Modelng Herarchcal Event Streams n System Level Performance Analyss IK Report 9 obas Ren, Ka Lampka, Lothar hele Computer Engneerng and Networks Laboratory Swss Federal Instsute of echnology (EH) Zurch, Swtzerland {ren lampka thele}@tk.ee.ethz.ch December, 8 Abstract he growng complexty of embedded real-tme systems requres elaborate methods n ther desgn process. Whle smulaton does not suffcently cover corner cases, analytc methods for system level performance analyss have been establshed n the past. her effcency n computng hard bounds on buffer szes, end-to-end delays or throughput has proven ther usefulness. One of the major drawbacks of these methods are the lmted system confguratons that can be analyzed wth hgh accuracy. In ths report we extend exstng methods for analyzng heterogeneous multprocessor systems such that dfferent types of data streams can be composed to a hgher level event stream wth multple herarches. Furthermore, we descrbe how such herarchcal event streams can be decomposed nto ther sub-streams. Addtonally, we show how these new methodologes are used to model the dstrbuted heterogeneous communcaton system of the COMBES case study provded by EADS. Introducton In the context of the COMBES project, we wll provde a formal framework for component based desgn of complex embedded systems. One of the mlestones to reach ths goal s the analyss of a case study provded by EADS Innovaton Works [4] n WP. hs case study concerns a dstrbuted Heterogeneous Communcaton System (HCS) contanng wred and wreless communcaton networks based on Ethernet. A common server s connected to dfferent devces, such as sensors (camera, smoke, pressure, etc.) and actuators (speakers, lght swtches, temperature control, sgns, etc.). he functons and applcatons of the Heterogeneous Communcaton System are runnng on common platforms that can be effcently scaled to ncorporate varous requrements. Amongst others, the server broadcasts audo streams to the devces, whch are equpped
wth speakers to play back the audo sgnals. o acheve clock synchronzaton the PP (Precson me Protocol, IEEE88) s used. Some key requrements of the systems are the followng: he system should provde up to 9 devces he system should be able to transmt up to audo streams broadcasted by the server he maxmal tme dfference between sendng a sgnal (e. g. speakng nto the mcrophone connected to the server) and playback of the sgnal at the devces should be less than. sec. he audo samples have to be played at all devces wth a maxmum jtter of. ms (e. g. the maxmal tme dfference of the playback tme between any two devces n the whole system s less than. ms) Clock synchronzaton as well as audo sgnal transmsson requre hard real-tme constrants such as guaranteed maxmum end-to-end delays. hs nvolves both communcaton and computaton. Because of the complexty of the overall system a compostonal system analyss s necessary. In addton, precse protocols for resource sharng are not yet known. hus, testng of varous system confguratons s necessary. For that reason we need a fast analyss that delvers hard bounds n terms of worst-case and best-case on varous system propertes. herefore, smulaton s not possble snce t s nether correct n terms of hard bounds nor fast. All these requrements ncludng the sze of the system ask for a component based desgn of the analyss model. In the feld of performance analyss of embedded realtme systems, dfferent technques have been ntroduced n the past. SymA/S [] and a technque denoted as modular performance analyss (MPA) [, ] are both tools whch perform compostonal system analyss. hey use local analyss technques for analyzng the ndvdual components, whch are nterconnected va event streams. As man contrbuton, these tools delver hard bounds on mnmum and maxmum end-toend delays, queue szes or resource utlzaton. A major problem of these methods s the restrcted scope. e. classes of systems that can be analyzed wth hgh precson. For classes not covered by the methods, smplfcatons are needed resultng n a loss of accuracy of the computed bounds or n a loss of the compostonalty of the methods. A system confguraton, whch s hghly relevant for practce occurs n communcaton systems, f multple data streams are combned n packets and transmtted over a communcaton channel. Accordng to [8] the combned data stream s denoted as herarchcal event stream. In the COMBES case study by EADS [4] the broadcast of dfferent audo streams by the server s such a scenaro. Once an audo packet has been successfully delvered and arrves at the end-pont of the communcaton channel, the data have to be de-packetzed. e. the herarchcal event stream has to be decomposed nto the ndvdual data streams. In ths report we ntroduce a major extenson to the aforementoned methods that allows jonng and forkng event streams wth hgh accuracy whle keepng the compostonalty of the methods. Related to the COMBES
