Two Methods to Release a New Real-time Task

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1 Two Methods to Release a New Real-tme Task Abstract Guangmng Qan 1, Xanghua Chen 2 College of Mathematcs and Computer Scence Hunan Normal Unversty Changsha, , Chna qqyy@hunnu.edu.cn Gang Yao 3 Sebel Center for Computer Scence 201 N. Goodwn Avenue Urbana, IL 61801, USA. yaog99@gmal.com In some real-tme systems, sometmes there s a need to nsert a new task at run tme. If the system s fully loaded, the scheduler has to compress one or more current tasks to free some bandwdth. Naturally, fndng the earlest tme to start the nserton wthout deadlne mssng s nterestng, whch s stll under nvestgaton. Instead of ths, ths paper dscusses how to do the compresson and nserton to make the nserton as early as possble, based on the conclusons n the publshed lterature wth the earlest deadlne frst algorthm(edf). By ntroducng the nstance remanng bandwdth(rb), an algorthm called maxmum remanng bandwdth compresson(mrbc) s provded to choose a proper task as the compressed one. An assocated theorem and ts lemma are also provded, whch declares that the safety adjustment can be mplemented mmedately upon request f the ncrease of the bandwdth due to nserton does not exceed the lowest utlzaton of ndvdual current tasks n the system and f every task s allowed to be compressed. Another algorthm named part utlzaton frst(puf) s also proposed, accordng to whch a new task s nserted wth a lower speed frst, then to ts full utlzaton after a certan tme. Keywords: remanng bandwdth; task nserton; real-tme system; deadlne. I. INTRODUCTION The utlzaton of a real-tme task s often referred to as ts bandwdth n perodc real-tme systems. Wth a fully loaded system scheduled wth EDF, the sum of the utlzatons of the tasks n the system s equal to 1 [1]. In ths case, f a new task requests to be nserted, the system wll become unschedulable,.e. deadlne mssng wll occur. Compressng one or more current tasks (decreasng ther utlzatons) to free some bandwdth s usually a way to tackle ths [2, 3]. In [2], Buttazzo presented an elastc schedulng model for the task set based on EDF, the problem of new tasks nserton was dscussed n detal [2, 4]. An expresson was gven to calculate the tme the nserton can be done wthout deadlne mssng. A deeper research was made and a modfed expresson was proved to evaluate an earler tme n [3], whch s the concluson to date wth the problem of nserton wth EDF. Fndng the earlest tme for startng the nserton wthout deadlne mssng s nterestng, whch s stll under nvestgaton. The purpose of ths paper s not on ths, but on the choce of a proper task for compressng and the way of the release of the new task to make the nserton as early as possble. By ntroducng the remanng bandwdth of an nstance, an algorthm s provded to choose a proper task as the compressed one. A useful theorem and ts lemma are also provded, whch declares that the safety adjustment can be mplemented mmedately upon request f the ncrease of the bandwdth due to nserton does not exceed the lowest utlzaton of ndvdual current tasks n the system and every task s allowed to be compressed. The part utlzaton frst algorthm s also proposed, accordng to whch a new task s nserted wth a lower speed frst, then to ts full utlzaton after a certan tme. In the next secton, two requrements wth task nserton are revewed. The concept of the remanng bandwdth s defned n Secton III, based on whch the algorthm MRBC s descrbed and the theorem for judgng mmedate nserton s provded. The PUF algorthm s also proposed n ths secton. The paper concludes n secton IV. Ths work has been supported by Hunan Provncal Natural Scence Foundaton of Chna under grant agreement No. 09JJ5040. ISSN : Vol. 3 No. 1 Feb-Mar

