Improved Worst-Case Response-Time Calculations by Upper-Bound Conditions

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1 Improved Worst-Case Response-Tme Calculatons by Upper-Bound Condtons Vctor Pollex, Steffen Kollmann, Karsten Albers and Frank Slomka Ulm Unversty Insttute of Embedded Systems/Real-Tme Systems Abstract Fast real-tme feasblty tests and analyss algorthms are necessary for a hgh acceptance of the formal technques by ndustral software engneers. Ths paper presents a possblty to reduce the computaton tme requred to calculate the worst-case response tme of a task n a fxed-prorty task set wth tter by a consderable amount of tme. The correctness of the approach s proven analytcally and expermental comparsons wth the currently fastest known tests show the mprovement of the new method. 1. Introducton Many embedded systems have real-tme constrants. To determne whether these real-tme constrants can be met or not, methods of the real-tme analyss are used. Durng the last 3 years varous approaches have been developed to verfy the real-tme behavor of embedded systems, partcularly wth regard to mprovng the run-tme complexty of real-tme and feasblty test algorthms. These test algorthms can be generally classfed nto exact tests and suffcent tests. The advantage of suffcent tests s that they delver ther results relatvely fast compared to the exact tests, but a dsadvantage s that certan task sets cannot be dentfed as feasble although they actually are. Ths acceptance problem of suffcent tests grows wth an ncreasng utlzaton of the consdered task set. Embedded system desgners want to acheve hgh utlzaton of ther embedded processors n order to mnmze the fnal system costs. However, f fast feasblty tests reect sutable task sets, a careful desgn space exploraton of the system s dffcult. A well known approach to exactly analyze the realtme behavour of an embedded system s the response tme analyss frst ntroduced by Lehoczky [4]. Certan commercal tools lke SymTA/S [7] are based on the response tme analyss to evaluate dstrbuted realtme systems by evaluatng the end-to-end deadlnes of chaned tasks. Because of the teratve formulaton of the response-tme analyss, ts run-tme may ncrease n a non lnear way. To counter ths, t s necessary to further reduce the run-tme complexty of the analyss. In ths paper we present such a reducton of runtme complexty. The approach s based on the dea of reducng the number of consdered obs by ntroducng an upper-bound condton to the suffcent real-tme test. We wll show that an algorthm based on ths approach leads to a vastly mproved performance of the test, especally when the utlzaton of the consdered task set s rather hgh. The paper s organzed as follows: Frst we gve an overvew over the related work and ntroduce the model used n lterature and n ths paper. Chapter 3 presents the mprovements we have explored. Both the theoretcal background and the algorthm are gven there. Expermental results showng the mpact of the method to response tme analyss are gven n chapter 4 followed by a concluson at the end. 2. Related Work 2.1. Overvew Many dfferent approaches exst to determne the feasblty of a real-tme system. Thereby, the approaches have to be as accurate as possble and as fast as possble. Much effort has been dedcated to reduce the run-tme complexty of such analyss methods. Lehoczky [4] has ntroduced the worst-case response tme analyss wth arbtrary deadlnes. Södn and Hansson [6] have proposed some lower-bound condtons for the startng ob of ths test to mprove the performance. A smlar approach has been ntroduced by Brl, Verhaegh and Pol [3]. Lu and Layland [5] have developed a suffcent feasblty test for strct perodc task sets scheduled by the /DATE9 29 EDAA

