Table 3 Definition of key mathematical notations

Low (L), Medium (M), High (H)

Experience level using in the placement problem

Light (L), Normal (M), Heavy (H)

Experience level using in the deployment problem

μ^L(w)

Membership function, μ^L(w)={w∈L,μ^L(w)∈[0,1]}

μ^M(w)

Membership function, μ^L(w)={w∈M,μ^M(w)∈[0,1]}

μ^H(w)

Membership function, μ^L(w)={w∈H,μ^H(w)∈[0,1]}

a

WLAN delay

b

MAN delay

c

Local edge VM utilization

d

Candidate edge VM utilization

x

WAN bandwidth

y

Length of the incoming application task

z

VM utilization on the edge server

t

Delay sensitivity of the related application

R_i

Fuzzy rule at index i^th

\(\mu ^{R_{i}}_{i}\)

The minimum (min) function to determine how the results of multiple rules are combined

within R_i. \(\mu ^{R_{i}}_{i} = {min}\{\mu _{a}^{R_{i}}(m), \mu _{b}^{R_{i}}(n),\mu _{c}^{R_{i}}(l),\mu _{d}^{R_{i}}(k)\}\)

{m,n,l,k}

The measured experiment value, as crisp data, is the input parameter to be fuzzified

μ_r

The maximum (max) function to determine how the results of multiple rules have the same decision r on the Fuzzy rules

ω

The center of gravity (COG) of the area under the curve by using the centroid defuzzifier method

T

Incoming task

O

Target of offload