text:
\section{Introduction}

\textbf{Edge computing}. As an emerging computing paradigm, \textit{edge computing}~\cite{satyanarayanan2017emergence,mao2017survey} pushes computing tasks
to the network edges such as base stations and access points. Both edge computing and cloud computing can support computation-intensive applications. Yet, since edge servers are closer to mobile devices than clouds, edge computing is more suitable to enable latency-critical applications
at mobile devices~\cite{hu2018wireless}. Many companies are now leveraging edge computing to ensure that their services are available at the edges with high speed~\cite{vaughan2019working}. In July 2019,  AT\&T signed  a deal of US\$2 billion with Microsoft to use the latter's capabilities related to edge and cloud computing~\cite{Microsoft}.

\textbf{Computation offloading for edge computing.}
An important topic studied in edge computing is \textit{computation offloading}~\cite{mach2017mobile}, where a mobile device offloads intensive computation tasks to the edge server with stronger computational resources. In the seminal model for computation offloading proposed by~\cite{chen2014decentralized,chen2015efficient} by Chen and his co-authors, mobile devices choose computation offloading to the edge cloud or local computing on the devices by comparing the utilities under the two settings, where the utility is computed as a weighted function of the time and energy required to complete the computation task. This model, which our paper will use, has been adopted (with possible refinements) in many studies~\cite{tao2017performance,mao2016dynamic}.

\textbf{Intelligent reflecting surfaces (IRSs) and IRS-aided communications}. An \textit{intelligent reflecting surface (IRS)} can intelligently control the wireless environment to improve signal strength received at the destination. This is vastly different from prior techniques which improve wireless communications via optimizations at the sender or receiver. Specifically, an IRS consists of many IRS units, each of which can  reflect the incident signal at a reconfigurable angle. In such \textit{IRS-aided communications}, the wireless signal travels from the source to the IRS, is optimized at the IRS, and then travels from the IRS to the destination. Such communication method is particularly useful when the source and destination have a weak  wireless channel in between  due to obstacles or poor environmental conditions, or they do not have direct line of sights.

Because of the capability of configuring wireless environments, IRSs are envisioned by many experts in wireless communications to play an important role in 6G networks. In November 2018, the Japanese mobile operator NTT DoCoMo and a startup MetaWave demonstrated the use of IRS-like technology for assisting wireless communications in 28GHz band~\cite{DOCOMO}. IRSs have been compared with the massive MIMO technology used in 5G communications. IRSs reflect wireless signals and hence consume little power, whereas massive MIMO transmits signals and needs much more power~\cite{hu2018beyond}.

\textbf{Problem studied in this paper: Computation offloading for
edge computing in IRS-aided communications.} We tackle the problem of mobile devices offloading computation tasks to a base station equipped with an edge server,
in the context of IRS-aided wireless communications. In the studied setting, mobile devices intend to send computation tasks to the base station via wireless channels, and communications between them are assisted by an IRS.
Fig.~\ref{fig-system} provides an illustration of the system model.

\textbf{Contributions.} The contributions of this paper are summarized as follows:
\begin{enumerate}

\item To the best of our knowledge, there exists no paper considering the computation offloading problem in the presence of IRS except that the very recent paper \cite{CunhuaPan_MEC_IRS} by Bai \emph{et al.} considers the latency minimization problem with the aid of IRS in edge computing system. In this paper, we extend the cost metric of mobile computing to a more comprehensive consideration, which subsumes the one used in \cite{CunhuaPan_MEC_IRS} as a special case. Besides we also take into account the rate constraints, which makes our problem more meaningful and much more challenging.

\item To solve the problem, We first develop an algorithm to detect the feasibility when given an arbitrary set of information rate constraints. The algorithm can also provide a feasible solution when we identify the problem as feasible. Then we develop an efficient solution which can maximize the earning of the edge server.

\item Through substantial numerical results, we demonstrate the significance of IRS devices. Our experimental results show that IRS can significantly boost the feasibility probability for information rate constraints. Besides, numerical results suggest that our algorithm converges fast and can effectively improve the edge server's earning.

