Non Pre-emptive Scheduling of Messages on SMTV Token-Passing Networks

Transcription

1 on Pre-emptve Schedulng of Messages on SM oen-passng etwors Eduardo ovar Department of Computer Engneerng, ISEP, Polytechnc Ittute of Porto, Portugal E-mal: Abstract Feldbus communcaton networs am to nterconnect seors, actuators and controllers wthn dstrbuted computer-controlled systems. herefore, they cottute the foundaton upon whch real-tme applcato are to be mplemented. A specfc class of feldbus communcaton networs s based on a smplfed verson of toen-passng protocols, where each staton may trafer, at most, one Sngle Message per oen st (SM). In ths paper, we establsh an analogy between non preemptve tas schedulng n sngle processors and schedulng of messages on SM toen-passng networs. Moreover, we clearly show that concepts such as blocng and nterference n non pre-emptve tas schedulng have ther counterparts n the schedulng of messages on SM toen-passng networs. Based on ths tas/message schedulng analogy, we provde pre-run-tme schedulablty condto for supportng real-tme messages wth SM toen-passng networs. We provde both utlsaton-based and respoe tme tests to perform the pre-run-tme schedulablty analyss of real-tme messages on SM toen-passng networs, coderng RM/DM and EDF prorty assgnment schemes.. Introducton Feldbus networs are wdely used as the communcaton support for dstrbuted computer-controlled systems (DCCS), n applcato rangng from process control to dscrete manufacturng. Usually, DCCS mpose real-tme requrements; that s, traffc must be sent and receved wthn a bounded nterval, otherwse a tmng fault s sad to occur. hs motvates the use of communcaton networs wthn whch the medum access control (MAC) protocol s able to schedule messages accordng to ther real-tme requrements. One of the MAC protocols used n wdely accepted feldbus networs s based on a smplfed verson of the toen-passng protocol. In these feldbus networs (PROFIBUS and P-E are ust two examples) there are typcally two types of networ nodes: masters and slaves. hs MAC protocol s based on a toen-passng procedure used by master stato to grant the bus access to each other, and a master-slave procedure used by master stato to communcate wth slave stato. Each tme a master Francsco asques Department of Mechancal Engneerng, FEUP, Unversty of Porto, Portugal E-mal: vasques@fe.up.pt receves the toen, t wll be able to perform, at most, one message cycle. We denote ths type of networs as Sngle Message per oen st (SM) toen-passng networs. A message cycle costs of a master's acton frame (request or send/request frame) and the assocated responder's acnowledgement or respoe frame. We assume that respoes from slaves are mmedate, wth a bounded turnaround tme. At each master node, requests generated at the applcaton process (AP) level are placed n the master's outgong queue. At each toen arrval, the hghest prorty message cycle wll be processed. We assume the followng message stream model: S = ( C,, D ) () S defnes a message stream n master ( =,..., n). A message stream s a temporal sequence of message cycles concernng, for tance, the remote readng of a specfc process varable. C s the longest message cycle duraton of stream S. hs duraton ncludes both the longest request and respoe tramsson tmes, and also the worst-case slave's turnaround tme. s the perodcty of stream S requests. In order to have a subsequent tmng analyss ndependent from the model of the tass at the applcaton process level, we assume that ths perodcty s the mnmum nterval between two coecutve arrvals of S requests to the outgong queue. Fnally, D s the relatve deadlne of a message cycle; that s, the maxmum admssble tme nterval between the tant when the message request s placed n the outgong queue and the tant at whch the related respoe s completely receved at the master's ncomng queue. Fnally, denotes the number of message streams assocated wth a master. In our model, the relatve deadlne of a message can be equal or smaller than ts perod (D ). hus, f n the outgong queue there are two message requests from the same message stream, ths mea that a deadlne for the frst of the requests was mssed. herefore, the maxmum number of pendng requests n the outgong queue wll be. We denote the worst-case respoe tme of a message stream S as R. hs tme s measured startng at the tant when the request s placed n the outgong queue, untl the tant when the respoe s completely receved at the ncomng queue. Bascally, ths tme span s made up of two components. One concer the tme spent by the request n the outgong queue, untl ganng access to the /00 $ IEEE

