Comments on Window-Constrained Scheduling

Transcription

1 Commens on Window-Consrained Scheduling Richard Wes Member, IEEE and Yuing Zhang Absrac This shor repor clarifies he behavior of DWCS wih respec o Theorem 3 in our previously published paper [1], and describes an alernaive approach o make guaranees for arbirary window-consrains. Index Terms Real-ime sysems, mulimedia, window-consrains, scheduling. 1 Inroducion In our previously published paper [1], Theorem 3 implies ha he DWCS algorihm holds for all possible window-consrains. DWCS can saisfy Theorem 3 under cerain condiions and he proof shows a paricular case, when here are wo classes of window-consrains x i /y i and x j /y j, such ha x i = y i 1, x j = y j 1, and x j /y j <x i /y i. While DWCS can be shown o saisfy Theorem 3 when each sream (or, equivalenly, job) J i has a window-numeraor x i = y i 1, regardless of he value of y i, he algorihm can fail o produce a feasible schedule for cerain oher window-consrains. Wihou loss of generaliy we define a window-consrain on J i as requiring ou of k i deadlines o be me, as opposed o allowing x i ou of y i deadlines o be missed in non-overlapping windows of k i deadlines (such ha k i = y i and = y i x i ). Algorihms such as DWCS updae he curren window-consrain m i/k i of each job J i by reducing m i by one each ime a job is serviced in is curren period, T i, and also by reducing k i by one each ime a new period sars. Furher deails regarding window-consrain adjusmens can be found in our earlier work on DWCS [1]. We can now sae he following heorem o show how i affecs feasibiliy of DWCS for general window-consrains. Theorem 1. Wih respec o Theorem 3 in our original paper and given ha we have arbirary iniial window-consrains: In each non-overlapping window of size q in he hyper-period, H, here can be more han q jobs ou of n wih curren window-consrain m i/k i = 1 a any ime, when U min = n C i 1 ( C i = 1, T i = q, i).

2 Proof. Suppose here are n jobs queued for service a ime = 0. Suppose also a ime aq (a < min i (k i ), q < n), here are q + 1 jobs each wih m i = k i 0. Le Γ j be he se of jobs ha have been serviced for j insances in period [0, aq), such ha Γ j = n j. Ou of each se Γ j le n j be he number of jobs whose curren window-consrains are m i = k i > 0. Finally, le j /k ij be he iniial window-consrain of each job J i in Γ j. In he inerval [0, aq), each of he n j jobs in Γ j 0 j a changes is window-consrain as follows: n 0 : (0, k i0 ) (0,aq) (0, k i0 a) n 0 : 0 = k i0 a > 0 n 1 : (1, k i1 ) (1,aq) (1 1, k i1 a) n 1 : 1 1 = k i1 a > 0 n a : (a, k ia ) (a,aq) (a a, k ia a) n a : a a = k ia a > 0 where, j=0 n j = n; j=0 n j = q + 1, j=0 jn j aq. Now, consider he following case: n 0 = n 0 = q + 1, n 1 = n 2 = = n a = 0. Then, for any job J i, here mus exis a value b j b j < a, such ha he jh insance of J i is serviced in he inerval [b j q, (b j + 1)q). Therefore, we have j (j 1) k ij b j 0 k i0 b j = k i0 a k i0 b j j k ij (k i0 a)(k ij b) + (k i0 b j )(j 1) (k i0 b j )k ij U min = k i0 a k i0 q (q + 1) + a j=1 n j i=0 j k ij q 1, a jn j aq j=0 Given ha a soluion o he above inequaliies exiss (e.g., a = 2, n 1 = 0, b 2 = 1, q = 4, k i0 = 3, 0 = 1, n 0 = 5, n 2 = 4, 2 = 35, k i2 = 64), i mus be ha he heorem holds. 2 Pfair-based Virual Deadline Scheduling (PVDS) To saisfy Theorem 3 in our original paper [1] for arbirary window-consrains, and hence show a feasible schedule is possible for 100% resource uilizaion (when C i = 1, T i = q, i), we propose an approach called PVDS. Firs, we define he virual deadline, V d i (), of a job J i a any ime such ha: V d i () = si + (l i + 1) k it i (l i = 0, 1,..., 1)