\end{enumerate}

\textbf{Notation.} Scalars are denoted by italic letters, while vectors and matrices are denoted by bold-face
lower-case and upper-case letters, respectively. $\mathbb{C}$ denotes the set of all complex numbers. For a vector $\boldsymbol{x}$, $\|\boldsymbol{x}\|$ denotes its Euclidean norm. For a matrix $\boldsymbol{M}$, its transpose and conjugate transpose are denoted by $\boldsymbol{M}^T$ and $\boldsymbol{M}^H$, while $\boldsymbol{M}_{i,j}$ means the element in the $i$-th row and $j$-th column of $\boldsymbol{M}$. For a vector $\boldsymbol{x}$, its transpose, conjugate transpose, and Euclidean norm are denoted by $\boldsymbol{x}^T$, $\boldsymbol{x}^H$, and $\|\boldsymbol{x}\|$, while $x_{i}$ means the $i$-th element of $\boldsymbol{x}$.

\begin{figure}[!t]
 \centering
\includegraphics[scale=0.35]{Presentation7.eps}
\caption{An Illustration of IRS-Aided Communication System} \label{fig-system}
\end{figure}

\section{Related Work} \label{sec-Related-Work}

Since our paper combines edge computing and wireless communications aided by intelligent reflecting surfaces (IRSs), we discuss both related studies in edge computing and IRS-aided communications.

\textbf{Edge computing and computation offloading}.
Edge computing pushes computations to the network edges in order to support  latency-critical applications at mobile devices~\cite{hu2015mobile}. In many studies~\cite{mao2017survey}, edge computing is also referred to as \textit{fog computing}, a term introduced by Bonomi~\textit{et~al.}~\cite{bonomi2012fog} of Cisco. In 2018, IEEE integrated the fog computing reference architecture developed by the \textit{OpenFog Consortium} (now the \textit{Industrial Internet Consortium}) as a standard~\cite{IEEEOpenFog}.

Among many references on computation offloading for edge computing, two studies most related to our work are the seminal work~\cite{chen2014decentralized,chen2015efficient} by Chen and his co-authors. Chen~\cite{chen2014decentralized} considers a network of many mobile device users and one base station equipped with edge cloud. Mobile devices select computation offloading to edge cloud or local computing on the devices by comparing the obtained utilities under the two settings. All communications in the network use the same wireless channel and interfere each other. Hence, the data uploading rate of a mobile device selecting computation offloading is related to the offloading decisions of other devices. Then the dynamics in offloading decision making of mobile devices are modeled as a game, which is analyzed using potential game theory. Improving~\cite{chen2014decentralized} to a more general setting, Chen~\textit{et~al.}~\cite{chen2015efficient} address a
wireless interference environment of multiple channels instead of only one channel. Thus, the data uploading rate of a mobile device selecting computation offloading is related to other devices which use the same wireless channel. Then a game is used to decide local/edge computing choices of mobile devices and their channel selection in the case of edge computing.

In addition to~\cite{chen2014decentralized,chen2015efficient} discussed above, many studies on computation  offloading  for  edge  computing have appeared recently. We discuss some of them below and refer readers to~\cite{tao2017performance,wang2019edge} for more comprehensive reviews. Mao~\textit{et~al.}~\cite{mao2016dynamic} consider computation offloading of an energy-harvesting device to the edge server, where the incorporation of energy harvesting makes the design of computation offloading policy challenging. Partial computation offloading instead of  full offloading is investigated in~\cite{wang2016mobile,ning2018cooperative,tao2017performance,bi2018computation}. Sardellitti~\textit{et~al.}~\cite{mao2016dynamic} tackle computation offloading for  edge  computing in MIMO multicell system so that the optimization variables to minimize mobile devices' energy consumption include the transmit precoding matrices of mobile devices in addition to computational resources assigned by the edge cloud to the devices. Chen and Hao~\cite{chen2018task} analyze computation offloading in the framework of software defined networking (SDN), where the SDN controller decides for each offloaded task which edge cloud and how much computing resource to use.

\textbf{Intelligent reflecting surface-aided communications}. Since IRSs can be controlled to reflect incident wireless signals in a desired way, IRS-aided communications have recently received much attention in the literature~\cite{Junginproceedings,jung2018performance,huang2019Reconfigurable,fu2019intelligent,yu2019miso,nadeem2019intelligent,mishra2019channel}. The studies include analyses of data rates~\cite{Junginproceedings,jung2018performance}, optimizations of power or spectral efficiency~\cite{huang2019Reconfigurable,yu2019miso}, and channel estimation~\cite{nadeem2019intelligent,mishra2019channel}. In these studies, IRSs are also referred to as \textit{large intelligent surface}~\cite{hu2018beyond,jung2018performance}, \textit{intelligent reflecting surface}~\cite{nadeem2019intelligent,jiang2019over,qingqing2019towards}, \textit{software-defined surface}~\cite{basar2019large}, and \textit{passive intelligent mirrors}~\cite{huang2018achievable}~\cite{di2019reflection}.

