Cost optimization of cloud-edge-fog federated systems with bidirectional offloading: one-hop versus two-hop

Abstract

Edge and fog computing technologies are akin to cloud computing but operate in closer proximity to users, offering similar services on a more widely distributed and localized scale. To enhance the computing environment and enable efficient offloading of computing requests, we propose a unified federation of these technologies, forming a federated cloud-edge-fog (CEF) system. Unlike current offloading models limited to single-hop and unidirectional vertical scenarios, our model facilitates two-hop, bidirectional (horizontal and vertical) offloading. The CEF model enables not only fog and edge devices to offload tasks to the cloud but also allows the cloud to offload tasks to the edges and fogs, creating a more dynamic and flexible computing ecosystem. To optimize this system, we formulate an optimization problem focused on minimizing the total cost while adhering to latency constraints. We employ simulated annealing as the solution approach. By adopting the proposed CEF model and optimization strategy, organizations can effectively leverage the strengths of cloud, edge, and fog computing while achieving significant cost reductions and improved task offloading efficiency. The findings from our study indicate that adopting a two-hop offloading approach can result in cost savings of 10–20% compared to the traditional one-hop method. Furthermore, when incorporating horizontal and bidirectional offloading, cost savings of approximately 12% and 20% can be achieved, respectively, in contrast to scenarios without horizontal offloading and only unidirectional vertical offloading. This advancement holds promise for optimizing computing resources and enhancing the overall performance of distributed systems in real-world applications.

Appendix A: Two-hop offloading

1.1 A.1: Communication latency

Two-hop offloading can be divided into first-hop and second-hop. For example, in an offloading scenario from $f_{j,i}$ through $e_{j}$ to C; first-hop is from $f_{j,i}$ to $e_{j}$, and second-hop is from $e_{j}$ to C. Since two-hop offloading is very similar to one-hop offloading, all offloading cases of one-hop are also available for two-hop offloading, along with five extra offloading options: (1) from $f_{j,i}$ through $e_{j}$ to C, (2) from $f_{j,i}$ through $e_{j}$ to $e_{j^\prime }$, (3) from $f_{j,i}$ through $e_{j}$ to $f_{j,i^\prime }$, (4) from $e_{j}$ through $e_{j^\prime }$ to $f_{j^\prime ,i}$, and (5) from C through $e_{j}$ to $f_{j,i}$. Since the estimation of first-hop communication latency is the same as one-hop offloading, the calculation of second-hop communication latency is discussed here. Let $D_{C}$ be the second-hop of communication latency from $f_{j,i}$ to C, which can be represented as

$$\begin{aligned} D_{C} = \frac{1}{r^\prime _{j} - \lambda _{j,i}\beta _{j,i}\beta ^\prime _{j}}, \quad r^\prime _{j} \ge \lambda _{j,i}\beta _{j,i}\beta '_{j}, \quad \forall j, i, \end{aligned}$$

(A1)

where $\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j}$ is the request rate offload from $f_{j,i}$ to C via $e_j$. Let $D_{j^\prime }$ be the second-hop of communication latency from $f_{j,i}$ to $e_{j^\prime }$, which can be represented as

$$\begin{aligned} D_{j^\prime }= & {} \frac{1}{r^\prime _{j,j^\prime } - \lambda _{j,i}\beta _{j,i}\beta ^\star _{j,j^\prime }}, \quad j \ne j^\prime , \nonumber \\{} & {} r^\prime _{j,j^\prime } \ge \lambda _{j,i}\beta _{j,i}\beta ^\star _{j,j'}, \quad \forall j, j^\prime , i, \end{aligned}$$

