Testing cyber-physical systems involves the execution of test cases on target-machines equipped with the latest release of a software control system. When testing industrial robots, it is common that the target machines need to share some common resources, e.g., costly hardware devices, and so there is a need to schedule test case execution on the target machines, accounting for these shared resources. With a large number of such tests executed on a regular basis, this scheduling becomes difficult to manage manually. In fact, with manual test execution planning and scheduling, some robots may remain unoccupied for long periods of time and some test cases may not be executed.
While OTS is closely related to FJSS, and efficient approaches to FJSS are known @cite_9 @cite_22 , there are some differences. First, in OTS, execution times are machine-independent. Second, each job in OTS consists of only one operation, while in FJSS one job can contain several operations, where there are precedences between the operations. Finally, some operations additionally require exclusive access to a global resource, preventing overlap with other operations.
@cite_9 @cite_22
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We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online non-clairvoyant setting. In this problem, a set of jobs J arrive over time to be scheduled on a set of M machines. Each job J has processing length pj, weight wj, and is processed at a rate of lij when scheduled on machine i. The online scheduler knows the values of wj and lij upon arrival of the job, but is not aware of the quantity pj. We present the first online algorithm that is scalable ((1+e)-speed O(1 2)-competitive for any constant e > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines. In this problem, each machine can be sped up by a factor of f-1i(P) when consuming power P, where fi is an arbitrary strictly convex power function. In particular, we get an O(γ2)-competitive algorithm when all power functions are of form sγ. These are the first non-trivial non-clairvoyant results in any setting with heterogeneous machines. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each job's utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms. We consider the classical machine scheduling, where n jobs need to be scheduled on m machines, with the goal of minimizing the makespan, i.e., the maximum load of any machine in the schedule. We study inefficiency of schedules that are obtained when jobs arrive sequentially one by one, and choose themselves the machine on which they will be scheduled. Every job is only interested to be on a machine with a small load (and does not care about the loads of other machines). We measure the inefficiency of a schedule as the ratio of the makespan obtained in the worst-case equilibrium schedule, and of the optimum makespan. This ratio is known as the sequential price of anarchy. We also introduce alternative inefficiency measures, which allow for a favorable choice of the order in which the jobs make their decisions. We first disprove the conjecture of Hassin and Yovel (OR Letters, 2015) claiming that for unrelated machines, i.e., for the setting where every job can have a different processing time on every machine, the sequential price of anarchy for m = 2 machines is at most 3. We provide an answer for the setting with m = 2 and show that the sequential price of anarchy grows at least linearly with the number of players. Furthermore, we show that for a certain order of the jobs, the resulting makespan is at most linearly larger than the optimum makespan. Furthermore, we show that if an authority can change the order of the jobs adaptively to the decisions made by the jobs so far (but cannot influence the decisions of the jobs), then there exists an adaptive ordering in which the jobs end up in an optimum schedule. To the end we consider identical machines, i.e., the setting where every job has the same processing time on every machine, and provide matching lower bound examples to the existing upper bounds on the sequential price of anarchy. We give a new and efficient approximation algorithm for scheduling precedence-constrained jobs on machines with different speeds. The problem is as follows. We are given n jobs to be scheduled on a set of m machines. Jobs have processing times and machines have speeds. It takes pj si units of time for machine i with speed si to process job j with processing requirement pj. Precedence constraints between jobs are given in the form of a partial order. If j?k, processing of job k cannot start until job j's execution is completed. The objective is to find a non-preemptive schedule to minimize the makespan of the schedule. Chudak and Shmoys (1997, “Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA),” pp. 581?590) gave an algorithm with an approximation ratio of O(log m), significantly improving the earlier ratio of O(m) due to Jaffe (1980, Theoret. Comput. Sci.26, 1?17). Their algorithm is based on solving a linear programming relaxation. Building on some of their ideas, we present a combinatorial algorithm that achieves a similar approximation ratio but runs in O(n3) time. Our algorithm is based on a new and simple lower bound which we believe is of independent interest. We present new approximation algorithms for the problem of scheduling precedence-constrained jobs on parallel machines that are uniformly related. That is, there arenjobs andmmachines; each jobjrequirespjunits of processing, and is to be processed on one machine without interruption; if it is assigned to machinei, which runs at a given speedsi, it takespj sitime units. There also is a partial order ? on the jobs, wherej?kimplies that jobkmay not start processing until jobjhas been completed. We consider two objective functions:Cmax=maxjCj, whereCjdenotes the completion time of jobj, and ?jwjCj, wherewjis a weight that is given for each jobj. For the first objective, the best previously known result is anOm-approximation algorithm, which was shown by Jaffe more than 15 years ago. We give anO(logm)-approximation algorithm. We also show how to extend this result to obtain anO(logm)-approximation algorithm for the second objective, albeit with a somewhat larger constant. These results also extend to settings in which each jobjhas a release daterjbefore which the job may not begin processing. In addition, we obtain stronger performance guarantees if there are a limited number of distinct speeds. Our results are based on a new linear programming-based technique for estimating the speed at which each job should be run, and a variant of the list scheduling algorithm of Graham that can exploit this additional information.