2 II. TWO REQUIREMENTS In Fgure 1, suppose there are two tasks n a real-tme system, 0 (8,16) and 1 (12,24). The task model s (C,T ), where C s ts computaton, T s ts perod, and utlzaton U = C /T. In ths example, we have U 0 =C 0 /T 0 =8/16=1/2, and U 1 =C 1 /T 1 =12/24=1/2. The total utlzaton of the task set s U =U 0 + U 1 =1, whch ndcates the system s schedulable based on EDF and fully loaded. Fgure 1 shows ts part of executon. If at some tme t r, a new task 2 (1,4) requests to run. Because U= U 0 + U 1 + U 2 =1+1/4>1 makes the new task set unschedulable, so certan bandwdth has to be freed from current tasks. Let us compress 0 (8,16) to 0 (8,32) so that U 0 =1/4 and U = U 0 + U 1 + U 2 =1. However, both [2] and [3] ponted out that even f U 1 after nserton, deadlne mssng may occur f the nserton of 2 s too early. The operaton of compresson nserton s anyway a type of nterference n the runnng system. Fgure 1. The schedulng of a smple task set It s not welcome that any deadlne s mssed for any task, even lower prorty tasks. An accurate schedulng algorthm should behave as what t declares. For example, f an Internet Servce Provder declares that the bandwdth allocated to a user s decreased, then the decreased bandwdth should be guaranteed. Therefore, generally, the frst requrement for the nserton of a new task s smooth,.e. no deadlne wll be mssed after nserton. A tme pont from whch on a smooth nserton can be mplemented s called a smooth nserton tme. Fgure 2. An example of smooth nserton Agan take the example n Fgure 1. Suppose t r =6 and the perod of 0 s prolonged from 16 to 32. The remanng computaton tme of the current nstance of 0 s equal to C 0 (t r )= C 0 (6)=2. Accordng to [2], a smooth nserton tme 0 =12 can be calculated. Comparatvely, an earler smooth nserton pont from [3] s = 0 =8. Fgure 2 shows the smooth nserton based on [3]. Although an earler smooth nserton tme may be evaluated from [3] than from [2], the accurate algorthm for the earlest smooth tme s stll under nvestgaton. Let us lst the expresson of n [3] as Equaton (1) for convenence. ' C ( tr ) δ d. ' U U ISSN : Vol. 3 No. 1 Feb-Mar

3 Where d represents the deadlne of the current nstance of the compressed task. C (t r ) s ts remanng computaton tme up to d. U and U are ts utlzatons before and after compresson respectvely. In our dscusson n ths paper, we assume d =T. In the example of Fgure 2, we have d =16, U =1/2, U =1/4, and C (t r ) =2. In [3], t s also ponted out that f from Equaton (1) s less than or equal to t r, then we take = t r, whch means an mmedate smooth nserton can be done at t r. An obvous observaton from Equaton (1) s that an nserton at the end pont of the current nstance of (as d =16 n Fgure 2) s defntely smooth. However, f a larger tme nterval between t r and d exsts, the new task has to wat mpatently. Reasonably we should nsert the new task as early as possble. Ths s our second requrement. As n Fgure 2, nsertng 2 at t = = 8 s normally welcome than at t = d =16, especally n urgent cases. III. THE MRBC AND PUF ALGORITHMS 3.1 The maxmum remanng bandwdth compressng(mrbc) algorthm In Fgure 2, f 1 s selected to be compressed nstead of 0,.e. from 1 (12,24) to 1 (12,48), then we have = 6, whch mples an mmedate smooth nserton. There s no earler than mmedate. To fnd ths reason, we convert (1) to Equaton (2). ' C ( tr ) U U. ' d The left sde ndcates the freed bandwdth from the compressed task. The rght s the utlzaton of the remanng computaton tme C (t r ) between and d. Defnton 1(the remanng bandwdth of the current nstance) If compressng starts at t r, C (t r ) s the remanng computaton tme of the current nstance, d s ts deadlne, then C (t r )/(d t r ) s called the remanng bandwdth of the current nstance of at t r,or the nstance remanng