# Constructing separable states in infinite-dimensional systems by operator
matrices

Jinchuan Hou and Jinfei Chai  Department of Mathematics, Taiyuan University of
Technology, Taiyuan 030024, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce a class of states so-called semi-SSPPT (semi super strong
positive partial transposition) states in infinite-dimensional bipartite
systems by the Cholesky decomposition in terms of operator matrices and show
that every semi-SSPPT state is separable. This gives a method of constructing
separable states and generalizes the corresponding results in [Phys. Rev. A
77, 022113(2008); J. Phys. A: Math. Theor. 45 505303 (2012)]. This criterion
is specially convenient to be applied when one of the subsystem is a qubit
system.

Keywords: infinite-dimensional bipartite quantum systems, separable states,
PPT states, Cholesky decomposition

PACS numbers: 03.65.Ud, 03.65.Db, 03.67.Mn

##  1\. Introduction and main result

Entanglement is an important resource in quantum information processing and
quantum computation [1]. However, the detection of entanglement is one of the
most difficult task in this area and much more effort had been paid to on this
research field [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Recall that,
mathematically, a quantum state $\rho$ (positive operator of trace 1) acting
on a separable complex Hilbert space $H=H_{A}\otimes H_{B}$ is called
_separable_ if it can be written as the form

(1)
$\displaystyle\rho=\sum_{i}p_{i}\rho_{i}^{A}\otimes\rho_{i}^{B},\quad\sum_{i}p_{i}=1,~{}p_{i}\geq
0$

or can be approximated in the trace-norm by the states of the above form,
where $\rho_{i}^{A}$ and $\rho_{i}^{B}$ are respectively quantum states of
subsystem A and B [13, 14]. Otherwise, $\rho$ is called inseparable or
_entangled_. A separable state with the form as in Eq.(1) is called _countably
separable_ [15, 16]. If $\dim H_{A}\otimes H_{B}<+\infty$, it is known that
all separable states $\rho$ acting on $H_{A}\otimes H_{B}$ are countably
separable [14]. But, in the infinite-dimensional case, there exists separable
states which are not countably separable [15].

One of the most famous and convenient criteria for detecting entanglement is
the positive partial transpose (PPT) criterion proposed by Peres and Horodecki
[17, 18] which asserts that if a quantum state $\rho$ acting on $H_{A}\otimes
H_{B}$ is separable, then its partial transposes are positive operators, that
is, $\rho^{T_{A/B}}\geq 0$. There are entangled PPT states except for those in
$2\otimes 2$ and $2\otimes 3$ systems [18]. So, it is important to know which
PPT states are separable.

In this paper we present a method to construct a class of PPT states on
infinite-dimensional bipartite systems by an operator-matrix trick and show
that such states are separable.

In a bipartite system A+B described by $H_{A}\otimes H_{B}$ with $\dim
H_{A}\otimes H_{B}=+\infty$, let $\\{|i_{a}\rangle\\}$ and
$\\{|j_{b}\rangle\\}$ be any orthonormal bases of $H_{A}$ and $H_{B}$,
respectively. Denote by $E^{a}_{kl}=|k_{a}\rangle\langle l_{a}|$ and
$E^{b}_{kl}=|k_{b}\rangle\langle l_{b}|$. Then any state $\rho$ acting on
$H_{A}\otimes H_{B}$ can be represented by

(2) $\displaystyle\rho=\sum_{k,l}^{\dim H_{A}}E^{a}_{kl}\otimes
B_{kl}=\sum_{k,l}^{\dim H_{B}}A_{kl}\otimes E^{b}_{kl},$

where $B_{kl}$s are trace class operators on $H_{B}$ and the series converges
in trace-norm [22]; that is,

(13)
$\displaystyle\rho=\left(\begin{array}[]{c|c|c|c|c}B_{11}&B_{12}&B_{13}&\cdots&\cdots\\\
\hline\cr B_{21}&B_{22}&B_{23}&\cdots&\cdots\\\
\hline\cr\vdots&\vdots&\ddots&\vdots&\cdots\\\ \hline\cr
B_{m1}&B_{m2}&B_{m3}&\cdots&\cdots\\\
\hline\cr\vdots&\vdots&\vdots&\cdots&\ddots\end{array}\right)=\left(\begin{array}[]{c|c|c|c|c}A_{11}&A_{12}&A_{13}&\cdots&\cdots\\\
\hline\cr A_{21}&A_{22}&A_{23}&\cdots&\cdots\\\
\hline\cr\vdots&\vdots&\ddots&\vdots&\cdots\\\ \hline\cr
A_{n1}&A_{n2}&A_{n3}&\cdots&\cdots\\\
\hline\cr\vdots&\vdots&\vdots&\cdots&\ddots\end{array}\right)$

under the given bases. Take operator sequences $\\{X_{i}\\}_{i=1}^{\dim
H_{A}}$ and $\\{S_{ij}\ :\ 1\leq i<j\leq{\dim H_{A}}\\}$ on $H_{B}$ so that
the operator matrix (infinite if $\dim H_{A}=\infty$) of the form

(20) $\displaystyle
X=\left(\begin{array}[]{c|c|c|c|c|c}X_{1}&S_{12}X_{1}&S_{13}X_{1}&\cdots&S_{1m}X_{1}&\cdots\\\
\hline\cr 0&X_{2}&S_{23}X_{2}&\cdots&S_{2m}X_{2}&\cdots\\\
\hline\cr\vdots&\vdots&\ddots&\vdots&\vdots&\cdots\\\ \hline\cr
0&0&0&X_{m-1}&S_{m-1,m}X_{m-1}&\cdots\\\ \hline\cr 0&0&0&0&X_{m}&\cdots\\\
\hline\cr\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)$

is a Hilbert-Schmidt operator on $H_{A}\otimes H_{B}$, that is,

${\rm Tr}(X^{\dagger}X)=\sum_{i}{\rm Tr}(X_{i}^{\dagger}X_{i})+\sum_{i<j}{\rm
Tr}(X_{i}^{\dagger}S_{ij}^{\dagger}S_{ij}X_{i})<\infty,$

then

(21) $\displaystyle\rho^{b}_{X}=\frac{1}{{\rm Tr}(X^{\dagger}X)}X^{\dagger}X$

is a bipartite state in $H_{A}\otimes H_{B}$. One can construct $\rho^{a}_{X}$
in the same way. For convenience, we call $\\{X_{i}\\}_{i=1}^{\dim H_{A}}$ and
$\\{S_{ij}\ :\ 1\leq i<j\leq{\dim H_{A}}\\}$ Cholesky operators, and say
Eq.(5) is a Cholesky decomposition $\rho_{X}^{a}$. These terminologies come
from the fact that every block matrix has a Cholesky decomposition of the form
in Eq.(5)

Definition 1. A state $\rho\in{\mathcal{S}}(H_{A}\otimes H_{B})$ is called a
semi-SSPPT state up to part B if it has a Cholesky decomposition as in Eq.(5)
and the associated Cholesky operators $\\{S_{ij}\ :\ 1\leq i<j\leq{\dim
H_{A}}\\}$ satisfying the condition

(22) $\displaystyle[S_{ki},S_{kj}^{\dagger}]=0,\quad k<i\leq j.$

The semi-SSPPT states up to part A are defined similarly. A state is called a
semi-SSPPT state if it is a semi-SSPPT states up to part B or a semi-SSPPT
states up to part A.

It is easily checked that every semi-SSPPT state is PPT. The following is our
may result.

Theorem 1. Let $\rho\in\mathcal{S}(H_{A}\otimes H_{B})$ be a state with $\dim
H_{A}\otimes H_{B}\leq\infty$. If $\rho$ is semi-SSPPT, then $\rho$ is
separable.

The terminology SSPPT (super strong positive partial transpose) comes from
[19] for finite-dimensional systems and [21] for infinite-dimensional systems,
where the additional assumption “every $S_{ij}$ is diagonalizable” is
required. The main result in [19, 21] shows that the SSPPT states are
countably separable. However, though the condition Eq.(6) ensures that every
$S_{ij}$ is a normal operator, we know that there are many normal operators on
infinite-dimensional Hilbert spaces that are not diagonalizable. Thus the
above theorem 1 generalizes the result in [19, 21] greatly.

The proof of theorem 1 will be presented in Appendix B. We point out, our
proof of theorem 1 needs new mathematical tools including introducing a
concept of SOT-separability for bounded positive operators and establishing a
Radon-Nikodym type theorem for spectral measure, which we present in Appendis
A.

##  2\. Corollaries and examples

Theorem 1 provides an easier way of constructing separable states in infinite-
dimensional bipartite systems.

Example 1. Assume $\dim H_{A}=n\leq\infty$ and $\dim H_{B}=m\leq\infty$. Let
$\\{X_{i}\\}_{i=1}^{n}$ and $\\{S_{i}\\}_{i=1}^{n}$ be two sequences of
operators on $H_{B}$ such that $\sum_{i=1}^{n}{\rm
Tr}(X_{i}^{\dagger}X_{i})=\frac{1}{2}$, $S_{i}$ normal and ${\rm
Tr}(X_{i}^{\dagger}S_{i}^{\dagger}S_{i}X_{i})\sum_{i<j}^{n}\frac{1}{2^{2j}}={\rm
Tr}(X_{i}^{\dagger}X_{i})$ for each $i\in\\{1,2,\ldots,n\\}$. Let
$S_{ij}=\frac{1}{2^{j}}S_{i}$ for $j>i$ and let $X$ be the operator matrix as
in Eq.(4). Then,

${\rm Tr}(X^{\dagger}X)=\sum_{i=1}^{n}{\rm
Tr}(X_{i}^{\dagger}X_{i})+\sum_{i=1}^{n}{\rm
Tr}(X_{i}^{\dagger}S_{i}^{\dagger}S_{i}X_{i})\sum_{i<j}^{n}\frac{1}{2^{2j}}=\frac{1}{2}+\frac{1}{2}=1.$

Thus, $\rho=X^{\dagger}X$ is a state in $H_{A}\otimes H_{B}$ and is separable
by theorem 1. In the case $\dim H_{A}=n=\infty$, as
$\sum_{i<j}^{\infty}\frac{1}{2^{2j}}=\frac{1}{3}\frac{1}{2^{2i}}$, one may
choose normal operator $S_{i}$ so that ${\rm
Tr}(X_{i}^{\dagger}S_{i}^{\dagger}S_{i}X_{i})=\frac{9}{2}$ for each $i$ to
ensure that $\sum_{i=1}^{\infty}{\rm
Tr}(X_{i}^{\dagger}S_{i}^{\dagger}S_{i}X_{i})\sum_{i<j}^{n}\frac{1}{2^{2j}}=\frac{1}{2}$.