We now first discuss IRS studies analyzing uplink communications, where IRSs help transmissions from mobile devices to base stations, since our work also focuses on uplinks. Jung\textit{~et~al.}~\cite{Junginproceedings,jung2018performance} analyze the impact of channel estimation errors on the uplink data rates.

For IRSs assisting communications between mobile devices and base stations, in addition to uplink studies discussed above, downlinks are investigated in~\cite{nadeem2019large,wu2018intelligent,wu2018intelligentfull,huang2019Reconfigurable,yu2019miso}. Regarding that there seems to be more downlink studies than uplink ones, a possible explanation is that in the former case, the base station can optimize the transmit beamforming in a centralize manner.

In addition to the above settings where IRSs aid communications between mobile devices and base stations, direct communications between mobile devices and IRSs are analyzed in~\cite{jung2019performance,hu2017potential,hu2018beyond,hu2018capacity}.

\section{System Model} \label{sec-System}

We consider a wireless network consisting of a multi-antenna base station (BS)/access point (AP) and $K$ single-antenna mobile devices (MDs). An illustration of the system model has been given in Fig.~\ref{fig-system}. The BS/AP hosts an edge server for providing edge computing via wireless communications. The mobile devices are numbered from $1$ to $K$. For $k \in \mathcal{K}\triangleq\{1,2,\ldots, K\}$, the $k$-th mobile device has a computation task $\mathcal{J}_k : = (b_k, d_k)$ which it intends to offload to the edge server, where $b_k$ denotes the data size of the task, and $d_k$ denotes the number of CPU cycles to execute the task.

We now introduce the computing model and communication model used in this paper. Mathematical details for them are deferred to Sections~\ref{subsec-Choices-Mobile-Users} and~\ref{subsec-Transmission}.

\textbf{Computing model.}
Our computing model mainly follows that of the seminal work~\cite{chen2014decentralized,chen2015efficient} discussed in  Section~\ref{sec-Related-Work}. The same as~\cite{chen2014decentralized,chen2015efficient}, mobile devices choose computation offloading to the edge cloud or local computing on the devices by comparing the utilities under the two settings, where the utility in each setting depends on the time and energy required to complete the computation task.
 Compared with~\cite{chen2014decentralized,chen2015efficient}, our model also requires the mobile device to issue a payment to the edge cloud if a weighted function of the time and energy required to complete the computation task has  a larger value under edge computing compared with local computing.
Formal details of the computing model will be presented in Section~\ref{subsec-Choices-Mobile-Users}.

 \textbf{Communication model.} The communications between mobile devices and the BS/AP are aided by IRSs. The signals also interfere each other when arriving at the base station. Formal details of the communication model will be presented in Section~\ref{subsec-Transmission}.

\section{Optimization Problem Formulation} \label{sec-Problem-Formulation}

\subsection{A Most Generic Model: Local Computing versus Edge Computing} \label{subsec-Choices-Mobile-Users}

\textbf{Utility of a mobile device under local computing.} Below we compute the utility of the $k$-th mobile device when it completes the task $\mathcal{J}_k$ under local computing, where $k \in \mathcal{K}$.

Let the $k$-th mobile device's computation capability be $c_k^{(m)}$ CPU cycles per unit time, where the superscript ``$(m)$'' throughout the paper is used for quantities with a mobile device. Then under local computing, the
computation execution time of the task $\mathcal{J}_k$ with $d_k$ CPU cycles is
\begin{align}
t_k^{(m)} : = \frac{d_k}{c_k^{(m)}}. \label{eq-tkm}
\end{align}

Let $\mu_k$ be the energy per CPU cycle. The energy to complete the task $\mathcal{J}_k$ with $d_k$ CPU cycles is
\begin{align}
e_k^{(m)} := \mu_k d_k. \label{eq-ekm}
\end{align}