(A2)

where $\lambda _{j,i}\beta _{j,i}\beta ^\star _{j,j^\prime }$ is the request rate offload from $f_{j,i}$ to $e_{j^\prime }$ via $e_j$. Let $D_{j,i^\prime }$ be the second-hop of communication latency from $f_{j,i}$ to $f_{j,i^\prime }$, which can be represented as

$$\begin{aligned} D_{j,i^\prime }= & {} \frac{1}{r_{j,i^\prime } - \lambda _{j,i}\beta _{j,i}\beta ^\prime _{j,i^\prime }}, \quad i \ne i^\prime , \nonumber \\{} & {} r_{j,i^\prime } \ge \lambda _{j,i}\beta _{j,i}\beta ^\prime _{j,i^\prime }, \quad \forall j, i, i^\prime , \end{aligned}$$

(A3)

where $\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j,i^\prime }$ is the request rate offload from $f_{j,i}$ to $f_{j,i^\prime }$ via $e_j$. Let $D^\prime _{j^\prime ,i}$ be the second-hop of communication latency from $e_{j}$ to $f_{j^\prime ,i}$, which can be represented as

$$\begin{aligned} D'_{j',i}= & {} \frac{1}{r_{j^\prime ,i} - \lambda ^\prime _{j}\beta ^\star _{j,j^\prime }\beta ^\prime _{j^\prime ,i}}, \quad j \ne j^\prime , \nonumber \\{} & {} r_{j^\prime ,i} \ge \lambda ^\prime _{j}\beta ^\star _{j,j^\prime }\beta ^\prime _{j^\prime ,i}, \quad \forall j, j^\prime , i, \end{aligned}$$

(A4)

where $\lambda '_{j} \beta ^\star _{j,j'} \beta '_{j',i}$ is the request rate offload from $e_{j}$ to $f_{j^\prime ,i}$ via $e_j$. Let $D''_{j,i}$ be the second-hop of communication latency from C to $f_{j,i}$, which can be represented as

$$\begin{aligned} D''_{j,i}= & {} \frac{1}{r_{j,i} - \lambda '' \beta ''_{j} \beta '_{j,i}}, \quad r_{j,i} \ge \lambda '' \beta ''_{j} \beta '_{j,i}, \nonumber \\{} & {} \forall j, i, \end{aligned}$$

(A5)

where $\lambda '' \beta ''_{j} \beta '_{j,i}$ is the request rate offload from C to $f_{j,i}$ via $e_j$.

1.2 A.2: Computation latency

The computation latency of two-hop offloading is also based on the M/M/c queuing model, and the methods of $l_{j,i}$, $l^\prime _{j}$, and $l^\prime {}^\prime$ are same as (9), (10), and (11), respectively. However, since we are considering the second-hop, the calculation of $R_{j,i}$, $R^\prime _{j}$, and $R^\prime {}^\prime$ are not same as given in (6), (7), and (8). $R_{j,i}$ in two-hop offloading can be represented as

$$\begin{aligned} R_{j,i}= & {} \lambda _{j,i} - \lambda _{j,i}\beta _{j,i} \nonumber \\{} & {} +\, \lambda ^\prime _j\beta ^\prime _{j,i} + \begin{matrix} \sum \limits _{i^\prime = 1, i^\prime \ne i}^{n_j} \lambda _{j,i^\prime }\beta _{j,i}\beta ^\prime _{j,i} \end{matrix} \nonumber \\{} & {} + \begin{matrix} \sum \limits _{j^\prime = 1, j^\prime \ne j}^{m} \lambda ^\prime _{j^\prime }\beta ^\star _{j^\prime ,j}\beta ^\prime _{j,i} \end{matrix} + \lambda ^\prime {}^\prime _{j}\beta ''_{j}\beta ^\prime _{j,i}, \end{aligned}$$

