Multi-Operation Multi-Machine Scheduling
|
|
- Wilfrid Shaw
- 6 years ago
- Views:
Transcription
1 Multi-Oeration Multi-Machine Scheduling Weizhen Mao he College of William and Mary, Williamsburg VA 3185, USA Abstract. In the multi-oeration scheduling that arises in industrial engineering, each job contains multile tasks (oerations) that require execution in different shos. It is assumed that in each sho there is only one machine to erform the required oerations. In this aer, a arallel model of multi-oeration scheduling is roosed, in which multile machines are available in each sho to erform the same tye of oerations. A multi-machine oen-sho scheduling roblem is studied in detail. 1 Introduction Multi-oeration scheduling as a research area is motivated by questions that arise in industrial manufacturing, roduction lanning, and comuter control. Consider a large automotive garage with secialized shos [?]. A car may require the following work: relace exhaust system, align wheels, and tune u. hese three tasks may be carried out in any order. However, since the exhaust system, alignment, and tune-u shos are in different buildings, it is imossible to erform two tasks for a car simultaneously. When there are many cars requiring services at the three shos, it is desirable to construct a service schedule that takes the least amount of total time. he examle above is in fact a secial case of the traditional multi-oeration scheduling [?,?], where m shos S 1,...,S m rovide different tyes of services to n jobs J 1,...,J n. Each sho S i contains exactly one machine M i that does the actual work. Each job J j contains m tasks j1,..., jm, with the required rocessing times j1,..., jm, resectively. ask ji must be executed by machine M i in sho S i, and no two tasks of the same job can be executed simultaneously. In addition, each machine can only execute one task at any time. he scheduling is called oen-sho scheduling (O) if the order in which a job asses through the shos is immaterial, flow-sho scheduling (F) if each job has the same sho ordering, e.g. (S 1,...,S m ), and job-sho scheduling (J) if the jobs may have different sho orderings. Multi-oeration scheduling roblems are otimization roblems and include various arameters. o define such roblems, we use the three-field classification α β γ [?], where α describes the sho and machine environment, β describes the job and task characteristics, and γ describes the otimality criteria. he automotive garage examle can be denoted by O3 C max, which is a three-sho (m = 3) oen-sho scheduling roblem of minimizing the maximum comletion time (makesan). he above multi-oeration model assumes that there is only one machine in each sho to execute tasks. However, this is certainly not the case in the arallel rocessing environment, where more than one machine (rocessor) are available to erform a certain tye of oerations. his motivates us to define the following multi-oeration 33
2 multi-machine model. Suose that there are m shos S 1,...,S m and n jobs J 1,...,J n. Sho S i consists of k i machines M i1,...,m iki to erform the same tye of oerations. Job J j consists of m tasks j1,..., jm to be executed on any machine in S 1,...,S m, resectively. he machines in a sho work in arallel, and deending on the secific environment, they may be identical (P), having the same seed, or uniform (Q), having different but fixed seeds, or unrelated (R), having different seeds for different tasks. A multi-oeration multi-machine scheduling roblem can also be defined by the α β γ classification. For instance, if the first field is OmPk, then the roblem is an oen-sho scheduling roblem, where there are m shos, each having k identical arallel machines. he automotive garage examle can be modified to include the arallel rocessing concet by having more than one worker (machine) in each of the three secialized shos. As in the traditional scheduling roblems, a multi-oeration multi-machine schedule can be nonreemtive or reemtive. We say that a schedule is nonreemtive if the execution of any task can not be interruted, and that a schedule is reemtive if the execution of a task can be interruted and be resumed later on. A reemtive schedule is desired when there is no enalty for an interrution. Since very few multi-oeration scheduling roblems can be solved in olynomial time, adding the multi-machine environment makes the resulting multi-oeration multi-machine scheduling even more difficult. In this aer, we study an oen-sho scheduling roblem with two shos and k identical arallel machines in each sho to minimize the makesan. We organize the aer as follows. In Section, we give an NP-comleteness roof of the nonreemtive version. In Section 3, we resent an efficient algorithm for the reemtive version. We conclude in Section 4. Nonreemtive Scheduling he roblem of interest in this section is the nonreemtive oen-sho scheduling of n jobs in two shos each with k identical machines to minimize the makesan. It is denoted by OPk C max. heorem 1. OPk C max is NP-comlete even when k =. Proof Consider the corresonding decision roblem, in which given a bound B, we are asked whether there is a multi-oeration multi-machine schedule with makesan C max B. his decision roblem of OP C max is certainly in NP. Now we show that it can be olynomially reduced from the NP-comlete PARIION [?]. Given any instance of PARIION, A = {a 1,a,...,a n } (ositive integers), we construct an instance of the decision roblem, in which there are n jobs J 1,...,J n. Each job J j consists of two tasks j1 and j with rocessing times a j and ε, resectively, where 0 ε 1 min j{a j }. Assume that there are two shos S 1 and S and that each sho S i has two identical machines M i1 and M i. ask j1 can be executed by machines M 11 and M 1 only, while task j by machines M 1 and M only. Finally, let B = j a j. We next show that there exists A A such that a j A a j = j a j if and only if there is a nonreemtive schedule with C max B for the instance defined. If there is A A with a j A a j = j a j, without loss of generality assume that A = {a 1,...,a h }. 34
