# A Cost-Aware Framework for QoS-Based and Energy-Efficient Scheduling in Cloud-Fog Computing
## 6 Service Management Framework
### System Architecture and Queuing System Model
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Notation** & **Definition** & & **Notation** & **Definition** \\ \hline \(a_{i}\) & Arrival time of a fog job \(j\), i.o the cloud layer & \(p\) & Service rate of a server \\ \(\hat{c}_{i}\) & Schedule ordering for a set of fog jobs & \(m\) & Index of a cloud resource \\ \(\hat{c}_{i,j}^{\ell}\) & The time spent by a fog job \(j\), i.e. the dispatcher’s queue & \(p_{i}\) & Bandwidth penalty mean \\ \(\alpha_{i,j}^{\ell}\) & The waiting time of a fog job \(j\), i.e. the queue \(\alpha_{i,j}^{\ell}\) & Inventory ordering \(\beta\) & Writing penalty mean \\ \(\alpha_{i,j}^{\ell}\) & In resource queues \(Q\) of the cloud environment & \(p_{i,j}\) & Utility execution mean \\ \(\hat{c}_{i}^{\ell}\) & Target completion time of a fog job \(j\), i.e. the queue \(\alpha_{i,j}^{\ell}\) & Utility cost & SLA penalty mean \\ \(\hat{c}_{i}\) & Density cost & \(p_{i,j}\) & SLA penalty mean \\ \(\hat{d}_{i}\) & The departure time of a fog job \(j\), i.e. the cloud layer & \(p_{i,j}^{\ell}\) & Maximum number of cloud resources \\ \(\hat{c}_{i}^{\ell}\) & Perceived service time of fog job \(j\), i.e. the cloud layer & \(p_{i,j}^{\ell}\) & Pushly cost of bandwidth usage per time unit of data \\ \(\alpha_{i}\) & Energy cost & \(p_{i,j}^{\ell}\) & Utility cost of waiting \(\hat{c}_{i}^{\ell}\) for a fog job \(j\), i.e. the cloud resource \(R_{m}\) \\ \(\alpha_{i,j}^{\ell}\) & Energy cost per time unit of bandwidth allocation & \(p_{i,j}^{\ell}\) & Family cost of serviceing a fog job \(j\), i.e. a cloud resource \(R_{m}\) \\ \(E_{i,j}\) & Total bandwidth energy cost of a fog job \(j\), i.e. the cloud \(\alpha_{i,j}^{\ell}\) & Utility cost of SLA violation of a fog job \(j\), i.e. a cloud resource \(R_{m}\) \\ \(\alpha_{i,j}^{\ell}\) & Energy cost per time unit of waiting \(\hat{c}_{i}^{\ell}\) in & \(Q\) & A set of cloud queues \\ \(E_{i,j}\) & Total cost of waiting energy of a fog job \(j\), i.e. the network queue & \(p_{i,j}\) & The nth cloud queue \\ \(\alpha_{i,j}^{\ell}\) & Energy cost per time unit of service \(\hat{c}_{i}\) in the & \(q\) & Communication bit-rate \\ \(E_{i,j}\) & Total cost of service energy of a fog job \(j\), i.e. the cloud resource & \(R_{m}\) & A set of cloud computing resources \\ \(\alpha_{i,j}\) & Energy cost per time unit of SLA violation \(\alpha_{i,j}^{\ell}\) with \(R_{m}\) & The nth cloud resource \\ \(E_{i,j}\) & The cloud service provider & \(\alpha_{i,j}^{\ell}\) & The response time of a fog job \(j\), i.e. the cloud \(\alpha_{i,j}^{\ell}\) & The response time of a fog job \(j\), i.e. the cloud \(\alpha_{i,j}\) \\ \(E_{i,j}\) & Total cost of SLA-violation energy of a fog job \(j\), i.e. the cloud \(\alpha_{i,j}^{\ell}\) & Across the cloud-fog environment \\ \(\hat{c}_{i}\) & Service cost per time unit of execution \(\hat{c}_{i}\) & \(t_{i,j}^{\ell}\) & The total waiting time of a fog job \(j\), i.e. the cloud \(\alpha_{i,j}^{\ell}\) & \(t_{i,j}^{\ell}\) \\ \(\hat{f}\) & Maximum number of allocations exist & \(u\) & Energy consumption per bit in the fog layer \\ \(\hat{F}\) & A set of fog nodes & \(v\) & Arbitrary scaling factor \\ \(\hat{k}_{i}\) & gth fog node & \(\hat{v}\) & Waiting cost per time unit of waiting \(t_{i,j}^{\ell}\) \\ \(\hat{k}_{i}^{\ell}\) & The waiting time of a fog job \(j\), i.e. the fog environment & \(\hat{\lambda}_{i}^{\ell}\) & Scaling factor on penalty cost of bandwidth usage \\ \(\hat{c}_{i}\) & Index of a fog node & \(\hat{\lambda}_{i}^{\ell}\) & Scaling factor on penalty cost of waiting \\ \(\hat{c}_{i}\) & Maximum number of fog nodes & \(\hat{\lambda}_{i}^{\ell}\) & Scaling factor on penalty cost of serviceing \\ \(i\) & Index of a fog job & \(\hat{\lambda}_{i}^{\ell}\) & Scaling factor of penalty cost SLA violation \\ \(J\) & A set of fog jobs & \(\hat{\lambda}_{i,j}\) & Allocation of a fog job \(j\), on queue \(Q_{i,j}\) of cloud resource \(R_{m}\) \\ \(\hat{l}_{i,j}\) & The fog job \(j\), i.e. the cloud \(\alpha_{i,j}^{\ell}\) & SLA cost not current time unit of SLA violation \(\alpha_{i,j}^{\ell}\) \\ \(\hat{k}_{i}\) & Measured cost factor for bandwidth allocation penalty & \(p_{i,j}^{\ell}\) & SLA violation for a fog job \(j\), i.e. the fog layer \\ \(\alpha_{i,j}^{\ell}\) & Measured cost factor for execution penalty & \(\Lambda_{i}\) & The cost of bandwidth usage incurred per data unit of a fog job \(j\), i.e. the cloud \(\alpha_{i,j}^{\ell}\) & Rate of energy consumption \\ \(\hat{k}_{i}\) & Maximum number of fog jobs in the stream & \(\lambda_{i,j}\) & Rate of energy cost \(t_{i,j}\) \\ \(\hat{c}_{i}\) & Service deadline of a fog job \(j\), i.e. the cloud \(\alpha_{i,j}\) & Rate of energy cost \(t_{i,j}\) \\ \(\hat{l}_{i,j}\) & Tendiness allowance for a fog job \(j\), i.e. the cloud \(\alpha_{i,j}^{\ell}\) & Rate of energy cost \(t_{i,j}\) \\ \hline \hline \end{tabular}
\end{table}