How much can lookahead help in online single machine scheduling

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1 JID:IPL AID:3753 /SCO [m3+; v 1.80; Prn:16/11/2007; 10:54] P.1 (1-5) Information Processing Letters ( ) How much can lookahead help in online single machine scheduling Feifeng Zheng a,,yinfengxu a,b,e.zhang c a School of Management, Xi an Jiao Tong University, Xi an, Shaanxi , China b The State Key Lab for Manufacturing Systems Engineering, Xi an, Shaanxi , China c School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai , China Received 6 March 2007; received in revised form 2 October 2007; accepted 17 October 2007 Communicated by F.Y.L. Chin Abstract This paper studies online single machine scheduling where jobs have unit length and the objective is to maximize the number of completed jobs. Lookahead is considered to improve the competitiveness of online deterministic strategies. For preemption-restart model, we will prove a lower bound of ( LD +2)/( LD +1) for the case where LD 1and3/2for the case where 0 LD < 1, in which LD is the length of time segment that online strategies can foresee at any time. For non-preemptive model, we mainly present a greedy strategy that has an optimal competitive ratio of 3/2when1 LD < 2 while its competitive ratio is bounded from above by 4/3 asld goes larger Elsevier B.V. All rights reserved. Keywords: Online strategies; Scheduling; Lookahead 1. Introduction Single machine online job scheduling problem has been extensively studied in various models, and one of the model objectives is to maximize the number of completed jobs. For the case of unit length and arbitrary deadline of job, Goldman et al. [5] studied nonpreemptive strategies, and proved an optimal competitive ratio of 2. To beat the lower bound of 2, they investigated the case where each job has a sufficiently large deadline. When the deadline is at least two times of job length, they proved a 3/2-competitive strategy. Chrobak et al. [2] considered the preemption-restart model where * Corresponding author. Tel.: ; fax: address: zhengff@mail.xjtu.edu.cn (F.-f. Zheng). a preempted job needs to be processed from the beginning to be satisfied and they presented an optimal 3/2- competitive deterministic strategy TightRestart (abbr. TR). Hoogeveen et al. [6] studied the preemption-restart model for another case where jobs have nonuniform lengths. They proposed a 2-competitive strategy and proved a matching lower bound. For the case of unit length of job, their strategy does not make use of abortion, and thus the lower bound applies to our nonpreemptive model. In the above literature, it is assumed that online strategies make processing decisions without any knowledge of future jobs at any time. Many authors study various semi-online models to improve the performance of online strategies in competitiveness. One of the models is to apply the function of lookahead. With lookahead, an online strategy can foresee the information of some future jobs or foresee /$ see front matter 2007 Elsevier B.V. All rights reserved. doi: /j.ipl

2 JID:IPL AID:3753 /SCO [m3+; v 1.80; Prn:16/11/2007; 10:54] P.2 (1-5) 2 F.-f. Zheng et al. / Information Processing Letters ( ) a finite time segment at any time. Lookahead has been comprehensively investigated in previous literature with different definitions (see Keskinocak [7]). Mao and Kincaid [8] studied a lookahead model of single machine scheduling to minimize the total completion time, where they defined that an online strategy A is in lookahead-k model if at any time t, A can foresee the next k jobs arriving after time t. Coleman and Mao [3] studied the same model in the scenario of multi-processor scheduling and considered non-preemptive strategies. Experiments show that lookahead does help. Without lookahead, it is well known that although EDF is optimal in preemptive model, this is not the case in non-preemptive model. Ekelin [4] used lookahead to improve the performance of non-preemptive EDF and gave experimental results. In this paper, we will introduce a different lookahead model where the lookahead parameter is defined as the length of time but not the number of future jobs, and will investigate the application of lookahead in both non-preemptive and preemption-restart models. To gauge the performance of an online strategy A, the competitive ratio analysis (see Borodin and Elyaniv [1]) is often used. Denote by Γ A (I) and Γ (I) the schedules produced by A and by an optimal off-line strategy OPT on a job input set I, respectively, and by Γ A (I) and Γ (I) the