3 A feasible schedule with C max B of the instance can be constructed by scheduling tasks 11,..., h1 on machine M 11 and the remaining tasks (h+1)1,..., n1 on machine M 1. Let s j1 and f j1 be the starting time and finishing (comletion) time of task j1 in sho S 1. As for each task j that must be executed in sho S, if f j1 + ε B, then schedule j in interval [ f j1, f j1 + ε] on either M 1 or M, otherwise schedule j in interval [s j1 ε,s j1 ] on either M 1 or M. See Fig. 1 for an examle. S S 1 M 11 M M M B B Fig. 1. A feasible schedule with C max B If there is a feasible schedule with C max B = j a j for the instance, the maximum comletion time of tasks in S 1 must be j a j. Since the schedule is nonreemtive, we define A = {a j j1 is executed by M 11 }. herefore, a j A a j = j a j. Q.E.D. he NP-comleteness of OPk C max indicates that no olynomial-time algorithm for the roblem is likely to be found and that research should focus on designing fast and good aroximation algorithms. 3 Preemtive Scheduling he roblem of interest in this section is the reemtive oen-sho scheduling of n jobs in two shos each with k identical machines to minimize the makesan. It is denoted by OPk mtn C max. We give an O(n ) algorithm for the roblem. Let i = 1,, j = 1,...,n, and l = 1,...,k. Let ji be the rocessing time of task ji of job J j. Note that task ji must be scheduled, with reemtion allowed, on any machine M il in sho S i. Define C = max{ k j j1, k j j,max j { j1 + j }}. C is obviously a lower bound on the makesan of any reemtive schedule. If we can construct a schedule with makesan C, the schedule must be otimal. able 1 shows an instance and its corresonding C. We first construct a schedule for tasks j1 in S 1 using McNaughton s wra-around rule [?]: just fill the machines successively, scheduling the tasks in any order and slitting a task whenever the bound C is met. he schedule in Fig. is one for tasks j1 in the examle given in able 1. Let C max (1) be the makesan of the schedule constructed, then C (1) max C and C (1) max < C only when j j1 < C. 35
4 able 1. An inatance of OP mtn C max k = 1 P J 1 J J 3 J 4 J 5 k j ji S S j1 + j C = 6 M S 1 M Fig.. A feasible schedule for S 1 We next construct a schedule for tasks j in S with makesan C () max C. o do so, we need to know whether a task can be scheduled in a time interval. Let us consider the interval (0,C) in the schedule for S 1 constructed in the revious ste. We say that t (0,C) is a turning oint if at time t at least one machine in S 1 changes its status from executing to idling or from executing a task to executing another task. Assume that there are h 1 turning oints t 1 < < t h 1 in (0,C). Clearly, h 1 n. Let t 0 = 0 and t h = C. We define an interval I r = (t r 1,t r ] for r = 1,...,h. Let D r be the set of indices of the jobs whose tasks may be scheduled in interval I r in S. Clearly, n k D r n. Let B j be the set of indices of the intervals in which j may be scheduled. Clearly, B j h and r B j (t r t r 1 ) = C j1. For the examle given in Fig., I 1 = (0,], D 1 = {,4,5}, and B 1 = {,3,4,5}. Let x r j be the length of time in I r that j of J j is being executed by any machine in S. If we can determine x r j for all r and j satisfying x r j = 0 for j D r, (1) r B j x r j = j for j = 1,...,n, () j D r x r j k(t r t r 1 ) for r = 1,...,h, (3) then we have a schedule for tasks j in S with C () max C. Constraint (1) ensures that j1 and j do not overla, constraint () ensures that the total amount of time for which j is rocessed is the required rocessing time j, and constraint (3) ensures that the total amount of time used in I r on all k machines is no larger than that available. We use the following algorithm to determine x r j for j D r. For notational simlicity, let δ r = j D r x r j k(t r t r 1 ). 1. x r j j C j1 (t r t r 1 ) for r = 1,...,h and j D r ;. while there exists I with δ > 0 36
5 3. locate I q with δ q < 0; 4. define 0 ε j x j for j D D q such that j D D q ε j = min{δ, δ q }; 5. x j x j ε j for j D D q ; 6. x q j x q j + ε j for j D D q ; o rove the correctness of the algorithm, we wish to show that constraints () and (3) are satisfied by the final values of x r j comuted. Consider the initial values of x r j in line 1. For each j = 1,...,n, r B j x r j = j C j1 r B j (t r t r 1 ) = j. In each iteration of the while loo, r B j x r j remains to be j since every time the value of some x j is decreased (line 5), the value of some x q j is increased by the same amount (line 6). herefore, constraint () is satisfied. As for constraint (3), if the while loo terminates eventually, then we must have δ r 0, hence by definition j D r x r j k(t r t r 1 ) for r = 1,...,h, which is constraint (3). Before we show that the loo always terminates, we need Lemmas?? and?? to justify lines 3 and 4, resectively. Lemma 1. At the beginning of each iteration of the while loo, if there exists I with δ > 0 (line ), then there exists I q with δ q < 0 (line 3). Proof Suose not, we must have