The following are some corollaries of theorem 1 which generalize the
corresponding results in [21] from finite-dimensional systems to infinite-
dimensional systems, and also illustrates the use of theorem 1 to detect the
separability of a state in the case when $\min\\{\dim H_{A},\dim H_{B}\\}=2$.
Note that every trace-class operator acting on an infinite-dimensional Hilbert
space can not be invertible; so the corollary in [21] is not applicable to
infinite-dimensional case.

Assume that $\dim H_{A}=2$ (or $\dim H_{B}=2$) and
$\rho\in\mathcal{S}(H_{A}\otimes H_{B})$. Then $\rho$ can be written in

(27) $\displaystyle\rho=\left(\begin{array}[]{cc}\rho_{11}&\rho_{12}\\\
\rho_{21}&\rho_{22}\end{array}\right)\quad{\rm(or}\
{\rho}=\left(\begin{array}[]{cc}\tilde{\rho}_{11}&\tilde{\rho}_{12}\\\
\tilde{\rho}_{21}&\tilde{\rho}_{22}\end{array}\right))$

up to part B (or, up to part A).

Corollary 1. Let $\rho$ be a state as in Eq.(7). If there is a Cholesky
decomposition $\rho=X^{\dagger}X$ up to part B/A with
$X=\left(\begin{array}[]{cc}X_{1}&S_{12}X_{1}\\\ 0&X_{2}\end{array}\right)$
such that $X_{1}$ has dense range and $\rho^{T_{A/B}}=Y^{\dagger}Y$ with
$Y=\left(\begin{array}[]{cc}X_{1}&S_{12}^{\dagger}X_{1}\\\
0&X_{2}\end{array}\right)$, then $\rho$ is separable.

Proof. Since
$(Y^{\dagger}Y)^{T_{A/B}}=(\rho^{T_{A/B}})^{T_{A/B}}=\rho=X^{\dagger}X$, one
gets
$X_{1}^{\dagger}S_{12}S_{12}^{\dagger}X_{1}=X_{1}^{\dagger}S_{12}^{\dagger}S_{12}X_{1}$,
which entails that $S_{12}S_{12}^{\dagger}=S_{12}^{\dagger}S_{12}$ as the
range of $X_{1}$ is dense. So, $\rho$ is semi-SSPPT and thus, by Theorem 1, is
separable. $\square$

Corollary 2. Let $\rho$ be a state as in Eq.(7). Then any one of the following
conditions implies that $\rho$ is separable.

(1) $\rho_{11}\geq\rho_{22}$ (or $\tilde{\rho}_{11}\geq\tilde{\rho}_{22}$).

(2) $\rho_{22}\geq\rho_{11}$ (or $\tilde{\rho}_{22}\geq\tilde{\rho}_{11}$).

Proof. Assume that $\rho$ satisfies the condition $\rho_{11}\geq\rho_{22}$. In
this case, $\dim H_{A}=2$ and
$\rho=\left(\begin{array}[]{cc}\rho_{11}&\rho_{12}\\\
\rho_{21}&\rho_{22}\end{array}\right)$ with $\rho_{ij}\in\mathcal{T}(H_{B})$
and $\rho_{22}\leq\rho_{11}$. We shall show that $\rho$ is semi-SSPPT and
hence is separable by Theorem 1.

Since $\rho\geq 0$ and $\rho_{22}\leq\rho_{11}$, there are contractive
operators $T$, $S$ on $H_{B}$ with $\ker T\cap\ker S\cap\ker
S^{\dagger}\supseteq\ker\rho_{11}$ such that
$\rho_{12}=\sqrt{\rho_{11}}T\sqrt{\rho_{22}}$ and
$\sqrt{\rho_{22}}=\sqrt{\rho_{11}}S=S^{\dagger}\sqrt{\rho_{11}}$ [23, Theorem
1.1], here, $\ker L$ denotes the null space of the operator $L$.

Let $S_{12}=TS^{\dagger}$ . Then we have
$\rho_{12}=\sqrt{\rho_{11}}S_{12}\sqrt{\rho_{11}}$. Note that

$\displaystyle\sqrt{\rho_{11}}S_{12}^{\dagger}S_{12}\sqrt{\rho_{11}}=\sqrt{\rho_{22}}T^{\dagger}T\sqrt{\rho_{22}}\leq\rho_{22}.$

Let

$\displaystyle
X_{2}=[\rho_{22}-\sqrt{\rho_{11}}S_{12}^{\dagger}S_{12}\sqrt{\rho_{11}}]^{\frac{1}{2}},$
$\displaystyle
X=\left(\begin{array}[]{cc}\sqrt{\rho_{11}}&S_{12}\sqrt{\rho_{11}}\\\
0&X_{2}\end{array}\right)\quad{\rm and}\quad
Y=\left(\begin{array}[]{cc}\sqrt{\rho_{11}}&S_{12}^{\dagger}\sqrt{\rho_{11}}\\\
0&X_{2}\end{array}\right).$

Then

$\displaystyle\rho=X^{\dagger}X=\left(\begin{array}[]{cc}{\rho_{11}}&\sqrt{\rho_{11}}S_{12}\sqrt{\rho_{11}}\\\
\sqrt{\rho_{11}}S_{12}^{\dagger}\sqrt{\rho_{11}}&\sqrt{\rho_{11}}S_{12}^{\dagger}S_{12}\sqrt{\rho_{11}}+X_{2}^{\dagger}X_{2}\end{array}\right)$

and

$\displaystyle\rho^{T_{A}}=$ $\displaystyle
Y^{\dagger}Y=\left(\begin{array}[]{cc}{\rho_{11}}&\sqrt{\rho_{11}}S_{12}^{\dagger}\sqrt{\rho_{11}}\\\
\sqrt{\rho_{11}}S_{12}\sqrt{\rho_{11}}&\sqrt{\rho_{11}}S_{12}S_{12}^{\dagger}\sqrt{\rho_{11}}+X_{2}^{\dagger}X_{2}\end{array}\right).$

Since $[\rho^{T_{A}}]^{T_{A}}=\rho$, we get

$\displaystyle\sqrt{\rho_{11}}S_{12}S_{12}^{\dagger}\sqrt{\rho_{11}}=\sqrt{\rho_{11}}S_{12}^{\dagger}S_{12}\sqrt{\rho_{11}}.$

On the other hand, $\ker T\cap\ker S\cap\ker
S^{\dagger}\supseteq\ker\rho_{11}$ ensures that $\ker
S_{12}\supseteq\ker\sqrt{\rho_{11}}$. This entails that
$S_{12}S_{12}^{\dagger}=S_{12}^{\dagger}S_{12}$, that is, $\rho$ is semi-
SSPPT.

Other cases con be dealt with similarly. $\square$

We give an example to illustrate how to apply Corollary 2.

Example 2. Assume $\dim H_{A}=2$ and $\dim H_{B}\leq\infty$. For any operators
$\rho_{11},D$ and $T$ acting on $H_{B}$ with $\rho_{11}$ a positive trace-
class operator, $\|D\|\leq 1$ and $\|T\|\leq 1$. Obviously, by the corollary 2
the state $\rho\in S(H_{A}\otimes H_{B})$ constructed by

$\rho=\frac{1}{{\rm
Tr}(\rho_{11}+\sqrt{\rho_{11}}DD^{\dagger}\sqrt{\rho_{11}})}\left(\begin{array}[]{cc}\rho_{11}&\sqrt{\rho_{11}}T[\sqrt{\rho_{11}}DD^{\dagger}\sqrt{\rho_{11}}]^{\frac{1}{2}}\\\
{[\sqrt{\rho_{11}}DD^{\dagger}\sqrt{\rho_{11}}]^{\frac{1}{2}}T^{\dagger}\sqrt{\rho_{11}}}&\sqrt{\rho_{11}}DD^{\dagger}\sqrt{\rho_{11}}\end{array}\right)$

is separable since

$\displaystyle\rho_{11}\geq\sqrt{\rho_{11}}DD^{\dagger}\sqrt{\rho_{11}}=\rho_{22}.$

##  3\. Conclusions

In terms of the Cholesky decomposition and local commutativity, we introduce a
notion of semi strongly super positive partial transpose (semi-SSPPT) states
and establish a criterion of separability: if a quantum state in an infinite-
dimensional bipartite system is semi-SSPPT, then it is separable. This
criterion generalizes the corresponding results in [19, 21] and gives a way of
constructing separable states by operator matrices. This criterion is
specially convenient to be applied when one of the subsystem is a qubit
system. To prove this criterion, we establish a Radon-Nikodym type theorem for
the spectral measure. However, our Radon-Nikodym type theorem is stated in
term of unbounded operators and the strong operator topology (SOT)
convergence. This forces us to introduce a notion of SOT-separability for
positive operators acting on tensor product of two Hilbert spaces, and show
that a state is separable if and only if it is SOT-separable. These results
together enable us to give a proof of the main criterion. Our discussion also
reveals that introduce and study separability for positive operators acting on
tensor product of Hilbert spaces are helpful for solving some problems raised
in quantum information theory.

Acknowledgments This work is partially supported by Natural Science Foundation
of China (11671294).

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##  Appendix A:
SOT-separability for positive operators and Radon-Nikodym type theorem for
spectral measure

To prove theorem 1, we need generalize the concept of separability from states
to bounded positive operators and Radon-Nikodym theorem from vector mesasures
in Hilbert space to the spectral measures.

Denote by ${\mathcal{B}}(H)$ and ${\mathcal{B}}_{+}(H)$ the set of all bounded
linear operators and the set of all positive bounded operators acting on a
complex Hilbert space $H$, respectively.