total number of completed jobs in Γ A (I) and Γ (I) respectively. The competitive ratio Γ of A is defined as r A = sup (I) I Γ A (I). This paper will study both preemption-restart and non-preemptive models with lookahead. For preemption-restart model, we will prove a lower bound of ( LD +2)/( LD +1) for the case that LD 1 and 3/2 for the case that 0 LD < 1. For non-preemptive model, we will put forward a greedy strategy that always tries to maximize its potential profit based on the job information with lookahead. It is shown that lookahead is helpless when LD < 1, and the greedy strategy has an optimal competitive ratio of 3/2 when 1 LD < 2 while the ratio is bounded from above by 4/3 for any other value of LD. 2. Basic definitions and a lookahead model For a job J, denote by a(j), p(j) and d(j) its arrival time, processing time and deadline, respectively. This paper will focus on the case of uniform length, i.e., p(j) = 1. If d(j) = a(j) + 1, J is of tight deadline. We say an online strategy A is in LK LD lookahead model if at any time t, A can foresee all the jobs that will arrive in time segment (t, t + LD], where the lookahead parameter LD is a non-negative real number. In particular, the model reduces to the one without lookahead when LD = Preemption-restart model In this section, we will mainly investigate the lower bound of competitive ratio for online strategies that make use of both preemption and lookahead. Theorem 1. In the LK LD model, if LD 1 and 0 LD < 1, no preemptive strategies have competitive ratio better than ( LD +2)/( LD +1) and 3/2, respectively. Proof. We will first prove the former case where LD 1. We will construct a job input sequence Γ to make an arbitrary preemptive strategy A behave poorly. Γ includes 2 LD +2 jobs in total. All the jobs in Γ except the first one J 0 are of tight deadline. J 0 with deadline LD + 3 δ is released at time 0, and J i (1 i LD ) will be released at time LD LD +θ +i 1 where 0 <δ θ and the exact value of θ will be determined later. LD LD +θ<1 holds so that J 1 is released strictly before time 1. Job I i (1 i LD ) will be released at time LD LD +i. The last job I LD +1 will be released strictly later than (LD LD +θ)+ld so that A cannot foresee the job at time a(j 1 ), preventing A from producing an optimal schedule. The exact value of a(i LD +1 ) depends on A s action. A has two selections on or before time a(j 1 ). (S1) A continues running J 0. I LD +1 will be released at time LD + 1 θ. After completing J 0, A will miss J 1, and it will finish at most LD of the rest of the jobs, either I 1,...,I LD or J 2,...,J LD together with I LD +1. OPT will first finish J 1,...,J LD and I LD +1, and then complete J 0 in time due to δ θ. Its total profit is LD +2. So, the ratio of profit obtained by OPT to that of A is equal to ( LD +2)/( LD +1) in this case. (S2) A aborts J 0 to start J 1 at time a(j 1 ). I LD +1 will then be released at time LD + 1. After finishing J 1, A may behave in two cases. (C1) A continually satisfies J 2,...,J LD and then either J 0 or I LD +1.Note that if A starts J 0 after finishing J LD, then I LD +1 is missed, vice versa. (C2) A does not start J 2 immediately after finishing J 1. Since I 1 is blocked by J 1, A will start I 2,...,I LD +1 in order. It will miss J 0 after finishing I LD +1. Hence, A finishes LD +1 jobs totally in either case. OPT will finish J 0 and then I 1,...,I LD +1 in order, finishing LD +2 jobs in total. The ratio of ( LD +2)/( LD +1) holds in this case.

3 JID:IPL AID:3753 /SCO [m3+; v 1.80; Prn:16/11/2007; 10:54] P.3 (1-5) F.-f. Zheng et al. / Information Processing Letters ( ) 3 Note that I LD +1 cannot be foreseen by A on or before the arrival of J 1. Otherwise A will produce an optimal schedule in Γ. So, we set LD + 1 θ>(ld LD +θ)+ LD or θ< 1 (LD LD ) 2. The first conclusion follows. For the case where 0 LD < 1, we construct the same job input sequence as in the case where 1 LD < 2, and obtain the same lower bound. The theorem is proved. Combining Theorem 1 with the upper bound of 3/2 in Chrobak et al. [2] for the case without lookahead, we conclude that lookahead is useless in preemptive model if LD < Non-preemptive model 4.1. A lower bound For non-preemptive model, Goldman et al. [5] gave an optimal competitive ratio of 2 without lookahead. The following theorem shows that lookahead is useless when LD < 1. Theorem 2. In the LK LD model, if LD < 1, nonpreemptive strategies cannot be better than 2-competitive. Proof. Suppose that LD = 1 ɛ where 0 <ɛ<1 can be arbitrarily small. For an arbitrary online strategy