r δ r > 0. So r j D r x r j k r (t r t r 1 ) = j j kc > 0, and hence C < 1 k j j. his is imossible. Q.E.D. Lemma. In each iteration of the while loo, if there exist I and I q with δ > 0 and δ q < 0, then there exist 0 ε j x j for j D D q such that j D D q ε j = min{δ, δ q } (line 4). Proof It is sufficient to show that j D s D r x s j δ s for any s with δ s > 0 and any r s. Let w b sr and δ b s be the values of j D s D r x s j and δ s, resectively, at the end of the b th iteration of the while loo. We wish to show that w b sr δ b s for each b. We induct on b. When b = 0, w 0 sr and δ 0 s are defined by the initial values of x s j (line 1). So w 0 sr = (t s t s 1 ) j j D s D r C j1 and δ 0 s = (t s t s 1 )( j j D s C j1 k). Since D s D s D r k and j C j1 1, then j j D s D r C j1 j j D s C j1 k. herefore, w 0 sr δ 0 s. Assume that w b 1 sr δ b 1 s for any s with δ b 1 s > 0 and any r s. Now consider b. Suose that I is the interval with δ b 1 > 0 chosen in line in the b th iteration and that I q is the interval with δ b 1 q < 0 chosen in line 3 in the b th iteration. After we move a total of min{δ b 1, δ b 1 q } from I to I q, we have w b q = w b 1 q w b r = w b 1 r w b sr = w b 1 sr min{δ b 1, δ b 1 q j D D q D r ε j w b 1 r } and δ b = δ b 1 min{δ b 1 min{δ b 1, δ b 1 q, δ b 1 and δ b s = δ b 1 s for any s with δ b s > 0 and any r s. q }. } for any r,q. So in summary, w b sr δ b s for any s with δ s > 0 and any r s. Q.E.D. Let d be the number of intervals I r with δ r 0 before the while loo is executed. We then have d h n+1. In each iteration, d is decreased by at least 1. So the loo must 37
6 terminate eventually. Constraint (3) is satisfied. It is clear that once we have all the x r j for r = 1,...,h and j = 1,...,n satisfying constraints (1), (), and (3), a schedule with C () max C for tasks j in S can be constructed easily. We have the following theorem. heorem. OPk mtn C max is solvable in O(n ) time. 4 Conclusions In this aer, we introduced a arallel model for multi-oeration scheduling, and studied a two-sho oen-sho scheduling roblem with multile identical machines available in each sho. We roved the NP-comleteness for the nonreemtive version and gave an efficient algorithm for the reemtive version. he multi-machine multi-oeration scheduling is a better and more ractical model than the traditional multi-oeration scheduling since it catures the essence of arallel rocessing that is being emloyed in various asects of industrial engineering. We wish that this aer will rovide a starting oint for future research in this area. References 1. Coffman, E. G. Jr.: Comuter and Job Sho Scheduling heory. John Wiley and Sons, New York (1976). Garey, M. R., Johnson, D. S.: Comuters and Intractability: A guide to the theory of NPcomleteness. Freeman, San Francisco (1979) 3. Gonzalez,., Sahni, S.: Oen sho scheduling to minimize finish time. J. ACM 3 (1976) Lawler, E. L., Lenstra, J. K., Rinnooy Kan, A. H. G., Shmoys, D. B.: Sequencing and scheduling: Algorithms and comlexity. Handbooks in Oerations Research and Management Science, Volume 4: Logistics of Production and Inventory, S. C. Graves, A. H. G. Rinnooy Kan and P. Zikin, ed., North-Holland (1990) 5. McNaughton, R.: Scheduling with deadlines and loss function. Management Sci. 6 (1959)
Dynamic-Priority Scheduling. CSCE 990: Real-Time Systems. Steve Goddard. Dynamic-priority Scheduling
CSCE 990: Real-Time Systems Dynamic-Priority Scheduling Steve Goddard goddard@cse.unl.edu htt://www.cse.unl.edu/~goddard/courses/realtimesystems Dynamic-riority Scheduling Real-Time Systems Dynamic-Priority
More informationNew Schedulability Test Conditions for Non-preemptive Scheduling on Multiprocessor Platforms
New Schedulability Test Conditions for Non-reemtive Scheduling on Multirocessor Platforms Technical Reort May 2008 Nan Guan 1, Wang Yi 2, Zonghua Gu 3 and Ge Yu 1 1 Northeastern University, Shenyang, China
More informationOnline Over Time Scheduling on Parallel-Batch Machines: A Survey
J. Oer. Res. Soc. China (014) :44 44 DOI 10.1007/s4030-014-0060-0 Online Over Time Scheduling on Parallel-Batch Machines: A Survey Ji Tian Ruyan Fu Jinjiang Yuan Received: 30 October 014 / Revised: 3 November
More informationAnalysis of M/M/n/K Queue with Multiple Priorities
Analysis of M/M/n/K Queue with Multile Priorities Coyright, Sanjay K. Bose For a P-riority system, class P of highest riority Indeendent, Poisson arrival rocesses for each class with i as average arrival
More informationPeriodic scheduling 05/06/
Periodic scheduling T T or eriodic scheduling, the best that we can do is to design an algorithm which will always find a schedule if one exists. A scheduler is defined to be otimal iff it will find a
More informationApproximating min-max k-clustering
Aroximating min-max k-clustering Asaf Levin July 24, 2007 Abstract We consider the roblems of set artitioning into k clusters with minimum total cost and minimum of the maximum cost of a cluster. The cost
More informationFinding Shortest Hamiltonian Path is in P. Abstract
Finding Shortest Hamiltonian Path is in P Dhananay P. Mehendale Sir Parashurambhau College, Tilak Road, Pune, India bstract The roblem of finding shortest Hamiltonian ath in a eighted comlete grah belongs
More informationPolynomial-Time Exact Schedulability Tests for Harmonic Real-Time Tasks
Polynomial-Time Exact Schedulability Tests for Harmonic Real-Time Tasks Vincenzo Bonifaci, Alberto Marchetti-Saccamela, Nicole Megow, Andreas Wiese Istituto di Analisi dei Sistemi ed Informatica Antonio