2 rate monotonc polcy. Snce then a lot of condtons for suffcent feasblty tests have been found. A good overvew s gven n [1]. Recently, Bn and Baruah [1] have presented an upper-bound condton for the worst-case response tme of a task. By means of ths bound condton a new suffcent feasblty analyss for perodc tasks has been ntroduced Run-Tme Improved Response-Tme Analyss Before we ntroduce the new analyss method of the worst-case response tme analyss, we wll gve an overvew to the model we use, whch s the same as n the related work. We assume a real-tme system whch has n ndependent tasks Γ {τ,...,τ n 1 } scheduled on a sngle processor by a fxed-prorty schedulng algorthm. Each task s characterzed by a perod T, whch s the nomnal tme between two consecutve obs n the absence of tter, a worst-case executon tme C of a task, whch denotes a maxmum amount of tme that s needed on a reference processor for a ob of the task to complete, a relatve deadlne D denotng the tme after the arrval of a ob, when the ob has to be completed, a tter J descrbng an nterval n whch the arrval of a ob can vary. We wll also use U to refer to the utlzaton of a task whch equals to U C T. Further we wll assume that n 1 U < 1, because f n 1 U 1 then the bound ntroduced by (1) wll not be usable for the proposed mprovement under certan condtons. We wll also assume that the tasks are sorted by decreasng prortes, meanng that τ has a hgher prorty than τ for all, : < < n. In a perodc statc-prorty task set wth tter I k 1 I k mn l N (Ik,l Ik,l (k 1) C,l 1 J C ) T (1) wth I, k s the completon-tme of the k-th ob of task τ as t was shown n [9]. Note that (1) s only vald for obs wthn the level- busy perod as defned n [4]. The response tme of a task s the dfference between ts completon tme and ts arrval tme. R k I k A k where A k max(k T J,). To calculate the worst case response tme of task τ, currently the response tmes of all obs wthn ts busy perod have to be evaluated [8]. The algorthm stops when the followng condton holds I k A k1, thus the worst-case response tme s: R max k...m Rk : m mn({l : I l A l1 }) (2) Södn and Hansson have presented n [6] several methods on how to reduce the run-tme complexty of the response-tme analyss. The man dea of ther presented methods s to defne lower-bound condtons where the analyss can start wthout changng the result of the analyss. Ths allows to skp every evaluaton up to these lower-bound condtons. Most of these methods reduce the number of teratons n the fx-pont teraton by tryng to start the fxpont teraton as close as possble to the fnal result. If the obs of a task are evaluated n successve order, you can start at I, k Ik 1 C [6]. If the tasks are beng evaluated n prorty order, the calculaton can start at I, k Im 1 (k 1) C. Another method s to reduce the number of obs to evaluate by startng wth the last ob that arrves at tme t.u.. For a detaled dscusson of the explaned methods see [6]. Bn and Baruah presented n [1] a suffcent feasblty test, n whch they characterze the completon tme as follows: I k X 1 ((k 1) C ) (3) where X (h) mn t {t : H (t) h}, wth H (t) t W (t) beng the worst-case dle tme and W (t) beng the worst-case workload of the hghest prorty tasks over an nterval of length t [1]. In 3.2 we wll extend the equatons to ncorporate task tter and derve a new upper-bound condton for the response tme. 3. Advanced Job Reducton for Response- Tme Analyss To reduce the number of obs whch have to be consdered, we propose two mprovements. In secton 3.1 we wll frst use the dea to start wth the latest ob for whch we can show that the response tme s greater or equal to the response tme of any of the prevous obs. Second, wth a much greater mpact, we ntroduce a condton whch allows us to stop evaluatng obs as early as possble. It s based on the upper-bound condton presented n [1] for a suffcent analyss by Bn and Baruah. We extend t to ncorporate task tter and use t to mprove the performance of the exact analyss Removng Startng Jobs by Lower-Bound Condton Södn and Hansson have shown n [6] that the evaluaton can be started at ob h max k N {k : A k } (4)