Definition A.1. A positive operator $T\in{\mathcal{B}}_{+}(H_{A}\otimes
H_{B})$ is called SOT-separable if there exist positive operators
$\\{A_{k}\\}\subset{\mathcal{B}}_{+}(H_{A})$ and
$\\{B_{k}\\}\subset{\mathcal{B}}_{+}(H_{B})$ such that

$None$ $T=\sum_{k}A_{k}\otimes B_{k}$

or if $T$ is the limit of the operators of the form as in Eq.(8) under the
strong operator topology (briefly, SOT). Otherwise, $T$ is said to be SOT-
inseparable.

We remark that, if the sum in Eq.(8) is a series, we mean that the series is
convergent under SOT. It is obvious that $0$ is SOT-separable and, if $\dim
H_{A}\otimes H_{B}<\infty$, then a nonzero positive operator $T$ is SOT-
separable if and only if $\frac{1}{{\rm Tr}(T)}T$ is a separable quantum
state.

Denote by ${\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ the set of all SOT-
separable operators, which is a SOT-closed convex cone in
${\mathcal{B}}(H_{A}\otimes H_{B})$.

The following is a SOT-separability criterion, which is similar to the
entanglement witness criterion for states.

Proposition A.1. A positive operator $T$ is SOT-inseparable if and only if
there is a self-adjoint operator $W\in{\mathcal{B}}(H_{A}\otimes H_{B})$ of
finite rank such that

(1) ${\rm Tr}(W(A\otimes B))\geq 0$ holds for any
$A\in{\mathcal{B}}_{+}(H_{A})$ and $B\in{\mathcal{B}}_{+}(H_{B})$;

(2) ${\rm Tr}(WT)<0$.

Proof. Since ${\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ is a SOT-closed
convex subset of ${\mathcal{B}}(H_{A}\otimes H_{B})$, by Hahn-Banach theorem,
$T\not\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ if and only if there
exists a SOT-continuous linear functional $\phi$ on
${\mathcal{B}}(H_{A}\otimes H_{B})$ and a real number $c$ such that ${\rm
Re}(\phi(S))\geq c$ for all $S\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$
but ${\rm Re}(\phi(WT))<c$. As $\phi$ is SOT-continuous, there are vectors
$x_{1},\ldots,x_{r};y_{1},\ldots,y_{r}\in H_{A}\otimes H_{B}$ with $r<\infty$
such that $\phi(X)=\sum_{i=1}^{r}\langle y_{i}|X|x_{i}\rangle$ holds for all
$X\in{\mathcal{B}}(H_{A}\otimes H_{B})$. Let
$E=\sum_{i=1}^{r}|x_{i}\rangle\langle y_{i}|$. Then
$E\in{\mathcal{B}}(H_{A}\otimes H_{B})$ is a finite-rank operator which
satisfies $\phi(X)={\rm Tr}(EX)$ for all $X$ (Ref., for example, [24]). If $X$
is self-adjoint, that is, if $X^{\dagger}=X$, then $\phi(X)^{*}={\rm
Tr}(EX)^{*}={\rm Tr}((EX)^{\dagger})={\rm Tr}(E^{\dagger}X)$ and hence ${\rm
Re}(\phi(X))={\rm Tr}(WX)$, where $W={\rm Re}(E)=\frac{1}{2}(E+E^{\dagger})$.
It follows that ${\rm Tr}(WS)={\rm Re}(\phi(S))\geq c$ for all
$S\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ and ${\rm Tr}(WT)={\rm
Re}(\phi(T))<c$. Note that $0\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$.
So we must have $c\leq 0$. We assert that ${\rm Tr}(WS)\geq 0$ holds for any
$S\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$. If, on the contrary, there
is $S\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ so that ${\rm
Tr}(WS)=a<0$. As ${\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ is a convex
cone, $tS\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ for any $t>0$. Thus
we have $ta={\rm Tr}(W(tS))\geq c$ for all $t>0$, which is a contradiction
since $ta\to-\infty$ when $t\to\infty$. Therefore, we have found a self-
adjoint operator $W$ of finite rank such that (1) and (2) hold.

Conversely, if there is some self-adjoint operator $W$ of finite rank such
that (1) and (2) hold, then ${\rm Tr}(WS)\geq 0$ holds for all
$S\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$ since $\phi:X\mapsto{\rm
Tr}(WX)$ is a SOT-continuous linear functional. Thus $\phi$ separates strictly
$T$ and ${\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$. So
$T\not\in{\mathcal{S}}_{\rm SOT}(H_{A}\otimes H_{B})$. $\Box$

Next we discuss the relationship between separability and SOT-separability for
a quantum state. Notice that, generally speaking, though a sequence
$\\{T_{n}\\}$ of positive trace-class operators converges to a state $\rho$
under SOT, one can not assert that $\\{T_{n}\\}$ converges to $\rho$ under the
trace-norm. For instance, let $\\{a_{n}\\}_{n=1}^{\infty}$ be a sequence of
positive numbers so that $\sum_{n=1}^{\infty}a_{n}=a<\infty$. For any $n$, let
$T_{n}={\rm diag}(t_{1},t_{2},\ldots,t_{k}\ldots)$ with $t_{k}=0$ if $k\leq n$
and $t_{n+m}=a_{m}$. Then $\\{T_{n}\\}_{n=1}^{\infty}$ is a sequence of
positive trace-class operators and $T_{n}\to 0$ in SOT. However, ${\rm
Tr}(T_{n})=a$ which does not converge to $0$. In addition, for a state $\rho$,
let $T^{\prime}_{n}=\rho+T_{n}$. Then $T_{n}^{\prime}\to\rho$ under SOT but
${\rm Tr}(T^{\prime}_{n})=1+a$ does not converge to ${\rm Tr}(\rho)=1$. Hence,
$\\{T^{\prime}_{n}\\}$ does not converge to $\rho$ in trace-norm. This
suggests that a SOT-separable state may not be separable. However, the
following result reveals surprisedly that this is not the case.

Proposition A.2. Let $\rho\in{\mathcal{S}}(H_{A}\otimes H_{B})$ be a state.
Then $\rho$ is separable if and only if $\rho$ is SOT-separable.

Proof. Clearly, $\rho$ is separable implies that $\rho$ is SOT-separable by
the definitions as the convergence in trace-class norm implies the convergence
in SOT.

Conversely, assume that $\rho$ is inseparable (i.e., entangled). We have to
show that $\rho$ is also SOT-inseparable. Let $\\{|i_{a}\rangle\\}$ and
$\\{|j_{b}\rangle\\}$ be arbitrarily given orthonormal bases for $H_{A}$ and
$H_{B}$, respectively. Let $P_{k}$ and $Q_{k}$ be finite-rank projections on
$H_{A}^{(k)}$ and $H_{B}^{(k)}$, the span of $\\{|i_{a}\rangle\\}_{i=1}^{k}$
and the span of $\\{|j_{b}\rangle\\}_{j=1}^{k}$, respectively. If $k\geq\dim
H_{A}$ (or $k\geq\dim H_{B}$), let $P_{k}=I_{A}$ (or $Q_{k}=I_{B}$) with
$I_{A}$ the identity operator on $H_{A}$. Let

$\rho_{k}=\frac{1}{{\rm Tr}((P_{k}\otimes Q_{k})\rho(P_{k}\otimes
Q_{k}))}(P_{k}\otimes Q_{k})\rho(P_{k}\otimes Q_{k}).$

Obviously, $\rho=\|\cdot\|_{\rm Tr}$-$\lim_{k\to\infty}\rho_{k}$. As $\rho$ is
inseparable, there exists infinitely many $k$ so that $\rho_{k}$ is
inseparable (otherwise, $\rho$ should be separable). Take a such $k$. Then
$\rho_{k}$ can be regarded as an inseparable state in the finite-dimensional
system $H_{A}^{(k)}\otimes H_{B}^{(k)}$. Thus, by the entanglement witness
criterion, there is a self-adjoint operator $W_{k}$ on $H_{A}^{(k)}\otimes
H_{B}^{(k)}$ such that ${\rm Tr}(W_{k}\rho_{k})<0$. Let $W=(P_{k}\otimes
Q_{k})W_{k}(P_{k}\otimes Q_{k})$. $W$ is a self-adjoint operator of rank $\leq
k^{2}<\infty$ and is an entanglement witness for $\rho$ because for any pure
states $P_{A}\in{\mathcal{S}}(H_{A})$ and $Q_{B}\in{\mathcal{S}}(H_{B})$,

${\rm Tr}(W(P_{A}\otimes Q_{B}))={\rm
Tr}(W_{k}((P_{k}P_{A}P_{k})\otimes(Q_{k}Q_{B}Q_{k})))\geq 0$

and

${\rm Tr}(W\rho)={\rm Tr}((P_{k}\otimes Q_{k})W_{k}(P_{k}\otimes
Q_{k})\rho)={\rm Tr}(W_{k}\rho_{k})<0.$

For any $A\otimes B\in{\mathcal{B}}_{+}(H_{A}\otimes H_{B})$, $(P_{k}\otimes
Q_{k})(A\otimes B)(P_{k}\otimes Q_{k})=(P_{k}AP_{k})\otimes(Q_{k}BQ_{k})$ is
either zero or a positive multiple of a finite rank separable state. So, we
still have

${\rm Tr}(W(A\otimes B))\geq 0.$

This implies by Proposition A.1 that $\rho$ is SOT-inseparable, as desired.
$\Box$.

Now, let us turn to the question of establishing Radon-Nikodym type theorem
for spectral measure. Let $\Omega$ be a nonempty set, $\mathcal{B}$ be a
$\sigma$-algebra of subsets of $\Omega$, $H$ be a Hilbert space. Recall that a
spectral measure for $(\Omega,{\mathcal{B}},H)$ is an operator-valued function
$E:{\mathcal{B}}\to{\mathcal{B}}(H)$ such that

(i) for each $\Delta$ in $\mathcal{B}$, $E(\Delta)$ is a projection;

(ii) $E(\emptyset)=0$ and $E(\Omega)=I$;

(iii) for $\Delta_{1},\Delta_{2}\in{\mathcal{B}}$,
$E(\Delta_{1}\cap\Delta_{2})=E(\Delta_{1})E(\Delta_{2})$.

(iv) if $\\{\Delta_{i}\\}\subset{\mathcal{B}}$ are pairwise disjoint sets,
then $E(\cup_{i}\Delta_{i})=\sum_{i}E(\Delta_{i})$, here the sum converges in
SOT (Ref. [24]).

