Edge computing task scheduling method based on user’s social relations: a construction and solution for Smart City Library

Social relationship computing

(1) Service assessment factors

The social relationship graph can store a social relationship score between user node i and service node j if they have had a direct interaction. This score is calculated by employing the service assessment of node i to node j and stored. Assuming the latest H service evaluation records of node i to node j are S_ij = {S_ij(1), S_ij(2), …, S_ij(k), …, S_ij(H)}, where S_ij(k) represents scores given by user node i to service node j, k ∈[1, H]. H represents the number of service assessments of user node i to service node j within a specific period to ensure the effectiveness of service assessment. The research employs the mathematical convolution concept (Wei, Yang & Liu, 2023) to derive Eq. (1) to compute the service assessment factor of user node i to service node j. To facilitate the processing of the proportion of each decision factor in the social relationship, service assessment factor scores are normalized by implementing Eq. (2). (1)${\displaystyle Q_{ij}=\frac{1}{H}\sum _{k=1}^H{S}_{ij}\left(k\right)\times \frac{1}{H-k+1}}$ (2)${\displaystyle Q_{ij}=\frac{Q_{ij}^{\prime }-min\left(Q_{ij}\right)}{max\left(Q_{ij}\right)-min\left(Q_{ij}\right)}=\frac{Q_{ij}^{\prime }}{5}}$

where Q_ij’ represents the service assessment calculated directly through historical evaluation. After normalization is run, Q_ij represents the service assessment factor, with max(Q_ij) and min(Q_ij), which are the maximum and minimum scores of a service evaluation, 0 and 5, respectively.

(2) Recommended trust factor

When there is no direct assessment between a user node and a device node, a user node can still calculate the social relationship score by referring to other nodes’ assessment of the service node since trust can be transferred. An intermediary node is a node assessed by user nodes and has assessments for service nodes.

Let the intermediary node set of user node i and service node j be V_ij, V_ij ={V_ij(1), V_ij(2), …, V_ij(k), …, V_ij(—V_ij—)}, where —V_ij— represents the number of intermediary nodes. The recommendation trust W_ij of the user node i to the device node j can be computed based on the service assessment of user node i to the intermediary node by employing Eq. (3). (3)${\displaystyle W_{ij}=\frac{1}{\mid V_{ij}\mid }\sum _{k=1}^{\mid V_{ij}\mid }{Q}_{ik}\times Q_{kj}}$

(3) Similar recommendation factor

When the user node i has not assessed the service node j, the user node i can also refer to the evaluation of the service node j by nodes with similar assessments to itself to judge the credibility of the nodes (Park & Tussyadiah, 2020). Assuming that node K_ij(η) has not only assessed the same node as user node i , but also assessed node j, then K_ij(η) represents a reference node. The set of reference nodes is denoted by K_ij ={ K_ij(1), K_ij(2), …, K_ij(η), …, K_ij(—K_ij—)} and the size of the set is denoted by —K_ij—. N_ξ denotes the intersection set of nodes serving user node i and reference node η, and the set size is represented by —N_ξ—. The research employs the generalized Jaccard similarity coefficient to attain the similarity score between user node i and reference node η on the same group of service nodes N_ξ, as shown in Eq. (4). (4)${\displaystyle J_{i\eta }=\frac{\sum _{\xi =1}^{\mid N_{\xi }\mid }min\left\{Q_{i\xi },Q_{\eta \xi }\right\}}{\sum _{\xi =1}^{\mid N_{\xi }\mid }max\left\{Q_{i\xi },Q_{\eta \xi }\right\}}}$

(4) Calculation of social relationship score

Users tend to trust their judgments more than others’ opinions. Therefore, when there is a direct assessment between user node i and service node j, the social relationship between them in the social relationship graph is determined by the service assessment, represented by P_ij = Q_ij.

However, when there is no direct assessment between user node i and service node j, the social relationship must be computed based on the recommendation trust factor W_ij and the similar recommendation factor A_ij. The impact of W_ij and A_ij on the social relationship P_ij can be gauged by the number of intermediary nodes —V_ij— and the number of reference nodes —K_ij—.

The social relationship score P_ij of user node i to service node j in a user social relationship matrix P can be obtained by implementing Eq. (5): (5)${\displaystyle P_{ij}=\begin{array}{cc}{Q}_{ij},\hfill & H\ne 0\hfill \\ \left\{\phi \times W_{ij}+\left(1-\phi \right)\times \right.A_{ij},\hfill & H=0,\mid V_{ij}\mid +\mid K_{ij}\mid \ne 0\hfill \\ 0.5,\hfill & \mid V_{ij}\mid +\mid K_{ij}\mid =0\hfill \end{array}.}$

Construction of social relationship graph

The process of constructing and implementing a user social relationship graph Gs is presented as follows: Firstly, the number of N user nodes and the number of M service nodes are obtained. Next, service rating information between mobile devices is obtained. Then, for each user node, a social relationship score between it and each service node is computed.