case study the server packetzes the audo streams and after successful transmsson the devces have to de-packetze the stream to play back the audo sgnals. Wthout the concept of the herarchcal event streams, the COMBES case study scenaro could not be modeled accurately and dfferent abstracton would be needed resultng n pessmstc worst-case Fgure bounds. Fgure (a) (a) Fgure Fgure (b) (b) Fgure Fgure (c) (c) BUS BUS C C Fgure (a) BUS C (a) Communcaton task actvated by multple nputs. Fgure Fgure Fgure / /,, AND AND, Fgure (b) BUS BUS C C BUS,, C, (b) Model of Fg. (a) for the MPA doman. Fgure (c) AND BUS C,,,, SYSEM SYSEM,,, (c) A system n whch the combned stream, may Fgure pass several tasks before beng decomposed. Fgure Fgure Fgure : Examples of tasks wth multple nput streams FIFO As an example, we consder a system wth a sngle communcaton task C actvated by two streams and as shown n Fg. (a). he task delvers FIFO the outgong FIFO streams and. Assocated wth the COMBES case study [4], the two streams and could be audo sgnals that are transmtted from one network component to the other. he characterstcs of the output streams depend on the ncomng data streams and as well as on the avalablty of the resource,. e. the avalablty of the bus Fgure 4a Fgure 4b n ths example. In the context of modular performance analyss, tasks actvated by multple nput streams are normally modeled as a combnaton module that combnes u the streams followed by the task usng the combned stream as nput as depcted n Fg. (b). Fgure Fgure For the4atask actvaton FIFOpattern n the combnaton Fgure Fgure 4b module 4b one manly dstngush between - and AND-actvaton [, 8]. In case of an -actvaton, each event on any of the nputs mples the executon of the task e. g. transmsson u u over the communcaton system. he events are put nto packets but are sent ndvdually. In the COMBES cast FIFO study FIFOsuch an -actvaton s realzed by combnng the n audo l streams. Every audo segment s packetzed from the server and broadcasted over the n network. Contrary, the AND-component only n executes f there n s an event present on each nput and each event s removed after processng. herefore, every event generated by an AND-actvated task, denoted as packet, contans one event of each nput stream. Whle n the context of MPAcombnng streams n an AND-actvated n n task l l s challengng but n detal descrbed n [], the decomposton of an AND-actvated combned n stream n the context n of MPA n s smple, snce n n we only consder n number of Fgure Fgure Fgure Event Count Curves γ, γ
events/packets n tme ntervals. As we have one event of each stream n one packet, the combned output stream does not dffer from the ndvdual output streams and therefore = =, holds. Snce AND-actvated tasks are not present n the COMBES case study, we only consder -actvated tasks n ths report. In the followng, we propose two new approaches to deal wth herarchcal event streams n modular performance analyss. he frst approach uses the well-know FIFO schedulng to manage event herarches as the second approach uses a new concept called Event Count Curves. Both methods are convenent to model the above mentoned scenaro wth hgh accuracy. In Secton we gve detals about the Real-tme Calculus [9], the framework we are usng for the MPA. Furthermore, we summarze the Herarchcal Event Model ntroduced by the authors of [7]. In Secton and 4 we present two dfferent approaches to deal wth herarchcal event streams. he frst approach s consderng FIFO schedulng whle the second uses a new concept denoted as Event Count Curves. Fnally, n Secton, we gve an example how herarchcal event streams can be appled and conclude our work n Secton 6. Related Work Recently, Rox et al. [8] ntroduced the concept of Herarchcal Event Streams (HES) that allow to embed dfferent types of streams n a hgher