2 bus (queung delay). he other concer the tme needed to process the message cycle, that s, to send the request and receve the related respoe (tramsson delay). hus, R = Q C (2) where Q s the worst-case queung delay of a message request from S. Analyss for the worst-case respoe tme can be performed f the worst-case toen rotaton tme s assumed for all toen cycles. Assume also that C M s the maxmum tramsson duraton of a message cycle. If a master uses the toen to perform a message cycle, we can defne the toen holdng tme as: H ρ C τ (3) = M where the symbol τ denotes the tme to pass the toen after a message cycle has been performed and the symbol ρ denotes the master s worst-case reacton tme. As the toen rotaton tme s the tme nterval between two coecutve toen vsts, the worst-case toen rotaton tme, denoted as, s: = n H (4) Coderng prorty-based dspatchng mechams, the worst-case respoe tme for a message request occurs when the request s placed n the master's queue ust after the toen arrval, hence not beng able to be processed n that toen vst. If there were any other message request pendng before the toen arrval, then the toen would have been used to tramt that message; otherwse, the master would not use the toen at all. herefore, the worstcase respoe tme of a message stream S wll be: R = Q C = Φ C (5) where the frst term denotes the message blocng, and the symbol Φ denotes the number of hgher-prorty messages (nterference) that can be scheduled ahead of a message from S. Fg. llustrates the case where Φ equals 0 (hghest prorty message cycle S ). he remander of ths paper s orgaed as follows. In Secton 2 we survey some relevant results for the prortybased schedulablty analyss of real-tme tass, both for fxed and dynamc prorty assgnment schemes. We gve emphass to the worst-case respoe tme analyss n non pre-emptve contexts, snce that analyss s of paramount mportance to the message schedulablty analyss n SM toen-passng communcaton networs. In Secton 3 we dscuss the analogy between tas schedulng n a sngle processor envronment and message schedulng n SM toen-passng networs. Based on ths tas/message schedulng analogy we provde utlsatonbased tests and respoe tme tests, n Secton 4 and Secton 5, respectvely, for SM toen-passng networs. In both cases, RM/DM and EDF prorty assgnment schemes are codered. Fnally, n Secton 6, some concluso are drawn. hghest prorty request from master released margnally after the toen arrval τ 2 H a lower prorty message nduces a prorty nverson wth length ρ req(s ) res(s ) 3 2 Fgure 2. Schedulablty Analyss of ass C M hghest prorty requests processed here Bus Master holdng the toen Real-tme computng systems wth tass dspatched accordng to a prorty-based polcy (we coder only RM/DM or EDF), must be tested a-pror n order to chec f, durng run tme, no deadlne wll be mssed. hs feasblty test s called the pre-run-tme schedulablty analyss of the tas set. here are manly two types of analytcal methods to perform pre-run-tme schedulablty analyss. One s based on the analyss of the processor utlsaton. he other s based on the respoe tme analyss for each ndvdual tas. In [], the authors demotrated that by coderng only the processor utlsaton, a test for the pre-run-tme schedulablty analyss could be obtaned. Contrarly, a respoe tme test must be performed n two stages. Frst, an analytcal approach s used to predct the worst-case respoe tme of each tas. he values obtaned are then compared, trvally, wth the relatve deadlnes of the tass. he utlsaton-based tests have a maor advantage: t s a smple computaton procedure, whch s appled to the overall tas set. By ths reason, they are very useful for mplementng schedulers that chec the schedulablty onlne. However, utlsaton-based tests have also mportant drawbacs, when compared wth ther respoetme counterparts. hey do not gve any ndcaton of the actual respoe tmes of the tass. More mportantly, and apart from partcular tas sets, they cottute suffcent but not necessary condto. hs mea that f the tas set passes the test, the schedule wll meet all deadlnes, but f t fals the test, the schedule may or may not fal at runtme (hence, there s a certan level of pessmsm). It s also worth mentonng that the utlsaton-based tests cannot be used for more complcated tas models [2]. 2.. Schedulablty ests: Fxed Prortes 2... Basc Utlsaton-Based est For the RM prorty assgnment, Lu and Layland ntroduced an utlsaton-based suffcent test: /00 $ IEEE

