## 3 Task Redirection Method
### Properties of the Task Redirection Method  
In [17], the load of each MECS was shown to converge to \(\vec{p}=\sum_{i\in N}\rho_{i}/|N|\) in polynomial time if each MECS repeats Algorithm 1.  
To state how the proposed method distributes tasks to MECSs, we first introduce the following definitions.  
**Definition 1**.: _A vector \(\vec{a}\in X\subseteq R^{n}\) is said to be a min-max fair vector on \(X\) if and only if we cannot decrease a component in \(a_{i}\in\vec{a}\) without increasing another component \(a_{j}\geq a_{i}\). Formally, for all \(\vec{b}\in X\), if there exists \(i=\{1,\ldots,n\}\) such that \(b_{i}\in\vec{b}<a_{i}\in\vec{a}\), then there exists \(j=\{1,\ldots,n\}\) such that \(b_{j}>a_{j}\geq a_{i}\)._  
**Definition 2**.: _Given a set of vectors \(A\subset\mathbf{R}^{n}\), a vector \(\vec{a}\in A\) is said to be leximax minimal if \(<\vec{a}>\) is lexicographically less than or equal to \(<\vec{b}>\) for any vector \(\vec{b}\in A\), where \(<\vec{a}>\) represents the vector obtained from vector \(\vec{a}\) by rearranging its elements in nonincreasing order._  
We say that a MECS load vector \(\vec{\rho}=(\rho_{1},\ldots,\rho_{n})\) is feasible if \(\rho_{i}<1\) for all \(i\in N\) and denote the set of feasible \(\vec{\rho}\)s as \(\Psi\subseteq R^{n}\). Then, we state the fairness of our task redirection method as Proposition 1.  
**Proposition 1**.: _The MECS load vector \(\vec{\rho}=(\rho_{1},\ldots,\rho_{n})=(\vec{\rho},\ldots,\vec{\rho})\in\Psi\) obtained by our task redirection method is min-max fair._  
Let us suppose that there is a min-max fair load vector \(\vec{a}=(a_{1},\ldots,a_{n})\in\Psi\) such that \(a_{i}\
eq a_{j}\) for some \(i,j(i\
eq j)\). Without loss of generality, we assumed that \(a_{i}\) is the smallest and \(a_{j}\) is the second smallest element in \(\vec{a}\). If we choose \(\epsilon\) such that \(0<\epsilon<a_{j}-a_{i}\), we have \(a_{i}+\epsilon<a_{j}\). Let us consider another load vector \(\vec{b}=(b_{1},\ldots,b_{n})\in\Psi\). If the loads of some MECSs decrease, the decreased load has to be accommodated by another MECSs. Specifically, for \(0<\delta_{i}<\epsilon\) and \(0\leq\delta_{k}<\epsilon\), the decrease in the load of MECS \(i\) from \(a_{i}\) to \(a_{i}-\delta_{i}\) induces variation in the loads of other MECSs from \(a_{k}\) to \(a_{k}+\delta_{k}\) and \(\sum_{k\
eq i}\delta_{k}=\delta_{i}\). When \(b_{i}=a_{i}-\delta_{i}\) and \(b_{j}=a_{j}+\delta_{j}\), \(b_{i}<a_{i}\) and \(b_{j}=a_{j}+\delta_{j}<a_{i}\), which contradicts that \(\vec{a}\) is a min-max fair vector.  
**Proposition 2**.: _The min-max fair MECS load vector \(\vec{\rho}=(\rho_{1},\ldots,\rho_{n})=(\vec{\rho},\ldots,\vec{\rho})\) is a unique teximax minimal load vector._  
It was shown in [18; 19] that if a max-min fair vector exists on a set \(X\subseteq R^{n}\), then it is the unique lexicographically maximal vector on \(X\). Since \(\vec{\rho}\) is min-max fair on \(\Psi\), \(-\vec{\rho}\) is a max-min fair load vector on \(-\Psi\), which makes \(-\vec{\rho}\) the unique lexicographically maximal vector on \(-\Psi\). Therefore, \(\vec{\rho}\) is the unique teximax minimal vector on \(\Psi\).  
According to Proposition 2, by redirecting tasks among MECSs in a distributed manner, our task redirection method makes \(\vec{\rho}\) lexicographically smallest given a set of tasks accommodated in a MEC system.