bandwdth, s represented by RB (t r ). If =t r, then the rght sde of Equaton (2) s the remanng bandwdth of the current nstance of at t r. Through careful observaton, t s found that the greater the remanng bandwdth at t r s, the earler the smooth nserton pont can be gotten. The earlest possble tme s of course t r. Defnton 2 At t r, for every task n the current task set, we calculate the value of the remanng bandwdth of every current nstance. The maxmum one s defned as the maxmum nstance remanng bandwdth, represented wth RB max (t r ). Smlarly, the mnmum one s called the mnmum nstance remanng bandwdth and named RB mn (t r ). Agan take the example n Fgure 2. The utlzaton of the new task s U 2 =1/4. The remanng bandwdth of 0 at t r =6 s RB 0 (6)=2/(16 6)=1/5, whle 1 has RB 1 (6)=12/(24 6)=2/3. Obvously, selectng 1 as the compressed task other than 0 possbly produces an earler smooth nserton tme. Here RB max (t r )=RB max (6)=RB 1 (6)=2/3, and RB mn (t r )=RB mn (6) =RB 0 (6) =1/5. Now we formally gve out the algorthm n Fgure 3 for selectng a current task to compress based on the remanng bandwdth. The Algorthm MRBC: maxmum remanng bandwdth compresson. Step 1. Wth t r, fnd out the current task to make RB (t r ) U U hold and go to Step 4. If no such a task s found, then choose the one that has the maxmum nstance remanng bandwdth as and go to Step 2. Step 2. Evaluate the smooth nserton tme wth U U = C (t r )/( d ) and go to Step 3. Step 3. Compress, nsert j from t= on, and fnsh the algorthm. Step 4. Compress, nsert j from t= t r on, and fnsh the algorthm. Fgure 3. The descrpton for the MRBC Algorthm. ISSN : Vol. 3 No. 1 Feb-Mar

4 Suppose there are n(n 1) current tasks n the system. In order to fnd that makes RB (t r ) U U hold or has the maxmum nstance remanng bandwdth, the step 1 of the algorthm may have to go through n nstances. The remanng computaton tme of every nstance must be evaluated, and thus n counters have to be mantaned at run tme. Ths s a type of expendture. The temporal complexty of the algorthm s apparently(n). Ths algorthm shows that the maxmum nstance remanng bandwdth RB max (t r ) s mportant to the task nserton. Theorem 1 dscovers a useful rule for judgng ths maxmum bandwdth. Theorem 1 Suppose arbtrary preempton s allowed n a perodc real-tme system at run tme based on EDF, and task mn has the lowest utlzaton U mn among all the tasks. Then, no matter whch task s selected to be compressed and when ths compresson starts, RB max (t r ) U mn s always true. Proof Consderng a current task (C,T ) at run tme. At the begnnng of an nstance, the nstance remanng bandwdth s U. At any other tme, accordng to the preempton before that tme, three possble cases exst. Case1. It has been preempted more by other tasks so that ts nstance remanng bandwdth s greater than U. Case2. It has preempted other tasks so that the nstance remanng bandwdth s less than U. Case3. The results from has been preempted and has preempted are the same up to now, and thus t has the nstance remanng bandwdth equal to U. For case 1 at t r, we have RB max (t r ) > U U mn. For case 2 at t r, suppose there s a task j (C j, T j) that has been preempted more by, and the utlzaton of j s U j, whch mples that the remanng bandwdth of the current nstance of j s greater than U j, and thus RB max (t r ) > U j U mn. For case 3 at t r, RB max (t r ) s at least not less than U, and greater than or equal to U mn. Therefore, n any case, RB max (t r ) U mn always holds. Ths proves Theorem 1. Lemma 1 If the utlzaton of a new task s not greater than the lowest utlzaton U mn of the current tasks, and any of them s permtted to be compressed, then ths new task can be mmedately nserted at any tme. Proof It s obvous from Theorem 1. Now wth Theorem 1 and Lemma 