Proposition A.3. (The Radon-Nikodym type theorem for spectral measure) Let $H$
be a complex Hilbert space, $\Omega$ be a nonempty set, $\mathcal{B}$ be a
$\sigma$-algebra of subsets of $\Omega$. Assume that $E$ is a spectral measure
for $(\Omega,\mathcal{B},H)$ and $\mu$ is a positive measure on
$(\Omega,{\mathcal{B}})$. If $E\ll\mu$, that is, if $\mu(\Delta)=0\Rightarrow
E(\Delta)=0$, then there is an operator-valued function
$D:\Omega\to{\mathcal{B}}(H)$ such that $\langle x|D(\omega)|x\rangle\geq 0$
a.e. $\mu$ and

$E(\Delta)x={\rm(B)}\int_{\Delta}D(\omega)x{\rm d}\mu_{\omega}$

holds for every $x\in H$ and $\Delta\in{\mathcal{B}}$, where
${\rm(B)}\int_{\Delta}$ means the Bochner integral.

We remark that $D(\omega)$ may take a unbounded operator.

Proof. Recall that a Banach space $X$ is said to have the Radon-Nikodym
Property (RNP) if for any finite positive measure space
$(\Omega,{\mathcal{F}},\mu)$ and vector-valued measure $F:{\mathcal{F}}\to X$,
if $F\ll\mu$, then there exists a Bochner integrable vector-valued function
$f:\Omega\to X$ such that $F(\Delta)={\rm(B)}\int_{\Delta}f(\omega){\rm
d}\mu_{\omega}$ holds for any $\Delta\in{\mathcal{F}}$. Not every Banach space
has RNP. However it is well-known that every Hilbert space has RNP (ref. [25,
26]).

Now let $(\Omega,{\mathcal{B}},\mu)$ be a finite positive measure space and
$(\Omega,{\mathcal{B}},H,E)$ be a spectral measure space so that $E\ll\mu$. We
remark here that we can not use the result in [26] because the spectral
measure is not $\sigma$-bounded. For any vector $x\in H$, it is clear that
$F_{x}:{\mathcal{B}}\to H$ defined by $F_{x}(\Delta)=E(\Delta)x$ is a
$H$-valued measure satisfying $F_{x}\ll\mu$ and $\sigma$-boundedness. As $H$
has RNP, there is a Bochner integrable vector-valued function $D_{x}:\Omega\to
H$ such that $E(\Delta)x=F_{x}(\Delta)={\rm(B)}\int_{\Delta}D_{x}(\omega){\rm
d}\mu_{\omega}$ holds for all $\Delta\in{\mathcal{B}}$. Note that, $D_{\alpha
x+y}(\omega)=\alpha D_{x}(\omega)+D_{y}(\omega)$ a.e. $\mu$. Hence there
exists an operator-valued function $D$ defined on $\Omega$ so that
$D(\omega)x=D_{x}(\omega)$ a.e. $\mu$ for each $x\in H$. And then
${\rm(B)}\int_{\Delta}D(\omega)x{\rm
d}\mu_{\omega}={\rm(B)}\int_{\Delta}D_{x}(\omega){\rm d}\mu_{\omega}$ for any
$x\in H$. Since, $\int_{\Delta}\langle x|D(\omega)|x\rangle{\rm
d}\mu_{\omega}=\langle x|E(\Delta)|x\rangle\geq 0$ for any Borel set $\Delta$,
one sees that $\langle x|D(\omega)|x\rangle\geq 0$ a.e. $\mu$ for each $x\in
H$. So, almost all $D(\omega)$ are (may unbounded) operators with domain $H$
satisfying $\langle x|D(\omega)|x\rangle\geq 0$ and

$None$ $E(\Delta)x={\rm(B)}\int_{\Delta}D(\omega)x{\rm d}\mu_{\omega}$

holds for all $x\in H$ and $\Delta\in\mathcal{B}$. $\Box$

Some times we denote the relation in Eq.(A.2) by

$None$ $E(\Delta)={\rm(SOT)}\int_{\Delta}D(\omega){\rm d}\mu_{\omega}$

holds for any $\Delta\in\mathcal{B}$.

##  Appendix B: Proof of main result

Now we are at a position to give a proof of the main result theorem 1.

Proof of Theorem 1. Assume that $\rho\in{\mathcal{S}}(H_{A}\otimes H_{B})$ is
a semi-SSPPT state. We have to show that $\rho$ is separable. We only need to
check the case that $\rho$ is semi-SSPPT up to part B since the proof for the
case of semi-SSPPT up to A is similar.

As $\rho$ is a semi-SSPPT state up to part B, we may write
$\rho=X^{\dagger}X$, where $X$ upper triangular operator matrices of the form
mentioned in Eq.(4) with respect to an orthonormal basis $\\{|i_{a}\rangle\\}$
of $H_{A}$. Let $C_{k}$ be the operator matrix with the same size as that of
$X$, which is induced from $X$ by replacing all entries by zero except for the
$k$th row of $X$, i.e.,

$\displaystyle
C_{k}=\left(\begin{array}[]{c|c|c|c|c|c|c|c|c}0&\cdots&0&0&0&0&\cdots&0&\cdots\\\
\hline\cr\vdots&\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&0&\cdots\\\
\hline\cr 0&\cdots&0&0&0&0&\cdots&0&\cdots\\\ \hline\cr
0&\cdots&0&X_{k}&S_{k,k+1}X_{k}&S_{k,k+2}X_{k}&\cdots&S_{km}X_{k}&\cdots\\\
\hline\cr 0&\cdots&0&0&0&0&\cdots&0&\cdots\\\
\hline\cr\vdots&\cdots&\vdots&\vdots&\vdots&\vdots&\cdots&\vdots&\ddots\end{array}\right),$

$k=1,2,\dots$. Then $C_{k}$ is a Hilbert-Schmidt operator and

(32) $\displaystyle\rho=\sum_{k}C_{k}^{\dagger}C_{k},$

Here the series converges in the trace-norm. If $C_{k}\not=0,$ write
$C_{k}^{\dagger}C_{k}=p_{k}\rho_{k}$ where $p_{k}={\rm
Tr}(C_{k}^{\dagger}C_{k})$. We will show that $\rho_{k}$ is separable for any
$k$ whenever $C_{k}\not=0$, and then, $\rho=\sum_{k}p_{k}\rho_{k}$ is
separable, too.

Consider the case when $k=1$. We have

$None$
$p_{1}\rho_{1}=(X_{1}^{\dagger}S_{1i}^{\dagger}S_{1j}X_{1})=\sum_{i,j}(|i_{a}\rangle\langle
j_{a}|)\otimes(X_{1}^{\dagger}S_{1i}^{\dagger}S_{1j}X_{1})$

with $\quad S_{11}=I_{B}$. Since $\rho$ is semi-SSPPT up to part B,
$\\{S_{1i}\\}$ is a commutative set of normal operators. Then there exists a
normal operator $N_{1}\in{\mathcal{B}}(H_{B})$ and bounded Borel functions
$\\{f_{1i}\\}$ such that $S_{1i}=f_{1i}(N_{1})$. Let
$N_{1}=\int_{\sigma(N_{1})}\omega{\rm d}E_{\omega}$ be the spectral
decomposition, where $(\sigma(N_{1}),{\mathcal{B}},H,E)$ is the spectral
measure of $N_{1}$, $\mathcal{B}$ is the $\sigma$-algebra of all Borel subsets
of $\sigma(N_{1})$, the spectrum of $N_{1}$. Thus, we have
$S_{1i}=\int_{\sigma(N_{1})}f_{1i}(\omega){\rm d}E_{\omega}$. Because $H_{B}$
is separable, there exists a probability measure, that is, the scalar spectral
measure, $(\sigma(N_{1}),{\mathcal{B}},\mu)$ so that, for any
$\Delta\in{\mathcal{B}}$, $E(\Delta)=0$ if and only if $\mu(\Delta)=0$ (Ref.
[24]). Then, by Proposition A.3, there exists an operator-valued function $D$
such that $\langle x|D(\omega)|x\rangle\geq 0$ a.e. $\mu$ for each $x\in H$
and

$\displaystyle E(\Delta)={\rm(SOT)}\int_{\Delta}D(\omega){\rm d}\mu_{\omega}$

holds for all $\Delta\in{\mathcal{B}}$. By Eq.(B.1) one gets, for each product
vector $x_{A}\otimes x_{B}\in H_{A}\otimes H_{B}$,

$\begin{array}[]{rl}&\rho_{1}(x_{A}\otimes x_{B})\\\
=&p_{1}^{-1}\sum_{i,j}(|i_{a}\rangle\langle
j_{a}|)\otimes(X_{1}^{\dagger}\int_{\sigma(N_{1})}f_{1i}(\omega)^{*}f_{1j}(\omega){\rm
d}E_{\omega}X_{1})(x_{A}\otimes x_{B})\\\
=&p_{1}^{-1}\sum_{i,j}(|i_{a}\rangle\langle
j_{a}|)x_{A}\otimes({\rm(B)}\int_{\sigma(N_{1})}f_{1i}(\omega)^{*}f_{1j}(\omega)X_{1}^{\dagger}D(\omega)X_{1}x_{B}{\rm
d}\mu_{\omega})\\\
=&\sum_{i,j}\\{{\rm(B)}\int_{\sigma(N_{1})}[p_{1}^{-1}f_{1i}(\omega)^{*}f_{1j}(\omega)|i_{a}\rangle\langle
j_{a}\rangle]x_{A}\otimes[X_{1}^{\dagger}D(\omega)X_{1}]x_{B}{\rm
d}\mu_{\omega}\\}.\end{array}$

Take an orthonormal basis $\\{|j_{b}\rangle\\}$ of $H_{B}$. For any $n$, let
$P_{n}$ be the $n$-rank projection onto span$\\{|i_{a}\rangle\\}_{i=1}^{n}$
and $Q_{n}$ be the $n$-rank projection onto
span$\\{|j_{b}\rangle\\}_{j=1}^{n}$. Then

