SCT-based priority-free conditionally-preemptive scheduling of modular real-time systems with exact task execution time

Abstract

This study presents a novel discrete-event systems (DES) modeling framework to address real-time system (RTS) with sporadic, periodic, and non-repetitive real-time tasks. Our approach is organized in three steps. First, the effect of individual timing parameters of each task, such as job arrival and deadlines, are represented by modular DES. Second, we choose the required modules for the specific RTS at hand to compose an overall model. Third, we utilize supervisory control to find all schedules that are consistent with the timing requirements of all tasks. In contrast to fixed task priorities, we address general preemption relations represented by a preemption matrix and thereby implement priority-free conditionally-preemptive (PFCP) real-time scheduling. As a particular feature of the closed-loop configuration, the schedules obtained refer to the actual job execution time as opposed to upper and lower bounds. We illustrate our approach by a real-world example in the context of an automated manufacturing system.

Appendix

For the running example RTS $\mathbb {S}$ referring to specification P₁, by the standard supcon procedure, we obtain the optimal close-loop behavior represented by 83 states and 116 transitions. The diagram of the calculated supervisor, denoted by SUPER, is given below.

SUPER = (SUPER, [mark, 0, 1, 6, 14, 46, 48, 52], [tran [0,l, 52], [0, γ₂, 27], [1, l, 52], [1, γ₁, 20], [1, γ₂, 39], [2, c₂, 4], [3, l, 15], [3, γ₂, 33], [4, α₁, 37], [4, β₂, 13], [5, β₁, 44], [7, γ₁, 38], [7, β₂, 14], [8, β₁, 33], [8, ρ₁, 9], [9, c₁, 11], [10, α₂, 44], [11, β₁, 10], [11, α₂, 5], [12, β₂, 35], [13, α₁, 40], [14, l, 46], [14, γ₁, 36], [15, l, 6], [15, γ₂, 10], [16, l, 23], [17, β₁, 15], [17, γ₂, 11], [18, β₁, 3], [18, ρ₁, 45], [18, γ₂, 8], [19, c₁, 18], [19, γ₂, 51], [20, α₁, 19], [20, γ₂, 41], [21, γ₁, 22], [21, β₂, 48], [21, ρ₂, 24], [22, β₂, 36], [22, ρ₂, 2], [23, l, 48], [24, γ₁, 2], [24, c₂, 7], [25, γ₁, 22], [25, β₂, 46], [25, ρ₂, 42], [26, c₂, 25], [27, α₂, 26], [28, β₁, 23], [29, β₁, 23], [29, ρ₁, 30], [30, c₁, 47], [31, β₂, 23], [31, ρ₂, 50], [32, c₂, 31], [33, α₂, 32], [34, β₁, 16], [34, ρ₁, 43], [35, c₁, 34], [36, α₁, 35], [37, β₂, 40], [38, α₁, 12], [38, β₂, 36], [39, γ₁, 41], [40, c₁, 29], [41, α₁, 51], [42, γ₁, 2], [43, c₁, 28], [44, c₂, 21], [45, c₁, 17], [45, γ₂, 9], [46, γ₁, 36], [47, β₁, 48], [48, l, 14], [48, γ₁, 36], [49, γ₁, 38], [49, β₂, 48], [50, c₂, 49], [51, c₁, 8], [52, γ₁, 20], [52, γ₂, 39], [6, l, 1], [6, γ₁, 20], [6, γ₂, 39]]) (53, 84)