1, for the problem of the task nserton, we fnd out the lowest value among all the sngle task s utlzatons of the current tasks, f the bandwdth of the new task n a perod s less than or equal to ths value and f every task s allowed to be compressed, then an mmedate nserton s smooth whenever t requests to jon n the system. Wth a fxed utlzaton of every task, we can even predct ths before runnng the system. Let us come back to the example n Fgure 1. U mn =U 0 = U 1 =1/2, and the task set ( 0, 1 ) has a hyperperod (least common multple perod) T LCM =48. Accordng to dfferent expressons, we can lst RB max (t) durng the frst hyperperod as follows. At tme t =0, RB max (0)=U mn =1/2. t, 0 t 16, RB max (t)= RB 1 (t) 1/2. At tme t =16, RB max (16)=U mn =1/2. t, 16 t 24, RB max (t)= RB 0 (t) 1/2. At tme t =24, RB max (24)=U mn =1/2. t, 24 t 32, RB max (t)= RB 1 (t) 1/2. At tme t =32, RB max (32)=U mn =1/2. t, 32 t 48, RB max (t)= RB 0 (t) 1/2. Totally, there are 4 ponts at whch RB max (t)=u mn =1/2, at any other tme RB max (t) > U mn s always satsfed. Actually, n many systems RB max (t) > U mn holds at any tme. Fgure 4 llustrates such an example. There are three tasks n the set. The hyperperod T LCM s equal to 16. U mn =U 0 =U 2 =1/4. Smlarly, RB max (t) s lsted as follows. t, 0 t (17 97) / 2, RB max (t)= RB 1 (t) U mn. ISSN : Vol. 3 No. 1 Feb-Mar

5 t, ( 17 97) / 2 t 8, RB max (t)= RB 0 (t) U mn. t, 8 t 14, RB max (t)= RB 1 (t) U mn. t, 14 t 16, RB max (t)= RB 2 (t) U mn. Fgure 4. An example of RBmax(t r ) Umn at any tme. 3.2 The part utlzaton frst(puf) algorthm Possbly, the remanng bandwdths wth n tasks have to be compared based on MRBC. If the new task can not start at t r, then the nserton has to be delayed. Actually, sometmes a real-tme task has a flexble perod, that s, t can vary from a mnmum perod T mn to a maxmum perod T max. Suppose the perod T j of a new task j (C j,t j ) ranges from T jmn to T jmax. The selected compressed task s stll represented by (C,T ). In order to make the system schedulable, we assume ' C j U U. (3) T j mn Ths mples that after the end of the current nstance of, the new task s allowed to run wth T jmn n full speed, but before that, t may be executed wth T jmax and thus part of ts full utlzaton to get a possble earler release of ts frst nstance. Ths s the dea of the PUF algorthm, whch s formally descrbed as n Fgure 5. The Algorthm PUF: part utlzaton frst. Step 1. If C j /T jmn s less than or equal to the lowest utlzaton U mn n the current task set and f every task s permtted to be compressed, then compress any of the current tasks, nsert the new task j wth T jmn from t= t r on, and fnsh the algorthm. Step 2. Wth t r, fnd out the current task to make RB (t r ) C j /T jmn hold and go to Step 5. If no such a task s found, then choose the one that has the maxmum nstance remanng bandwdth as and go to Step 3. Step 3. Calculate wth Cj /T jmax = C (t r )/( d ). If tr, then take = t r. Go to Step 4. Step 4. If <d, then compress, release every nstance of j startng before d wth T jmax, run ts nstances startng after d wth T jmn, and fnsh the algorthm. If =d, then compress, nsert j wth T jmn from t =d on and fnsh the algorthm. Step 5. Compress, nsert j wth T jmn from t = t r on and fnsh the algorthm. There are three ponts that should be notced. Fgure 5. The descrpton for the PUF algorthm The concluson from Theorem 1 and Lemma 1 s ncluded n Step 1 of PUF. If the full utlzaton C j /T jmn s less than or equal to the lowest utlzaton U mn n the system, then Step 2 to Step 5 are not needed. If a task s found wth RB (t r ) C j /T jmn, then the new task j can be nserted mmedately from t r on. Otherwse, the smooth tme s evaluated wth ts part utlzaton C j /T jmax. ISSN : Vol. 3 No. 1 Feb-Mar