level structure and proposed a Herarchcal Event Model (HEM) to model such herarchcal streams. he key of ther approach are the so-called nner event streams that are embedded n the HES boundng the event occurrences of events related to an ndvdual nput stream. When an operaton s appled to the HES,. e. the stream s processed n a task, an nner update functon keeps track of the status of the nner event streams. o decompose the HES nto the dfferent nput streams, one only needs to extract the nner event streams from the HES. hs model allows to accurately analyzng the processng and communcaton on the combned as well as on the embedded ndvdual streams. he authors demonstrated sgnfcant mprovements n terms of tghtness of the worst case response tme by usng HES nstead of flat event stream models. Nevertheless, n the processng step of herarchcal event streams n a component, the HEMs have to be deeply processed and therefore each component has to be able to handle such herarchcal streams. Albers et al. [] used a herarchcal data structure to descrbe repettvely occurrng patterns wthn event streams. Snce such scenaros refer to the combnaton of streams, the authors also referred to ther approach as herarchcal event stream, whch s also justfed snce the pattern are gven as PJD-models, so that the overall model can be vewed as herarchcal (or nested) PJD-model. Nevertheless, the topc faced n our report s not related to ther work, snce t does not assume any pattern of event occurrence. he approach presented here takes arbtrary nput patterns. In ths report, we use MPA-RC ntroduced by Chakraborty et al. [] as the bass for our analyss. In ths compostonal approach, a system s modeled as a set of components nterconnected by streams representng event or resource streams. However, contrary to other technques the streams are not defned on a straght tme-lne, but for tme ntervals of length. Formally, we defne the boundng functon ( ) = 4
( u ( ), l ( ) ), also commonly denoted as arrval curve, whch gves the maxmum and mnmum number of events arrvng n any tme nterval of length. In practce such functons can be obtaned from formal specfcatons, traffc models such as perodc wth jtter (PJD-models), or from smulaton traces. Analogous to the upper and lower arrval curves, one may also defne an upper and lower bound for the avalablty of resources ( ) = ( u ( ), l ( ) ). For computng the boundng functons on outgong events ( ) and remanng resources ( ) the prncples of the Real-me Calculus [9] are appled on the nput curves. he method s based on the Network Calculus [] and uses the concept of the mn-plus and max-plus algebra whch defne the mn-plus convoluton ( ) and mn-plus deconvoluton ( ) as well as the analogons max-plus convoluton ( ) and max-plus deconvoluton ( ). he most common component used n the MPA-RC s the greedy processng component (GPC). hs component nstantates a fully preemptable task at every event arrval and actve tasks are sequentally processed n a greedy manner. Snce the resource avalablty s lmted, an nput queue n form of a buffer s needed. he outgong event stream and the remanng resource of the component are gven by the followng relatons: u GP C = mn{(u u ) l, u } l GP C = mn{(l u ) l, l } u GP C = (u l ) l GP C = (l u ) () For a system wth components that are actvated by multple event streams as depcted n Fg. (b), we also need to know how to combne two or more event streams. In [], the authors gve a detaled dervaton and come out wth the followng formula for an -actvated task: u = u ; l = l () FIFO schedulng. System Composton In ths secton, we descrbe how to use FIFO schedulng to deal wth herarchcal event streams. he key of ths approach s that we keep track of the ndvdual nput streams nstead of evaluatng the combned stream. he method reles on the fact, that an -actvaton does not change the order of the arrvng events and forwards them mmedately n a FIFO manner nto the output stream. For ths approach, we omt the step of combnng the streams completely but keep the streams separate as a bundle of streams. Each tme ths bundle s gong to be processed n a task we replace the task wth a FIFO component usng the resource nput of the task and the streams contaned n the bundle as nputs. hs FIFO component s justfed snce also the combned stream would behave lke ths accordng to the -actvaton as stated before. In Fg.