We consider that the $k$-th mobile device combines the time and energy to complete the task $\mathcal{J}_k$ in a weighted manner to quantify the cost. Specifically, with the time and energy weighted by coefficients $w_k^{(t)}$ and $w_k^{(e)}$ respectively, the combined cost for the $k$-th mobile device is $w_k^{(t)} t_k^{(m)} + w_k^{(e)} e_k^{(m)}$.
Let the benefit (without subtracting the cost) of completing the task $\mathcal{J}_k$ be $f_k(b_k, d_k)$. Then the gross utility  for the $k$-th mobile device to complete the task $\mathcal{J}_k$ under local computing is
\begin{align}
 U_k^{(m)}(b_k, d_k)& := f_k(b_k, d_k) - w_k^{(t)} t_k^{(m)} - w_k^{(e)} e_k^{(m)}.  \label{eq-lc-utilty-1}
\end{align}
Substituting Eq.~(\ref{eq-tkm}) and Eq.~(\ref{eq-ekm}) into  Eq.~(\ref{eq-lc-utilty-1}), we obtain
\begin{align}
 U_k^{(m)}(b_k, d_k)& = f_k(b_k, d_k) - w_k^{(t)} \frac{d_k}{c_k^{(m)}} - w_k^{(e)} \mu_k d_k. \label{eq-lc-utilty-2}
\end{align}

\textbf{Utility of a mobile device under edge computing.} We now compute the utility of the $k$-th mobile device when its task $\mathcal{J}_k$ is completed under edge computing, where $k \in \mathcal{K}$.

First, we analyze the time needed for the $k$-th mobile device to send the task $\mathcal{J}_k$ with $b_k$ bytes to the edge server. This depends on the transmission rate, which further depends on which of the other $K-1$ users choose edge computing and hence are transmitting. Thus, we introduce an indicator variable $a_k$ to represent the $k$-th mobile device's choice of edge computing or local computing; specifically,
\begin{subnumcases}{\hspace{-22pt}a_k \hspace{-1.5pt}=}
\hspace{-5pt}1, &\hspace{-16pt}if the $k$-th mobile device selects edge computing, \\
\hspace{-5pt}0, &\hspace{-16pt}if the $k$-th mobile device selects local computing.
\end{subnumcases}

Let $[-k]$ be the indices of mobile devices other than the $k$-th one; i.e., $[-k]\triangleq\mathcal{K}\setminus\{k\}$.
Let a vector $\boldsymbol{a}_{-k}$ represent the choices made by mobile devices in $[-k]$; i.e.,
$\boldsymbol{a}_{-k}\triangleq[a_i: i\in \mathcal{K}\setminus\{k\}]^T$.

For an IRS with $N$ IRS units, the IRS phase shift matrix as a diagonal matrix $\boldsymbol{\Phi}\in \mathbb{C}^{N \times N}$ is given by
\begin{align}
 & \boldsymbol{\Phi} := \sf{Diag}(\boldsymbol{\phi}) , \textup{ for }
\boldsymbol{\phi}\triangleq[\kappa_1e^{j \theta_1}, \ldots, \kappa_Ne^{j \theta_N}]^T ,  \label{eq-def-Phi-matrix}
\end{align}
with $\kappa_n$ and $\theta_n$ denoting the reflecting amplitude and angle of the $n$-th IRS unit respectively, for $n \in \mathcal{N}\triangleq\{1,2,\ldots, N\}$. To fully exploit the reflecting surface's ability to adjust the wireless environment, here we assume that the reflecting amplitude and reflecting angle could take values within  $[0,1]$ and $[0,2\pi)$ respectively. The data transmission rate of the $k$-th mobile device depends on $\boldsymbol{a}_{-k}$ and the IRS reflecting matrix $\boldsymbol{\Phi}$ (or the vector $\boldsymbol{\phi}$), and we denote it by  $R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})$. The specific expression of $R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})$ will be clear in Section~\ref{subsec-Transmission}. Then the time to transmit the data of $b_k$ bytes to the edge server is
\begin{align}
 & t_{k,send}^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi}) = \frac{b_k}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})}, \label{eq-time-send-cloud}
\end{align}
where the superscript ``$(e)$'' throughout the paper is used for quantities associated with the edge server.

In the transmission of the $k$-th mobile device for choosing edge computing,
let $\nu_k$ be the energy consumed per unit time. Then after time $t_{k,send}^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi}) $, an amount of $\nu_k t_{k,send}^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi}) $ energy is consumed. The same as~\cite{chen2015efficient}, we also consider that the $k$-th mobile device commits a tail energy of $L_k$ after transmitting the data of $b_k$ bytes. Then the total energy spent by the $k$-th mobile device is
\begin{align}
 & e_k^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi}) = \nu_k t_{k,send}^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi}) + L_k.