More informationImproved Bounds on Bell Numbers and on Moments of Sums of Random Variables
Imroved Bounds on Bell Numbers and on Moments of Sums of Random Variables Daniel Berend Tamir Tassa Abstract We rovide bounds for moments of sums of sequences of indeendent random variables. Concentrating
More information1. INTRODUCTION. Fn 2 = F j F j+1 (1.1)
CERTAIN CLASSES OF FINITE SUMS THAT INVOLVE GENERALIZED FIBONACCI AND LUCAS NUMBERS The beautiful identity R.S. Melham Deartment of Mathematical Sciences, University of Technology, Sydney PO Box 23, Broadway,
More informationThe Graph Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule
The Grah Accessibility Problem and the Universality of the Collision CRCW Conflict Resolution Rule STEFAN D. BRUDA Deartment of Comuter Science Bisho s University Lennoxville, Quebec J1M 1Z7 CANADA bruda@cs.ubishos.ca
More informationOn the Chvatál-Complexity of Knapsack Problems
R u t c o r Research R e o r t On the Chvatál-Comlexity of Knasack Problems Gergely Kovács a Béla Vizvári b RRR 5-08, October 008 RUTCOR Rutgers Center for Oerations Research Rutgers University 640 Bartholomew
More informationOn generalizing happy numbers to fractional base number systems
On generalizing hay numbers to fractional base number systems Enriue Treviño, Mikita Zhylinski October 17, 018 Abstract Let n be a ositive integer and S (n) be the sum of the suares of its digits. It is
More information4 Scheduling. Outline of the chapter. 4.1 Preliminaries
4 Scheduling In this section, e consider so-called Scheduling roblems I.e., if there are altogether M machines or resources for each machine, a roduction sequence of all N jobs has to be found as ell as
More informationCHAPTER-5 PERFORMANCE ANALYSIS OF AN M/M/1/K QUEUE WITH PREEMPTIVE PRIORITY
CHAPTER-5 PERFORMANCE ANALYSIS OF AN M/M//K QUEUE WITH PREEMPTIVE PRIORITY 5. INTRODUCTION In last chater we discussed the case of non-reemtive riority. Now we tae the case of reemtive riority. Preemtive
More informationThe Value of Even Distribution for Temporal Resource Partitions
The Value of Even Distribution for Temoral Resource Partitions Yu Li, Albert M. K. Cheng Deartment of Comuter Science University of Houston Houston, TX, 7704, USA htt://www.cs.uh.edu Technical Reort Number
More informationMATH 2710: NOTES FOR ANALYSIS
MATH 270: NOTES FOR ANALYSIS The main ideas we will learn from analysis center around the idea of a limit. Limits occurs in several settings. We will start with finite limits of sequences, then cover infinite
More informationUniversity of Twente. Faculty of Mathematical Sciences. Scheduling split-jobs on parallel machines. University for Technical and Social Sciences
Faculty of Mathematical Sciences University of Twente University for Technical and Social Sciences P.O. Box 217 7500 AE Enschede The Netherlands Phone: +31-53-4893400 Fax: +31-53-4893114 Email: memo@math.utwente.nl
More informationResearch Article An Iterative Algorithm for the Reflexive Solution of the General Coupled Matrix Equations
he Scientific World Journal Volume 013 Article ID 95974 15 ages htt://dxdoiorg/101155/013/95974 Research Article An Iterative Algorithm for the Reflexive Solution of the General Couled Matrix Euations
More informationASPECTS OF POLE PLACEMENT TECHNIQUE IN SYMMETRICAL OPTIMUM METHOD FOR PID CONTROLLER DESIGN
ASES OF OLE LAEMEN EHNIQUE IN SYMMERIAL OIMUM MEHOD FOR ID ONROLLER DESIGN Viorel Nicolau *, onstantin Miholca *, Dorel Aiordachioaie *, Emil eanga ** * Deartment of Electronics and elecommunications,
More informationEnd-to-End Delay Minimization in Thermally Constrained Distributed Systems
End-to-End Delay Minimization in Thermally Constrained Distributed Systems Pratyush Kumar, Lothar Thiele Comuter Engineering and Networks Laboratory (TIK) ETH Zürich, Switzerland {ratyush.kumar, lothar.thiele}@tik.ee.ethz.ch
More informationFor q 0; 1; : : : ; `? 1, we have m 0; 1; : : : ; q? 1. The set fh j(x) : j 0; 1; ; : : : ; `? 1g forms a basis for the tness functions dened on the i
Comuting with Haar Functions Sami Khuri Deartment of Mathematics and Comuter Science San Jose State University One Washington Square San Jose, CA 9519-0103, USA khuri@juiter.sjsu.edu Fax: (40)94-500 Keywords:
More informationSome results of convex programming complexity
2012c12 $ Ê Æ Æ 116ò 14Ï Dec., 2012 Oerations Research Transactions Vol.16 No.4 Some results of convex rogramming comlexity LOU Ye 1,2 GAO Yuetian 1 Abstract Recently a number of aers were written that
More informationMath 4400/6400 Homework #8 solutions. 1. Let P be an odd integer (not necessarily prime). Show that modulo 2,
MATH 4400 roblems. Math 4400/6400 Homework # solutions 1. Let P be an odd integer not necessarily rime. Show that modulo, { P 1 0 if P 1, 7 mod, 1 if P 3, mod. Proof. Suose that P 1 mod. Then we can write
More informationMonopolist s mark-up and the elasticity of substitution
Croatian Oerational Research Review 377 CRORR 8(7), 377 39 Monoolist s mark-u and the elasticity of substitution Ilko Vrankić, Mira Kran, and Tomislav Herceg Deartment of Economic Theory, Faculty of Economics
More informationUniformly best wavenumber approximations by spatial central difference operators: An initial investigation
Uniformly best wavenumber aroximations by satial central difference oerators: An initial investigation Vitor Linders and Jan Nordström Abstract A characterisation theorem for best uniform wavenumber aroximations
More informationLinear diophantine equations for discrete tomography