6 t = d s used as a crtcal tme. If the release of the frst nstance of j s wth T jmax, then other nstances of ths task startng before d, f any, wll also be wth T jmax. The nstances after d can defntely be released wth T jmn because Equaton (1) declares d. Fgure 6 s an example of PUF. Suppose 0 (6,16) s the only task allowed to be compressed. The freed utlzaton s U 0 -U 0 =6/16-6/48=1/4. The request tme s t r =4. The remanng bandwdth of the current nstance of 0 s (6-4)/(16-4)=1/6, whch s less than the full utlzaton 1/4 of the new task 2 (1,4). The perod of 2 ranges from T jmn = T 2mn =4 to T jmax = T 2max =6. If the frst nstance of 2 starts at t=4 wth T 2mn =4, then the deadlne at t=16 wll be mssed snce up to t=16 the sum of the processor demands by 2 and 1 exceeds the length of the nterval, that s, 3+10=13>12, whch makes the system unschedulable [5, 6]. Fgure 6. An example of the PUF algorthm. Therefore, n order to start the frst nstance early at t=4 and make every deadlne satsfed, we nsert the new task 2 n two steps. The frst two nstances are wth T 2max =6, whle other nstances startng after t=16 are released wth T 2mn =4, whch s shown n Fgure 6. IV. CONCLUSIONS How to nsert a new real-tme task as early as possble? Two approaches are provded n ths paper. The frst approach relates manly wth compresson. Theorem 1 s very smple and useful snce t says the new task can be nserted mmedately at any tme f the utlzaton of the new task does not exceed the lowest utlzaton of ndvdual current tasks n the system and f every task s allowed to be compressed. The MRBC algorthm chooses a task to compress that has the maxmum remanng bandwdth, based on whch the earlest nserton can be mplemented, but t has the temporal complexty (n). The second approach relates wth nserton. Wth an accepted varaton of ts perod to some extent, the new task can be nserted n two steps, frst wth a longest perod T jmax then a shortest one T jmn. In ths way, not only the schedulablty s guaranteed, but also the nserton can be done as quckly as possble. ACKNOWLEDGMENT Guangmng Qan thanks Hunan Provncal Natural Scence Foundaton of Chna under grant agreement No. 09JJ5040 for fnancal support to ths research. REFERENCES [1] C.L.Lu, J.W.Laylan, Schedulng Algorthms for Multprogrammng n a Hard Real Tme Envronment, J.ACM, vol.20, no.1, pp.40-61, [2] G.C.Buttazzo, G.Lpar, M.Caccamo, and L. Aben, Elastc Schedulng for Flexble Workload Management, IEEE.Trans.Computers, vol. 51, no.3, pp , [3] Q.Guangmng, An Earler Tme for Insertng and/or Acceleratng Tasks, Sprnger. Real Tme Systems, vol.41, no.3, pp , [4] G.C.Buttazzo, G.Lpar, and L.Aben, Elastc Task Model for Adaptve Rate Control, In Proc.19th IEEE Real-Tme Systems Symp., pp , Madrd, Span, Dec [5] S.Kbarauh, R.R.Howell, and L.E.Roser, Algorthms and Complexty Concernng the Preemptve Schedulng of Perodc Real-Tme Tasks on One Processor, Sprnger. Real Tme Systems, vol.2, pp ,1990. [6] K.Jeffay, and D.L.Stone, Accountng for Interrupt Handlng Costs n Dynamc Prorty Task Systems, In Proc.14th IEEE Real-Tme Systems Symp., pp , Ralegh Durham, NC, Dec ISSN : Vol. 3 No. 1 Feb-Mar

7 AUTHORS PROFILE Guangmng Qan s a professor at College of Mathematcs and Computer Scence, Hunan Normal Unversty, Chna(P.R.C). Hs current research nterests nclude real-tme schedulng and network swtchng. Xanghua Chen s a postgraduate student of Hunan Normal Unversty. Gang Yao s a Postdoctoral Research Collaborator at the Unversty of Illnos at Urbana Champagn. He receved a Ph.D. n Computer Engneerng from the Scuola Superore Sant'Anna of Psa, Italy, n 2010, the BE and ME degrees from Tsnghua Unversty, Bejng, Chna. Hs man nterests nclude real-tme schedulng algorthms, safety-crtcal systems and shared resource protocols. ISSN : Vol. 3 No. 1 Feb-Mar