Smiths Rule In Stochastic Scheduling Caroline Jagtenberg Uwe - PowerPoint PPT Presentation

Intro Related Work Preliminaries The Result End Smith’s Rule In Stochastic Scheduling Caroline Jagtenberg Uwe Schwiegelshohn Marc Uetz Utrecht University Dortmund University University of Twente Aussois 2011 Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End The (classic) setting Problem n jobs, nonpreemptive, processing times p j and weights w j m identical, parallel machines C j = completion time of job j goal: minimize total weighted completion time, � w j C j P | | � w j C j (thanks JKL) Complexity The problem is (strongly) NP-hard [ Bruno et al. 1974 ] PTAS exists [ Skutella and Woeginger, 2000 ] Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End WSPT a.k.a. Smith’s rule a.k.a. Photographer’s Rule WSPT Schedule jobs in order of non-increasing ratios w j / p j Performance On 1 machine WSPT is optimal [ Smith, 1956 ] √ For identical, parallel machines WSPT is a 1+ 2 ≈ 1 . 207- 2 approximation; this is tight [ Kawaguchi and Kyan, 1986 ] Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Step to stochastic scheduling Stochastic Scheduling processing times P = ( P 1 , . . . , P n ) unknown in advance P j ’s are random variables, known distribution solution no schedule, but scheduling policy Π for any policy Π: � w j C j (Π , P ) is a random variable Minimize expected performance E ( � w j C j (Π , P )) Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Complexity of (general) stochastic scheduling In general, optimal policies are NP-hard to find Calculating the objective value of a given policy can be # P complete [ Hagstrom 1988 ] Optimal policy may require deliberate idleness [ U. 2003 ] Question Does it become (significantly) easier if we restrict e.g. to only exponentially distributed processing times, i.e., P j ∼ exp( λ j )? i.e., P j ’s are memory-less, P [ P j > x + t | P j > t ] = P [ P j > x ] Open Problem 1 Does there exist an optimal policy without deliberate idleness? Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Intuition Quote ”Scheduling: Theory, Algorithms, and Systems” [ Pinedo, 2002 ] Example: P || C max is NP-hard for deterministic scheduling, but for P j ∼ exp( λ j ), LEPT is optimal [ Weiss and Pinedo, 1980 ] Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Most natural & simple scheduling policy: WSEPT WSEPT or Smith’s rule Greedily schedule jobs in order of decreasing w j / E ( P j ) = w j λ j . Facts about WSEPT for minimizing E [ � w j C j ] For one machine WSEPT is optimal [ Rothkopf, 1966 ] For parallel machines WSEPT is optimal if ordering exists w. w 1 ≥ ... ≥ w n and w 1 λ 1 ≥ ... ≥ w n λ n [ K¨ ampke, 1987 ] For parallel machines WSEPT is a (2 − 1 m )-approximation [ M¨ ohring, Schulz, U. 1999 ] Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Our (Counterintuitive?) Result Theorem Performance of WSEPT is not better than 1 . 243 OPT. That is, ∃ instances where in expectation E [ � w j C WSEPT ] > 1 . 243 E [ � w j C OPT ] j j Counterintuition: This is even worse than WSPT in deterministic scheduling, which is at most 1 . 207 OPT. Proof Follows from analysis and adaptation of the instance given by Kawaguchi and Kyan. Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Kawaguchi & Kyan (deterministic) example n · m small jobs with p j = w j = 1 x · m big jobs with p j = w j = p , n � w j C j = ( p 2 xm ) + ( 1 1 Left schedule: 1 − x m ) + o(1) 2 Right schedule: � w j C j = ((1 + p ) pxm ) + ( 1 2 m ) + o(1) √ √ 1 2 gives a maximal ratio of 1+ 2 p = 1 + 2 and x = ≈ 1 . 