 \label{eq-energy-send-cloud}
\end{align}

After the data of $b_k$ bytes is fully transmitted to the edge server, with the edge server computation capability denoted by $c^{(e)}$ CPU cycles per unit time, the time for the edge server to complete the task $\mathcal{J}_k$ with $d_k$ CPU cycles is
\begin{align}
 & t_{k,exe}^{(e)}  = \frac{d_k}{c^{(e)}} . \label{eq-time-exe-cloud}
\end{align}

As in~\cite{chen2014decentralized,chen2015efficient}, we ignore the time required for the base station to send back the result to a mobile device. Hence, our study includes only uplinks but not downlinks in Fig.~\ref{fig-system}. Then under edge computing, the total time to complete the task $\mathcal{J}_k$ is $t_{k,send}^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi}) + t_{k,exe}^{(e)} $. With $w_k^{(t)}$ and $w_k^{(e)}$ being the weights for the time and energy in the cost computation, the combined cost is $ w_k^{(t)} \cdot\left[ t_{k,send}^{(e)} (\boldsymbol{a}_{-k}, \boldsymbol{\phi}) + t_{k,exe}^{(e)}  \right] + w_k^{(e)} \cdot e_k^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi})$. Under edge computing, we consider that the $k$-th mobile device also needs to have a payment of $P_k$ to the edge server. Then with the benefit (without subtracting the cost) of completing the task $\mathcal{J}_k$ being $f_k(b_k, d_k)$, the gross utility  for the $k$-th mobile device to complete the task $\mathcal{J}_k$ under edge computing is
\begin{align}
 & U_k^{(e)} (\boldsymbol{a}_{-k}, \boldsymbol{\phi}, b_k, d_k) \nonumber \\ &= f_k(b_k, d_k) -  w_k^{(t)} \cdot\left[ t_{k,send}^{(e)} (\boldsymbol{a}_{-k}, \boldsymbol{\phi}) + t_{k,exe}^{(e)}  \right] \nonumber \\ & \quad - w_k^{(e)} \cdot e_k^{(e)}(\boldsymbol{a}_{-k}, \boldsymbol{\phi}) - P_k . \label{eq-utility-cloud-1}
\end{align}

Substituting Eq.~(\ref{eq-time-send-cloud})~(\ref{eq-energy-send-cloud})~(\ref{eq-time-exe-cloud}) into Eq.~(\ref{eq-utility-cloud-1}), we obtain
\begin{align}
 & U_k^{(e)} (\boldsymbol{a}_{-k}, \boldsymbol{\phi}, b_k, d_k) \nonumber \\ &= f_k(b_k, d_k) -   w_k^{(t)}  \cdot \left[\frac{b_k}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} + \frac{d_k}{c^{(e)}} \right] \nonumber \\ & \quad -  w_k^{(e)} \cdot \left[\frac{\nu_k b_k}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} + L_k\right] - P_k . \label{eq-utility-cloud-2}
\end{align}

In our studied model, for the $k$-th mobile device, if the gross utility obtained under edge computing is at least the gross utility obtained under local computing, then the $k$-th mobile device will have $a_k =1$ to choose edge computing over local computing; otherwise, $a_k =0$. Formally,
\begin{align}
a_k = \begin{cases}
1, & \textup{if $U_k^{(e)} (\boldsymbol{a}_{-k}, \boldsymbol{\phi}, b_k, d_k) \geq  U_k^{(m)}(b_k, d_k)$,} \\
0, & \textup{otherwise.}
\end{cases} \label{eq-ak10v1}
\end{align}

From Eq.~(\ref{eq-lc-utilty-2}) and Eq.~(\ref{eq-utility-cloud-2}),
the inequality $U_k^{(e)} (\boldsymbol{a}_{-k}, \boldsymbol{\phi}, b_k, d_k) \geq U_k^{(m)}(b_k, d_k)$ is equivalent to
\begin{align}
 P_k \leq \, & f_k(b_k, d_k) - w_k^{(t)}  \cdot \left[\frac{b_k}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} + \frac{d_k}{c^{(e)}} \right] \nonumber  \\ &  - w_k^{(e)} \cdot \left[\frac{\nu_k b_k}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} + L_k\right]  \nonumber  \\ &  - \left[f_k(b_k, d_k) - w_k^{(t)} \frac{d_k}{c_k^{(m)}} - w_k^{(e)} \mu_k d_k\right]. \label{eq-Pk-inequ}
\end{align}