C /, AND C, re Fgure a smple example wth only one task s shown on the left. If we want to model ths system usng FIFO schedulng, we remove the -actvaton and replace the task wth a FIFO component as shown n Fg. on the rght. Fgure, SYSEM, bundle FIFO Number of events related to Fgure : A smple system wth an -actvated task on the left and how t s modeled wth FIFO Fgure schedulng 4a on the rght. Fgure 4b u o evaluate propertes related to the combned stream such as requred buffer szes or end-to-end delays of the packets of the combned stream we have to perform a separate analyss wth the orgnal system. e. we have to combne the streams usng () and FIFO to evaluate the performance analyss wth ths combned stream and the orgnal tasks.. FIFO Component In ths secton, we show how a FIFO component can be mplemented n the context of MPA-RC. n A FIFO component n as depcted n n Fg. (a) modelsa set of tasks n a real-tme system that share an avalable resource n a FIFO manner. As nput, the FIFO component has one resource stream and n event streams,..., n. he remanng resource of the FIFO-component can be calculated usng a varaton of the formula () for a GPC. As nput event stream we us the sum of all the nput streams,..., n accordng to () Fgure n l u = (u F IF O SYSEM l = (l F IF O l ) u ) () o determne the bounds of the output event streams γ, γ,..., n, we analyze the mnmum and maxmum avalable resource for every nput stream, respectvely. Hence, Event Count Curves we consder two cases for every stream : a best-case, where task assocated wth stream has the hghest prorty and a worst-case, where the same task has the lowest prorty. In the frst case, we observe that the maxmum avalable resource u for task s the upper bound u of the total resource nput of the FIFO component. he mnmum resource l avalable for task s the remanng resource after all other tasks as shown n Fg. (b). Usng ths polcy, we can derve the followng formulas to lmt the avalable resource for every stream : 6
Fgure Fgure FIFO FIFO u = u ; l = ( l j u ) (4) j n Fgure 4a n Fgure 4a FIFO FIFO n n (a) Model of the FIFO Component n the Real-me Calculus. n Fgure 4b u Fgure Gven the bounds u, l on the avalable resource for every task, the outgong streams of the FIFO component can then be calculated usng a greedy processng components wth and as ts nputs. Wth () the outputs are gven by SYSEM u F IF O = mn{( u u ), l u } l F IF O = mn{( l u ), l } l () γ, γ 4 Event Count Curves 4. Defnton Event Count Curves n l (b) Best-case (left) and worst-case (rght) scenaro for an event stream. Fgure : Model of the FIFO Component Unlke the Herarchcal Event Model [8, 7] and the approach usng FIFO schedulng ntroduced n the prevous secton, the method usng Event Count Curves (ECC) does not keep track of the ndvdual nput streams. herefore, t can be denoted as compostonal and herarchcal analyss method as a flattenng of the herarchcal event stream s not necessary for beng processed n a tasks. Wth the ECCs, the focus les on the composton and decomposton of the combned stream as shown n Fg. 4. he basc dea of ths approach reles on the fact, that the order of events n an event stream remans the same whatever operaton s appled to the event stream,. e. the sequence of the events before and after a task s the same. herefore, f we store the 7
pattern how the nput streams are combned at the -actvaton or packetzaton unt, we can later on use ths nformaton to decompose the stream nto ts ndvdual substreams. he man advantage of ths method s that besdes the composton and the decomposton of the stream, the methodologes for analyzng the system do not have to be changed or modfed due to the exstence of herarchcal event streams. Furthermore, the sub-system between the -actvaton and ts correspondng decomposton s not lmted to a few components but can be of arbtrary complexty. As mentoned earler, an event stream s gven by an upper and lower bound, whch lmt the occurrences of events n a gven tme nterval. One may note that we consder the tme nterval doman and not the tme doman. Usng such a model, t s not possble to evaluate the exact pattern of the composton of the combned stream after an actvaton of several event streams, but we