3 Lemma 1: The startng ob as formulated n (4) s equal to h J T. Proof: Assume k J T. Because A k s monotoncally non-decreasng, we only have to show that A k by showng that k T J and A k 1 > by showng that (k 1)T J >. k T J T (k J T ) (k 1)T J T (k 1 J T ) > We wll now mprove the lower-bound condton by followng lemma: Lemma 2: The response tme of ob k J C T of task τ s greater or equal to max l..k 1 R l. Proof: Assume k J C T. Frst we wll show that k J C T k T J C A k C (5) Because of the nature of A k we know that A k 1 and A k 1 > C. If A k then k would be the soluton to (4), where R k > max k...k 1 R k follows. Now we wll consder the case where < A k C. Accordng to [6], the completon tme of ths ob s by at least C greater compared to the completon tme of the prevous ob I k I k 1 C (6) Wth (5) and (6) we can formulate the followng equaton R k R k 1 I k where R k A k max k...k 1 R k follows. I k 1 C A k (7) Snce only up to one ob per task can be reduced, the new lower bound condton s ust a small mprovement compared to prevous methods [6]. In Fg. 1 the new approach s formulated as an algorthm Removng Remanng Jobs by Upper- Bound Condton After mprovng the lower-bound condton of the startng ob of the tasks, we wll now derve a new condton whch wll allow us to skp the evaluaton of all remanng obs from the pont where the new condton holds. The new approach s based on the upper-bound condton for the response tme gven n 1 I 1 # q 2 f o r t o n 1 / n number o f t a s k s / 3 R max 4 k J C T 5 I # I #q 1 kc 6 do 7 I # I #q C 8 q 9 do 1 I #q1 (k 1)C 1 I#q J ( T C ) 11 q q 1 12 whle ( I #q > I #q 1 ) 13 R max max(i #q A k,rmax ) 14 k k 1 15 whle (I #q > A k ) 16 end Fgure 1. Algorthm by Södn and Hansson wth the startng ob mprovement gven n secton 3.1 [1]. Ths upper-bound condton s adapted to ncorporate task tter by extendng the upper-bound condton of the workload. The upper-bound condton of the workload s a lnear functon gven by w o (t) U t x. To calculate the upper-bound condton for a gven task t s necessary to compute the constant x, whch s performed by solvng the equaton w o (kt J C ) (k 1) C gvng us the followng value for x J U C (1 U ), whch results n followng equaton for the upper-bound condton of the workload w o (t) U t J U C (1 U ) (8) Transformng (8) as n [1] H lb (t) t W ub (t) ( 1 U ) (U t J U C (1 U )) l (J l U l C l (1 U l )) h (J U C (1 U )) X ub (h) 1 U where W ub (t) s the upper bound of the workload W (t), H lb (t) the lower bound of the dle tme H (t) and X ub (h) the upper bound of X (h) as mentoned n 2.2, leads to followng upper-bound condton for the completon tme ι k (k 1)C 1 (J U C (1 U )) (9) 1 1 U

4 therefore the response tme s bounded by ρ k ι k A k (1) In the followng sectons of the paper t s assumed that k J. We wll now show that the T U 1 1 U response tme s bounded from above by R ub (k 1)C 1 (J U C (1 U )) (11) 1 1 U by formulatng the followng lemma Lemma 3: ρ k has ts maxmum at k k. Proof: Assume k N. Frst we wll show that A k 1 k > by showng k > J T 1 and A k 1 T 1. showng k J J k T J T U 1 1 U U 1 1 U J T J 1 T by > J 1 (12) T J T 1 (13) From (12) and (13) we can see that ρ k s monotoncally non-ncreasng for all k > k and monotoncally nondecreasng for all k < k. Because A k has the property of beng monotoncally nondecreasng, we now know that A k > for all k > k and that A k for all k < k, thus ρ k1 ρ k C 1 1 U T < k > k (14) ρ k ρ k 1 C 1 1 U > k < k (15) Now we have to show that ρ k where t s assumed that A k >, because n the case that A k the result s the same as n (15) ρ k ρ k 1 ρ k 1 C 1 1 U k T J T ( J T U 1 1 U k ) and addtonally we have to show that ρ k 1 ρ k. Now we wll assume that A k otherwse t would be lke (14) ρ k 1 ρ k C 1 1 U (k 1)T J T ( J T U 1 1 U (k 1)) < Drectly from lemma 3 we can formulate the followng suffcent feasblty test: Corollary 1: A task set scheduled by statc prortes s feasbly f for all tasks the upper-bound condton of the response tme s less or equal to the deadlne R ub (k 1)C 1 J U C (1 U ) 1 1 U D (16) Note that n the case that all tasks do not have a tter : J, ths suffcent test s equvalent to the test presented n [1]. Now we can fnally formulate the condton whch wll help us reduce the number of obs whch have to be consdered by followng corollary Corollary 2: If max l...k Rl ρk1 then R max l...k Rl. Proof: In the case that k < k 1 the condton max l...k R l ρ k1 does not hold, hence only the case where k k 1 has to be consdered. From lemma 3 we know that at k k, ρ k has ts maxmum and for any successve ob ρ k does not ncrease, meanng that at ρ k for any k k t s the maxmum response tme of all the followng obs. Now f the response tme of all followng obs s less or equal to the currently computed maxmum response tme, the currently computed maxmum response tme s the actual worst-case response-tme of the analyzed task. To mplement ths, smply change lne 15 of Fg. 1 from whle (I #q > A k ) to whle (I#q > A k ) (ρk > R max ) From corollary 2 we can also formulate an exact feasblty test for a task Corollary 3: If the currently analyzed ob k k 1 and the currently computed maxmum response tme max l...k R l D and the maxmum response tme of all followng obs ρ k1 D then τ s feasble. See Fg. 2 for an example on how to ncorporate ths nto an exact feasblty test for a task set. 4. Experments The new worst-case response tme analyss and exact feasblty test as developed n secton 3 s compared wth the work of Södn and Hansson [6]. For ths comparson we have generated random task sets wth perods dstrbuted unformly rangng from 1 t.u. up to 1,, t.u., tter was also generated by unformly dstrbutng t n the range between and 5 tmes ts perod, J < 5 T, utlzaton was dstrbuted wth the UUnFast algorthm as proposed n [2]. Wth the generated utlzaton, the worst-case

5 1 I 1 # q 2 f o r t o n 1 / n number o f t a s k s / 3 k J C T 4 I # I #q 1 kc 5 do 6 I # I #q C 7 q 8 do 9 I #q1 (k 1)C 1 I#q J ( T 1 q q 1 11 f (I #q A k > D ) 12 return n o t f e a s b l e 13 whle ( I #q > I #q 1 ) 14 k k 1 15 whle (I #q > A k ) (ρk > D ) 16 end 17 return f e a s b l e ) C Fgure 2. An exact feasblty test average run-tme [ms] n 1, < J < 5 T Soedn / Hansson utlzaton [%] Fgure 3. Run-tme vs. utlzaton 1 4 n 1, < J < 5 T executon tme was set to C U T. Fnally, deadlnes were set to twce the perod D 2 T. Experments conducted wth the startng ob as shown n 3.1 resulted n overall slower computaton tmes compared to startng at the ob proposed by Södn and Hansson n [6]. We assume ths s due to the uncertanty of ts arrval tme. If the evaluaton starts at k J C T the value of A k has to be computed. If the evaluaton starts at k J T then A k and does not have to be computed. The proposed new startng ob does save us to evaluate some obs, but the addtonal computaton tme needed to compute the arrval-tme outweghs the gans from the saved obs. Because of ths we started to evaluate at k J T n our experments. Frst we sampled the utlzaton axs n steps of.1% startng at.1% up to 99.9%. For each step 1 random task sets were generated wth 1 tasks each. Fg. 3 and 4 are showng the average run-tme and the average number of obs evaluated per task set each as a functon of the utlzaton of a task set. Note