3 where si is he sar ime of he curren window of size, which is. l i is he number of job insances ha have already me heir deadlines in he curren window before. In ordering jobs for service, he one wih he earlies virual deadline a ime has highes prioriy. If an insance of J i is no serviced in is reques period, T i, is virual deadline says he same, whereas if a job insance is serviced, is virual deadline increases by k it i. If insances of J i are serviced in he curren window, J i is ineligible for service unil he sar of is nex window, unless all oher jobs have received heir minimum service requiremens. Lemma 1. There is no idle ime before he failure of a synchronized job se where U min = n C i = 1 (C i = 1, T i = q, i), under pfair-based virual deadline scheduling. Proof. Observe ha a synchronized job se is one in which all he jobs in he se are queued for service a he same ime. For simpliciy, we can assume hese jobs are all ready for service a ime = 0. Now, suppose [, + a) is he firs idle period before a leas one of he jobs in he se fails o mee is service consrains. According o he algorihm, if here is idle ime, each job J i has eiher saisfied is windowconsrain in he curren window, or has finished service in he curren period, T i = q. Henceforh, le us define wo ses of jobs: (1) Γ 0 is he se of jobs ha have saisfied heir window-consrains in he curren window, by he sar of he curren period, and, (2) Γ 1 is he se of jobs ha each have been serviced l i (l i ) imes in he curren window, wih he l i h insance of J i serviced in he curren period. Le J ik be he job J i in he se Γ k. For each and every job J i0 in Γ 0, is virual deadline (or sar ime of is nex non-overlapping window) a ime,, saisfies he consrain + a. Similarly, since each and every job in Γ 0 has been serviced by he sar of he curren period, he previous virual deadline of each job J i0 mus be less han he previous virual deadline of each and every job, J i1, in Γ 1. In wha follows, le Γ 0 = n 0, Γ 1 = n 1, and l i1 be he number of insances of each job J i1 in Γ 1. Therefore, i 0, i 1 k i1 T i1 + k i 1 T i1 l i1 k i1 T i1 1 Le k p T p = max i0 ( ) k p T p i 1, k i1 T i1 + k i 1 T i1 l i1 k p T p k i1 T i1 1 k p T p 1 + l i1 1 k p T p (1) k i1 T i1 k p T p k i1 T i1

4 n 0 = ( 0 ) + n 0 F rom (1), (2) > ( 0 ) + n 0 > ( 0 ) + i 0, + a > ( 1 + l i1 ) (2) k i1 T i1 ( 1 k p T p ) k p T p k i1 T i1 1 k i1 T i1 = ( ) = This implies > which is impossible, hereby yielding a conradicion o he supposiion here is idle ime before a failure. Lemma 2. A job se is schedulable by pfair-based virual deadline scheduling when U min = n C i = 1 (C i = 1, T i = q, i). Proof. Assume a schedule fails a ime for job J i, where = a + bt i (0 b < k i, a 0). Le l i be he number of insances of job J i serviced in he curren window before, so ha l i 1 = k i b. Therefore, + k it i (l i + 1) = a + k it i (m k i + b) a bt i = k it i ( k i + b b) = k it i (k i ( 1) + b(1 )) k i k i k i = k it i (1 )(b k i ) 0 + k it i (l i + 1) (3) k i Since job J i fails a ime, and is virual deadline is less han, J i is no serviced in he inerval [ T i, ). Now, le l j be he number of insances of job J j serviced in he curren window before. I follows ha T i = q jobs oher han J i will be served in he inerval [ T i, ). Each of hese oher jobs, J j, has is mos recen virual deadline a k + l j T j j, which is less han or equal o job J i s curren virual deadline a + (l i + 1) k it i. Moreover, all jobs oher han J i mus have mos recen virual deadlines ha are less han or equal o J i s curren virual deadline. If his were no he case, J i would be able o service l i + 1 insances before. Therefore,

5 j i, + l j + (l i + 1) k it i + l j ( + (l i + 1) k it i ) (4) From Lemma1, we know here is no idle before he firs failure a ime. Therefore, F rom (4), (5) + l i + j i = + l i + j i ( + l j ) (5) (( + (l i + 1) k it i ) ) + l i + ( + (l i + 1) k it i ) j i + l i + ( + (l i + 1) k it i )(1 ) + k it i (l i + 1) 1 (6) Equaions (3) and (6) imply ha + 1, which is impossible, so he assumpion ha a schedule fails a ime canno hold. Exending he previous lemma, he following Theorem can be shown o hold (alhough we omi he proof for breviy): Theorem 2. There exiss a feasible pfair-based virual deadline schedule for a synchronized job se Γ, where U min = n C i References 1 (C i = 1, T i = q, i). [1] R. Wes, Y. Zhang, K. Schwan, and C. Poellabauer. Dynamic window-consrained scheduling of real-ime sreams in media servers. IEEE Transacions on Compuers, 53(6): , June 2004.