207 √ 2 2+ Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Stochastic version of Kawaguchi & Kyan example x · m i.i.d. big jobs each with P j ∼ exp( λ ), and w j := E [ P j ] = 1 λ := p n · m i.i.d. small jobs each with P j ∼ exp( n ), and w j := E [ P j ] = 1 n Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Scheduling m jobs with P j ∼ exp( λ ) Lemma Say we start at time t = 0 m i.i.d. jobs with P j ∼ exp( λ ), the expected number of available machines at time t is at least f ( t ) := m (1 − e − t λ ) − 1 . Interpretation Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Behaviour of parallel jobs with P j ∼ exp( λ ) When scheduling in parallel m jobs with i.i.d. processing times P j ∼ exp( λ ), the first completion is expected at time 1 / ( m λ ). As P j ’s are memory less, E [ P j − t | P j > t ] = E [ P j ] = 1 /λ , the second completion is expected time 1 / (( m − 1) λ ) later. j 1 etc., so j th completion is expected at time t j = � ( m − i + 1) λ i =1 m 1 � i ≥ ln ( m ) + 0 . 58, find that t j ≤ 1 λ ln ( m Using H ( m ) = m − j ), i =1 so # free machines at t : ≥ ⌊ m (1 − e − t λ ) ⌋ ≥ m (1 − e − t λ ) − 1 Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Stochastic version of (worst case) WSPT schedule Remember Kawaguchi and Kyan’s (worst case) schedule Machines finish processing short jobs “more or less” at t = 1 � m − 1 E [difference] ≤ 1 1 i ≈ 0 (as we have n > m ) i =1 n Each long job completes in expectation at time ≈ (1 + 1 λ ) Hence, E [ � w j C j ] ≈ to the deterministic case. Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Stochastic version of (optimal) WSPT schedule The expected optimal schedule of the stochastic variant: Contribution of long jobs is the same as in the deterministic case. What about the small jobs? � T Compute time T such that 0 f ( t ) dt ≥ total expected processing volume of small jobs. How? Numerically, T = 1 . 2933 suffices. Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End We can now approximate location of small jobs. But how much do they contribute to the objective value E [ � w j C j ] ? Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Contribution of Small Jobs Lemma Consider nT jobs with i.i.d. processing times P j ∼ exp( n ) and weights w j = 1 / n, scheduled on a single machine. Then for all ε > 0 there exists n large enough so that � T E [ � j w j C j ] ≤ t dt + ε . 0 Proof. � T n nT 1 / n + T 2 T 2 + 1 j w j C j ] = 1 = 1 0 t dt + 1 E [ � 2 n T = 2 n T 2 Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Contribution of Small Jobs Generalization We can generalize this lemma for parallel machines. Let m ( t ) be the number of machines available at time t, then � T E [ � j w j C j ] ≤ m ( t ) t dt + ε 0 Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Comparing the objective values E [ � w j C j ] Ingredients 1 long jobs’ contribution same as in deterministic case 2 machines w. small jobs finish at ≈ equal times (“sand”) � # available machines ≥ f ( t ) = m (1 − e − t λ ) − 1 3 for OPT � T small jobs contribute E [ � j w j C j ] ≤ 0 f ( t ) t dt Putting all that together, we get WSEPT is an α -approximation, E ( P j w j C j [ B ]) with α ≥ E ( P j w j C j [ A ]) ≥ 1 . 229 ( n → ∞ , m → ∞ ) Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End The result Optimizing over # and length E [ P j ] of the long jobs The result above was for √ 1 2 m ≈ 0 . 29 m long jobs with E [ P j ] = 1 + 2 ≈ 2 . 4 √ 2+ Taking for example: 0 . 43 m long jobs with E [ P j ] ≈ 1 . 8 yields α > 1 . 243 Theorem For jobs with exponentially distributed processing times, WSEPT is no better than a 1 . 243 - approximation. Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Conclusions What we’ve found With P j ∼ exp( λ j ), WSEPT can be factor > 1 . 243 away from optimal policy (in expectation); worse than tight bound for deterministic scheduling, ≈ 1 . 207 [ → WAOA 2010 proceedings] Open Problems 2 Instance(s) where WSEPT performs even worse? 3 I’d rather go and improve the upper bound (2 − 1 / m ) ! 4 Stochastic scheduling for P j ∼ exp( λ j ), hard at all? 5 And the complexity of computing E [ � j w j C WSEPT ]? j Marc Uetz Smith’s Rule in Stochastic Scheduling

Intro Related Work Preliminaries The Result End Smith’s Rule = Photographer’s Rule Group photos. . . Put short and important people first Back Marc Uetz Smith’s Rule in Stochastic Scheduling