3 ( 2 ) C (6) = hs utlsaton-based test s vald for perodc ndependent tass, wth relatve deadlnes equal to the perod, and for pre-emptve systems. In [3], the authors provde an exact analyss: l mn U n,, (7) = l (, l ) R where R = {(,l)} wth and l =,..., /. It s clear that nequalty (7) s not an easy to use test, hence loosng one of the advantages nherent to the more basc formulato: ts smplcty Extended Utlsaton-Based est In [4], the authors update the utlsaton-based test (6) to nclude blocng perods, durng whch hgher-prorty tass are bloced by lower-prorty ones, to solve the problem of non-ndependence of tass: (, ), C B = 2 (8) where B s the maxmum blocng a tas can suffer. hs analyss can be extended to a non pre-emptve context, snce n ths case a hgher-prorty tas can also be "bloced" by a lower-prorty tas. Assumng that the tass are completely ndependent, the maxmum blocng tme a tas can suffer s gven by: B = 0, f P = mn { P } = { } =,.., { } (9) B max C, f P mn P lp( ) =,.., where lp() denotes the set of lower-prorty tass (than tas ). herefore, nequalty (8) can be used as an utlsaton-based test for a set of non pre-emptable ndependent tass, wth the blocng for each tas as gven by (9). Moreover, acceptng an ncreased level of pessmsm, nequalty (8) can be smplfed to: = C B max, ( 2 ) (0) ote that f all tass have the same computaton tme, (0) coders that each tas may be bloced at the rate of the hghest-prorty tas Respoe me ests: Pre-emptve Context In [5] the authors proved that the worst-case respoe tme R of a tas s found when all tass are synchronously released (crtcal tant) at ther maxmum rate. R s defned as: R = I C () where I s the maxmum nterference that tas can experence from hgher-prorty tass n any nterval [t, t R ). Wthout loss of generalty, t can be assumed that all processes are released at tme tant 0. Coder a tas wth hgher-prorty than tas. Wthn the nterval [0, R ), t wll be released R / tmes. herefore, each release of tas wll mpose an nterference C. he worstcase respoe tme R of a tas τ s then: R = R C hp( ) C (2) where hp() denotes the set of hgher-prorty tass (than tas ). Equaton (2) embodes a mutual dependence, snce R appears n both sdes of the equaton. he easest way to solve such equaton s to form a recurrence relatohp [6]: m m W W = C C hp( ) (3) he recurson ends when W m = W m = R and can be solved by successve terato startng from W 0 = C. Indeed, t s easy to show that W m s non-decreasng. Coequently, the seres ether converges or exceeds (case of RM) or D (case of DM). If the seres exceeds (or D ), the tas τ s not schedulable Respoe me ests: non Pre-emptve Context In [6] the authors updated the analyss of [5] to nclude blocng factors ntroduced by perods of non pre-empton, due to the non-ndependence of the tass. he worst-case respoe tme s updated to: R R = B C C hp( ) (4) whch may also be solved usng a smlar recurrence relatohp. B s also as gven by equaton (9). Some care must be taen usng equaton (4) for the evaluaton of the worst-case respoe tme of non preemptable ndependent tass. In the case of pre-emptable tass, wth equaton (2) we are fndng the processor's level- busy perod precedng the completon of tas ; that s, the tme durng whch tas and all other tass wth a prorty level hgher than the prorty level of tas stll have processng remanng. For the case of non preemptve tass, there s a slght dfference, snce for the evaluaton of the processor's level- busy perod we cannot nclude tas tself; that s, we must see the tme tant precedng the executon start tme of tas. herefore, equaton () can be used to evaluate the tas's respoe tme of a tas set n a non pre-emptable context and ndependent tass, where the nterference s: ( ) I I = B C (5) hp /00 $ IEEE

4 ote also that a re-defnton for the crtcal tant must be made. he maxmum nterference occurs when tas and all other hgher-prorty tass are synchronously released ust after the release of the longest lower-prorty tas (than tas ) Schedulablty ests: Dynamc Prortes Basc Utlsaton-Based est For the EDF prorty assgnment, Lu and Layland also ntroduced an utlsaton-based pre-run-tme schedulablty test (6), vald for non pre-emptve, ndependent and perodc tass, for whch the relatve-deadlne s equal to the perod. C (6) = Inequalty (6) can be easly updated to nclude blocng perods due to the non-ndependence of the tass. In [7], the author updated nequalty (6) to: C B =,, (7) where B s the maxmum blocng a tas