4.1.1 Time-varying context data sequence of a cloudlet
Suppose there are p candidate cloudlets. The time-varying sequences of cloudlet load L, CPU utilization C and network conditions D in q consecutive moments are denoted as \(L_i^j\), \(C_i^j\), \(D_i^j\), \(i\in \{1,2,...,p \}\), \(j\in \{1,2,...,q \}\). The load and CPU utilization can be viewed by top and vmstat commands respectively in Linux environment. The network conditions are measured by network latency. The time-varying mobility property is characterized by the expected user residence time within the cloudlet. By calculating the expected residence time of mobile users at q moments within p cloudlets respectively, we can obtain the time-varying data sequences, denoted as \(M_i^j\), \(i\in \{1,2,...,p\},j\in \{1,2,...,q\}\). The time-varying contextual data of \(cloudlet_i\) are collectively denoted as a quaternion vector \(CON(i)=(L_i, C_i, D_i, M_i)\).
After normalizing the contextual original data, the four time-varying contextual data sequences of
p candidate cloudlets at
q moments can be expressed as matrix
\(CON(cloudlet_i^j)_{p \times 4}, i\in \{1,2,...,p\}\),
\(j\in \{1,2,...,q\}\), which is formulated as follows:
$$\begin{aligned} \begin{gathered} CON\left( cloudlet_{i}^{j}\right) _{p \times 4}= \left( L_{(p \times q)}, C_{(p \times q)}, D_{(p \times q)}, M_{(p \times q)}\right) =\\ {\left[ \begin{array}{cccc} \left( L_{1}^{1},\ \!\!\! L_{1}^{2},\ \ \!\!\!\ldots ,\ \!\!\! L_{1}^{q}\right) &{} \!\!\!\left( C_{1}^{1},\ \!\!\! C_{1}^{2},\ \ \!\!\!\ldots ,\ \!\!\! C_{1}^{q}\right) &{} \!\!\!\left( D_{1}^{1},\ \!\!\! D_{1}^{2},\ \ \!\!\!\ldots ,\ \!\!\! D_{1}^{q}\right) &{} \!\!\!\left( M_{1}^{1},\ \!\!\! M_{1}^{2},\ \ \!\!\!\ldots ,\ \!\!\! M_{1}^{q}\right) \\ \left( L_{2}^{1},\ \!\!\! L_{2}^{2},\ \ \!\!\!\ldots ,\ \!\!\! L_{2}^{q}\right) &{} \!\!\!\left( C_{2}^{1},\ \!\!\! C_{2}^{2},\ \ \!\!\!\ldots ,\ \!\!\! C_{2}^{q}\right) &{} \!\!\!\left( D_{2}^{1},\ \!\!\! D_{2}^{2},\ \ \!\!\!\ldots ,\ \!\!\! D_{2}^{q}\right) &{} \!\!\!\left( M_{2}^{1},\ \!\!\! M_{2}^{2},\ \ \!\!\!\ldots ,\ \!\!\! M_{2}^{q}\right) \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \left( L_{p}^{1},\ \!\!\! L_{p}^{2},\ \ \!\!\!\ldots ,\ \!\!\! L_{p}^{q}\right) &{} \!\!\!\left( C_{p}^{1},\ \!\!\! C_{p}^{2},\ \ \!\!\!\ldots ,\ \!\!\! C_{p}^{q}\right) &{} \!\!\!\left( D_{p}^{1},\ \!\!\! D_{p}^{2},\ \ \!\!\!\ldots ,\ \!\!\! D_{p}^{q}\right) &{} \!\!\!\left( M_{p}^{1},\ \!\!\! M_{p}^{2},\ \ \!\!\!\ldots ,\ \!\!\! M_{p}^{q}\right) \! \end{array}\right] } \end{gathered} \end{aligned}$$
(10)
The i-th row of the matrix
\(CON(cloudlet_i^j)\),
CON(
i) corresponds to the context data sequence of the
\(cloudlet_i\), expressed as follows:
$$\begin{aligned} \begin{gathered} CON(i)=\left( L(i), C(i), D(i), M(i)\right) \\ ={\left( (L_i)_{(1*q)},(C_i)_{(1*q)},(D_i)_{(1*q)},(M_i)_{(1*q)}\right) } \end{gathered} \end{aligned}$$
(11)
4.1.2 The SVNS representation of the time-varying context
Most of the existing work on analyzing the time-varying features of mobile edges adopt the average value based scheme, which is difficult to accurately characterize the inconsistent and uncertain fuzzy information. In this paper, we consider the characterization tool of fuzzy information,
single-valued neutrosophic set (SVNS). SVNS [
24] is a branch of neutrosophic set (NS) theory [
25], which expands the traditional fuzzy set, adopts three measures of truth-membership, indeterminacy-membership and falsity-membership to characterize fuzzy decision-making information. It has the ability to delicately and accurately describe the fuzzy nature of objective things. At the same time, the neutrosophic set has independent unique operations, which have been widely used in multi-attribute decision making [
26], artificial intelligence [
27] and service evaluation [
28].
