Mantanng temporal valdty of real-tme data on non-contnuously executng resources Tan Ba, Hong Lu and Juan Yang Hunan Insttute of Scence and Technology, College of Computer Scence, 44, Yueyang, Chna Wuhan Martme Communcaton Research Insttute, 4379, Wuhan, Chna Abstract. Ths paper studes the problem of temporal valdty mantenance on non-contnuously-executng resources. Response tme bounds for sensor transacton schedulng are derved n the context of the hybrd exted multprocessor perodc resource model. Then two deadlne and perod assgnment schemes are proposed to mantan the temporal valdty of real-tme data. The calculaton based scheme (DPA-C) uses the response tme bounds to compute deadlnes and perods. The chec based scheme (DPA-A) assgns deadlnes and perods drectly. It then checs the feasblty of the assgnment based on the response tme bounds. Experments are conducted to evaluate the performance of the proposed schemes. The results show that DPA-C performs better than DPA-A n terms of the schedulng success rato and the mposed update worload. Introducton Cyber-physcal systems (CPS) are wdely used n many applcaton areas to montor the envronmental status and mae response to crtcal events. Typcal applcaton areas nclude flght control, health montorng and ndustral process control. In CPS, the real-tme data objects are defned to model the current status of external enttes. Each data object s assocated wth a valdty nterval whch specfy the lfetme of the object s values. A realtme data object s temporally vald f ts current value s valdty nterval doesn t expre []. It s very mportant to guarantee the real-tme data valdty, snce otherwse CPS would mae wrong control decsons based on the nvald data objects. In recent years, there has been a number of wors on temporal valdty mantenance. The More-Less (ML) scheme uses sensor transactons to update the values of real-tme data objects perodcally [, ]. The deferrable schedulng algorthm for fxed prorty transactons (DS- FP) adopts the sporadc tas model [3, 4]. It defers the release tmes of transacton nstances as much as possble by explotng the semantcs of valdty constrants. The earlest deadlne frst (EDF) scheme was adopted n [5, 6] for schedulng sensor transactons. Han et al. proposed two schemes to mantan valdty constrants durng mode swtches [7]. The problem of co-schedulng sensor transactons and applcaton transactons was studed n [8]. L et al. studed the temporal valdty mantenance problem on parttoned multprocessors [9]. Current studes for mantanng the temporal valdty of real-tme data assume the processng resource s contnuous avalable for transacton executon. Ths assumpton, however, s not true n many cases. For example, a complex real-tme system usually conssts of several subsystems. Some subsystems may run on one physcal platform. Each of them can only get a fracton of the resource. As another example, n order to acheve thermal reslency, real-tme systems wll change between dfferent power modes dynamcally. Runnng n a low power mode can be vewed as runnng on a platform wth dscontnuous resource supply. In ths paper, we study how to mantan temporal valdty of real-tme data on platforms where the resource supply s not contnuous. We derves the response tme bounds of sensor transactons under global fxed prorty schedulng scheme n the context of the hybrd exted multprocessor perodc resource model. Based on the response tme bounds, we then present two schemes to assgn relatve deadlnes and perods to transactons so that the real-tme data valdty s properly mantaned. The calculaton based scheme (DPA-C) uses the bounds to compute deadlnes and perods. The chec based scheme (DPA-A) maes the assgnment drectly. It then checs whether the transacton set s feasble under the assgnment. Experments are conducted to evaluate the performance of these two schemes. The results show that compared wth DPA-A, DPA-C can acheve hgher schedulng success rato whle mpose lower update worload. The rest of the paper s organzed as follows. Secton presents real-tme data model and resource model. Secton 3 presents the response tme bounds of sensor transactons and two schemes for deadlne and perod assgnment. The expermental results are gven n secton 4 and n secton 5 we gve the concluson. The Authors, publshed by EDP Scences. Ths s an open access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense 4. (http://creatvecommons.org/lcenses/by/4./).