Defining $C_k$ and $A_k(b_k)$ by
\begin{align}
C_k  :=  w_k^{(t)} \frac{d_k}{c_k^{(m)}} + w_k^{(e)} \mu_k d_k - \frac{w_k^{(t)} d_k}{c^{(e)}} - w_k^{(e)} L_k  . \label{eq-def-Ck}
\end{align}
and
\begin{align}
A_k(b_k) := w_k^{(t)} b_k+w_k^{(e)}\nu_k b_k, \label{eq-def-Ak}
\end{align}
we write the above Inequality~(\ref{eq-Pk-inequ}) as
\begin{align}
 P_k \leq C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})},\label{eq-Pk-inequ2}
\end{align}
which converts Eq.~(\ref{eq-ak10v1}) into
\begin{align}
a_k = \begin{cases}
1, & \textup{if $P_k \leq C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})}$,} \\
0, & \textup{otherwise.}
\end{cases} \label{eq-ak10v2}
\end{align}

From Eq.~(\ref{eq-ak10v2}), we have the following:
\begin{itemize}[leftmargin=10pt]
\item If $C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} \geq 0$, the edge server can set $P_k \in [0, C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})}]$, and the $k$-th mobile device will have $a_k =1$ to choose edge computing over local computing.
\item If $C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} < 0$, in case that the edge server does not accept a negative payment from the $k$-th mobile device (i.e., the edge server does not compensate the mobile device for choosing edge computing), then the $k$-th mobile device will have $a_k =0$ to choose local  computing over edge computing, and the edge server will receive no payment from the $k$-th mobile device so that $P_k =0$.
\end{itemize}

Besides, the information rate for each MD should be no smaller than a requirement $r_k$ to ensure that each MD could fulfill a basic communication and/or computation task.

From the above analysis, the maximal payment that the edge server can get from the $k$-th mobile device is \mbox{$\max\left\{C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})},0\right\}$.}  Then to maximize the total payment from all $K$ mobile devices, the edge server solves the following optimization Problem~(P1):
\begin{subequations} \label{opt-p1}
\begin{alignat}{2}
\textup{(P1):}\max_{\boldsymbol{\phi}}&\  \sum_{k=1}^K \max\left\{C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})},0\right\} \label{opt-p1-objective} \\
\sf{s.t.} &\ a_k = \begin{cases}
1, &  \textup{if $C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} \geq 0$}, \nonumber \\
0, & \textup{otherwise},
\end{cases} \forall k \in\mathcal{K}, \label{opt-p1-constraint1} \\
& R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})\geq r_k, \forall k, \\
& | \boldsymbol{\phi}_n |\leq1, \forall n\in\mathcal{N}.\label{opt-p1-constraint2}
\end{alignat}
\end{subequations}
The above constraint~(\ref{opt-p1-constraint2}) holds since $\boldsymbol{\phi}_n=\kappa_ne^{j \theta_n}$ from Eq.~(\ref{eq-def-Phi-matrix}).

\subsection{A Simplified Case} \label{subsec-BS-profit}

Maximizing the non-concave objective function in~(\ref{opt-p1-objective}) even without considering the constraints~(\ref{opt-p1-constraint1}) and (\ref{opt-p1-constraint2}) is difficult. Hence, we consider that if $C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})}<0$, the edge server accepts a negative payment from the $k$-th mobile device (i.e., the edge server compensates the mobile device for choosing edge computing). Then regardless of the relationship between $C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})} $ and $0$, the maximal payment that the edge server can get from the $k$-th mobile device is always $C_k - \frac{A_k(b_k)}{R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})}$. Also, the $k$-th mobile device for each $k\in\mathcal{K}$ will always have $a_k =1$ to choose edge computing over local computing. Now with $a_i=1$ for $i\in\mathcal{K}$, $\boldsymbol{a}_{-k}$ becomes a vector of all $1$s, and we write $R_k(\boldsymbol{a}_{-k}, \boldsymbol{\phi})$ as $R_k(\boldsymbol{\phi})$ for notation simplicity. Thus, to maximize the total payment from all $K$ mobile devices, the edge server solves the following optimization Problem~(P2):
\begin{subequations}\label{opt-p2}
\begin{alignat}{2}
\textup{(P2):} \quad \max_{\boldsymbol{\phi}} & \qquad \sum_{k=1}^K \left[C_k - \frac{A_k(b_k)}{R_k(  \boldsymbol{\phi})}\right] \label{opt-p2-objective} \\
\textup{s.t.} & \ R_k(\boldsymbol{\phi})\geq r_k, \forall k\in\mathcal{K}, \label{opt-p2-rate_cnstr} \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}.\label{opt-p2-constraint2}
\end{alignat}
\end{subequations}
Later we will show that the problem (P2) can be efficiently solved. Hence, Problem~(P2) of (\ref{opt-p2}) significantly reduces the difficulty of solving Problem~(P1) of (\ref{opt-p1}).