that the y-axs s scaled logarthmcally. As can be seen the hgher the utlzaton the more effcent the condton becomes. In ths experment the mprovement of the run-tme over all evaluated task sets amounts to almost 5%. If we only consder task sets wth an utlzaton greater or equal to.9 then the mprovement rses to almost 66%. In addtonal experments the utlzaton s set to.8 and the number of tasks was vared from 5 up to 1 tasks per task set n ncrements of 1 task. For each sample we generated 1 random task sets. Fg. 5 and 6 are agan showng the average runtme and the average number of obs evaluated per task set ths tme as a functon of the number of tasks average number of obs evaluated per task set 1 3 Soedn / Hansson utlzaton [%] Fgure 4. Jobs vs. utlzaton n a task set. The experments gve the result that wth an ncreasng number of tasks n a task set, our new approach s more effcent than the prevous one presented by Södn and Hansson. Fg. 7 and 8 show the same relatonshps as Fg. 5 and 6 wth the dfference that the exact feasblty test algorthms were used, see also Fg Concluson In ths paper the mprovements of [1] and [6] are combned and extended to formulate a new faster analyss algorthm for the response-tme analyss. The extenson of the upper-bound condton for the worstcase response tme gven to the response-tme analyss to consder tter and explotng ts propertes, mproves the response-tme analyss dramatcally. Ths result s gven because the new algorthm formulated n ths paper reduces the number of obs consdered by the analyss. The mprovement s compared expermentally wth the work of Södn and Hansson. The experments show that by usng the new algorthm, larger task sets

6 6 U.8, < J < 5 T 16 U.8, < J < 5 T, D 2 T average run-tme [ms] Soedn / Hansson average run-tme [ms] Soedn / Hansson Fgure 5. Run-tme vs. number of tasks Fgure 7. Run-tme vs. number of tasks (feasblty test) 3 U.8, < J < 5 T 14 U.8, < J < 5 T, D 2 T average number of obs evaluated per task set Soedn / Hansson average number of obs evaluated per task set Soedn / Hansson Fgure 6. Jobs vs. number of tasks Fgure 8. Jobs vs. number of tasks (feasblty test) and task sets wth a rather hgh utlzaton are evaluated faster. In future work we wll explore the effect of the run-tme mprovement n the context of dstrbuted systems. References [1] Enrco Bn and Sanoy K. Baruah. Effcent computaton of response tme bounds under fxed-prorty schedulng. In Proceedngs of the 15th Internatonal Conference on Real- Tme and Network Systems, pages 95 14, March 27. [2] Enrco Bn and Gorgo C. Buttazzo. Measurng the performance of schedulablty tests. Real-Tme Systems, 3(1-2): , 25. [3] Render J. Brl, Wm F. J. Verhaegh, and Evert-Jan D. Pol. Intal values for onlne response tme calculatons. In Proceedngs of 15th Euromcro Conference on Real-Tme Systems, pages 13 22, 23. [4] John P Lehoczky. Fxed prorty schedulng of perodc task sets wth arbtrary deadlnes. In Proceedngs of the 11th IEEE Real-Tme Systems Symposum, pages 21 29, December 199. [5] C. L. Lu and James W. Layland. Schedulng algorthms for multprogrammng n a hard-real-tme envronment. J. ACM, 2(1):46 61, [6] Mkael Södn and Hans Hansson. Improved response-tme analyss calculatons. In IEEE Real-Tme Systems Symposum, pages , [7] Symtavson. [8] K. W. Tndell. An extendble approach for analysng fxed prorty hard real-tme tasks. The Journal of Real-Tme Systems, 6: , [9] Ken Tndell and John Clark. Holstc schedulablty analyss for dstrbuted hard real-tme systems. Mcroprocessng and Mcroprogrammng, 4: , Aprl [1] Jana Wu, Jyh-Charn Lu, and We Zhao. On schedulablty bounds of statc prorty schedulers. In RTAS 5: Proceedngs of the 11th IEEE Real Tme on Embedded Technology and Applcatons Symposum, pages , Washngton, DC, USA, 25. IEEE Computer Socety.