The value of the membership function of the SVNS is a real number. Given a domain X, a SVNS A on X includes truth-membership function
\(T_A (x)\), indeterminacy-membership function
\(I_A (x)\) and falsity-membership function
\(F_A (x)\), denoted as follow:
$$\begin{aligned} A=\left\{ \left\langle X, T_{A}(x), I_{A}(x), F_{A}(x)\right\rangle \mid x \in X\right\} \end{aligned}$$
where
\(T_A (x),I_A (x),F_A (x)\in [0,1]\), and
\(0 \le T_A (x)+I_A (x)+F_A (x) \le 3\).
In order to use SVNS to characterize the time-varying context, it is necessary to establish three membership functions for each time-varying context factor of
\(cloudlet_i\), that is, to obtain the three membership values for each context factor of
\(cloudlet_i\). Converting the contextual data sequence expressed in Eq. (
11) to a SVNS, it can be expressed as follows:
$$\begin{aligned} \begin{gathered} \forall con\in \{L,C,D,M\}, CON(i) \Rightarrow \{T_i^{con} (x),\ I_i^{con} (x),\ F_i^{con} (x)\} \end{gathered} \end{aligned}$$
(12)
4.1.3 The context SVNS generation using backward cloud model
In order to establish the three membership functions of the time-varying context,
cloud model [
29](CM) is used to complete the transformation expressed in Eq. (5). The cloud model is a cognitive model that implements the duplex transformation of qualitative concepts and quantitative data. The contextual data sequence have been acquired in section 4.1, so the current problem is one that goes from quantitative (data sequence) to qualitative (degree of membership).
The cloud model has three numerical features: expectation
\(\hat{E} x\), entropy
\(\hat{E} n\), and super-entropy
\(\hat{H} e\).
\(\hat{E} x\) is the expectation of the spatial distribution of cloud drops in the theoretical domain, which represents the basic certainty of the qualitative concept;
\(\hat{E} n\) represents the measure of uncertainty of the qualitative concept, which reflects the range of values that can be accepted by this concept in the theoretical domain.
\(\hat{H} e\) is the entropy of the entropy
\(\hat{E} n\), which is the uncertainty of the entropy and expresses the deviation of the cloud model. Therefore,
\(\hat{E} x\) can be taken as the truth-membership
\(T_A\) of SVNS,
\(\hat{E} n\) as the indeterminacy-membership
\(I_A\), and
\(\hat{H} e\) as the falsity-membership
\(F_A\), that is,
$$\begin{aligned} T_A(x)= & {} \hat{E} x(x) \end{aligned}$$
(13)
$$\begin{aligned} I_A(x)= & {} \hat{E} n(x) \end{aligned}$$
(14)
$$\begin{aligned} F_A(x)= & {} \hat{H} e(x) \end{aligned}$$
(15)
In this paper, hybrid cloud and cloud models both refer to the word ‘cloud’. To distinguish them, hybrid cloud is used to refer to nodes or clusters in distributed computing. It is a computational paradigm. However, cloud, cloud model, cloud droplet and other related words are all terms in cloud theory, which involves uncertain concepts in the artificial intelligence field.