$\begin{array}[]{rl}&(P_{n}\otimes Q_{n})\rho_{1}(P_{n}\otimes
Q_{n})(x_{A}\otimes x_{B})\\\
=&{\rm(B)}\int_{\sigma(N_{1})}[p_{1}^{-1}\sum_{i,j=1}^{n}f_{1i}(\omega)^{*}f_{1j}(\omega)|i_{a}\rangle\langle
j_{a}\rangle]\otimes[Q_{n}X_{1}^{\dagger}D(\omega)X_{1}Q_{n}]{\rm
d}\mu_{\omega}(x_{A}\otimes x_{B})\end{array}$

holds for any $x_{A}\otimes x_{B}\in H_{A}\otimes H_{B}$, which entails that

$\begin{array}[]{rl}&(P_{n}\otimes Q_{n})\rho_{1}(P_{n}\otimes Q_{n})\\\
=&{\rm(B)}\int_{\sigma(N_{1})}[p_{1}^{-1}\sum_{i,j=1}^{n}f_{1i}(\omega)^{*}f_{1j}(\omega)|i_{a}\rangle\langle
j_{a}\rangle]\otimes[Q_{n}X_{1}^{\dagger}D(\omega)X_{1}Q_{n}]{\rm
d}\mu_{\omega}.\end{array}$

Let
$A_{n}(\omega)=\sum_{i,j=1}^{n}p_{1}^{-1}f_{1i}(\omega)^{*}f_{1j}(\omega)|i_{a}\rangle\langle
j_{a}\rangle]$ and $B_{n}(\omega)=Q_{n}X_{1}^{\dagger}D(\omega)X_{1}Q_{n}$. It
is easily seen that $A_{n}(\omega)$ is a rank one positive operator on $H_{A}$
for each $\omega\in\sigma(N_{1})$. As $B_{n}(\omega)$ is bounded and satisfies
$\langle x_{B}|B_{n}(\omega)|x_{B}\rangle\geq 0$ for any $x_{B}\in H_{n}$,
$B_{n}(\omega)$ must be a positive operator. Therefore,

$\begin{array}[]{rl}\sigma_{n}=&\frac{1}{{\rm Tr}((P_{n}\otimes
Q_{n})\rho_{1}(P_{n}\otimes Q_{n}))}(P_{n}\otimes Q_{n})\rho_{1}(P_{n}\otimes
Q_{n})\\\ =&\frac{1}{{\rm Tr}((P_{n}\otimes Q_{n})\rho_{1}(P_{n}\otimes
Q_{n}))}{\rm(B)}\int_{\sigma(N_{1})}A_{n}(\omega)\otimes B_{n}(\omega)\ {\rm
d}\mu_{\omega}\end{array}$

is a separable state. Since $\rho_{1}=\mbox{\rm
SOT-}\lim_{n\to\infty}(P_{n}\otimes Q_{n})\rho_{1}(P_{n}\otimes Q_{n})$, we
see that $\rho_{1}$ is SOT-separable. Then the proposition A.2 ensures that
$\rho_{1}$ is a separable state, as desired.

Similarly, one can check that $\rho_{k}$ is separable for each $k$, $k\geq 1$.
Hence, we see that $\rho$ is a separable state, as desired. $\Box$


11institutetext: The University of Sydney, Sydney NSW 2000, Australia
11email<EMAIL_ADDRESS>
#  Maximizing Consumer Satisfaction
of IoT Energy Services

Amani Abusafia  Athman Bouguettaya  Abdallah Lakhdari

###### Abstract

We propose a novel Quality of Experience (QoE)-aware framework to crowdsource
IoT energy services efficiently. The proposed framework leverages the
provisioning of energy services as an auxiliary to increase consumers’
satisfaction. A novel QoE model is developed as a metric to assess the
consumers’ satisfaction with the provisioning of energy services. Two novel
composition algorithms, namely, Partial-Based (PB) and Demand-Based (DB)
approaches, are proposed to ensure the highest QoE for consumers. Both
approaches leverage the providers’ flexibility and shareable nature of energy
services to efficiently allocate services and optimize the QoE. A set of
extensive experiments is conducted to evaluate the proposed approaches’
efficiency and effectiveness.

###### Keywords:

Quality of ExperienceIoT ServicesEnergy ServicesEnergy Sharing;
CrowdsourcingIncentiveIoT

##  1 Introduction

Internet of Things (IoT) is a paradigm that enables everyday objects (i.e.,
_things_) to connect to the internet and exchange data. IoT devices, such as
smartphones and wearables, usually have augmented capabilities including
sensing, networking, and processing [1]. Abstracting the capabilities of these
IoT devices using the _service paradigm_ may yield to multitude of novel _IoT
services_ [2]. These IoT services may be exchanged between IoT devices as
crowdsourced IoT services. For example, an IoT device may offer WiFi hotspots
or wireless energy services to charge other IoT devices [2]. These
crowdsourced IoT services present a convenient and cost-effective solutions
[2]. Our focus is on wireless energy sharing services among IoT devices.

_Energy-as-a-Service (EaaS)_ is the abstraction of the wireless delivery of
energy among nearby IoT devices [3][2]. EaaS is an IoT service where energy is
delivered from an energy provider (e.g., a smart shoe or smartphone) to an
energy consumer (e.g., a smartphone) through wireless means. EaaS may be
deployed through the newly developed “Over-the-Air” wireless charging
technologies [4][5]. Several companies, including Xiaomi111mi.com,
Energous222energous.com, and ossia333ossia.com, are currently developing
wireless charging technologies for IoT devices over a distance. For example,
Energous developed a device that can charge up to 3 Watts power within a
5-meter distance.

The crowdsourced EaaS ecosystem is a dynamic environment that consists of
providers and consumers congregating in microcells. A microcell is any
confined area where people may gather (e.g., coffee shops). In this ecosystem,
IoT devices may share energy with nearby IoT devices. A key aspect to
unlocking the full potential of the EaaS ecosystem is to design an end-to-end
Service Oriented Architecture (SOA) to share crowdsourced energy. We identify
three key components of the SOA: energy service provider, energy service
consumer, and super-provider. In this architecture, providers advertise
services, consumers submit requests, and super-provider (i.e., microcell’s
owner) manage the exchange of energy services between providers and consumers.
This paper focuses on managing energy sharing from the super-provider
perspective.

Super-provider typically focus on ensuring that customers keep coming back to
their businesses. Their revenue is usually directly related to foot traffic
[6]. Customer satisfaction is therefore paramount as a strategy to either
maintain or increase the business target revenue [7]. A key objective is to
ensure that customers have the best experience. We propose to use energy
sharing as a key ingredient to provide customers with the best quality of
experience when visiting the business. For example, a case study showed that
“Sacred”, a cafe in London, had a noticeable increase in foot traffic after
installing wireless charging points444air-charge.com.

We define a Quality of Experience (QoE) metric to represent the level of
satisfaction across energy consumers over a period of time in a specific
microcell. Note that QoE is different from Quality of Service (QoS). QoE uses
QoS as a base to express satisfaction of a service over a period of time. QoE
has traditionally been used in domains that assess how users perceive a
service [8][9][10]. Our proposed environment requires the use of a different
type of QoE. In particular, we identify the following three aspects that shape
the new QoE definition: (1) crowdsourced environment resources are usually
limited and cannot fulfill all consumers’ requirements. Hence, assessing
consumers’ satisfaction should consider the limited available resources.
Energy services may be provided partially due to the limited resources and the
shareable nature of energy services, e.g., a single service may be split into
smaller services and provided among multiple consumers. In a limited resource
environment, consumers’ experience with partial services differs from complete
services. (3) Consumers’ satisfaction with energy services will indirectly
impact their experience with the super-provider’s microcell. Therefore,our
research focuses on the  super-provider’s perspective of QoE.

Assessing the QoE from a super-provider’s perspective usually entails
measuring the aggregated satisfaction of consumers over time. Consumer’s
satisfaction is defined as meeting or exceeding a set of expected service
goals [11]. In this context, we define consumer satisfaction as receiving the
requested energy or part of it. We focus on optimizing the QoE by efficiently
provisioning and fulfilling the consumers’ energy requirements.

The limited availability of energy is a key challenge that may hinder the
super-provider from optimizing the consumers’ QoE [2]. For instance, an energy
consumer might not find their requested energy at a certain time in the
microcell, resulting in an unsatisfying experience. In this context, using
traditional resource allocation algorithms may incur uneven energy sharing for
some consumers. Therefore, we propose a QoE-driven service provisioning
framework to satisfy energy consumers in a crowdsourced IoT environment. The
framework requires prior knowledge of providers’ temporal preferences and the
microcell energy demands. The proposed framework leverages the shareable
nature of energy services to split the energy between consumers if the
required energy is more than the available energy[3]. Intuitively, the super-
provider may prefer to offer part of the required services to all consumers
than offering it to some of them. Hence, we propose a heuristic Partial-Based
(PB) approach which splits services among consumers in the case of low energy
availability. Another possible solution is to leverage flexible providers that
offer services on multiple time slots by allocating their services to the most
demanding slots. Intuitively, this may ensure a better distribution of the
available services. Therefore, we additionally propose a heuristic Demand-
Based (DB) approach. The DB approach extends the PB approach by prioritizing
the allocation of services based on the highest demanding time slots. The main
contributions of this paper are:

  * •

A novel Quality of Experience (QoE) model for crowdsourced energy services.

  * •

A framework for QoE-driven composition of IoT energy services.

  * •

An experimental analysis with two implementations of the proposed QoE-driven
energy composition framework.

Figure 1: IoT energy services environment

###  1.1 Motivating Scenario

We describe a scenario in a confined place (i.e., microcell) where people
congregate, e.g., cafes,and restaurants (see Fig.1 (A)). Each microcell may
have several IoT devices acting as energy providers or consumers (see Fig.1
(B)). The super-provider aims to leverage the crowdsourced energy services as
a tool to enhance the consumers’ experience. We assume all local energy
services and requests are submitted and managed at the edge, e.g., a router in
the microcell (see Fig.1(B)). We assume that the super-provider offers
incentives to encourage energy sharing in the form of credits. These would be
used to receive more energy when the providers act as consumers in the future
[2] We assume the super-provider has a prior knowledge of the Microcell Energy
Demand ($\mathcal{MED}$) in the microcell over a period of time (T) (see Fig.2
(A)). The $\mathcal{MED}$ may be estimated based on previous history [12]. The
$\mathcal{MED}$ is represented in terms of the requested energy in each time
slot, e.g., 700 mAh at time slot $t_{1}$. The granularity of the time slots
can also be estimated based on the previous history of the microcell [2].