A, we will construct a job input sequence with two jobs to make A be at best 2-competitive. a(j 0 ) = 0 and d(j 0 ) = 3 θ where 0 <θ<ɛ. The second job J 1 is of tight deadline, and the value of a(j 1 ) depends on the behavior of A, which has two selections at time 0. Case 1. A starts J 0 at time 0. J 1 will arrive at time 1 θ. A cannot foresee J 1 at time 0 since LD = 1 ɛ<1 θ. A satisfies J 0 but misses J 1, while OPT will complete both jobs. The ratio of what OPT gains to that of A is equal to 2. Case 2. A starts J 0 at some time t>0. J 1 will then arrive at time 1 + t/2. Again, A will miss J 1 while OPT still completes both jobs. In both cases, the ratio of 2 holds and the theorem follows A greedy strategy and its competitiveness In this section, we will present an online strategy GD l (a greedy looking ahead l units of time). By Theorem 2, we will focus on the case where LD 1. GD l has relationship with strategy TR that was put forward by Chrobak et al. [2]. Before describing GD l, we will give some auxiliary definitions, redescribe and discuss TR as well. AjobJ i keeps pending after arrival until either it is started to be processed or it misses its deadline. A pending job J i misses its deadline or expires if it has not been started at time d(j i ) 1. Let S t be the job set that includes all those pending jobs arriving before time t, and Q t be the set that includes those arriving at time t. Once a pending job expires at time t, it is excluded from S t. Assume that J i is being processed by strategy A at time t and it was started at time t (t 1,t]. For each J k S t Q t, we say it is an urgent job to A at time t if d(j k )<t + 2, i.e., J k will expire before A completes J i.ifa is idle at time t, J k is said urgent to A given that d(j k )<t+ 2. Let Q t = Q 1 t Q 2 t where Q 1 t includes those urgent jobs and Q 2 t includes the other ones in Q t. We describe the TR strategy as follows. When it is idle at time t, TR will start a job J i S t Q t by EDF (Earliest Deadline First) rule given that S t Q t, otherwise it keeps idle. Suppose that during the processing of J i, there come a set of jobs at time t (t, t + 1) and Q 1 t is not empty. TR will abort J i and start an arbitrary job J k Q 1 t if all the jobs in {J i } S t Q 2 t can be satisfied by TR after it completes J k, i.e., after time t + 1. Otherwise TR continues J i. Lemma 1. TR does not make two consecutive abortions in preemptive model without lookahead. Proof. Assume otherwise that TR produces a schedule σ = (J 0,...,J k,j k+1,...,j n ) where J k and J k+1 are two aborted jobs. Denote by s(j i ) the time at which TR starts J i, and s(j i )<s(j i+1 ) holds for 0 i<n.by construction of TR, J k+1 aborts J k since J k+1 belongs to Q 1 a(j k+1 ) on its arrival so that d(j k+1) <s(j k ) + 2. For the next abortion where J k+2 aborts J k+1,itisrequired that J k+1 can be satisfied by TR after the completion of J k+2. That is, d(j k+1 ) s(j k+2 ) + 2. This contradicts that d(j k+1 )<s(j k ) + 2 <s(j k+2 ) + 2. Hence, TR does not make two consecutive abortions. The lemma is proved. In lookahead model, TR can be treated as one that ignores any future information when it makes processing decisions. So, its competitive ratio is still 3/2 when LD Description of GD l GD l replaces the use of abortion with lookahead and looks ahead l units of time. Let Rt l be the job set

4 JID:IPL AID:3753 /SCO [m3+; v 1.80; Prn:16/11/2007; 10:54] P.4 (1-5) 4 F.-f. Zheng et al. / Information Processing Letters ( ) that includes the jobs arriving during time segment (t, t + l] where 1 l LD is a real number. Note that R l t is known to GD l at time t. GD l is triggered at any time t if GD l is idle and S t Q t is not empty. It first produces virtually a feasible processing schedule δ t for the jobs in S t Q t R l t such that the number of jobs in δ t is maximized. If the first job J in δ t is in S t Q t, then J will be started by GD l at once. Ties are broken by EDF rule. Otherwise if J R l t, GD l will keep idle until the next job arrives. In fact, GD l represents a kind of non-preemptive greedy strategy. Given a job input sequence, for different values of l [1, LD], GD l makes processing decisions with different job information. In particular, when l = 1, GD 1 looks ahead exactly one unit of time Competitiveness analysis We first discuss the case that l = 1, and then give a lower bound of competitive ratio for GD l with general l ( 1). Theorem 3. GD 1 is 3/2-competitive in the LK LD model with LD 1. Proof. Given an arbitrary job input sequence Γ,ifGD 1 is proved completing the