Consider the problem of non-migratively scheduling a set of implicit-deadline sporadic tasks to meet all deadlines on a
two-type heterogeneous multiprocessor platform. We ask the following question: Does there exist a phase transition
behavior for the two-type heterogeneous multiprocessor scheduling problem? We also provide some initial observations
via simulations performed on randomly generated task sets

Andersson, Björn

Bletsas, Konstantinos

Raravi, Gurulingesh

Repositório Científico do Instituto Politécnico do Porto

        Two-type Heterogeneous Multiprocessor Scheduling: Is there a Phase Transition? (Extended Abstract)     www.hurray.isep.ipp.pt Technical Report HURRAY-TR-110503 Version:  Date: 05-30-2011 Gurulingesh Raravi Björn Andersson Konstantinos Bletsas  Technical Report HURRAY-TR-110503  Two-type Heterogeneous Multiprocessor Scheduling:                                                                                             Is there a Phase Transition? (Extended Abstract) © IPP Hurray! Research Group www.hurray.isep.ipp.pt   1  Two-type Heterogeneous Multiprocessor Scheduling: Is there a Phase Transition? (Extended Abstract) Gurulingesh Raravi, Björn Andersson, Konstantinos Bletsas IPP-HURRAY! Polytechnic Institute of Porto (ISEP-IPP) Rua Dr. António Bernardino de Almeida, 431 4200-072 Porto Portugal Tel.: +351.22.8340509, Fax: +351.22.8340509 E-mail:  http://www.hurray.isep.ipp.pt  Abstract Consider the problem of non-migratively scheduling a set of implicit-deadline sporadic tasks to meet all deadlines on a two-type heterogeneous multiprocessor platform. We ask the following question: Does there exist a phase transition behavior for the two-type heterogeneous multiprocessor scheduling problem? We also provide some initial observations via simulations performed on randomly generated task sets.  Two-type Heterogeneous Multiprocessor Scheduling: Is there a Phase Transition?Gurulingesh Raravi∗, Bjo¨rn Andersson†∗ and Konstantinos Bletsas∗∗CISTER-ISEP Research Center, Polytechnic Institute of Porto, Portugal†Software Engineering Institute, Carnegie Mellon University, Pittsburgh, USAEmail: ∗{ghri, baa, ksbs}@isep.ipp.pt, †baandersson@sei.cmu.eduI. INTRODUCTIONConsider the problem of non-migratively scheduling aset of implicit-deadline sporadic tasks to meet all deadlineson a two-type heterogeneous multiprocessor platform.A. System Model and AssumptionsThe system is as follows:• Computing Platform (denoted as Π): The com-puting platform consists of m processors; Of those,m1 ≥ 1 are of type-1, and m2 ≥ 1 are of type-2, i.e.,m1 +m2 = m. A processor is denoted as pij ∈ Π,where j ∈ {1, · · · ,m}.• Task Set (denoted as τ ): The task set comprises nimplicit-deadline sporadic tasks (i.e., for each task, itsdeadline is equal to its minimum inter-arrival time).A task is denoted as τi ∈ τ , where i ∈ {1, · · · , n}.• Utilization (denoted as U ): The utilization of a taskτi on a processor pij is given by uji , a non-negativereal number.The following assumptions are made:• No job parallelism: A job can be executing on atmost one processor at any given time instant• Independent tasks: The execution of jobs are in-dependent, i.e., they neither share any resources norhave data dependency and• No migration: All the jobs released by a task mustexecute on the same processor to which the task isassigned.B. Phase TransitionA behavior in which a given system transitions fromone state to another is known as phase transition be-havior. During the phase transition, certain propertiesof the system change “drastically”. In context of real-time scheduling, one way to relate this concept is toreason about the difficulty of scheduling problems. Wecan say that when a scheduling problem satisfies certainproperty (i.e., when the system is in a certain phase),it is almost certain to schedule the task set and uponchanging the property (the system enters a new phase),it is hardly possible to schedule the task set. For example,such a behavior has been observed for a uniprocessornon-preemptive scheduling problem. It has been shownthat there exists a utilization threshold U∗ such that, forlarge task sets, task sets with utilization U < U∗ canalmost surely be scheduled and task sets with utilizationU > U∗ almost surely cannot be scheduled [3]. It isalso believed that such a behavior exists for identicalmultiprocessor scheduling problem [4]. We are interestedin finding whether such a behavior exists for two-typeheterogeneous multiprocessor platforms.II. OPEN PROBLEMDoes there exist a phase transition behavior for the two-type heterogeneous multiprocessor scheduling problem?III. SOME INSIGHTSIn a quest to find an answer to the question, weperformed some simulations and did not observe the phasetransition behavior in our simulations. We understand thatthese simulations/observations are not enough to answerthe question and