can suffer, coderng the stac resource protocol (SRP). he ey dea behnd the SRP s that when a ob needs a resource whch s not avalable, t s bloced at the tme t attempts to pre-empt, rather than later. hs maes nequalty (7) vald for sets of non pre-emptable tass, dspatched accordng to the EDF scheme. Smlarly to the updatng of (8) to (0), (7) can be updated to a smpler (but more pessmstc) test: = C B max (8), where B s defned as: B = max C (9) { } Another relevant result from [7] s that (7) can also be extended to tas sets wth relatve deadlnes smaller than perods: C B = D, (20), D As a corollary, nequalty (6) can be extended for tas sets wth D : C (2) D = hese smple utlsaton-based tests ((8) and (20)) are however qute pessmstc. Less pessmstc utlsatonbased tests wll now be addressed n Secto and 2.2.3, for pre-emptve and non pre-emptve tass, respectvely. Later, n Secto and 2.2.5, recent results on respoe tme analyss wll be addressed, for pre-emptve and non pre-emptve tass, respectvely Extended Utlsaton ests: Pre-emptve Context In [8] the author extends the results of Lu and Layland n order to coder sporadc tass, where nequalty (6) s updated to: = t D C t, t 0 (22) wth x = 0 f x < 0. hs formulaton has advantages over (20), n the see that t tur out to be a necessary and suffcent condton. Inequalty (22) can not be classfed as a smple test when compared to (20). It has also an addtonal problem, snce t must be checed over an nfnte contnuous tme nterval [0, ). A smplfcaton to the schedulablty test can be made coderng that the rght sde of nequalty (22) does only change at D tme tants, and thus the nequalty does only need to be checed for these tme tants. Dfferent authors have addressed the problem of fndng an upper lmt for t. It s possble to prove that f the total utlsaton of the processor s (condton (6)), t exsts a pont t max, such that =,..,n( (t D )/ C ) t always hold, t tmax. Coequentely, nequalty (22) can be re-wrtten as follows: = t D C t, t wth S = = S, U{ D, ℵ} [ 0, tmax ) (23) In [9] the authors demotrated that t max could be gven by (U/(-U)) max =,...,.{(-D)}, where U represents the overall processor's utlsaton ( =,...,. (C / )). hs result was further mproved n [0], where the upper bound for t s defned as t max =(( =,...,. (-D / ) C )/(-U). Although ths last formulaton gves a smaller value for t max, t stll suffers from the same dsadvantage: as the overall utlsaton approaches, ts value becomes very large. Another approach s codered n [0] and [], where the authors demotrate that t max = L (synchronous processor's busy perod). he synchronous processor's busy perod s defned as the tme nterval from the crtcal tant up to the frst tant when there are no more pendng tass n the system: L L = C (24) = Equaton (24) may be solved by recurrence, startng wth L 0 = =,..,C Extended Utls. ests: non Pre-emptve Context For the non pre-emptve context, a smlar test was presented n [8] and [2]: /00 $ IEEE

5 t D C max { C } t, t D,,..., mn = (25) = wth D mn = mn =,..., { D } Comparng to the test for the pre-emptve context (22), the ncluson of the blocng factor s ntutve (see Secton 2.2..). However, n [3] the authors dscuss the pessmsm nherent to the nequalty (25). he man argument s that n ths nequalty t s codered that the cost of possble prorty nverso s always ntated by the longest tas and, moreover, t s effectve durng the entre nterval under analyss. o reduce ths level of pessmsm, t s suggested the followng modfcaton: = t D C max =,..., D > t wth max =,..., D > t { C } { C } t, = 0 f / t S, : D > t (26) hat s, the blocng factor s only ncluded f ts deadlne occurs after t Respoe me ests: Pre-emptve Context he worst-case respoe tme analyss for pre-emptve EDF schedulng was frst ntroduced n [4]. In hs wor, Spur demotrated that the worst-case respoe tme of a tas s found n the processor's deadlne- busy perod (analogous to the processor's level- busy perod n the case of fxed prortes). However, the longest processor's deadlne- busy perod may occur when all tass but tas (contrarly to the case of fxed prorty assgnment) are synchronously released and at ther maxmum rate. hs mea that, n order to fnd the worst-case respoe tme of tas, we need to examne multple scenaros wthn whch, whle tas has an tance released at tme a, all other tass are synchronously released at tme t = 0. hus, gven a value of a, the respoe of an tance of tas s: R ( a) = { C, L ( a) a} max (27) where L (a) s the length of the deadlne- busy perod, whch starts at tme tant