MATEC Web of Conferences System model Consder a set of real-tme data objects X { X } n and a set of sensor transacton { } n. The valdty nterval of data object X s V. Sensor transacton update the value of X. The worst-case executon tme of s C. Defnton X s temporally vald at tme t, f the samplng tme of X s current value ts( X ) plus V s not less than t, that s ts( X ) V t []. The transacton set s executed on a non-contnuouslyexecutng resource. In ths paper we ntroduce the hybrd exted multprocessor perodc resource model (HEMPR). The model s characterzed by Ω= (П, Δ, Θ, m, m). It specfes that for a resource that conssts of m processors, m processors of t collectvely provde Θ unt of resource n every perod П wthn the deadlne Δ (Δ П). The rest of the processors each can guarantee full resource supply. The HEMPR model can be vewed as a combnaton of the MPR model [] and the dedcated recourse model (DR). If we set m = m and Δ = П, then we get MPR. If m s set to be zero, then we get DR. Thus our model s more flexble than MPR and DR. Notce that the processors n HEMPR can be ether physcal or vrtual processors. The vrtual processors themselves should be allocated to proper physcal processors. There are two approaches to schedule the sensor transactons on multprocessors: parttoned approach and global approach. In the parttoned approach, a transacton wll be allocated to a processor and can be executed only on ths processor. In the global approach, transactons are allowed to mgrate between dfferent processors durng ther executon. In ths paper the global fxed prorty schedulng approach (GFP) s consdered. We assume the transactons are sorted accordng to nondecreasng order of ther valdty ntervals. Hgher prortes are assgned to transactons wth lower valdty ntervals. A transacton nstance s allowed to be executed on one processor at any tme. 3 Temporal valdty mantenance on noncontnuously executng resources 3. Response tme bounds The worst-case resource supply pattern for the HEMPR model can be obtaned by mang some modfcatons to the reasonng for MPR model n []. In our pattern, the resources provded by m processors n the frst perod are allocated from tme zero. For subsequent perods, the resources are allocated as late as possble wthout volatng the deadlne constrant. Then the supply bound functon of HEMPR s gven by: N( mm ) t ( L) t a sbf () t () ( m m ) t otherwse In (), a / m, N ( ta)/, Lmn{, ta N }. ( L) s calculated as follows: La ( L) max{, ( L ) m} b otherwse In (), b / m. () In order to derve the upper bound of a sensor transacton s response tme, we ext Nan s dea of busy perod extenson that s desgned for dedcated resources [].Consder a transacton nstance l, wth release tme rl, and fnsh tme f l,. Let t o denote the earlest tme nstant before r l, such that all resource supply of HEMPR n tme nterval [ to, rl, ) are consumed by hgher prorty nstances. There are at most m sensor transactons wth carry-n nstance. Let t denote the length of the nterval, then the upper bound of total worload of hgher prorty transactons n the nterval s: n NC CI NC m largest CI NC I t and I (,) t (3) W () t I (,) t ( I (,) t I (,)) t The calculaton of (,) s gven by equatons (5)-(8) n [] and s omtted here for brevty. The next problem s to obtan an upper bound for nterference that l, suffers n [ to, f l, ). Let I () t denote the bound. For a dedcated resource, t s W ( t)/ m. But for HEMPR, t s not the case snce the resource supply may not be contnuous. In order to obtan I () t, we compute the parallel supply functons for HEMPR and then use these functons to calculate the bound. A level- supply functon Y () t s the mnmum resource supply n every tme nterval of length t wth parallelsm at most []. Based on the worst-case resource supply pattern, we then have: Y () t t mm (4) sbf ( t, ) otherwse In (4), sbf (,) t s obtaned by replacng m n sbf () t wth and wth ( m)( a) max{ b,}. Let Dt (, j) denote the duraton over an nterval of length t durng whch the tme s provded by j processors n parallel, then : Ymm b() t Ymm () bt j m Ymm() t Ymm() t Dtm (, ) jmm b Dt (, j) (5) t Dtm (, ) Dtm (, m b) j m m otherwse Then accordng to [], I () t s gven by: I() t Dtm (, m) I ( m) { m} S mn{ Dts (, ( )), max{( W ( t) Dts (, ( j)) s( j)),} / s ( ) } j (6) In (6), I{ m} ( m) s an ndcator functon to tae the case m m nto consderaton. S { mm, mm b, m} / {}. s () s -th element of S. Let R denote the mnmal value that satsfes the equaton below: I() t C t (7) Then R rl, tomust be an upper bound of the response tme of for ths partcular t o. The reason s that I () t s an upper bound of nterference on l, n [ to, f l, ), and there are no unused resource unts n [ to, r l, ), so t s not possble to obtan a larger response tme than R rl, to. Snce rl, s not earler than t o, we can see that R s a response tme bound of for arbtrary t o.
