LRE-TL: an optimal multiprocessor algorithm for sporadic task sets with unconstrained deadlines

Abstract

This article presents a detailed discussion of LRE-TL (Local Remaining Execution-TL-plane), an algorithm that schedules hard real-time periodic and sporadic task sets with unconstrained deadlines on identical multiprocessors. The algorithm builds upon important concepts such as the TL-plane construct used in the development of the LLREF algorithm (Largest Local Remaining Execution First). This article identifies the fundamental TL-plane scheduling principles used in the construction of LLREF. These simple principles are examined, identifying methods of simplifying the algorithm and allowing it to handle a more general task model. For example, we identify the principle that total local utilization can never increase within any TL-plane as long as a minimal number of tasks are executing. This observation leads to a straightforward approach for scheduling task arrivals within a TL-plane. In this manner LRE-TL can schedule sporadic tasks and tasks with unconstrained deadlines. Like LLREF, the LRE-TL scheduling algorithm is optimal for task sets with implicit deadlines. In addition, LRE-TL can schedule task sets with unconstrained deadlines provided they satisfy the density test for multiprocessor systems. While LLREF has a O(n 2 ) runtime per TL-plane, LRE-TL's runtime is O(n log n) per TL-plane.

FPSL, FPCL and FPZL schedulability analysis

“…For sporadic tasksets with implicit-deadlines, the utilisation bound for LRE-TL (Funk and Nadadur, 2009) is 100%, hence all of the implicit-deadline tasksets generated in our experiments are feasible (as their utilisation does not exceed m). For each of the algorithms / schedulability tests examined, an approximate value for the Optimality Degree can therefore be obtained by simply counting the total number of schedulable tasksets over the full range of utilisation values, and dividing this number by the total number of tasksets generated.…”

“…For example, the LLREF scheduling algorithm , which is optimal for periodic tasksets with implicit deadlines, and the LRE-TL scheduling algorithm (Funk and Nadadur, 2009) which is optimal for sporadic tasksets with implicit deadlines, divide the timeline into intervals that start and end at task releases and deadlines (referred to as TL-planes by Cho et al (2006)). In each interval, LLREF and LRE-TL ensure that each active task i τ executes for at least t U i , where i U is the task's utilisation, and t is the length of the time interval.…”

Minimizing scheduling overhead in LRE-TL real-time multiprocessor scheduling algorithm

“…As a global scheduler able to migrate tasks between processors, Pfair can successfully schedule any task set whose execution requirement does not exceed processor capacity. However, recently, a number of proposed algorithms have exploited the concept of deadline partitioning (dividing the time into time slices wherein all tasks share the same deadline) to achieve optimality while greatly reducing the number of required preemptions and migrations, such as in LLREF [16] and LRE-TL [14].…”

FPZL Schedulability Analysis

“…For example, the LLREF scheduling algorithm [13], which is optimal for periodic tasksets with implicit deadlines, and the LRE-TL scheduling algorithm [19] which is optimal for sporadic tasksets with implicit deadlines, divide the timeline into intervals that start and end at task releases/deadlines (referred to as TL-planes in [13]). In each interval, LLREF and LRE-TL ensure that each active task i τ executes for at least t U i , where i U is the task's utilisation, and t is the length of the time interval.…”

“…The RHS of (19) gives an upper bound on the interference from higher priority tasks and lower priority tasks executing in the zerolaxity state in an interval of length…”