t = 0. L (a) can be evaluated by the followng teratve computaton: L ( a) = D a D L mn ( a) a C a D D, C (28) Equaton (28) can be solved by recurrence, startng wth L 0 (a) = 0. Obvously, n equaton (28), the computatonal load only coders tass that have deadlnes earler than D. Fnally, n the general case, the worst-case respoe tme for a gven tas s: R a 0 { R ( a) } = max (29) he remanng problem s how to determne the values of a. Loong to the rght-hand sde of equaton (28), we can easly understand that ts value only changes at D D steps. { D D, ℵ 0 } [ 0 L[ a U, (30) = wth L as gven by equaton (24) Respoe me ests: non Pre-emptve Context he worst-case respoe tme analyss for the non preemptve EDF schedulng was ntroduced n [3]. he man dfference from the analyss for the pre-emptve case s that a tas tance wth a later absolute deadlne can cause a prorty nverson. hus, and smlarly to the fxed prorty case (Secton 2.2.4), tead of analysng the deadlne- busy perod precedng the completon tme of tas, we must analyse the busy perod precedng the executon start tme of the tas s tance. Coequently, the respoe tme of the τ s tance released at tme a s: R ( a) = { C, L ( a) C a} max (3) where L (a) s now the length of the busy perod (precedng executon). hus, R (a) can be evaluated by mea of the followng teratve computaton: L ( a) = max { C } D a D D > a D ( a) L mn a D D, a C C (32) 3. Analoges to Message Schedulng n SM In ths secton we dscuss the analogy between tas schedulng n a sngle processor envronment and message schedulng n SM toen-passng networs. hs analogy wll later enable the formulaton of feasblty tests for the pre-run-tme schedulablty analyss of message stream sets n SM toen-passng networs. able as Computaton me (C) Perod () A 0 60 B 0 80 C 0 00 D 0 00 In the schedulablty analyss of tass n the non preemptve context, the concept of processor's busy perod denotes the tme nterval wthn whch the processor s not dle (see Secton 2..4). Coder the tas set example (D=) of able /00 $ IEEE

6 When ths request appears, there s no pendng tas as D Maxmum blocng for a hgher prorty tas (as all Cs are equal) me utlsaton of the shared resource (processor) by a tas nterval. hs mea that the contrbuton of each hgherprorty message cycle to the overall queung delay of a lower-prorty message cycle s always equal to. as C as B Message D Maxmum blocng for any-prorty message cycle me utlsaton of the shared resource (toen/networ) by a message cycle, as seen by the local staton as A Message C processor busy perod release of tas Message B Fgure 2 Assumng a RM prorty assgnment polcy n a non pre-emptve context, Fg. 2 llustrates a tme-lne coderng that the frst tance of tas D (lower-prorty tas) s released margnally after tme tant 0, and before all other tances of hgher-prorty tass. ote that the blocng of a tas n a non pre-emptve context s equal to the maxmum executon length of a lower-prorty tas (see equaton (9)). Coder now the followng message stream set example (where D=) to be scheduled n a SM toenpassng networ: able 2 Message Message Cycle Length (C) Perod () A 2 60 B 2 80 C 2 00 D 2 00 hs case wll be shown to be loosely equvalent to the prevous tas schedulng example, when the toen cycle tme s equal to the tass' executon tme ( = C tas ). Coder now Fg. 3, whch llustrates the tme-lne for a message schedulng on a SM toen-passng networ, coderng a messages' release pattern (arrval of requests to the outgong queue) smlar to the prevous tass' release pattern. It s clear that the message blocng tme s equal to the toen cycle tme. However, ths blocng term s ndependent of the prorty orderng of message trafers. herefore, the blocng problem n the tas schedulng theory can only be codered to be loosely equvalent to the blocng problem n SM toen-passng networs, snce the prorty orderng property s not preserved. It s also clear that tests avalable for the schedulablty analyss of non pre-emptable tass n sngle processor systems can be adapted to the message schedulng n SM toen-passng networs, coderng that the blocng term s equal to the toen cycle tme, ndependently of the message prorty. herefore, the computaton tme of a tas can be codered equvalent to the toen cycle tme, snce n a SM toen-passng networ the shared resource (networ access/toen) s avalable once n every Message A toen busy perod Arrval of the request to the queue Fgure 3 Itant of toen arrval Fnally, n able 3 we summarse the analoges between tas schedulng n non pre-emptve sngle processor envronments