data aggregaton are dscussed n [7][0][], prmarly rom the expermental perspectve, whle [] gves an overall ramework or QoI ocusng on attrbutes such as completeness and condence. Ths paper provdes a theoretcal vew along the latency-accuracy axes usng pont processes and complments the above work. Ecent algorthms or computng aggregaton unctons have been nvestgated or network montorng purposes usng a herarchcal model [4][7] or the computaton, a lat (-ter model [3], as well as ully decentralzed approaches [] based on gosspng. The emphass s placed ether on the mantenance o the spannng tree o the management overlay that computes the aggregate [4], the trade-os between accuracy and communcaton cost [3][7][9], or the convergence o the algorthms to the aggregaton values and reslence to nodes alures []. In contrast, n ths work, we study the dstrbuton o event arrval tmes (tmelness at the processng center and ther eect on accuracy, not the communcaton costs ncurred n the uson network or ts topologcal assumptons. In [6], the problem o aggregaton s consdered n the context o peer-to-peer network, n whch nodes can dynamcally on and leave durng the computaton o the aggregate. The ocus s on the condtons that need to be satsed n order to obtan vald results or the aggregaton unctons. Fnally, recent work n [8] studes the problem o computng aggregate unctons such as sum, count, mean and medan on data streams wth unrelable and nosy data and proposes algorthms or contnuous computaton o estmate o aggregates. Our work explores perodc computaton o aggregates, as the montored events accumulate over a tme perod speced as a desgn parameter o the montorng system. VI. COCLUSIO Data aggregaton o network measurements s an mportant uncton o network montorng systems, as t summarzes perormance ndcators o the network nto metrcs that are easly assmlated by network admnstrators and are urther used as a bass or SLAs wth customers. For ths reason, the ablty to characterze the accuracy o the aggregate computaton when measured samples are mssng or delayed becomes o paramount mportance, especally n large, dstrbuted network montorng nrastructures where the collecton and processng o measurements are perormed at dsparate stes. Ths paper provdes an analytcal study o the error that s ntroduced nto the calculaton o typcal aggregaton unctons (sum, count, average and mn/, as they re computed perodcally usng measurements that mght arrve late or computaton at the network management center or processng. A model based on pont processes s used or analyzng the arrval process o montored events, and, based on ths model, closed-orm solutons and error bounds are provded or calculatng the expected value and varance o the error. Smulaton results very our analyss and provde urther nsghts on the accuracy-tmelness tradeos as we change network varables such as rate o events and desgn parameters such as aggregaton perod. REFERECES [] D. Verma, B. Ko, P. Zeros, K. Lee,. Duggan, K. Stewart, B. Rvera and A. Swam, "Understandng the Qualty o anagement n Computer etworks", IB techncal report RC4560, avalable rom [] H..Taylor and S.Karln, An Introducton to Stochastc odelng, Rev. ed. ed San Dego ; London : Academc Press, 994. [3] Leo A.Goodman, "On the Exact Varance o Products," Journal o the Amercan Statstcal Assocaton, vol. 55, no. 9, pp , Dec.960. [4] AT&T anaged Internet Servce (IS, [5] TT Communcatons Global IP etwork Servce Level Agreement (SLA [6] A. Delgannaks, Y. Kotds and. 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