Problem~(P2) provides a lower bound for Problem~(P1). Moreover, considering Problem~(P2) also has practical interest. As discussed above, we   consider that  the edge server is willing to compensate a mobile device for choosing edge computing if providing edge computing to the mobile device for  some computation task does not bring immediate earning. Doing so may encourage the mobile device to adopt edge computing for future computation tasks, and the edge server can profit from the mobile device in these future tasks. In addition, the edge server's compensation may help the edge server to increase the user base of its edge computing and hence improve the long-term earning.

\subsection{Signal Model}
\label{subsec-Transmission}

In this paper we consider deploying the IRS to assist a typical wireless communication system, where a multi-antenna AP is communicating with $K$ single-antenna MDs. Suppose that the AP has $M$ antennas and the IRS equipment has $N$ reflecting elements. We use $\boldsymbol{h}_{\textup{r},k}^H \in \mathbb{C}^{1 \times N}$ to denote the wireless channel from the $k$-th MD to the IRS. Let $\boldsymbol{G} \in \mathbb{C}^{M \times N} $ be the channel from the IRS to the AP. Define $\boldsymbol{h}_{\textup{d},k}^H\in \mathbb{C}^{1 \times M} $ to be the channel from the $k$-th MD to the AP.

Here we consider the system experiences quasi-static flat-fading channel. Therefore the AP can obtain the channel status information (CSI) $\boldsymbol{h}_{\textup{r},k}^H, \boldsymbol{G} ,\boldsymbol{h}_{\textup{d},k}^H $ via the standard channel estimation and feedback technique (e.g. in a classical time-division duplexing (TDD) system).
Based on knowledge of $\boldsymbol{h}_{\textup{r},k}^H, \boldsymbol{G} ,\boldsymbol{h}_{\textup{d},k}^H$, the AP manipulates the amplitude and phase of the IRS elements to improve its earnings. That is AP tries to solve the Problem~(P2) with respect to $\boldsymbol{\phi}$.

In view of $\boldsymbol{\Phi}=\textup{diag}(\boldsymbol{\phi})$, we define the effective channel $\boldsymbol{h}_k(\boldsymbol{\phi})\in \mathbb{C}^{M \times 1}$ of the $k$-th MD by
\begin{align}
 &\boldsymbol{h}_k(\boldsymbol{\phi}):=  \boldsymbol{G} \textup{diag}(\boldsymbol{\phi}) \boldsymbol{h}_{\textup{r},k} + \boldsymbol{h}_{\textup{d},k}. \label{eq-def-hk}
\end{align}
The received signal at AP can be represented as
\begin{align}
\boldsymbol{r}(\boldsymbol{\phi}) = \sum_{j}\sqrt{q}_j\boldsymbol{h}_j(\boldsymbol{\phi})s_j+\boldsymbol{n},  \label{eq-rcvd_sig}
\end{align}
where the vector $\boldsymbol{n}\in\mathbb{C}^M$ denotes the local thermal noise at the receiver and is usually modeled as Gaussian distribution $\mathcal{CN}(\boldsymbol{0},\sigma^2\boldsymbol{I}_M)$.