(A6)

where $\lambda _{j,i^\prime }\beta _{j,i^\prime }\beta ^\prime _{j,i}$ is the request rate offloaded from $f_{j, i^\prime }$ to $f_{j,i}$, $\lambda ^\prime _{j^\prime }\beta ^\star _{j^\prime ,j}\beta ^\prime _{j,i}$ is the request rate offloaded from $e_{j^\prime }$ to $f_{j,i}$, and $\lambda ^\prime {}^\prime \beta ''_{j} \beta ^\prime _{j,i}$ is the request rate offloaded from C to $f_{j,i}$. The $R^\prime _{j}$ in two-hop offloading can be represented as

$$\begin{aligned} R'_j= & {} \lambda '_j - \lambda '_j \beta '_j \nonumber \\{} & {} +\, \lambda ''\left( 1 - \begin{matrix} \sum \limits _{i = 1}^{n_j}\beta '_{j,i} \end{matrix} \right) \beta ''_{j} - \begin{matrix} \sum \limits _{i = 1}^{n_j}\lambda '_j\beta '_{j,i} \end{matrix} \nonumber \\{} & {} +\, \begin{matrix} \sum \limits _{i = 1}^{n_j}\lambda _{j,i}\left( 1 - \beta '_j - \sum \limits _{i' = 1, i' \ne i}^{n_j}\beta '_{j,i'} \right. \\ \left. - \sum \limits _{j'= 1, j' \ne j}^{m}\beta ^\star _{j,j'} \right) \beta _{j,i} \end{matrix} \nonumber \\{} & {} -\, \begin{matrix}\sum \limits _{j' = 1, j' \ne j}^{m}\lambda '_{j}\beta ^\star _{j,j'} \end{matrix} \nonumber \\{} & {} +\, \begin{matrix}\sum \limits _{j' = 1, j' \ne j}^{m}\lambda '_{j'}\left( 1 - \sum \limits _{i = 1}^{n_j}\beta '_{j,i} \right) \beta ^\star _{j',j} \end{matrix} \nonumber \\{} & {} +\, \begin{matrix}\sum \limits _{j' = 1, j' \ne j}^{m}\begin{matrix}\sum \limits _{i = 1}^{n_j}\lambda _{j',i}\beta _{j',i}\beta ^\star _{j',j}, \end{matrix} \end{matrix} \end{aligned}$$

(A7)

where $\lambda _{j^\prime ,i}\beta _{j^\prime ,i}\beta ^\star _{j^\prime ,j}$ is the request rate offloaded from $f_{j^\prime ,i}$ to $e_j$. $\lambda ^\prime {}^\prime \left( 1 - \begin{matrix} \sum \nolimits _{i = 1}^{n_j}\beta ^\prime _{j,i} \end{matrix}\right) \beta ^\prime {}^\prime _j$, $\lambda _{j,i}(1 - \beta ^\prime _j - \sum \nolimits _{i^\prime = 1, i^\prime \ne i}^{n_j}\beta ^\prime _{j,i^\prime } - \sum \nolimits _{j^\prime = 1, j^\prime \ne j}^{m}\beta ^\star _{j,j^\prime })\beta _{j,i}$, and $\lambda ^\prime _{j^\prime }\left( 1 - \sum \nolimits _{i = 1}^{n_j}\beta ^\prime _{j,i}\right) \beta ^\star _{j^\prime ,j}$ are request rate received by $e_j$ from C, $f_{j,i}$, and $e_{j^\prime }$, respectively.

The $R^\prime {}^\prime$ in two-hop offloading can be represented as

$$\begin{aligned} R^\prime {}^\prime= & {} \lambda ^\prime {}^\prime - \begin{matrix} \sum \limits _{j = 1}^{m}\lambda ^\prime {}^\prime \beta ^\prime {}^\prime _{j} \end{matrix} + \begin{matrix} \sum \limits _{j = 1}^{m}\lambda ^\prime _{j}\beta ^\prime _{j} \end{matrix} \nonumber \\{} & {} +\, \begin{matrix} \sum \limits _{j = 1}^{m} \begin{matrix} \sum \limits _{i = 1}^{n_j}\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j}, \end{matrix} \end{matrix} \end{aligned}$$

(A8)

where $\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j}$ is the request rate offloaded from $f_{j,i}$ to C.