Journal of X-Ray Science and Technology 10 001 59 66 59 IOS Press Linear diohantine euations for discrete tomograhy Yangbo Ye a,gewang b and Jiehua Zhu a a Deartment of Mathematics, The University of Iowa,
More information#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS
#A64 INTEGERS 18 (2018) APPLYING MODULAR ARITHMETIC TO DIOPHANTINE EQUATIONS Ramy F. Taki ElDin Physics and Engineering Mathematics Deartment, Faculty of Engineering, Ain Shams University, Cairo, Egyt
More informationHENSEL S LEMMA KEITH CONRAD
HENSEL S LEMMA KEITH CONRAD 1. Introduction In the -adic integers, congruences are aroximations: for a and b in Z, a b mod n is the same as a b 1/ n. Turning information modulo one ower of into similar
More informationVarious Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various Families of R n Norms and Some Open Problems
Int. J. Oen Problems Comt. Math., Vol. 3, No. 2, June 2010 ISSN 1998-6262; Coyright c ICSRS Publication, 2010 www.i-csrs.org Various Proofs for the Decrease Monotonicity of the Schatten s Power Norm, Various
More informationA Competitive Algorithm for Minimizing Weighted Flow Time on Unrelated Machines with Speed Augmentation
Cometitive lgorithm for Minimizing Weighted Flow Time on Unrelated Machines with Seed ugmentation Jivitej S. Chadha and Naveen Garg and mit Kumar and V. N. Muralidhara Comuter Science and Engineering Indian
More informationOn the performance of greedy algorithms for energy minimization
On the erformance of greedy algorithms for energy minimization Anne Benoit, Paul Renaud-Goud, Yves Robert To cite this version: Anne Benoit, Paul Renaud-Goud, Yves Robert On the erformance of greedy algorithms
More informationCryptanalysis of Pseudorandom Generators
CSE 206A: Lattice Algorithms and Alications Fall 2017 Crytanalysis of Pseudorandom Generators Instructor: Daniele Micciancio UCSD CSE As a motivating alication for the study of lattice in crytograhy we
More informationOn Wald-Type Optimal Stopping for Brownian Motion
J Al Probab Vol 34, No 1, 1997, (66-73) Prerint Ser No 1, 1994, Math Inst Aarhus On Wald-Tye Otimal Stoing for Brownian Motion S RAVRSN and PSKIR The solution is resented to all otimal stoing roblems of
More informationReal-Time Computing with Lock-Free Shared Objects
Real-Time Comuting with Lock-Free Shared Objects JAMES H. ADERSO, SRIKATH RAMAMURTHY, and KEVI JEFFAY University of orth Carolina This article considers the use of lock-free shared objects within hard
More informationGOOD MODELS FOR CUBIC SURFACES. 1. Introduction
GOOD MODELS FOR CUBIC SURFACES ANDREAS-STEPHAN ELSENHANS Abstract. This article describes an algorithm for finding a model of a hyersurface with small coefficients. It is shown that the aroach works in
More informationOXFORD UNIVERSITY. MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE WEDNESDAY 4 NOVEMBER 2009 Time allowed: hours
OXFORD UNIVERSITY MATHEMATICS, JOINT SCHOOLS AND COMPUTER SCIENCE WEDNESDAY 4 NOVEMBER 2009 Time allowed: 2 1 2 hours For candidates alying for Mathematics, Mathematics & Statistics, Comuter Science, Mathematics
More informationIdeal preemptive schedules on two processors
Acta Informatica 39, 597 612 (2003) Digital Object Identifier (DOI) 10.1007/s00236-003-0119-6 c Springer-Verlag 2003 Ideal preemptive schedules on two processors E.G. Coffman, Jr. 1, J. Sethuraman 2,,
More informationMODELING THE RELIABILITY OF C4ISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL
Technical Sciences and Alied Mathematics MODELING THE RELIABILITY OF CISR SYSTEMS HARDWARE/SOFTWARE COMPONENTS USING AN IMPROVED MARKOV MODEL Cezar VASILESCU Regional Deartment of Defense Resources Management
More informationAnalysis of some entrance probabilities for killed birth-death processes
Analysis of some entrance robabilities for killed birth-death rocesses Master s Thesis O.J.G. van der Velde Suervisor: Dr. F.M. Sieksma July 5, 207 Mathematical Institute, Leiden University Contents Introduction
More informationResearch Article Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inputs
Abstract and Alied Analysis Volume 203 Article ID 97546 5 ages htt://dxdoiorg/055/203/97546 Research Article Controllability of Linear Discrete-Time Systems with Both Delayed States and Delayed Inuts Hong
More informationEstimation of the large covariance matrix with two-step monotone missing data
Estimation of the large covariance matrix with two-ste monotone missing data Masashi Hyodo, Nobumichi Shutoh 2, Takashi Seo, and Tatjana Pavlenko 3 Deartment of Mathematical Information Science, Tokyo
More informationIntroduction Consider a set of jobs that are created in an on-line fashion and should be assigned to disks. Each job has a weight which is the frequen
Ancient and new algorithms for load balancing in the L norm Adi Avidor Yossi Azar y Jir Sgall z July 7, 997 Abstract We consider the on-line load balancing roblem where there are m identical machines (servers)
More informationAdditive results for the generalized Drazin inverse in a Banach algebra
Additive results for the generalized Drazin inverse in a Banach algebra Dragana S. Cvetković-Ilić Dragan S. Djordjević and Yimin Wei* Abstract In this aer we investigate additive roerties of the generalized
More informationResearch Article An iterative Algorithm for Hemicontractive Mappings in Banach Spaces
Abstract and Alied Analysis Volume 2012, Article ID 264103, 11 ages doi:10.1155/2012/264103 Research Article An iterative Algorithm for Hemicontractive Maings in Banach Saces Youli Yu, 1 Zhitao Wu, 2 and
More informationStrong Matching of Points with Geometric Shapes