We also assume that the super-provider has prior knowledge of the providers
preferences in terms of time and energy service attributes. An incentive model
is employed to predict the amount of energy that would be available for
consumption [13]. For instance, provider 1 in Fig.2 (A) wants to offer the
energy service $S1$ with 500 mAh at time $t_{1}$. Another example, provider 2
wants to offer $S2$ at time slots $t_{1}$ , $t_{2}$, or $t_{3}$. We assume the
provider would stay for the full-time slot. We also assume that the provider’s
service amount is fixed and can be split among multiple time slots. For
instance, provider 2 may share part of their service $S2$ on $t_{1}$, e.g.,
300 mAh, and the other part at $t_{2}$ or $t_{3}$. We also assume a single
energy provider may share their spare energy with multiple energy consumers,
within _a specific time interval_. The super-provider uses rewards to
encourage providers to share energy. Rewards may come in the form of stored
credits to providers. A provider receives a reward based on an incentive model
[13].

Figure 2: (A) Microcell energy demand and providers services (B) Greedy energy
provisioning approach

The super-provider will allocate services to time slots to serve as many
consumers as possible to maximize their quality of experience in the
microcell. However, it is challenging to fulfill multiple energy requirements
with limited energy services [2]. For example, in Fig.2 (A), the total energy
demand ($\sum R$) is 3200 mAh, and the total available energy services ($\sum
S$) is 2300 mAh. The available services may fulfill 71.9% of the energy
requests which cannot be fully provisioned with the temporal constrains of
services and requests. Therefore, satisfying all consumers with their under-
provisioned requests is more challenging.

Fig.2 (B) presents the outcome of a greedy FCFS, i.e., first come first
served, allocation strategy for the available energy [14]. In greedy, the time
slots and services will be scheduled based on their start time. For instance,
in Fig.2 (A) even though $S2$ can be offered in $t_{1}$, $t_{2}$ and $t_{3}$,
$S1$ will be allocated to $t_{1}$ because it comes first in terms of time. The
greedy strategy does not leverage the shareable nature of energy services or
the providers’ flexibility which may affect the energy allocation efficiency
and impact the consumers’ experience. Therefore, the greedy strategy may not
be a good fit in this context. For example, in Fig.2 (B), the greedy-based
approach could only fulfill 1300 mAh from the total demand which is equivalent
to 40.6%. Moreover, the total number of consumers ($\sum C$) in Fig.2(A) is 14
and the greedy approach could offer energy to 7 consumers which is equivalent
to 50%. In this context, we consider the size of fulfilled requests and the
number of fulfilled consumers in assessing the quality of experience. In this
example, using the greedy approach resulted in 45.3% of consumers’ QoE.

Allocating the limited available energy with the time constraints of both
services and requests represents critical challenges for efficient and QoE-
aware provisioning of IoT energy services. We propose a framework that will
compose the energy services to maximize the consumers’ experience. Our
framework leverages leverage the providers’ flexibility and shareable nature
of energy services to efficiently allocate services and optimize the QoE.

##  2 Preliminaries

We consider the scenario of energy sharing in a microcell $M$ during a time
interval _T_. _T_ is divided into a set of $\\{t_{1},...,t_{n}\\}$ where
$t_{i}$ is a predefined time period, e.g., one hour. We use the below
definitions to formalize the problem.

Definition 1: Energy-as-a-Service (EaaS). We adopt the definition of $EaaS$ in
[3]. An energy service ($EaaS$) is a tuple of $<E_{id},E_{pid},F,Q>$, where
$E_{id}$ is an energy service ID, $E_{pid}$ is a provider ID, $F$ is the
function of sharing wireless energy, $Q$ is a set of non-functional ($QoS$)
attributes, including:

  * •

$p_{ae}$ is the amount of energy shared by the provider.

  * •

$p_{loc}$ is the location of the energy provider $<x,y>$.

  * •

$p_{t}$ is the set of time intervals $<t_{s},t_{e}>$ a provider may offer
their energy.

Definition 2: Energy Service Request (ER). We adopt the definition of ER in
[13]. An $ER$ is a tuple of $<E_{id},E_{cid},F,QR>$, where $ER_{i}d$ is an
energy request ID, $E_{cid}$ is a consumer ID, $F$ is the function of
receiving energy wirelessly by an IoT device, $QR$ is a set of non-functional
attributes, including:

  * •

$c_{re}$ is the amount of requested energy.

  * •

$c_{loc}$ is the location of the energy consumer $<x,y>$.

  * •

$c_{t}$ is the time interval $<t_{s},t_{e}>$ of requiring energy.

Definition 3: Microcell Energy Demand $\mathcal{MED}$. $\mathcal{MED}$ is the
total amount of requested energy during a time interval $T$ (See Fig.2). $T$
is divided into time slots. We define $\mathcal{MED}$ by aggregating the
amount of required energy per time slot. Therefore, the definition of
$\mathcal{MED}=\\{t_{1},t_{2},...,t_{n}\\}$ where $t$ is a tuple of
$<d,rwd,re,nc,ER>$. Here $d$ is a predefined time in the time interval of the
microcell $T$, e.g., [9:00 AM -10:00 AM]. $rwd$ is the reward of providing the
required energy $re$. We compute $rwd$ using the incentive model proposed by
[13]. We assume that the super-provider will use the microcell history to
compute the energy demand in advance. $nc$ is the number of consumers in the
microcell at time slot $t$. $ER$ is the set of available requests in the
microcell at time slot $t$.

Definition 4: Quality of Experience (QoE). $QoE$ is defined as an objective
function to measure consumers’ satisfaction with energy provisioning in a
microcell $M$ within a predefined time interval $T$. The function definition
is:

$QoE(M)=F(T,\mathcal{ES},\mathcal{MED})$ (1)

where $\mathcal{ES}$ is the set of energy services and $\mathcal{MED}$ is the
microcell energy demand.

###  2.1 Problem Definition

Given a set of $n$ energy services $\mathcal{ES}=\\{EaaS_{1},EaaS_{2},....,$
$EaaS_{n}\\}$ and a set of $m$ energy requests
$\mathcal{ER}=\\{ER_{1},ER_{2},....,$ $ER_{m}\\}$ in a microcell $M$. The
super-provider advertise the microcell energy demand $\mathcal{MED}$. Energy
providers register their services in terms of: (1) the amount of energy
$p_{ae}$ (2) the time slots $t_{i}$ to offer their services. The super-
provider uses the providers preferences to allocate their services to time
slots. The allocation approach aims at fulfilling the maximum number of
requests and thereby maximize the QoE. We formulate the service composition
problem to a time-constrained optimization problem as follows:

  * •

Maximize $QoE(M)=F(T,\mathcal{ES},\mathcal{MED})$,

Subject to:

  * •

$t_{i}.re>0$ for each $t_{i}\in T$,

  * •

$EaaS_{j}.P_{t}\subset t_{i}.d$ for each $EaaS_{j}\in\mathcal{ES}$.

Where $P_{t}$ is the time interval $<t_{s},t_{e}>$ a provider of $EaaS_{j}$
may offer their energy, $t_{i}.d$ is the duration of a time slot $i$ in the
time interval of the microcell $T$, and$t_{i}.re$ is the required energy $re$
at time slot $i$.

The goal of the composition is to efficiently allocate the available energy
services to time slots. The objective function attempts to optimally assign
energy services according to their spatio-temporal features, providers’
preferences and required energy in time slots. The spatial aspect in energy
service focuses on a geographical cell. The temporal aspect focuses on the
times of energy service provisioning.

We use the following assumptions to formulate the problem.

  * •

Providers energy size is fixed during composition.

  * •

Providers are available in all their selected time slots.

  * •

Providers may offer partial services to multiple consumers at the same time.

  * •

Consumers’ time windows do not overlap with time slots.

  * •

Providers and consumers have fixed location during energy sharing.

  * •

The microcell has multiple providers and multiple consumers.

  * •

There is no energy loss in sharing. As the technology matures, we anticipate
that the devices will be able to share more energy, and the energy loss of
sharing will become minimal [2].

  * •

The exact amount of required energy for a microcell is given [15].

  * •

A reward system is used to incentivize providers to offer their service [13].

  * •

A trust framework is used to preserve the privacy of the IoT devices [16].

##  3 Quality of Experience Model

The Quality of Experience (QoE) in a microcell is measured based on the number
of satisfied consumers and the amount of fulfilled requests. Recall, the time
interval of the microcell is divided into time slots. Therefore, QoE for each
time slot $t_{i}$ will be computed using the following attributes:

  * •

Satisfaction Ratio: We define the Satisfaction Ratio ($\mathcal{SR}$) as the
number of consumers who received their requested energy or part of it. We
compute $\mathcal{SR}$ per time slot $t$ as follows:

$\mathcal{SR}=\frac{|\\{ER\in\mathcal{ER}\;|\;ER\;is\;completed\;\&\;c_{t}\in
d\\}|}{|\mathcal{ER}|}$ (2)

Where $\mathcal{ER}$ is the set of all requests in time slot $t$, $|.|$ is the
cardinality of the set, $c_{t}$ is the request time, and $d$ is the time
duration of $t$.

  * •

Fulfillment Ratio: The satisfaction ratio is not enough to measure QoE. For
example, if we have a set of energy requests in mAh $\mathcal{ER}$= {10, 20,
20, 70}, serving the first 3 consumers is not equal to serving the last 3 due
to the different amount of requested energy. Therefore, We define the
Fulfillment Ratio ($\mathcal{FR}$) based on the percentage of fulfillment for
each request. We compute $FR$ per time slot $t$ as follows:

$\mathcal{FR}=\sum_{i=1}^{n}\left(w_{i}\times\frac{Received\\_Energy_{i}}{Requested\\_Energy_{i}}\right)$
(3)

where n is the number of all energy requests in $t$, and $w_{i}$ is the weight
of the request over the total amount of requested energy in $t$.