3. Deadlne and perod assgnment schemes In order to mantan the temporal valdty constrants, the sum of the relatve deadlne D and the perod T of must be no larger than the valdty nterval V. In addton, the transacton set must be schedulable by GFP on HEMPR. In ths secton two schemes are proposed for perod and deadlne assgnment. Calculaton based scheme (DPA-C) uses the response tme bounds derved n subsecton 3. to calculate deadlnes and perods drectly. Ths calculaton s carred out from to n. For, f R s larger than V /, s consdered as unschedulable. Snce I () t and W () t are both non-deceasng functons of t, R can be obtaned n an teratve way. That s: R I( R) C (8) In (8), R denotes the value of R n -th teraton. R s set to zero. DPA-C s presented n Algorthm. Notce that when m =m, R s set to a C whch s the length of the nterval wthout resource supply. For, the tme to compute W () t s On ( ), then I () t can be calculated n constant tme based on W () t snce there are at most three dfferent parallel levels to be consdered. There are at most V / C teratons to get R. Let V max max{ V n}, then the tme complexty of DPA-C s OnV ( ). max Algorthm Calculaton based Assgnment Input: and Ω Output: deadlnes and perods of f s schedulable, otherwse report falure for = to n = f m < m R C else R a C whle true f R V / Г s unschedulable, return falure else f R R brea compute R usng (8) D R T V D The teratve computaton of DPA-C could be tmeconsumng for transactons wth large valdty ntervals. Dfferent from DPA-C, the chec based scheme (DPA-A) adopts a very smple method to assgn deadlnes and perods. For, the scheme sets D V / ( F ) and T V D. F s the scalng factor of DPA-A. Its value should be not less than and s chosen by system desgners. A schedulablty condton for s obtaned from (7): I( D) C D (9) CI Here W( D) s computed usng only I (, D) to guarantee a safe bound snce the teraton s not used. For each transacton, DPA-A checs whether (9) s satsfed. If not, the transacton set s consdered as nfeasble. Otherwse DPA-A contnues to test subsequent transactons. For each transacton, t s not hard to see that the tme to chec (9) s On ( ). So the tme complexty of DPA-A s On ( ). It s lower than that of DPA-C. However, as one can see from the expermental results, ths reducton comes wth the cost of lower schedulng success rato and hgher mposed worload. 4 Performance evaluaton In ths secton the results of the expermental evaluaton on DPA-C and DPA-A are presented. There are two performance metrcs. The frst metrc s the schedulng success rato. It s defned as the rato of the schedulable transacton sets under the partcular scheme to the total number of transacton sets. The second metrc s the update worload mposed by sensor transactons defned as u / (( ( m m ) ) / ( m )). Ths defnton taes n the dscontnuty of the resource supply nto account. The parameters and ther default settngs are gven n table. The resource deadlne of HEMPR s set to be the same as the resource perod. The rato between Θ and m П s chosen randomly n [.7, ]. The rato between m and m s chosen randomly n [.5, ]. The value of Θ and m can then be obtaned based on the ratos. Table. Parameters and settngs. Parameters Meanng Value V C Valdty nterval Computaton tme N Number of p processors Resource perod [,3] [5,] [,8] Fg. shows the schedulng success rato of DPA-C and DPA-A when the densty of the transacton set s vared from.5 to.5. The densty s defned as the sum of the rato of a transacton s computaton tme and ts valdty nterval. In the experment, N p s set to 4 and the number of dedcated processors s set to be half of N p. The rato between Θ and m П s fxed at.8. The scalng factor for DPA-A s set to. It can be seen that the schedulng success rato of DPA-C s constantly no lower than that of DPA-A. For example, when the densty s about.9, almost all transacton sets can t be accepted by DPA-A whle DPA-C stll can accept any transacton set. The reason s DPA-A uses looser bounds for feasblty test whch wll lead to hgher reject rate of transactons. Fg. shows the update worload generated from two 3