Strong Matching of Points with Geometric Shaes Ahmad Biniaz Anil Maheshwari Michiel Smid School of Comuter Science, Carleton University, Ottawa, Canada December 9, 05 In memory of Ferran Hurtado. Abstract
More informationComplex Analysis Homework 1
Comlex Analysis Homework 1 Steve Clanton Sarah Crimi January 27, 2009 Problem Claim. If two integers can be exressed as the sum of two squares, then so can their roduct. Proof. Call the two squares that
More informationElementary Analysis in Q p
Elementary Analysis in Q Hannah Hutter, May Szedlák, Phili Wirth November 17, 2011 This reort follows very closely the book of Svetlana Katok 1. 1 Sequences and Series In this section we will see some
More informationFig. 21: Architecture of PeerSim [44]
Sulementary Aendix A: Modeling HPP with PeerSim Fig. : Architecture of PeerSim [] In PeerSim, every comonent can be relaced by another comonent imlementing the same interface, and the general simulation
More informationA generalization of Amdahl's law and relative conditions of parallelism
A generalization of Amdahl's law and relative conditions of arallelism Author: Gianluca Argentini, New Technologies and Models, Riello Grou, Legnago (VR), Italy. E-mail: gianluca.argentini@riellogrou.com
More informationA STUDY ON THE UTILIZATION OF COMPATIBILITY METRIC IN THE AHP: APPLYING TO SOFTWARE PROCESS ASSESSMENTS
ISAHP 2005, Honolulu, Hawaii, July 8-10, 2003 A SUDY ON HE UILIZAION OF COMPAIBILIY MERIC IN HE AHP: APPLYING O SOFWARE PROCESS ASSESSMENS Min-Suk Yoon Yosu National University San 96-1 Dundeok-dong Yeosu
More informationTHE 2D CASE OF THE BOURGAIN-DEMETER-GUTH ARGUMENT
THE 2D CASE OF THE BOURGAIN-DEMETER-GUTH ARGUMENT ZANE LI Let e(z) := e 2πiz and for g : [0, ] C and J [0, ], define the extension oerator E J g(x) := g(t)e(tx + t 2 x 2 ) dt. J For a ositive weight ν
More information2-D Analysis for Iterative Learning Controller for Discrete-Time Systems With Variable Initial Conditions Yong FANG 1, and Tommy W. S.
-D Analysis for Iterative Learning Controller for Discrete-ime Systems With Variable Initial Conditions Yong FANG, and ommy W. S. Chow Abstract In this aer, an iterative learning controller alying to linear
More information4. Score normalization technical details We now discuss the technical details of the score normalization method.
SMT SCORING SYSTEM This document describes the scoring system for the Stanford Math Tournament We begin by giving an overview of the changes to scoring and a non-technical descrition of the scoring rules
More informationSystem Reliability Estimation and Confidence Regions from Subsystem and Full System Tests
009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 0-, 009 FrB4. System Reliability Estimation and Confidence Regions from Subsystem and Full System Tests James C. Sall Abstract
More informationUncorrelated Multilinear Principal Component Analysis for Unsupervised Multilinear Subspace Learning
TNN-2009-P-1186.R2 1 Uncorrelated Multilinear Princial Comonent Analysis for Unsuervised Multilinear Subsace Learning Haiing Lu, K. N. Plataniotis and A. N. Venetsanooulos The Edward S. Rogers Sr. Deartment
More informationMobius Functions, Legendre Symbols, and Discriminants
Mobius Functions, Legendre Symbols, and Discriminants 1 Introduction Zev Chonoles, Erick Knight, Tim Kunisky Over the integers, there are two key number-theoretic functions that take on values of 1, 1,
More informationWhen do the Fibonacci invertible classes modulo M form a subgroup?
Annales Mathematicae et Informaticae 41 (2013). 265 270 Proceedings of the 15 th International Conference on Fibonacci Numbers and Their Alications Institute of Mathematics and Informatics, Eszterházy
More informationDETC2003/DAC AN EFFICIENT ALGORITHM FOR CONSTRUCTING OPTIMAL DESIGN OF COMPUTER EXPERIMENTS
Proceedings of DETC 03 ASME 003 Design Engineering Technical Conferences and Comuters and Information in Engineering Conference Chicago, Illinois USA, Setember -6, 003 DETC003/DAC-48760 AN EFFICIENT ALGORITHM
More informationConvex Optimization methods for Computing Channel Capacity
Convex Otimization methods for Comuting Channel Caacity Abhishek Sinha Laboratory for Information and Decision Systems (LIDS), MIT sinhaa@mit.edu May 15, 2014 We consider a classical comutational roblem
More informationA continuous review inventory model with the controllable production rate of the manufacturer
Intl. Trans. in O. Res. 12 (2005) 247 258 INTERNATIONAL TRANSACTIONS IN OERATIONAL RESEARCH A continuous review inventory model with the controllable roduction rate of the manufacturer I. K. Moon and B.
More informationDiscrete-time Geo/Geo/1 Queue with Negative Customers and Working Breakdowns
Discrete-time GeoGeo1 Queue with Negative Customers and Working Breakdowns Tao Li and Liyuan Zhang Abstract This aer considers a discrete-time GeoGeo1 queue with server breakdowns and reairs. If the server
More informationCOMPARISON OF VARIOUS OPTIMIZATION TECHNIQUES FOR DESIGN FIR DIGITAL FILTERS
NCCI 1 -National Conference on Comutational Instrumentation CSIO Chandigarh, INDIA, 19- March 1 COMPARISON OF VARIOUS OPIMIZAION ECHNIQUES FOR DESIGN FIR DIGIAL FILERS Amanjeet Panghal 1, Nitin Mittal,Devender
More informationwhere x i is the ith coordinate of x R N. 1. Show that the following upper bound holds for the growth function of H:
Mehryar Mohri Foundations of Machine Learning Courant Institute of Mathematical Sciences Homework assignment 2 October 25, 2017 Due: November 08, 2017 A. Growth function Growth function of stum functions.