and message schedulng n SM toenpassng networs. able 3 as Schedulng B = max C Maxmum blocng (except for { } the lowest prorty tas/message) lp( ) Maxmum blocng (lowest prorty tas/message) Resource usage tme for the hgher-prorty tass/messages Resource usage tme for the tas/message tself Message Schedulng 0 Coderng these analoges between tas schedulng and message schedulng, a set of (toen) utlsaton-based and respoe tme tests can be developed for the schedulablty analyss of SM toen-passng networs. In the followng secto we present both types of tests. C C 4. (oen) Utlsaton-Based ests In ths secton, we derve utlsaton-based schedulablty tests for both fxed and dynamc prorty assgnment schemes. Such schedulablty tests, whch can be qute pessmstc, provde a tool to evaluate the schedulablty of the overall message set wth a reduced complexty. 4.. Case of Rate Monotonc Prorty Assgnment Coderng the analoges to the blocng and tass' computaton tme drawn n the prevous secton, the schedulablty test (0) for the RM dspatched tass can be adapted to encompass the characterstcs of the SM toen-passng protocol, as follows: = max n 2, C (33) /00 $ IEEE

7 As the worst-case toen cycle tme () s cotant, equaton (33) can be re-wrtten as: = mn { } 2, (34) ote that, as we are coderng SM toen-passng networs, the nterference from other masters s only reflected on the evaluaton of the parameter. Coder the followng example, whch hghlghts the use of the proposed schedulablty test (34): Stream S able 4 Perod Coderng that the worst-case toen rotaton tme s =, t follows that the schedulablty test s: = herefore the message stream set of able 4 s schedulable by the RM algorthm n a SM toenpassng networ. In Fg. 4, we present a possble tme-lne for the message schedulng, assumng that all messages are requested ust after the frst toen arrval. In ths way, we represent a blocng term at the begnnng of the tme-lne. hs example hghlghts the pessmsm assocated to the utlsaton-based tests, snce, although the schedulablty test s ust margnally true, none of the message cycles s scheduled close to ts deadlne Case of EDF Prorty Assgnment Coderng agan the analoges to the blocng and tass' computaton tme drawn n Secton 3, the schedulablty test (8) for the EDF dspatched tass can also be adapted to encompass the characterstcs of the SM toenpassng protocols, as follows:, { } = mn (35) n Coder now the example of able 5, where perods are codered to be margnally smaller than multples of the worst-case toen cycle tme ( = ). he applcaton of the schedulablty test (35) to ths message stream set s: RUE = 4 Hence, ths message stream set s schedulable coderng the EDF prorty assgnment scheme, whle wth the RM assgnment scheme would not pass the schedulablty test (34): s FALSE. Requests at the AP level Message cycle processed Prorty queue (RM) S S low h able 5 Stream S Perod Fgure 4 5. Respoe me ests In ths secton we derve respoe tme schedulablty tests for both fxed and dynamc prorty assgnment schemes. Such schedulablty tests, compared to the (toen) utlsaton-based tests are more complex, but also much less pessmstc, as t wll be shown n ths secton. hs s an expected result, as respoe tme tests for tas schedulng are suffcent and necessary condto, whle the utlsaton-based tests are generally only suffcent condto (refer to Secton 2). It s also mportant to note that for the case of D < there are no smple utlsatonbased tests for the case of fxed prortes. 5.. Respoe me ests: Fxed Prorty Based on the analoges between tas schedulng and message schedulng on SM toen-passng networs, the worst-case respoe tme analyss for the non pre-emptve context (refer to Secton 2..4) can be adapted to encompass the characterstcs of the SM toen-passng protocols. he worst-case message respoe s defned n equaton (2), where Q s: Q Q Q = = (36) hp ( ) hp ( ) ote that ths queung delay s the equvalent to the tas's nterference n a non pre-emptve context (5) /00 $ IEEE

