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 k∈S 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 |