Quality of Experience: As previously stated, We define the QoE in a microcell
based on the satisfaction ratio $\mathcal{SR}$ and fulfillment ratio
$\mathcal{FR}$ of each time slot $t$. Therefore, we compute the $QoE(M)$ as
the following:

$QoE(M)=\alpha\times\left(\sum_{i=1}^{m}SR_{i}\times\beta_{i}\right)+(1-\alpha)\times\left(\sum_{i=1}^{m}FR_{i}\times\gamma_{i}\right)$
(4)

where m is the number of time slots in the microcell’s time interval $T$.
$\mathcal{SR}_{i}$ is the satisfaction ratio of a time slot computed by Eq.2.
$\beta_{i}$ is the weight of a time slot $t_{i}$ which is its number of
consumers over the total number of consumers in $T$. $\mathcal{FR}_{i}$ is the
ratio of fulfillment of the time slot computed by Eq.3. $\gamma_{i}$ is the
weight of a time slot $t_{i}$ which is its total required energy over the
total amount of required energy in $T$. $\alpha$ is a user-defined weight
between zero and one to define the weight of $\mathcal{SR}_{i}$ and
$\mathcal{FR}_{i}$ in $QoE$.

Figure 3: Quality of experience driven service composition framework

##  4 Quality of Experience Framework

We introduce a quality of experience composition framework for managing energy
services to enhance consumers’ QoE (See Fig.3). The framework is divided into
three phases: (1) Microcell energy advertisement, (2) Composing energy
services, and (3) Quality of experience assessment. In the first phase, the
super-provider will advertise the energy demand of the microcell and receives
providers’ preferences. In the second phase, the super-provider will compose
energy services to maximize the QoE. In the last phase, the super-provider
will assess the QoE for the resulted composition.

###  4.1 Microcell Energy Demand Advertisement

In this phase, the super-provider computes the reward for each time slot based
on the amount of required energy using the incentive model in [13]. Then, the
system will announce the required energy and rewards for the whole microcell
using Definition 3. Energy providers will register based on their preferences
in terms of their energy amount and the time slots they will be available (See
Fig.2).

###  4.2 Energy Services Composition

This phase aims to compose energy services to maximize the QoE. We propose two
heuristic approaches to compose energy services: Partial-Based (PB) and
Demand-Based (DB). The PB composition is inspired by the FCFS resource
allocation algorithm [14]. The PB approach, splits services among consumers if
the required energy is more than the available energy. Intuitively, offering
part of the services will satisfy more consumers than offering it to some of
them. The DB composition is inspired by the priority allocation algorithm
[14]. The DB approach extends the PB approach by prioritizing slots with the
highest demanding to ensure services availability. We discuss each approach
below.

Algorithm 1 Partial-Based Composition of Services

1:$\mathcal{MED}$, $\mathcal{ES},threshold$

2:$energy\\_comp$

3:for $t_{i}$ in $\mathcal{MED}$ do

4: $selectedES=\\{\\}$

5: $demand$ = $t_{i}.re$

6: for $es_{j}$ in $t_{i}.\mathcal{ES}$ do

7: if $demand>$ 0 then

8: $demand$ = $demand$ \- $es_{j}.p_{ae}$

9: $energy\\_comp.add(t_{i},es_{j})$

10: $selectedES.add(es_{j})$

11: if $demand<$ 0 then

12: $es_{j}.p_{ae}$ =$demand$ * -1

13: $demand=0$

14: else

15: Remove_Service($es_{j},\mathcal{MED}$)

16: if $demand$ = 0 then

17: Assign_Energy($t_{i}.\mathcal{ER},selectedES$)

18: else

19: Assign_Partial_Energy($t_{i}.\mathcal{ER},selectedES,nc,threshold$)

20:return $energy\\_comp$

####  4.2.1 Partial-Based Energy Services Composition

The Partial-Based (PB) composition aims at maximizing the QoE by composing
services for each time slot based on the first come first served approach. For
example, if a provider offers their services on two-time slots, the algorithm
will assign the service for the earlier time slot. If the time slot did not
need the service, the service will be assigned to the next time slot.
Moreover, PB chunks services between energy consumers if the available energy
services are not enough to fulfill the total required energy in the time slot.
Intuitively, offering part of the required services to all consumers is more
satisfying than offering it only to some of them.

Algorithm 1 presents the PB service composition. For every time slot $t_{i}$,
the algorithm retrieves the total required amount of energy (Line 3). Then,
for each registered service $es$ in $t$, the algorithm keeps adding services
to the set of selected services until the required energy is fulfilled or all
the available services have been selected (Lines 4 - 13). Note that if a
service was partially needed, then the service available amount will be
updated to be used by other registered time slots (Lines 9 - 11). Moreover, if
a service was fully used by a time slot, then it will be removed from other
registered time slots (Lines 12 - 13). After processing all services, if the
energy demand of the slot is zero, the algorithm assigns the selected services
to requests (Lines 14 - 15). If the energy demand is not fulfilled, the
algorithm distributes the available services among available requests (Line
17). If the service chunks are smaller than the threshold, consumers will be
removed and the service will be shared among the rest. The threshold prevents
dividing services into small neglectable chunks. The composition of the
selected services will be returned in Line 18.

####  4.2.2 Demand-Based Energy Services Composition

The Demand-Based (DB) composition goal is to maximize QoE by giving priority
to time slots with higher energy demand. The intuitive idea of the DB approach
is that high-demanding time slots will require more services. Thus, services
should be assigned to them prior to less demanding time slots which may ensure
a better distribution of the available services. For instance, if a provider
offers their service on two-time slots, the algorithm will assign the service
to the more demanding time slot. If that time slot does not need the service,
the service will be assigned to the next time slot. This indicates that the
order of time slots in composing services matters because if a service is used
in a time slot, it will be removed from others. Removing a service from a time
slot may affect the amount of available energy and thus the number of served
and satisfied consumers. Moreover, DB approach maximizes the QoE by chunking
services between energy consumers if the available services are not enough.

Algorithm 2 Demand-Based Composition of Services

1:$\mathcal{MED}$, $\mathcal{ES},threshold$

2:$energy\\_comp$

3:$\mathcal{SMED}$ = sort($\mathcal{MED},nc:descending,re:descending$)

4:for $t_{i}$ in $\mathcal{SMED}$ do

5: $selectedES=\\{\\}$

6: $demand$ = $t_{i}.re$;

7: $sortedES$ = sort($t_{i}.\mathcal{ES},nt:ascending$)

8: for $es_{j}$ in $sortedES$ do

9: if $demand>$ 0 then

10: $demand$ = $demand$ \- $es_{j}.p_{ae}$

11: $energy\\_comp.add(t_{i},es_{j})$

12: $selectedES.add(es_{j})$

13: if $demand<$ 0 then

14: $es_{j}.p_{ae}$ =$demand$ * -1

15: $demand=0$

16: else

17: Remove_Service($es_{j},\mathcal{SMED}$)

18: if $demand$ = 0 then

19: Assign_Energy($t_{i}.\mathcal{ER},selectedES$)

20: else

21: Assign_Partial_Energy($t_{i}.\mathcal{ER},selectedES,nc,threshold$)

22:return $energy\\_comp$

Algorithm 2 presents the DB service composition. The algorithm starts by
sorting the time slots in a descending order based on the number of consumers
$nc$, then the amount of requested energy $re$ (Line 1). The goal of sorting
is to start composing services for the most demanding time slots. As some
services may be registered in multiple services, using these services for the
most demanding time slots may offer a better experience. Line 4 retrieves the
total required amount of energy for each time slot $t_{i}$. Then, for every
time slot, the registered services will be sorted in ascending order based on
the number of time slots a service was registered in. This sort will allow us
to start with the least connected services. In other words, using such
services may impact less number of time slots than using services that are
registered in many time slots. Then, for each registered service $es$ in $t$,
the algorithm keeps adding services to the set of selected services until the
required energy is fulfilled or all the available services have been selected
(Lines 6 - 15). Similar to the PB approach, if a service was partially needed,
then the service available amount will be updated to be used by other
registered time slots (Lines 11 - 13). Moreover, if a service was fully used
by a time slot, then it will be removed from other registered time slots
(Lines 14 - 15). After processing all services, if the energy demand of the
slot is zero, the algorithm assigns the selected services to requests (Lines
16 - 17). If the energy demand is not fulfilled, the algorithm distributes the
available services among available requests (Line 19). If the service chunks
are smaller than the threshold, consumers will be removed and the service will
be shared among the rest. The threshold prevents dividing services into small
neglectable chunks. Line 20 returns the composition of the selected services.

###  4.3  Assessing Quality of Experience

The super-provider assesses the QoE of each proposed composition in this
phase. The QoE is computed using the model discussed in Section 3. The
assessment of QoE gives an indicator of consumers’ satisfaction in the
microcell.

##  5 Evaluation

We compare the proposed composition approaches, Partial-Based composition
(PB), and Demand-Based Composition (DB), with the resource allocation
algorithms, namely, first come first served allocation (Greedy), and Max-Min
Fair allocation (Max-Min) [14][2]. The Greedy approach is a modified FCFS
algorithm where the time slots and services will be scheduled based on their
start time. The Max-Min is a modified Max-Min Fair allocation where services
that can be offered in multiple time slots will be split among these time
slots using the a Max-Min technique. We evaluate the effectiveness and the
efficiency of each approach.

###  5.1 Dataset Description

We used a real dataset generated from the developed app in [17]. The dataset
consists of energy transfer records between a provider (smartphone) and a
consumer (smartphone). The records attributes are the provider ID, consumer
ID, transaction date, time, energy services’ and requests’ amount, and
transfer duration. We use the energy dataset to generate the QoS parameters
for the energy services and requests. For instance, the amount of a wireless
charging transfer in mAh is used to define the amount of requested/provided
energy. In addition, the energy dataset records of a wireless charging
transfer duration are used to define the end time of each request/service.

We augmented the dataset of the energy sharing to mimic the behavior of the
crowd within microcells by utilizing a dataset published by IBM for a coffee
shop chain with three branches in New York city555https://ibm.co/2O7IvxJ. The
dataset consists of transaction records of customers purchases in each coffee
shop for one month. Each coffee shop consists of, on average, 560
transnational records per day and 16,500 transaction record in total. We use
the IBM dataset to simulate the spatio-temporal features of energy services
and requests. Our experiment uses the consumer ID, transaction date, time,
location, and coffee shop ID from each record in the dataset to define the
spatio-temporal features of energy services and requests, e.g., start and
location of energy service or a request. We ran a total of 7000 experiments
with 6-time slots each time slot was an hour long. In each run, the providers’
temporal provision preferences were registered randomly to [1-3] time slots.
In addition, the number of services and requests varied between 300 to 2000
per run depending on the experiments’ setting. For each run, we used the
proposed approaches to compose energy services. We then measured the QoE for
each composition. Table 1 presents the experiments parameters.