same number of jobs as TR does, then the theorem follows. Based on Lemma 1, assume that TR produces a schedule σ = (...,J i,r k,j i+1,...,j n ) for Γ where R k (1 k m) is the kth aborted job, and J i (0 i n) is a completed one. If m = 0orthereisno aborted job in σ, then GD 1 will produce the same processing schedule as TR does since both strategies select jobs by EDF rule at any time. Otherwise, on the arrival of each R k, GD 1 will keep idle until the arrival of J i+1. The reasoning is as follows. By construction of TR, all the jobs in {R k } S a(ji+1 ) Q 2 a(j i+1 ) can be satisfied after TR completes J i+1, that is, those jobs together with J i+1 consist of a feasible schedule at time a(j i+1 ). Hence, J i+1 instead of R k becomes the first jobintheδ a(rk ) produced by GD 1 at time a(r k ). So, GD 1 completes the same number of jobs as TR does, implying that GD 1 is 3/2-competitive. The theorem is proved. Since non-preemptive strategies are candidates of preemptive ones, the conclusion of Theorem 1 applies to non-preemptive strategies. Together with Theorem 3, we conclude that GD 1 is an optimal strategy when 1 LD < 2. Theorem 4. In the LK LD model with LD 1, GD l cannot be better than 4/3-competitive for 1 l LD. Proof. Assume otherwise that GD l is (4/3 δ)-competitive where δ>0 can be an arbitrarily small real number. Let k = l. We will construct a job input sequence Γ to make GD l behave poorly and lose. All the jobs in Γ have tight deadlines except those released at the beginning. At time 0, a set S 0 of 2n jobs with uniform deadline 4n are released, where n is a large natural number to be determined later. For 1 i n, jobi i will be released at time 2i 1 θi, where 0 <θ<1/n and then θn < 1. For 1 i 2n k, jobj i will be released at time 2n + k 1 + i. Note that a(j 1 ) a(i n ) = k θn>ld so that GD l cannot foresee J 1 at time a(i n ). By construction of GD l, it will first complete I i (1 i n) since all the jobs in S 0 can be satisfied after time 2n θn. After finishing I n, GD l will satisfy the jobs in S 0 one by one in the following 2n units of time. It completes the last job in S 0 at time a(i n ) n = (2n 1 θn) n = 4n θn>4n 1, and then misses J i for 1 i 2n k. Hence, GD l finishes totally n + 2n = 3n jobs. OPT will first complete the 2n jobs in S 0 during time segment [0, 2n) and then satisfy J i for 1 i 2n k, finishing 4n k jobs in total. The ratio of the profit gained by OPT to that of GD LK is equal to 4n k 3n = 4 3 3n k. Given that k/(3n) < δ or n>k/(3δ), A cannot be (4/3 δ)-competitive. The theorem follows. 5. Conclusion This paper discusses online single machine scheduling to maximize the number of completed jobs in lookahead model. We prove that lookahead is useless for preemptivestrategiesif the lookaheadparameter LD is less than 2, the same conclusion holds for non-preemptive strategies when LD < 1. We also give a lower bound of ( LD +2)/( LD +1) for preemptive strategies when LD 1, and propose a greedy non-preemptive strategy, which is optimal for the case where 1 LD < 2, while its competitive ratio cannot be less than 4/3 asld goes larger. Acknowledgements We would like to thank anonymous referees for their helpful suggestions. This work was partially supported by NSF of China under Grants , , , and

5 JID:IPL AID:3753 /SCO [m3+; v 1.80; Prn:16/11/2007; 10:54] P.5 (1-5) F.-f. Zheng et al. / Information Processing Letters ( ) 5 References [1] A. Borodin, R. El-yaniv, Online Computation and Competitive Analysis, Cambridge University Press, Cambridge, [2] M. Chrobak, W. Jawor, J. Sgall, T. Tichy, Online scheduling of equal-length jobs: randomization and restarts help, in: 31st International Colloquium on Automata Languages and Programming, vol. 3142, Springer, Berlin, 2004, pp [3] B. Coleman, W. Mao, Lookahead scheduling in a real-time context, in: Proceedings of the Sixth International Conference on Computer Science and Informatics, Durham, NC, USA, 2002, pp [4] C. Ekelin, Clairvoyant non-preemptive EDF scheduling, in: 18th Euromicro Conference on Real-time Systems, Dresden, Germany, 2006, pp [5] S.A. Goldman, J. Parwatikar, S. Suri, On-line scheduling with hard deadlines, Journal of Algorithms 34 (2000) [6] H. Hoogeveen, C.N. Potts, G.J. Woeginger, On-line scheduling on a single machine: Maximizing the number of early jobs, Operations Research Letters 27 (2000) [7] P. Keskinocak, Online algorithms with lookahead: A survey, ISYE working paper, [8] W. Mao, R.K. Kincaid, A look-ahead heuristic for scheduling jobs with release dates on a single machine, Computers and Operations Research 21 (1994)