1.3 A.3: Communication cost

Here, we calculated the communication cost between tiers. Since two-hop offloading has an extra second-hop offloading, its estimation is slightly different from the one-hop offloading. The $S^\prime _{C,E}$ in two-hop offloading can be represented as

$$\begin{aligned} S^\prime _{C,E}= & {} s^\prime _{C,E} \Bigg (\begin{matrix} \sum \limits _{j=1}^{m}\lambda '' \beta ''_{j} \end{matrix} + \begin{matrix} \sum \limits _{j=1}^{m}\lambda ^\prime _j\beta ^\prime _{j} \end{matrix} \nonumber \\{} & {} +\, \begin{matrix} \sum \limits ^{m}_{j=1}\begin{matrix}\sum \limits ^{n_j}_{i=1}\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j}\Bigg ),\end{matrix}\end{matrix} \end{aligned}$$

(A9)

where $\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j}$ is the second-hop of the offloading from $f_{j,i}$ to C. The $S^\prime _{E,E^\prime }$ in two-hop offloading can be represented as

$$\begin{aligned} S^\prime _{E,E^\prime }= & {} s^\prime _{E,E^\prime }\Bigg (\begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{j^\prime =1, j^\prime \ne j}^{m}\lambda ^\prime _j\beta ^\star _{j,j^\prime } \end{matrix}\end{matrix} \nonumber \\{} & {} +\, \begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{i=1}^{n_j}\begin{matrix} \sum \limits _{j^\prime =1, j^\prime \ne j}^{m}\lambda _{j,i}\beta _{j,i}\beta ^\star _{j,j^\prime }\Bigg ), \end{matrix} \end{matrix}\end{matrix} \end{aligned}$$

(A10)

where $\lambda _{j,i}\beta _{j,i}\beta ^\star _{j,j^\prime }$ is the second-hop of the offloading from $f_{j,i}$ to $e_{j^\prime }$. The $S_{E,F}$ in two-hop offloading can be represented as

$$\begin{aligned} S_{E,F}= & {} s_{E,F}\Bigg (\begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{i=1}^{n_j}\lambda ^\prime _j\beta ^\prime _{j,i} \end{matrix}\end{matrix} \nonumber \\{} & {} +\, \begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{i=1}^{n_j}\lambda _{j,i}\beta _{j,i} \end{matrix}\end{matrix} \nonumber \\{} & {} +\, \begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{i=1}^{n_j}\begin{matrix}\sum \limits _{i^\prime =1, i^\prime \ne i}^{n_j}\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j,i^\prime } \end{matrix}\end{matrix}\end{matrix} \nonumber \\{} & {} +\, \begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{j^\prime =1, j^\prime \ne j}^{m}\begin{matrix}\sum \limits _{i=1}^{n_j}\lambda ^\prime _{j}\beta ^\star _{j,j^\prime }\beta ^\prime _{j^\prime ,i} \end{matrix}\end{matrix}\end{matrix}\nonumber \\{} & {} +\, \begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{i=1}^{n_j}\lambda ^\prime {}^\prime \beta ''_j \beta ^\prime _{j,i}\Bigg ), \end{matrix}\end{matrix} \end{aligned}$$

(A11)

where $\lambda _{j,i}\beta _{j,i}\beta ^\prime _{j,i^\prime }$ is the second-hop of the offloading from $f_{j,i}$ to $f_{j,i^\prime }$, $\lambda ^\prime _{j}\beta ^\star _{j,j^\prime }\beta ^\prime _{j^\prime ,i}$ is the second-hop of the offloading from $e_j$ to $f_{j^\prime ,i}$, and $\lambda ^\prime {}^\prime \beta ''_{j}\beta ^\prime _{j,i}$ is the second-hop of the offloading from C to $f_{j,i}$.

1.4 A.4: Computation cost

The evaluation of computing cost in two-hop offloading is the same as one-hop offloading as shown in (15) for the cloud tier, (16) for the edge tier, and (17) for the fog tier.