MATEC Web of Conferences Fgure. Schedulng success rato comparson. schemes. The parameter settngs are the same as those descrbed for fg.. It can be seen that the update worload produced by DPA-C s lower than that of DPA- A. Both schemes update worload becomes hgher when the densty ncreases. For temporal valdty mantenance schemes, the update worload s an mport metrc. Reducng the worload of sensor transactons wll leave more resources to other types of transactons, thus the overall system performance s mproved. Fgure 3. Schedulng success rato comparson (densty=.75). named calculaton based scheme (DPA-C) and chec based scheme (DPA-A), are proposed to mantan the valdty constrants. DPA-C calculates deadlnes and perods by usng the response tme bounds. DPA-A maes the assgnment drectly and then checs whether the assgnment s feasble based on the derved bounds. The expermental results show that DPA-C acheves hgher schedulng success rato and mposes lower update worload than DPA-A does. Ths paper only consders the sensor transacton schedulng problem. In the furfure we wll study how to smultaneously schedule sensor transactons and user transactons on non-contnuouslyexecutng resources. References Fgure. Update worload comparson. Fg.3 shows the schedulng success rato of two schemes under dfferent ratos between Θ and m П, whle the densty s fxed at.75. Other parameter settngs reman unchanged. It s observed that the success rato of DPA-C s constantly hgher than that of DPA-A. In addton, both schemes success rato wll ncrease when the recourse capacty ncreases. More experments are carred out by varyng parameter settngs. The results are smlar to what we have presented here and are omtted. 5 Concluson In ths paper, we study the problem of mantanng temporal valdty on platforms where the resource supply s not contnuous. The response tme bounds of sensor transactons scheduled by GFP for HEMPR model are derved. Two deadlne and perod assgnment schemes,. M. Xong, K. Ramamrtham, Dervng deadlnes and perods for real-tme update transactons, IEEE Trans Comput,53(5), 567 583(4). J. Wang, S. Han, K. Y. Lam, et al., Mantanng data temporal consstency n dstrbuted real-tme systems, Real-Tme Syst, 48(4), 387 49() 3. S. Han, D. Chen, M. Xong, et al., Schedulablty Analyss of Deferrable Schedulng Algorthm for Mantanng Real-Tme Data Freshness, IEEE Trans Comput, 63(4), 979 994(4) 4. M. Xong, S. Han, D. Chen, et al., Desh: overhead reducton algorthms for deferrable schedulng, Real- Tme Syst, 44(), 5() 5. M. Xong, Q. Wang, K. Ramamrtham, On earlest deadlne frst schedulng for temporal consstency mantenance, Real-Tme Syst, 4(), 8 37(8) 6. J. L, M. Xong, V. C. S. Lee, et al., Worload- Effcent Deadlne and Perod Assgnment for Mantanng Temporal Consstency under EDF, IEEE Trans Comput,6(6), 55 68(3) 7. S. Han, D. Chen, M. Xong, et al., Onlne schedulng swtch for mantanng data freshness n flexble realtme systems, Proceedng of Real-Tme Systems Symposum, IEEE, Washngton D.C., 5 4(9) 8. S. Han, K. Y. Lam, J. Wang, et al., On Co- Schedulng of Update and Control Transactons n Real-Tme Sensng and Control Systems: Algorthms, 4
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