8 Coderng agan the message stream set example of able 5, the worst-case respoe tme for each message stream wll be as shown n able 6 (wth C = 0.2, ). able 6 Stream Respoe S Coderng the message stream, the terato for evaluatng ts queung delay are as follows: Q ( 0) ( ) ( 2) ( 3) ( 4) ( 5) 4 = ; Q4 = 4; Q4 = 5; Q4 = 6; Q4 = 7; Q4 = (5) (4) and terato stop, snce Q 4 = Q 4 = 7. herefore, from equaton (2) R 4 = = 7.2, whch s smaller than ts relatve deadlne (ts perod), and thus, the message stream set s RM schedulable. Coderng the same message stream set, the (toen) utlsaton-based test (34) gves , whch s equvalent to state that ths message stream set may or not be schedulable. herefore, t tur out that the respoe tme test s much less pessmstc than the (toen) utlsaton-based test. he tme-lne presented n Fg. 5 llustrates the above results. S 7 that s, a message request concernng stream S wll be delayed by all message requests of other streams havng earler or equal absolute deadlnes than the absolute deadlne for S (absolute deadlnes are the dfference between the relatve deadlne, D, and the begnnng of the evaluaton nterval - assumed at tme tant 0). ote that whle ( Q / ) requests havng relatve deadlnes smaller or equal to D can be placed n the AP queue, from those requests, only a maxmum of (D - D )/ wll have absolute deadlnes earler than D. We llustrate ths effect wth the followng example. Stream S able 7 Perod If we coder the synchronous release pattern for message streams (able 7), a tme-lne for the EDF schedule may be as llustrated n Fg. 6. Requests at the AP level S Requests at the AP level Message cycle processed Message cycle processed Prorty queue (EDF) low low h Prorty queue (RM) S h Fgure 6 S Fgure Respoe me ests: Dynamc Prorty Based on the analoges between tas schedulng and message schedulng on SM toen-passng networs, the worst-case respoe tme analyss for the non pre-emptve context (refer to Secton 2.2.5) can also be adapted to encompass the characterstcs of the SM toen-passng protocols. he worst-case message respoe tme s, obvously, gven by equaton (2). However, a maor dfference exsts for the defnton of the queung delay, whch for the EDF case must be defned as: Q D D Q = mn, (37) D D As t can be seen from Fg. 6, there s a request for S arrvng to the queue before the processng of the frst request for S 4. However, as that request for S has an absolute deadlne whch s later than the absolute deadlne for S 4, t wll be processed only after the request for S 4. hs behavour of the EDF scheduler s effectvely tralated by equaton (37), as can be seen by the followng successve terato ( = ): ( 0) Q = mn, mn, mn, Q { } { } { } 4 { 2,} mn{,} mn{,} 4 4 = ( ) 4 = mn = () (0) and terato stop, as Q 4 = Q 4 = 4. he maxmum queung delay for a request of stream S 4, coderng that the streams have a synchronous release pattern, s thus as shown n Fg. 6. ote however that the worst-case respoe tme for EDF dspatched messages s not necessarly found wth a /00 $ IEEE

9 synchronous release pattern (refer to Secto and 2.2.5). herefore, equaton (37) must be updated to: Q ( a) = B ( a) ( ) Q a a D D a mn, (38) D a D where B s defned as follows:, a = 0 B ( a) = (39), a 0 : D > a D ote that whle wth the RM/DM approach (Secton 5.) the blocng term s and effectve for all the message streams, wth the EDF approach, we must only coder (f a 0) a blocng f t exsts a message stream S ( ) wth an absolute deadlne later than the relatve deadlne of the tance of S released at tme tant a. A man dfference exsts n comparson to the analogous formulaton for tas schedulng (32), snce n the case of the SM toen-passng model, for a = 0 there s always a blocng wth the value. Smlarly to the case of tas schedulng, a belongs to the followng set of values: a 0, l l 0 l = { Ψ D D, Ψ ℵ } [ 0, L[ where the (toen) synchronous busy perod s gven by: L L = = he queung delay s thus: Q a { 0 Q ( a) a} (40) (4) = max, (42) snce the computaton Q (a) may occasonally gve a value smaller than a (for tance, when the value of a corresponds to more than one request of S durng the nterval under analyss, the nterval [0,Q (a)]. Fnally, substtutng equaton (42) bac n equaton (2), the worst-case respoe tme of a message stream dspatched accordng to the EDF scheme s: a { 0 Q ( a) a} C R = max, (43) he analyss outlned wll be now llustrated for the stream set example of table 7. he results presented were obtaned usng the followng exact charactersaton of the message stream set of table 7: able 8 Stream S C D For ths message stream set, the value for L (upper bound for a) s: L = 9. herefore, the values of a that must be tested for each message stream are: able 9 Stream a = 0 a a2 a3 a4 a5 a6 a7 S In order to evaluate the queung delay for each release pattern, equaton (7.8) must be evaluated for each a value. he results for (Q (a) - a) are: able 0 Stream a = 0 a a2 a3 a4 a5 a6 a7 S In ths table, for each message stream the value of max{0, Q (a) - a} s hghlghted. he worst-case respoe tmes for the message streams are presented n able. able Stream Respoe a S = = = = herefore, the message stream set s EDF schedulable, snce R (D ),, whle t would not be schedulable by RM. In fact, stream S 4, and usng equaton (36), wll have the followng worst-case queung delay: ( 0) ( ) ( 2) ( 3) ( 4) ( 5) Q4 = ; Q4 = 4; Q4 = 5; Q4 = 6; Q4 = 7; Q4 = 7 and thus R 4 = = 7.2, whch s larger than 4 (D 4 )= Fg. 7 puts ths to evdence. Requests at the AP level Message cycle processed Prorty queue (RM) S S S2 low h Fgure 7 wo message requests belongng to the same stream: a deadlne was mssed /00 $ IEEE