In cloud model, a cloud consists of cloud drops. A cloud drop is a realization of a qualitative concept, and a certain number of cloud drops can express a cloud. Here, the value of one-dimensional time-varying context of
\(cloudlet_i\) at moment
j can be regarded as a cloud drop. Its sequence of values describing the dimensional context at
q moments can characterize the cloud model which is denoted as
\({CM}^{con}(i)\). That is,
\({CM}^{con}(i)=(con_i^1,con_i^2,...,con_i^q )\). For
\(cloudlet_i\), use the membership calculation scheme shown in Eq. (
13) - Eq. (
15). That is, the backward cloud generator is selected to establish a context cloud model for each dimension as follows:
$$\begin{aligned} T_i^{con}(x)= & {} \hat{E} x_{i}^{c o n}=\bar{X}=\frac{1}{q} \sum _{j=1}^{q} c o n_{i}^{j} \end{aligned}$$
(13*)
$$\begin{aligned} I_i^{con}(x)= & {} \hat{E} n_{i}^{c o n}=\sqrt{\frac{\pi }{2}} \times \frac{1}{q} \sum _{j=1}^{q}\left| c o n_{i}^{j}-\hat{E} x\right| \end{aligned}$$
(14*)
$$\begin{aligned} F_i^{con}(x)= & {} \widehat{H} e_{i}^{c o n}=\sqrt{S^{2}-\hat{E} n^{2}}, S^{2}=\frac{1}{q-1} \sum _{k=1}^{q}\left( c o n_{i}^{j}-\bar{X}\right) ^{2} \end{aligned}$$
(15*)
where
\(con\in \{L,C,D,M\}\),
\(con_i^j\) denotes the time-varying context sequence of a certain context of
\(cloudlet_i\),
\(i\in \{1,2,...,p\}\),
\(j\in \{1,2,...,q\}\).
A element in the SVNS A is called single-valued neutrosophic number (SVNN), which is expressed as \(\{T_A,I_A,F_A\}\). For the \(cloudlet_i\), the original time-varying context data sequences of each dimension generates three numeric features through the backward cloud generator, that is, three SVNNs are generated.
So far, the transformation from the time-varying context data sequence of the candidate cloudlets to the context SVNN has been completed, which can be expressed as follow:
$$\begin{aligned} CON(i)=(con_i^1,\ con_i^2,\ ...,\ con_i^q )=CM_i^{con} \Rightarrow \\ \{ T_i^{con} (x), I_i^{con} (x), F_i^{con} (x)\}=\{\hat{E} x_i^{con},\ \hat{E} n_i^{con},\ \hat{H} e_i^{con} \} \end{aligned}$$
(12*)
Therefore, the time-varying contextual numerical sequence matrix
\(CON(cloudlet_i^j)_{p \times 4}\) of the
p candidate cloudlets is transformed into the following SVNS context matrix as follows:
$$\begin{aligned} {SVNS}\left( cloudlet_{i}^{con}\right) _{p \times 4}=\left\{ T_{i}^{con}, I_{i}^{con}, F_{i}^{con}\right\} _{p \times 4}= \\ {\left[ \begin{array}{cccc} \left\{ T_{1}^{L}, I_{1}^{L}, F_{1}^{L}\right\} &{} \!\!\!\!\left\{ T_{1}^{C}, I_{1}^{C}, F_{1}^{C}\right\} &{} \!\!\!\!\left\{ T_{1}^{D}, I_{1}^{D}, F_{1}^{D}\right\} &{} \!\!\!\!\left\{ T_{1}^{M}, I_{1}^{M}, F_{1}^{M}\right\} \\ \left\{ T_{2}^{L}, I_{2}^{L}, F_{2}^{L}\right\} &{} \!\!\!\!\left\{ T_{2}^{C}, I_{2}^{C}, F_{2}^{C}\right\} &{} \!\!\!\!\left\{ T_{2}^{D}, I_{2}^{D}, F_{2}^{D}\right\} &{} \!\!\!\!\left\{ T_{2}^{M}, I_{2}^{M}, F_{2}^{M}\right\} \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ \left\{ T_{p}^{L}, I_{p}^{L}, F_{p}^{L}\right\} &{} \!\!\!\!\left\{ T_{p}^{C}, I_{p}^{C}, F_{p}^{C}\right\} &{} \!\!\!\!\left\{ T_{p}^{D}, I_{p}^{D}, F_{p}^{D}\right\} &{} \!\!\!\!\left\{ T_{p}^{M}, I_{p}^{M}, F_{p}^{M}\right\} \nonumber \end{array}\right] } \end{aligned}$$
where
\(i\in \{1,2,...,p\}, con\in \{L,C,D,M\}\). Each row vector of the matrix
SVNS(
i) represents the four-dimensional context SVNNs of
\(cloudlet_i\), and the four column vectors respectively correspond to the SVNS of all candidate cloudlets for the four-dimensional context.
The method of transformation from the original context time-varying sequence to the context SVNS proposed in this paper is extensible. Under specific scenario, other customization decision-making factors can also be extended to take into account.