More informationSampling and Distortion Tradeoffs for Bandlimited Periodic Signals
Samling and Distortion radeoffs for Bandlimited Periodic Signals Elaheh ohammadi and Farokh arvasti Advanced Communications Research Institute ACRI Deartment of Electrical Engineering Sharif University
More informationMODEL-BASED MULTIPLE FAULT DETECTION AND ISOLATION FOR NONLINEAR SYSTEMS
MODEL-BASED MULIPLE FAUL DEECION AND ISOLAION FOR NONLINEAR SYSEMS Ivan Castillo, and homas F. Edgar he University of exas at Austin Austin, X 78712 David Hill Chemstations Houston, X 77009 Abstract A
More informationEvaluating Circuit Reliability Under Probabilistic Gate-Level Fault Models
Evaluating Circuit Reliability Under Probabilistic Gate-Level Fault Models Ketan N. Patel, Igor L. Markov and John P. Hayes University of Michigan, Ann Arbor 48109-2122 {knatel,imarkov,jhayes}@eecs.umich.edu
More informationCombining Logistic Regression with Kriging for Mapping the Risk of Occurrence of Unexploded Ordnance (UXO)
Combining Logistic Regression with Kriging for Maing the Risk of Occurrence of Unexloded Ordnance (UXO) H. Saito (), P. Goovaerts (), S. A. McKenna (2) Environmental and Water Resources Engineering, Deartment
More informationBy Evan Chen OTIS, Internal Use
Solutions Notes for DNY-NTCONSTRUCT Evan Chen January 17, 018 1 Solution Notes to TSTST 015/5 Let ϕ(n) denote the number of ositive integers less than n that are relatively rime to n. Prove that there
More informationA New Method of DDB Logical Structure Synthesis Using Distributed Tabu Search
A New Method of DDB Logical Structure Synthesis Using Distributed Tabu Search Eduard Babkin and Margarita Karunina 2, National Research University Higher School of Economics Det of nformation Systems and
More information19th Bay Area Mathematical Olympiad. Problems and Solutions. February 28, 2017
th Bay Area Mathematical Olymiad February, 07 Problems and Solutions BAMO- and BAMO- are each 5-question essay-roof exams, for middle- and high-school students, resectively. The roblems in each exam are
More informationLilian Markenzon 1, Nair Maria Maia de Abreu 2* and Luciana Lee 3
Pesquisa Oeracional (2013) 33(1): 123-132 2013 Brazilian Oerations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/oe SOME RESULTS ABOUT THE CONNECTIVITY OF
More information#A47 INTEGERS 15 (2015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS
#A47 INTEGERS 15 (015) QUADRATIC DIOPHANTINE EQUATIONS WITH INFINITELY MANY SOLUTIONS IN POSITIVE INTEGERS Mihai Ciu Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit No. 5,
More informationAN OPTIMAL CONTROL CHART FOR NON-NORMAL PROCESSES
AN OPTIMAL CONTROL CHART FOR NON-NORMAL PROCESSES Emmanuel Duclos, Maurice Pillet To cite this version: Emmanuel Duclos, Maurice Pillet. AN OPTIMAL CONTROL CHART FOR NON-NORMAL PRO- CESSES. st IFAC Worsho
More information8 STOCHASTIC PROCESSES
8 STOCHASTIC PROCESSES The word stochastic is derived from the Greek στoχαστικoς, meaning to aim at a target. Stochastic rocesses involve state which changes in a random way. A Markov rocess is a articular
More informationNotes on duality in second order and -order cone otimization E. D. Andersen Λ, C. Roos y, and T. Terlaky z Aril 6, 000 Abstract Recently, the so-calle
McMaster University Advanced Otimization Laboratory Title: Notes on duality in second order and -order cone otimization Author: Erling D. Andersen, Cornelis Roos and Tamás Terlaky AdvOl-Reort No. 000/8
More informationSums of independent random variables
3 Sums of indeendent random variables This lecture collects a number of estimates for sums of indeendent random variables with values in a Banach sace E. We concentrate on sums of the form N γ nx n, where
More informations v 0 q 0 v 1 q 1 v 2 (q 2) v 3 q 3 v 4
Discrete Adative Transmission for Fading Channels Lang Lin Λ, Roy D. Yates, Predrag Sasojevic WINLAB, Rutgers University 7 Brett Rd., NJ- fllin, ryates, sasojevg@winlab.rutgers.edu Abstract In this work
More informationShadow Computing: An Energy-Aware Fault Tolerant Computing Model
Shadow Comuting: An Energy-Aware Fault Tolerant Comuting Model Bryan Mills, Taieb Znati, Rami Melhem Deartment of Comuter Science University of Pittsburgh (bmills, znati, melhem)@cs.itt.edu Index Terms
More informationMatching Partition a Linked List and Its Optimization
Matching Partition a Linked List and Its Otimization Yijie Han Deartment of Comuter Science University of Kentucky Lexington, KY 40506 ABSTRACT We show the curve O( n log i + log (i) n + log i) for the
More informationON THE LEAST SIGNIFICANT p ADIC DIGITS OF CERTAIN LUCAS NUMBERS
#A13 INTEGERS 14 (014) ON THE LEAST SIGNIFICANT ADIC DIGITS OF CERTAIN LUCAS NUMBERS Tamás Lengyel Deartment of Mathematics, Occidental College, Los Angeles, California lengyel@oxy.edu Received: 6/13/13,
More informationAn Analysis of Reliable Classifiers through ROC Isometrics
An Analysis of Reliable Classifiers through ROC Isometrics Stijn Vanderlooy s.vanderlooy@cs.unimaas.nl Ida G. Srinkhuizen-Kuyer kuyer@cs.unimaas.nl Evgueni N. Smirnov smirnov@cs.unimaas.nl MICC-IKAT, Universiteit
More informationThe Thermo Economical Cost Minimization of Heat Exchangers