10 As a fnal remar, t s mportant to note that the stream set of able 8 does not emphasse the mportance of parameter a. hs s only due to the specfc characterstcs of the partcular stream set. In fact, the results n ables 0 and show that coderng a = 0 corresponds vrtually to the actual worst-case respoe tme. he followng example wll better llustrate the mportance of parameter a n the evaluaton of the queung delay. he only dfference to the prevous example s n the value of D 2. Stream S C able 2 D For ths stream set example, the set of values for a would be (L = 9): able 3 Stream a = 0 a a2 a3 a4 a5 a6 a7 S Usng the resultng values for each Q (a), the dfference (Q (a) - a) s: able 4 Stream a = 0 a a2 a3 a4 a5 a6 a7 S As t can be seen, for stream, wth a = 0.09, the queung delay (as compared to the case of a = 0) ncreases from.00 to.9. hs s an understandable result, as ts "absolute deadlne" wll then be = 3.99, and therefore, S wll be scheduled earler. 6. Concluso he man contrbuton of ths paper was the adaptaton, by provdng the convenent analoges, of the feasblty tests avalable for non pre-emptve tas schedulng to the schedulng of messages n SM toen-passng networs. We reasoned on how the blocng effect (resultng from non pre-empton) n the schedulablty analyss of tass could be mapped to each case of prorty scheme used to schedule messages. We showed how the worst-case executon tme of tass could be tralated to the upper bound of the toen rotaton tme n SM toen-passng networs. More mportant, we demotrated how the smple utlsaton-based feasblty tests for non preemptve ndependent tass could be easly adapted to be used as (toen) utlsaton-based tests. However, as these tests can be qute pessmstc, we developed respoe-tme tests whch were also adapted from the well now respoe tme tests used for RM/DM scheduled non pre-emptable ndependent tass and we also adapted the more recently developed respoe tme tests for EDF scheduled non preemptable ndependent tass. Acnowledgements hs wor was partally supported by ISEP, FLAD, DEMEGI and FC. References [] Lu, C. and Layland, J. (973). Schedulng Algorthms for Multprogramng n Hard-Real-me Envronment. In Journal of the ACM, ol. 20, o., pp [2] ndell, K. (992). An Extendble Approach for Analysng Fxed Prorty Hard Real-me ass. Department of Computer Scence, Unversty of Yor, echncal Report YCS-89. [3] Lehoczy, J. (990). Fxed Prorty Schedulng of Perodc as Sets wth Arbtrary Deadlnes. In Proceedngs of the th IEEE Real-me Systems Symposum, pp [4] Sha, L., Raumar, R. and J. Lehoczy (990). Prorty Inhertance Protocols: an Approach to Real-me Synchroaton. In IEEE raacto on Computers, ol. 39, o. 9, pp [5] Joseph, M. and Pandya, P. (986). Fndng Respoe mes n a Real-me System. In he Computer Journal, ol. 29, o. 5, pp [6] Audsley,., Bur, A., Rchardson, M., ndell, K and Wellngs, A. (993). Applyng ew Schedulng heory to Statc Prorty Pre-emptve Schedulng. In Software Engneerng Journal, ol. 8, o. 5, pp [7] Baer,. (99). Stac-Based Schedulng of Real-me Processes. In Real-me Systems, ol. 3, o., pp [8] Zheng, Q. (993). Real-me Fault-olerant Communcaton n Computer etwors. PhD hess, Unversty of Mchgan. [9] Baruah, S., Howell, R., Roser, L. (990). Algorthms and Complexty Concernng the Pre-emptve Schedulng of Perodc Real-tme ass on One Processor. In Real-me Systems, 2, pp [0] Rpoll, I., Crespo, A., Mo, A. (996). Improvement n Feasblty estng for Real-tme Systems. In Real-me Systems,, pp [] Spur, M. (995). Earlest Deadlne Schedulng n Real-tme Systems. PhD hess, Scuola Superore Santa Anna, Psa. [2] Zheng, Q., Shn, K. (994). On the Ablty of Establshng Real-me Channels n Pont-to-Pont Pacet-Swtched etwors. In IEEE raacto on Communcato, ol. 42, o. 2/3/4, pp [3] George, L., Rverre,., Spur, M. (996). Preemptve and on-preemptve Real-me Un-Processor Schedulng. echncal Report o. 2966, IRIA. [4] Spur, M. (996). Analyss of Deadlne Scheduled Real- me Systems. echncal Report o. 2772, IRIA /00 $ IEEE