hence more work needs to be done in thisregard. We brief our simulation setup and observations inthis section.We randomly generated the problem instances compris-ing the task set (with an upper bound of 15 tasks) and thecomputing platform (with an upper bound of 2 processorsof each type). We then formulated the task assignmentproblem as Zero-One Integer Linear Program (ILP) asdiscussed in [1]. This formulation is shown in Figure 1.Here Z denotes the maximum capacity of any processorMinimize Z subject to the following constraints :C1. ∑mj=1 xji = 1 (i = 1, 2, · · · , n)C2. ∑ni=1(xji · uji)≤ Z (j = 1, 2, · · · ,m)C3. xji is a non-negative integer (i = 1, 2, · · · , n);(j = 1, 2, · · · ,m)Figure 1. ILP formulation – ILP-Feas(τ,Π)that is used and is set as the objective function (to beminimized). Z ≤ 1 implies that the sum of utilization oftasks assigned to any processor is less than or equal tothe available capacity on that processor – hence, Z ≤ 1indicates that the task set is feasible on the platform. Thevariable xji (referred to as indicator variable) indicate theassignment of task τi to processor pij , i.e., xji = 1 impliesthat τi is (entirely) assigned to processor pij , xji = 0implies that τi is not assigned to processor pij . The firstconstraint (C1) indicates that every task must be assignedto processors. The second constraint (C2) indicates thatno processor capacity should be used more than Z. Thethird constraint (C3) indicates that the indicator variablesmust be non-negative integers.We extracted only those problem instances that arefeasible (i.e., a problem instance in which the task setcould be assigned on the platform without missing anydeadlines when EDF [2] is used to schedule the taskson each processor – ILP solver returns Z ≤ 1 for suchtask sets). For each of the feasible problem instances, wecomputed the success ratio which is defined as follows:success ratio = NsuccNvalidwhere, Nsucc denotes the number of assignments that meetall the deadlines of the tasks and Nvalid denotes the totalnumber of possible valid assignments. A valid assignmentis one in which (i) no task is left unassigned and (ii) thetask assignment in one valid assignment is different fromother valid assignments. All the possible valid assignmentsare generated using exhaustive enumeration.We then plotted our observations for 10000 feasible tasksets with Z on X-axis and ‘average success ratio’ (foreach Z) on Y-axis as shown in Figure 2. As we can seefrom the graph, there is a gradual decrease in the value of‘average success ratio’ and hence no sharp threshold on aparticular value of Z where ‘average success ratio’ reducessignificantly. The fluctuation in the ‘average success ratio’in the initial half of the graph (where 0 < Z ≤ 0.45)can be attributed to the fact that our task set generatorgenerated very few task sets for which ILP solver gavethe output 0 < Z ≤ 0.45 — to be precise, among 10000task sets, only 250 were in this category. Hence, we believethat with a more balanced task set generator, we will notobserve those fluctuations.                    Figure 2. Average success ratio of randomly generated 10000 (feasible)task setsFrom the observations made, we tend to believe thatwith the parameters that we have chosen, it may not bepossible to observe the phase transition behavior if there isone. However, these are initial observations and we needto carry out more work to answer the following questions:1) Does there exist a phase transition behavior? If thereis one, then with what parameters we can observesuch a behavior?2) If there is no phase transition, then what is itsimplication considering the fact that such a behaviorhas been observed in the past for a uniprocessorscheduling problem [3][4]?ACKNOWLEDGMENTSThis work was partially supported by the REHEAT project,ref. FCOMP-01-0124-FEDER-010045, funded by FEDER fundsthrough COMPETE (POFC - Operational Programme ThematicFactors of Competitiveness), National Funds (PT) through FCT -Portuguese Foundation for Science and Technology and REJOINproject of FLAD (Luso-American Development Foundation).REFERENCES[1] S. Baruah, Task partitioning upon heterogeneous multi-processor platforms, 10th IEEE International Real-Timeand Embedded Technology and Applications Symposium(2004).[2] C. L. Liu and J. W. Layland, Scheduling Algorithms forMultiprogramming in a Hard Real-Time Environment, Jour-nal of the ACM (1973).[3] S. Gopalakrishnan, M. Caccamo and L. Sha, Sharp Thresh-olds for Scheduling Recurring Tasks with Distance Con-straints, IEEE Transactions on Computers (2008).[4] S. Gopalakrishnan, A sharp threshold for rate monotonicschedulability of real-time tasks, 1st International Real-Time Scheduling Open Problems Seminar (2010).

Two-type heterogeneous multiprocessor scheduling: Is there a phase transition? (Extended Abstract)

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