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Table 1 Notation table

From: Real-time semi-partitioned scheduling of fork-join tasks using work-stealing

Symbols Description
τ Set of n tasks
D i Relative deadline of task τ i
T i Period of task τ i
a i,j Arrival time of job j of task τ i
d i,j Absolute deadline of job j of task τ i
\(S_{i}=\left [s_{i}^{1}, s_{i}^{2}, \ldots, s_{i}^{n_{i}}\right ]\) Sequence of n i segments, \(n_{i} \in \mathbb {N}\)
\(s_{i}^{k}\) Segment kS i
\(t_{s_{i}^{k}}^{q}\) Sub-task q belonging to segment \(s_{i}^{k}\)
v k Number of sub-tasks belonging to segment \(s_{i}^{k}\)
\(e_{s_{i}^{k}}^{q}\) Execution time of sub-task \(t_{s_{i}^{k}}^{q}\)
C i Total execution requirement of task τ i
P i Minimum execution requirement of task τ i
\(c_{s_{i}^{k}}\) Worst-case execution time among the sub-tasks of segment \(s_{i}^{k}\)
U i Utilization factor of task τ i
λ i Density factor of task τ i
U τ Total utilization factor of the task set τ
λ τ Total density of the task set τ
π Set of m homogeneous cores
\(\tau ^{\pi _{j}}\phantom {\dot {i}\!}\) Set of tasks assigned to core π j
\(\tau _{\text {NM}}^{\pi _{j}}\) Subset of non-migrating tasks assigned to π j
\(\tau _{\mathrm {M}^{\pi _{j}}}\) Subset of migrating tasks assigned to π j
k i Number of frames of a migrating task τ i
H Least common multiple of the periods of all the tasks in τ
σ Job-to-core assignment sequence
M[i,j] Matrix of integers of the current job-to-core assignment
s Number fo intervals of length k i ·T i
\(\ell _{i}^{j}\) Number of frames out of k i that were successfully assigned to π j
nb i (t) Upper-bound on the number of non-zero frames in the interval [s·k i ·Ti,t]
\(w_{i}^{j}\) Workload in the interval [s·k i ·Ti,t]
dbf j (τ i ,t) Demand-bound function for task τ i in π j in the interval [0,t]
ϕ n Fork point for segment n
μ n Synchronization point segment n
ω n Stealing window n
slack (ϕ n ) Slack of the job at time instant ϕ n
\(d_{s}^{(n)}\) Intermediate deadline of the n th parallel segment
m s Number of sub-tasks spawned in segment n
\(c_{s_{i}^{(n)}}\) Worst-case execution time among the tasks that belong to segment n
\(AV_{\tau _{i}}\) Gain per task of Approach-S over Approach-NS
AV τ Average gain per task in the task set