To determine whether to add an edge between a user node and a service node in the social relationship graph, the social relationship score P between them is compared to a set threshold δ. An edge is added if P is greater than or equal to δ. Otherwise, the social relationship graph is not modified, and the next service device is monitored.

Calculation of communication cost

Under the control of the BS, any user equipment can construct a D2D connection with a serving device. Since tasks are typically time-sensitive, the BS only constructs and maintains one D2D connection per mobile device. A D2D communication is adopted among mobile devices, assuming orthogonal spectrum resources and no interference between devices. According to Shannon’s theorem, a D2D communication transmission rate between user equipment i and service equipment j can be calculated by Eq. (7): (7)${\displaystyle R_{ij}=B{log}_2\left(\frac{P_i^t\mid D_{ij}{\mid }^2{L}_{ij}^{-\gamma }}{{\Sigma }_{U_{i^{\prime }}\in \mathrm{U},i^{\prime }\ne \mathrm{i}}{P}_i\mid D_{ij}{\mid }^2{L}_{ij}^{-\gamma }+{\sigma }^2}+1\right)}$

where B, PT i, and D_ij represent the available bandwidth of the transmission channel, the transmission power of the user equipment i, and the channel parameter of the D2D transmission. The path loss between user device i and serving device j is L −γ ij, where L, γ, σ2 represent the physical distance between mobile devices, the path loss exponent, and the noise variance.

In a task transmission, the mobile device can only construct one D2D connection at a time, and the mobile device, as the server, has relatively limited computing power and can only receive one task to do a calculation at a time. The time it takes for user equipment i to transmit a task to service equipment j is the data transmission delay Td ij. The transmission delay caused by task scheduling of user equipment to service equipment can be computed. When tasks are scheduled to service nodes for calculation, two parts of energy consumption will be generated. One is the energy consumption caused by user equipment i scheduling tasks to service equipment j, and the other is the energy consumption led by service equipment j sending task execution outcomes back to user equipment i. Total energy consumption, the sum of the energy consumption of these 2 parts, constitutes the energy consumption Ed ij of scheduling transmission. Equations (8) and (9) are used to do calculations as follows:

(8)${\displaystyle T_{ij}^d=\frac{I_i}{R_{ij}}+\frac{O_i}{R_{ji}}}$ (9)${\displaystyle E_{ij}^d=\frac{I_i}{R_{ij}}{P}_i^t+\frac{O_i}{R_{ji}}{P}_j^t.}$

The cost calculation of a task execution

User devices have two options for task execution: local execution or scheduling tasks on other mobile devices. When a task is executed locally on a user device, the cost of task execution includes the energy consumption and computational delay produced by the device itself. On the other hand, when a user device schedules a task to be executed on a service node, the cost of the energy consumption of task execution includes the energy consumption for task transmission, the energy consumption for computing tasks on the service device, the standby energy consumption of the user device waiting for the task to return, and the delay cost, which includes task transmission and task computation delays.

To execute locally, Eqs. (10) through (16) are used:

(10)${\displaystyle T_i^{loc}=\frac{I_i{C}_i}{F_i}}$ (11)${\displaystyle E_i^{loc}=P_i^c{T}_i^{loc}.}$

To execute scheduling, the relevant equations are given as follows:

(12)${\displaystyle T_{ij}^c=\frac{I_i{C}_i}{F_j}}$ (13)${\displaystyle E_{ij}^c=P_i^c{T}_{ij}^c}$ (14)${\displaystyle T_{ij}^{sch}=T_{ij}^d+T_{ij}^c}$ (15)${\displaystyle E_i^w=T_{ij}^{sch}\times E_o}$ (16)${\displaystyle E_{ij}^{sch}=E_{ij}^d+E_{ij}^c+E_i^w.}$

Task scheduling model

Let λ_ij set as a binary decision variable of task scheduling. If a task of a user node i is offloaded to the service node j, then λ_ij = 1; otherwise, λ_ij = 0. It means the task is not executed on the service node. The task-device bipartite graph is represented by G = {N, ɛ}, where eij represents the associated edge between user task i and device node j, which satisfies the social and communication conditions required for task scheduling. To sum up, this research aims to minimize the execution delay and energy consumption of all tasks in the entire scenario, and the objective function is shown in Eq. (17): (17)${\displaystyle \underset{\lambda }{min}\sum _{i=1}^N\left[\left(1-\sum _{j=1}^M{\lambda }_{i,j}\right)Z_i^{loc}+\sum _{j=1}^M{\lambda }_{i,j}{Z}_{ij}^{sch}\right]=\underset{\lambda }{min}\frac{1}{2}\sum _{i=1}^N\left[\left(1-\sum _{j=1}^M{\lambda }_{i,j}\right)\left(T_i^{loc}+E_i^{loc}\right)+\sum _{j=1}^M{\lambda }_{i,j}\left(T_{ij}^{sch}+E_{ij}^{sch}\right)\right].}$