Table 2 The summarization of main notations.
From: An optimal workflow scheduling in IoT-fog-cloud system for minimizing time and energy
Notations | Definition | Notations | Definition |
|---|---|---|---|
W | A DAG workflow | tij,k | Time interval of \({t}_{j}\) to \({vm}_{k}\) |
Edgei,j | Communication cost between the tasks of i-th and j-th in the workflow w | pred (tj) | Predecessors of task \({t}_{j}\) |
ti | i-th task in workflow w | \({\lambda }_{k}\) | Constant failure rate per time unit of a \({vm}_{k}\) |
\({vm}_{k}\) | k-th virtual machine | \({\alpha }_{j}\) | Circuit-dependent constant |
\({f}_{j,k}\) | Frequency \({t}_{j}\) on the \({vm}_{k}\) | \({et}_{j, k}\) | Execution time or the calculation time of the j-th task in workflow on the k-th VM |
\({m}_{j}\) | Dynamic power exponent | \({st}_{j}\) | Start time of the task \({t}_{j}\) |
\({\alpha }_{j}\) | Circuit-dependent constant | \({ft}_{{t}_{j}, {f}_{k}}\) | Finish time of task \({t}_{j}\) assigned on \({vm}_{k}\) at frequency level \({f}_{k}\) |
\({\alpha }_{r}\) | Reliability ratio | \({st}_{{t}_{j}, {f}_{k}}\) | Start time of task \({t}_{j}\) assigned on \({vm}_{k}\) at frequency level \({f}_{k}\) |
avail [\({vm}_{k}\)] | Earliest Available Time (EAT) of \({vm}_{k}\) in order to execute the desired task | \({et}_{worst}\) | Worst execution time of task \({t}_{j}\) on \({vm}_{k}\) at frequency \({f}_{k}\) |
\({aet}_{j}\) | Mean computation time of task \({t}_{j}\) on all VMs | \(succ({t}_{j})\) | Successors of task \({t}_{j}\) |
\({Rel}_{max}\)(w) | Maximum reliability of a workflow w | \({rank}_{u}\)(\({t}_{j})\) | Upward rank value of a task |
\({Rel}_{max}^{j}\) | \({Rel}_{max}^{j}\) is maximum reliability of j-th task | \({t}_{o(j)}\) | Task with priority j |
\({R}_{Req}^{o(j)}\) | Required reliability of task \({t}_{o\left(j\right)}\) | \({RT}_{req}\)(\({t}_{j}\), \({vm}_{k}\)) | MST requirement of the task \({t}_{j}\) on the VM’s CPU \({vm}_{k}\) |
\(EST({t}_{j},{vm}_{k}\)) | Earliest Start Time (EST) of task \({t}_{j}\) on \({vm}_{k}\) | \(AFT({t}_{j},{vm}_{k}\)) | Actual Finish Time (AFT) of task \({t}_{j}\) on \({vm}_{k}\) |
\({t}_{entry}\) | input task in the workflow w | \({t}_{exit}\) | Output task in the workflow w |
AST(\({t}_{i1}\)) | Actual Start Time (AST) of task \({t}_{i1}\) | Λ[\({vm}_{k}\)] | End spot of the available time of \({vm}_{k}\) |