###  5.2 Evaluation of the Composition Framework

We ran six experiments to determine the effectiveness and efficiency of the
proposed approaches. The experiments evaluated the approaches in terms of
their satisfaction rate, fulfillment rate, quality of experience, impact of
thresholds and computation cost. We run the approaches in different settings
by changing the ratio of services to requests in the time interval $T$. We
gradually increased the ratio from 15% to 90%. We repeated the experiment 1000
times at each point and considered the average value for each approach.

Table 1: Experiments Variables

Variables | Value
---|---
Energy dataset for coffee shop 8 in April | 16830
Number of services & requests | [300-2000]/run
Number of time slots | 6
Provided energy | 5 - 100%
Requested energy | 5 - 100%
Time interval | 6 hours
Service registration | [1-3] time slots / service

Figure 4: The average of satisfaction ratio

####  5.2.1 Quality of Experience Evaluation

As previously stated, we compute the QoE based on $\mathcal{SR}$ and
$\mathcal{FR}$ (See Sec.3). In this subsection, we study the impact of each
ratio, then we evaluate the QoE.
The first experiment compares the $\mathcal{SR}$ of the proposed approaches PB
and DB, against Greedy and Max-Min. As previously stated, $\mathcal{SR}$
represents the number of consumers who received energy fully or partially.
Therefore, a high $\mathcal{SR}$ of a composition ensures a higher number of
satisfied consumers and thereby a better QoE. The $\mathcal{SR}$ of a time
slot is computed using Eq.2 and then averaged for the microcell similar to the
first part of Eq.4. Fig.4 presents the average $\mathcal{SR}$ in the microcell
for each approach. The x-axis in Fig.[4-8] represents the ratio of the number
of energy services to requests. In Fig.4, the $\mathcal{SR}$ increases when
the number of available services increases for all the composition approaches.
For instance, when the ratio of services to requests is 80%, all approaches
provide a higher $\mathcal{SR}$ compared to the ratio is 20%. This observation
can be explained by the availability of services to offer energy. The more
services available, the more requests can be fulfilled. The proposed approach
PB performs better than Greedy as it splits the available energy between the
consumers as partial services, unlike the Greedy approach which fulfills a
request fully before serving the next request. For the same reason PB also
performs better than Max-Min. Even though, Max-Min has a better energy
utilization by splitting energy services fairly between time slots (See
Fig.6), a fair distribution of energy does not necessarily result in equally
satisfied consumers as in the time slots. This is due to the different energy
requirements of consumers. In addition, the proposed approach DB gives the
best results as it prioritizes the time slots that have the highest demand in
terms of the number of consumers and amount of required energy. Recall the
order of time slots in composing services is crucial because if a service is
used in a time slot, it will be removed from others. Removing a service from a
time slot may affect the amount of available energy and thus the number of
served consumers. Prioritizing the most demanding time slots allows DB to have
more services to use, and therefore increases $\mathcal{SR}$ by increasing the
number of fulfilled consumers.

The second experiment compares the $\mathcal{FR}$ of each approach. As
previously stated, $\mathcal{FR}$ presents the rate of fulfillment for each
request. Therefore, a high $\mathcal{FR}$ of a composition ensures a higher
level of satisfaction for consumers and thereby a better QoE. The
$\mathcal{FR}$ of a time slot is computed using Eq.3 and then averaged for the
microcell similar to the second part of Eq.4. Fig.6 represents the average
$\mathcal{FR}$ in the microcell for each approach. In Fig.6, the
$\mathcal{FR}$ increases when the number of available services increases for
all the approaches. This observation can be explained by the availability of
services to offer energy. PB performs similar to Greedy in terms of
$\mathcal{FR}$. This is an expected behaviour since both approaches start with
the same time slots and, therefore, have the same set of available services.
The difference between both approaches is in the way they share energy among
consumers, i.e., complete services in Greedy and partial services in PB.
Moreover, Max-Min has a better $\mathcal{FR}$ because it has better energy
utilization (see Fig.6). A higher energy utilization is achieved by splitting
energy services fairly between time slots. DB gives the best results as it
prioritizes the time slots that have the highest demand as discussed in the
previous experiment.

Figure 5: The average of energy utilization

Figure 6: The average of fulfillment ratio

The third experiment compares the QoE using all approaches. As previously
stated, the QoE presents the overall satisfaction of consumers across time.
Therefore, a high QoE of a composition indicates a higher level of
satisfaction for consumers. The QoE is computed using Eq.4. Note that we used
$\alpha=0.5$ to give equal weight for both $\mathcal{SR}$ and $\mathcal{FR}$.
Fig.8 presents the average QoE using each approach. In Fig.8, similar to the
previous experiments, the QoE increases when the availability of services
increase. PB approach performs better than Greedy in terms of QoE due to its
higher $\mathcal{SR}$ as discussed in the first experiment. Additionally, PB
preforms better than Max-Min when the number of energy services is less than
the request. This is because in a limited resources environment PB will
satisfy more consumers (higher $\mathcal{SR}$) by partially fulfilling their
requests. However, when there is enough services, Max-Min will better utilize
the energy to completely fulfill requests (higher $\mathcal{FR}$). Moreover,
the DB approach gives the best results due to its higher $\mathcal{SR}$ and
$\mathcal{FR}$.

####  5.2.2 Threshold Impact Evaluation

The following two experiments study the impact of thresholds on the PB
approach. Recall that PB and DB approaches split energy between consumers
based on a defined threshold. The threshold prevents dividing services into
small neglectable chunks. The experiments of both PB and DB gave the same
behavior. Thus, we are only presenting the results of PB.

Figure 7: The average of quality of experience

Figure 8: The average of {$QoE,\mathcal{SR},\mathcal{FR}$} using PB
composition

Figure 9: The average of QoE using PB composition with various thresholds in a
microcell

Figure 10: The average execution time of all composition

Fig.8 represents the impact of the threshold on the three previously tested
attributes: $\mathcal{SR}$, $\mathcal{FR}$, and QoE. We tested the PB approach
with a 99% ratio of services to requests. The x-axis in Fig.8 represents the
threshold of partial services. $FR$ does not change as the threshold
increases, because it relies on the order of time slots and not the size of
distribution (threshold) as discussed in the previous experiment. Also, both
$\mathcal{SR}$ and QoE decrease as the threshold increases due to the
thresholds’ size. When the threshold’s size increases, fewer consumers will be
served. A lower number of fulfilled consumers results in low $\mathcal{SR}$
and thereby a low QoE.

The fifth experiment compares the impact of the threshold on the QoE with
different ratios of services to requests. We tested the PB approach with
thresholds of {10, 30, 50, 70, 90}. In Fig.10, the QoE increases when the
number of available services increases for all threshold values. Additionally,
the QoE for threshold 10 is the highest among all due to the threshold’s size.
When the size of the threshold is small, more consumers will be served. A
higher number of fulfilled consumers results in high $\mathcal{SR}$ and
thereby a high QoE.

####  5.2.3 Computation Efficiency Evaluation

The execution time for all approaches increases with the increase in services’
availability (See Fig.10). This is due to the increase in processing time to
assign these services.

##  6 Related Work

Energy sharing services have been introduced as an alternative ubiquitous
solution to charge IoT devices [18]. Several studies have addressed challenges
related to fulfilling the requirements of energy consumers [3][16][19]. A
temporal composition algorithm was proposed to compose energy services to
fulfill a consumer’s energy requirement [3]. The algorithm proposed the use of
fractional knapsack to maximize the provided energy. An elastic composition
was proposed to address the reliability of highly fluctuating energy providers
[16]. The composition uses the concepts of soft and hard deadlines to extend
the stay of a consumer and select more reliable services. The intermittent
behavior of energy services was addressed by a fluid approach [19]. The
approach uses the mobility patterns of the crowd to predict the intermittent
disconnections in energy services then replace or tolerate theses
disconnections. Other studies tackled challenges from a provider’s perspective
[13][20]. An context-aware incentive model was proposed to address the
resistance in providing energy services [13]. Another article addresses the
commitment of energy consumers to receive their initiated requests [20].
Existing literature in energy services addresses issues from a consumer or a
provider perspective [18]. To the best of our knowledge, challenges related to
the microcell perspective such as the QoE are yet to be addressed.

Quality of experience (QoE) has several definitions in the literature based on
the field of research [8][21][22]. However, all existing definitions focus on
assessing the quality of an application or a service based on the perception
of the end-users. In addition, most of the literature focuses on assessing the
QoE for multimedia applications. For instance, A method was proposed to gauge
gaming QoE under system influencing factors such as delay, packet loss, and
frame rates [23]. Another study proposed “Kaleidoscope” as an automated
solution to evaluate Web features [10]. As previously stated, the existing
research focuses on assessing the QoE of a service based on the perception of
the end-users. To the best of our knowledge, assessing the QoE in energy
services is not explored yet. In addition, using energy services as a tool to
enhance QoE in other microcell-based services is yet to be addressed.

##  7 Conclusion

We proposed an energy service composition framework that evaluates QoE in a
microcell. A new QoE based-assessment was proposed to capture the overall
satisfaction across consumers over a period of time. A two QoE-driven
composition of energy service were proposed. The Partial-Based (PB) approach
uses partial services to maximize the number of satisfied consumers and
thereby increase the QoE. The Demand-Based (DB) approach uses partial services
and prioritizes the most demanding time slots to maximize the number of
satisfied consumers and their level of fulfillment and thereby increase the
QoE. Experimental results show that DB outperforms all the evaluated
approaches. The efficiency of the proposed approaches was investigated against
a Greedy approach. Future direction is to consider the probability of change
in the microcell energy demand.

## Acknowledgment

This research was partly made possible by LE220100078 and LE180100158 grants
from the Australian Research Council. The statements made herein are solely
the responsibility of the authors.

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