can gve hard bounds for the number of events related to a specfc nput stream wth respect to the number of events n the combned stream. Defnton (Event Count Curve γ). For a fxed number of consecutve events n the combned stream, the Event Count Curve γ (n) = ( γ u (n), γl (n)) bounds the number of events related to nput stream. he Event Count Curves (ECCs) are a powerful model to descrbe the composton of a combned event stream. hey are related to the workload curves ntroduced by Maxagune et al. [6] but clearly dffer n the usage and meanng. In partcular, the ECCs can be used to decompose the combned stream nto ts sub-streams as follows: heorem (Decomposton wth Event Count Curves). If s the combned output stream that needs to be decomposed and γ the ECC related to events of stream, the sub-stream s gven by the concatenaton of these two curves. ( ) = γ ( ( )) (6) Proof. he evaluaton of ( ) gves the mnmum or maxmum number of events for a gven nterval, respectvely. hs number can be translated usng the ECC γ whch gves then the number of events related to sub-stream for the gven nterval. Example (Event Count Curves). Consder the system gven n Fg. 4 wth the two strct perodcal nput streams and wth perod p = and p = (jtter j = j = ) whch may represent perodc audo sgnals or synchronzaton sequences n the COMBES case study. Example traces of the ndvdual streams as well as of the combned streams are depcted n Fg. (a). he Event Count Curves derved at the -actvaton are shown n Fg. (b). he meanng of the ECC s the followng; f you look at 4 consecutve events n the combned streams, we have at least and at most events from stream whch s expressed by the followng noton: γ l (4) =,γ u (4) =. 4. Jon and Fork In ths secton we descrbe how Event Count Curves can be derved from two nput streams ( ) and ( ) n the context of the RC. We gve a closed soluton how to 8
n n n Fgure SYSEM γ γ, Event Count Curves Fgure 4: System used n Ex.. compute the ECCs, show a detaled dervaton and proof the correctness of the formula. Frst we ntroduce the δ u,l (e) curves that ndcate the maxmum or mnmum tme nterval n whch at most or at least e events occur, respectvely. hese curves are strongly related to the arrval curves u,l ( ). Le Boudec et al. [] defned the pseudonverse functon whch exactly descrbes the relatonshp between u,l ( ) and δ u,l (e): δ l (e) := nf { : u ( ) e} δ u (e) := sup{ : ( ) l e} (7) Wth ths defnton, we can gve the desred closed formula to determne the ECCs. heorem (Event Count Curves n the RC). If and are two arrval curves, then the ECC γ u,l related to stream can be computed as follows γ l (e) = nf{e : e + u (δ u (e )) e} γ u (e) = sup{e : e + l (δ l (e )) e} (8) where δ s the pseudo-nverse of defned n (7). Proof. Frst, we defne the ɛ u,l (e) γ u,l (e ) as pseudo-nverse curves of the event count curves ɛ l (e ) := nf{e : γ u (e) e } ɛ u (e ) := sup{e : γ l (e) e } (9) where e s the number of events n the combned stream and e s the number of events related to nput stream. ɛ u,l (e ) gve the maxmum or mnmum number of consecutve events n the combned stream so that at most or at least e events n ths sequence are related to nput stream a, respectvely. Snce the pseudo-nverse s reversble we can express γ u,l (e) usng ɛ u,l (e ) n the followng way 9
BUS entatve of ned stream the C, t Fgure 6 representatve of stream representatve of stream Fgure 7 t t (a) Example traces used n Ex.. representatve of combned stream the t Event Count Curve γ Event Count Curve γ Event Count Curve γ 4 6 7 Number of events n the combned stream BUS C 4 Upper Curves usng ECC Number of events related to Number of events related to 4 4 6 7 8 Number of events n the combned stream Fgure 4 7 Event Count Curve γ 4 6 7 Number of events n the combned stream Number of events related to 4 4 6 7 8 Number of events n the combned stream Event Count Curve γ 4 6 7 Number of events n the combned stream For the proof, Fgure 8we are consderng events related to stream. γ u,l (e) bounds the number of events e related to as follows BUS γ(e) l e γ u (e) () whch can be wrtten wth the pseudo-nverse ɛ u,l as follows: C 4 6 me nterval (b) Event Count Curves from Ex. where dots ndcate the upper and crosses the lower bounds of the ECCs Fgure : Example traces and event count curves of Ex. Number of events related to γ l (e) := nf{e : ɛ u (e ) e} 4 γ u (e) := sup{e : ɛ l (e ) e} () 4 ɛ l (e ) e ɛ u (e ) () We also need nformaton about the combned stream. We know, that the total number of events e s the sum of the two nput streams and therefore e = e + e ()