1.5 A.5: Objective and constraints

The objective function in two-hop offloading is the same as one-hop offloading, as shown in (18). The constraint of two-hop offloading will be more complicated because of the second-hop compared to the one-hop offloading, and the objective function must meet the (22)–(27), along with the following constraints.

$$\begin{aligned}{} & {} \hat{\beta } l'' + \begin{matrix} \sum \limits _{j=1}^{m}\beta ''_j\left( 1- \begin{matrix} \sum \limits _{i=1}^{n_j}\beta ^\prime _{j,i} \end{matrix} \right) \left( l^\prime _j + T''_j \right) \end{matrix} \nonumber \\{} & {} \quad +\, \begin{matrix} \sum \limits _{j=1}^{m}\begin{matrix} \sum \limits _{i=1}^{n_j}\beta ''_j\beta ^\prime _{j,i}\left( l_{j,i} + D''_{j,i} \right) \le \mathcal {L}_{max}, \end{matrix} \end{matrix} \end{aligned}$$

(A12)

$$\begin{aligned}{} & {} \hat{\beta }_j l^\prime _j + \beta ^\prime _j\left( l^\prime {}^\prime + T^\prime _j \right) + \begin{matrix} \sum \limits _{i=1}^{n_j}\beta ^\prime _{j,i}\left( l_{j,i} + T^\prime _{j,i} \right) \end{matrix} \nonumber \\{} & {} \quad + \begin{matrix} \sum \limits _{j^\prime =1, j^\prime \ne j}^{m}\beta ^\star _{j,j^\prime }\left( 1 - \begin{matrix} \sum \limits _{i=1}^{n_j}\beta ^\prime _{j^\prime ,i} \end{matrix} \right) \left( l^\prime _{j^\prime } + T^\star _{j,j^\prime } \right) \end{matrix} \nonumber \\{} & {} \quad +\,\begin{matrix} \sum \limits _{j^\prime =1, j^\prime \ne j}^{m}\begin{matrix} \sum \limits _{i=1}^{n_j}\beta ^\star _{j,j^\prime }\beta ^\prime _{j^\prime ,i}\left( l_{j^\prime ,i} + D'_{j',i}\right) \end{matrix} \end{matrix} \le \mathcal {L}_{max}, \end{aligned}$$

(A13)

$$\begin{aligned}{} & {} \hat{\beta }_{j,i} l_{j,i} + \beta _{j,i}(1 - \beta ^\prime _j - \begin{matrix} \sum \limits _{i^\prime =1, i^\prime \ne i}^{n_j}\beta ^\prime _{j,i^\prime } \end{matrix} \nonumber \\{} & {} \quad -\, \begin{matrix} \sum \limits _{j^\prime =1, j^\prime \ne j}^{m}\beta ^\star _{j,j^\prime } \end{matrix} ) \left( l^\prime _j + T_{j,i} \right) \nonumber \\{} & {} \quad +\, \begin{matrix} \sum \limits _{i^\prime =1, i^\prime \ne i}^{n_j}\beta _{j,i}\beta ^\prime _{j,i^\prime }\left( l_{j,i^\prime } + D_{j,i^\prime } \right) \end{matrix} \nonumber \\{} & {} \quad +\, \begin{matrix} \sum \limits _{j^\prime =1, j^\prime \ne j}^{m}\beta _{j,i}\beta ^\star _{j,j^\prime }\left( l^\prime _{j^\prime } + D_{j^\prime } \right) \end{matrix} \nonumber \\{} & {} \quad +\, \beta _{j,i}\beta ^\prime _{j}\left( l^\prime {}^\prime + D_C \right) \le \mathcal {L}_{max}. \end{aligned}$$

(A14)

The constraints in (A12), (A13), and (A14) ensure that the total communication latency plus the total computation latency of the cloud, edge, and fog in the case of two-hop offloading do not exceed the maximum latency limit.

An appendix contains supplementary information that is not an essential part of the text itself but which may be helpful in providing a more comprehensive understanding of the research problem or it is information that is too cumbersome to be included in the body of the paper.