# HARD REAL-TIME SCHEDULING: THE DEADLINE-MONOTONIC APPROACH 1. Department of Computer Science, University of York, York, YO1 5DD, England.

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3 I τ 1 :T 1 =4;C 1 =1 τ 2 :T 2 =5;C 2 =1 τ 3 :T 3 =6;C 3 =2 τ 4 :D 4 =10 C 3 C 1 C 2 C 1 Release τ 3 Release τ 2 Release τ 1 Release τ 1, τ 2 and τ (D 4 ) Release τ 1 Fig 1 Interference Increases" In Fig 1 there are three processes with higher priority than τ 4 We see that as the higher priority processes are released, I 4 increases monotonically with respect to t The graph is stepped with plateaus representing intervals of time in which no higher priority processes are released It is obvious that only one equation need be evaluated for each plateau as the interference does not change To maximise the time available for the execution of τ i we choose to evaluate at the rightmost point on the plateau Therefore, one possible reduction in the number of equations to evaluate schedulability occurs by testing τ i at all points in [0, ] that correspond to a higher priority process release Since as soon as one equation identifies the process set as schedulable we need test no further equations Thus, the effect is to evaluate equations only in [0, t i ] The number of equations has been reduced in most cases We note that no reduction will occur if for each point in time in [0, ) a higher priority process is released with τ i meeting its deadline at The number of equations is reduced yet further by considering the computation times of the processes Consider Fig 2 C s τ 1 :T 1 =4;C 1 =1 τ 2 :T 2 =5;C 2 =1 τ 3 :T 3 =6;C 3 =2 C 1 τ 4 :D 4 = 10; C 4 =1 C 3 C 2 C 1 t=0: Release τ 1 τ 2 τ 3 τ 4 t=4: Release τ 1 t=5: Release τ 2 t=6: Release τ 3 t=8: Release τ 1 t (D 4 ) Fig 2 Process Computation Time In Fig 2 the total computation requirement of the process set (C s ) is plotted against time At the first point in time when the outstanding computation is t equal to the time elapsed, we have found t 4 In the above diagram this point in time coincides with the deadline of τ 4 Considering Fig 2, there is little merit in testing the schedulability of τ i in the interval [0, ) Also, since time 0 corresponds with a critical instant (a simultaneous release of all processes) the first point in time that τ i could possibly complete is: = Σ i C j This gives a schedulability constraint of: where I y x = + 1 y 1 Σ z =1 x C T z z Since the value of assumes that only one release of each process occurs in [0, ), the constraint will fail if there have been any releases of higher priority processes within the interval (0, ) The exact amount of work created by higher priority processes in this interval is given by: The next point in time at which τ i may complete execution is: = + This gives a schedulability constraint of: + 1 Again, the constraint will fail if releases have occurred in the interval [, ) Thus, we can build a series of equations to express the schedulability of τ i (1) + 1 where = Σ i C j (2) + 1 where = + (k) and where tk + t k 1 tk where t k = tk 1 I y x = y 1 Σ z =1 + x C T z z If any of the equations hold, τ i is schedulable Obviously, the equations terminate if t k > for process τ i and equation k At this point τ i is unschedulable

6 Lehoczky, JP, L Sha, and Y Ding (1989) The Rate-Monotonic Scheduling Algorithm: Exact Characterization and Average Case Behaviour Proc IEEE Real-Time Sys Symp, pp Leung, JYT, and J Whitehead (1982) On the Complexity of Fixed-Priority Scheduling of Periodic, Real-Time Tasks Perf Eval (Netherlands), 2, pp Liu, CL and JW Layland (1973) Scheduling Algorithms for Multiprogramming in a Hard Real-Time Environment J ACM, 20, pp Sha, L, and JB Goodenough (1988) Real- Time Scheduling Theory and Ada CMU/SEI-88-TR-33, SEI, Carnegie- Mellon University Sha, L and JP Lehoczky (1989) Aperiodic Task Scheduling for Hard Real-Time Systems J of Real-Time Sys, 1, pp Stankovic, JA (1988) Real-Time Computing Systems: The Next Generation COINS Tech Rep 88-06, Dept of Comp and Inf Sci, Univ of Massachusetts Tindell, K (1990) Allocating Real-Time Tasks (An NP-Hard Problem made Easy) YCS 147, Dept of Comp Sci, Univ of York

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