must be hold for all tme. If we consder ɛ (e ), the number of events e s fxed and therefore we can defne a tme nterval that satsfes tatve of the d stream t δ l (e ) δ u (e ) (4) hs nterval ndcates the tme n whch exactly e occur n stream. Wth ths nterval we can bound the occurrences of e by the followng bounds Event Count Curve γ 4 6 7 mber of events n the combned stream Number of events related to l ( ) e u ( ) l (δ l (e )) e u (δ u (e )) () Combnng (), () and () we fnd a closed formula for ɛ u,l (e ) 4 Fgure 7 Event Count Curve γ ɛ l (e ) = e + (δ l (e l )) ɛ u (e ) = e + u (δ u (e )) (6) Wth ths formula and the defnton of γ u,l (e ) n () we can derve the Event Count Curve by concatenatng these equatons. 4 6 7 8 Number of events n the combned stream Number of events related to 4 Event Count Curve γ 4 6 7 8 Number of events n the combned stream Case Study US C 4 SERVER NAC NAC NAC DEV NAC DEV DEV Upper Curves usng ECC 4 6 me nterval Fgure 6: Dstrbuted multmeda system wth a server broadcastng audo streams to several devces (DEV) va network access controllers (NAC). In the followng secton, we show how the concept of the herarchcal event streams can be appled to the COMBES case study by EADS [4] ntroduced n Secton. he system descrbed n the case study consder a dstrbuted multmeda cabn system wth a server broadcastng dfferent audo streams to varous playback devces as depcted n Fg. 6. he server s connected to several network access controllers (NACs) that forward the streams ether to another NAC or to the playback devce tself. Hence, we have a dasy-chan lke backbone topology n the cabn. here exsts also other
4 6 7 nts n the combned stream 4 6 7 8 Number of events n the combned stream 4 6 7 8 Number of events n the combned stream 4 NAC NAC NAC SERVER NAC traffc e. g. status and control messages for emergency DEV sgns or smokedev detectors and tme synchronzaton packets that are routed n the same way as the audo streams through the network. he devces are equpped wth a speaker (output for headphones) and control logc so that the user can swtch between the ndvdual audo streams he DEV wants to lsten to. Upper Curves usng ECC SERVER BUS NAC DEV audo C audo audo audo other traffc C other traffc 4 6 me nterval Fgure 7: MPA model of the dstrbuted multmeda system shown n Fg. 6 usng herarchcal event streams. Fg. 7 shows the MPA model of the dstrbuted multmeda system descrbed n the EADS case study [4] wth two audo streams only. Snce these audo streams are broadcast streams t s feasble to combne them to a herarchcal event stream. hat means, we do not have to model a task for every audo stream ndvdually but only one for the herarchcal event stream. herefore, the streams are combned n the server, transmtted over the network and decomposed n the playback devces. On the communcaton channel, modeled as bus, as well as n the network access controllers (NACs) we only see one task for all the audo streams. hs approach s hghly scalable n terms of number of audo streams snce only the server and the playback devces need to be changed f we ncrease or decrease the number of audo sgnals. 6 Concluson In ths report, we showed how FIFO schedulng, a well-known technque, can be used to handle herarchcal event streams n the context of system-level performance analyss based on the MPA-RC. In addton, we ntroduced the concept of the Event Count Curves whch s more modular and also enables one to model herarchcal event streams. Wth an example system we showed the applcablty of herarchcal event streams on a real-world applcaton. References [] K. Albers, F. Bodmann, and F. Slomka. Herarchcal event streams and event dependency graphs: a new computatonal model for embedded real-tme systems. Real-me Systems, 6. 8th Euromcro Conference on, pages pp., -7 July 6.
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