he hermo Economical ost Minimization of Heat Exchangers Dr Möylemez Deartment of Mechanical Engineering, niversity of Gaziante, 7310 sait@ganteedutr bstract- thermo economic otimization analysis is resented
More informationSolution sheet ξi ξ < ξ i+1 0 otherwise ξ ξ i N i,p 1 (ξ) + where 0 0
Advanced Finite Elements MA5337 - WS7/8 Solution sheet This exercise sheets deals with B-slines and NURBS, which are the basis of isogeometric analysis as they will later relace the olynomial ansatz-functions
More informationAge of Information: Whittle Index for Scheduling Stochastic Arrivals
Age of Information: Whittle Index for Scheduling Stochastic Arrivals Yu-Pin Hsu Deartment of Communication Engineering National Taiei University yuinhsu@mail.ntu.edu.tw arxiv:80.03422v2 [math.oc] 7 Ar
More informationScheduling Parallel Jobs with Linear Speedup
Scheduling Parallel Jobs with Linear Speedup Alexander Grigoriev and Marc Uetz Maastricht University, Quantitative Economics, P.O.Box 616, 6200 MD Maastricht, The Netherlands. Email: {a.grigoriev, m.uetz}@ke.unimaas.nl
More informationA Public-Key Cryptosystem Based on Lucas Sequences
Palestine Journal of Mathematics Vol. 1(2) (2012), 148 152 Palestine Polytechnic University-PPU 2012 A Public-Key Crytosystem Based on Lucas Sequences Lhoussain El Fadil Communicated by Ayman Badawi MSC2010
More informationNode-voltage method using virtual current sources technique for special cases
Node-oltage method using irtual current sources technique for secial cases George E. Chatzarakis and Marina D. Tortoreli Electrical and Electronics Engineering Deartments, School of Pedagogical and Technological
More informationMulti-instance Support Vector Machine Based on Convex Combination
The Eighth International Symosium on Oerations Research and Its Alications (ISORA 09) Zhangjiajie, China, Setember 20 22, 2009 Coyright 2009 ORSC & APORC,. 48 487 Multi-instance Suort Vector Machine Based
More informationLINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL
LINEAR SYSTEMS WITH POLYNOMIAL UNCERTAINTY STRUCTURE: STABILITY MARGINS AND CONTROL Mohammad Bozorg Deatment of Mechanical Engineering University of Yazd P. O. Box 89195-741 Yazd Iran Fax: +98-351-750110
More informationSolvability and Number of Roots of Bi-Quadratic Equations over p adic Fields
Malaysian Journal of Mathematical Sciences 10(S February: 15-35 (016 Secial Issue: The 3 rd International Conference on Mathematical Alications in Engineering 014 (ICMAE 14 MALAYSIAN JOURNAL OF MATHEMATICAL
More informationAnalysis of execution time for parallel algorithm to dertmine if it is worth the effort to code and debug in parallel
Performance Analysis Introduction Analysis of execution time for arallel algorithm to dertmine if it is worth the effort to code and debug in arallel Understanding barriers to high erformance and redict
More informationAlgorithms for Air Traffic Flow Management under Stochastic Environments
Algorithms for Air Traffic Flow Management under Stochastic Environments Arnab Nilim and Laurent El Ghaoui Abstract A major ortion of the delay in the Air Traffic Management Systems (ATMS) in US arises
More informationDesign of NARMA L-2 Control of Nonlinear Inverted Pendulum
International Research Journal of Alied and Basic Sciences 016 Available online at www.irjabs.com ISSN 51-838X / Vol, 10 (6): 679-684 Science Exlorer Publications Design of NARMA L- Control of Nonlinear
More informationIteration with Stepsize Parameter and Condition Numbers for a Nonlinear Matrix Equation
Electronic Journal of Linear Algebra Volume 34 Volume 34 2018) Article 16 2018 Iteration with Stesize Parameter and Condition Numbers for a Nonlinear Matrix Equation Syed M Raza Shah Naqvi Pusan National
More informationMultiplicative group law on the folium of Descartes
Multilicative grou law on the folium of Descartes Steluţa Pricoie and Constantin Udrişte Abstract. The folium of Descartes is still studied and understood today. Not only did it rovide for the roof of
More informationOn Load Shedding in Complex Event Processing
On Load Shedding in Comlex Event Processing Yeye He icrosoft Research Redmond, WA, 98052 yeyehe@microsoft.com Siddharth Barman California Institute of Technology Pasadena, CA, 906 barman@caltech.edu Jeffrey
More informationPower Aware Wireless File Downloading: A Constrained Restless Bandit Approach
PROC. WIOP 204 Power Aware Wireless File Downloading: A Constrained Restless Bandit Aroach Xiaohan Wei and Michael J. Neely, Senior Member, IEEE Abstract his aer treats ower-aware throughut maximization
More informationAgent Failures in Totally Balanced Games and Convex Games
Agent Failures in Totally Balanced Games and Convex Games Yoram Bachrach 1, Ian Kash 1, and Nisarg Shah 2 1 Microsoft Research Ltd, Cambridge, UK. {yobach,iankash}@microsoft.com 2 Comuter Science Deartment,
More informationGENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS
International Journal of Analysis Alications ISSN 9-8639 Volume 5, Number (04), -9 htt://www.etamaths.com GENERALIZED NORMS INEQUALITIES FOR ABSOLUTE VALUE OPERATORS ILYAS ALI, HU YANG, ABDUL SHAKOOR Abstract.
More information