Assume that linear receiver $\boldsymbol{u}_k^H\in \mathbb{C}^{1\times M}$ is utilized at the AP to decode signals associated with the $k$-th MD. Then the signal-to-interference-plus-noise ratio (SINR) is a function of $\boldsymbol{\phi}$ given as
\begin{align}
\gamma_k\big(\boldsymbol{\phi}\big) = \frac{q_k|\boldsymbol{u}_k^H \boldsymbol{h}_k(\boldsymbol{\phi})|^2}{\boldsymbol{u}_k^H \boldsymbol{W}_k( \boldsymbol{\phi}) \boldsymbol{u}_k }. \label{eq-gamma-k}
\end{align}
with interference-plus-noise spatial covariance matrix
\begin{align}
\boldsymbol{W}_k( \boldsymbol{\phi}) = \sigma^2 \boldsymbol{I}_{M}+\sum_{i\neq k} q_i \boldsymbol{h}_i(\boldsymbol{\phi}) \boldsymbol{h}_i^H(\boldsymbol{\phi}). \label{eq-Zk-matrix}
\end{align}
To improve receiving quality, the optimal $\boldsymbol{u}_k^\star$ \cite{schubert2005iterative,monzingo1980introduction}
\begin{align}
\boldsymbol{u}_k^\star = \frac{[{\boldsymbol{W}_k( \boldsymbol{\phi})}]^{-1}  \boldsymbol{h}_k(\boldsymbol{\phi})}{\|[{\boldsymbol{W}_k( \boldsymbol{\phi})}]^{-1} \boldsymbol{h}_k(\boldsymbol{\phi})\|_2}  , \label{eq-mu-k-optimal}
\end{align}
should be  used to yield the maximum $\gamma_k^{\star}\big(\boldsymbol{\phi}\big)$ given as
\begin{align}
\gamma_k^\star\big(\boldsymbol{\phi}\big)   = q_k\boldsymbol{h}_k^H(\boldsymbol{\phi}) [{\boldsymbol{W}_k( \boldsymbol{\phi})}]^{-1}  \boldsymbol{h}_k(\boldsymbol{\phi}). \label{eq-gamma-k-optimal}
\end{align}

Assuming that unit bandwidth is used, then by Shannon's formula~\cite{shannon1948mathematical} the allowed information rate of the $k$-th MD $R_k( \boldsymbol{\phi})$ is evaluated as \footnote{Here we somewhat abuse the notation $R_k(\cdot)$. Our $R_k\big(\boldsymbol{\phi}\big)$ is actually defined as the information rate associated with SINR maximizing receiving vector, not for generic receiving vectors.}
\begin{align}
 &R_k( \boldsymbol{\phi}) = \log(1+\gamma_k^{\star}(\boldsymbol{\phi}) )  . \label{eq-Rk-rate}
\end{align}
Considering result in (\ref{eq-gamma-k-optimal}) we obtain
\begin{align}
R_k( \boldsymbol{\phi})& =\log(1+ q_k\boldsymbol{h}_k^H(\boldsymbol{\phi}) [{\boldsymbol{W}_k( \boldsymbol{\phi})}]^{-1}  \boldsymbol{h}_k(\boldsymbol{\phi})) \label{eq-obj-1-Rk} .
\end{align}

\subsection{Problem Formulation} \label{subsec-optimizing-Mobile-IRS}

Putting the above discussions together, our goal is to solve (P2) in (\ref{opt-p2}), where $R_k( \boldsymbol{\phi})$ is given by (\ref{eq-obj-1-Rk}).

In~(\ref{opt-p2-objective}), since $b_k$ does not depend on the optimization variable $\boldsymbol{\phi}$, we just abbreviate $A_k(b_k)$ to $A_k$ in the following discussion. Maximizing the term in~(\ref{opt-p2-objective}) is the same as minimizing $\sum_{k=1}^K  \frac{A_k}{R_k(\boldsymbol{\phi}) }$, so we can write Problem~(P2) as the following Problem~(P3):
\begin{subequations}
\begin{alignat}{2}
\textup{(P3):} \quad \min_{\boldsymbol{\phi}} & \qquad \sum_{k=1}^K  \frac{A_k}{R_k(\boldsymbol{\phi}) } \\
\textup{s.t.} & \ R_k(\boldsymbol{\phi})\geq r_k, \forall k\in\mathcal{K}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall k\in\mathcal{N}.
\end{alignat}
\end{subequations}
We will discuss the feasibility and solution of (P3) in the subsequent sections in details.

\section{Feasibility Problem} \label{Feasibility-Check}

The problem (P3) is difficult due to both of its complex objective as well as the group of rate constraints, which are all nonconvex. Putting aside the highly nonconvex objective, one first question comes to our mind is the feasibility of the problem. That is, with an arbitrary set of rate constraints $\{r_k\}$ assigned, is the problem (P3) feasible? Determining the feasibility of (P3) is itself hard but at the same time highly meaningful. Since it is the base for scheming a reasonable rate requirement and serves as a starting point to solve (P3) (as will become clear in the next section). In this section we develop an iterative evaluation procedure, which i)
can serve as a sufficient condition for the feasibility of (P3) and ii) can provide a feasible solution when (P3) is identified as feasible.

Noticing the relation in (\ref{eq-Rk-rate}), the feasibility check of (P3), can be equivalently written as the following problem:
\begin{subequations}
\begin{alignat}{2}
\textup{(P4):} \quad \sf{Find}\ & {\boldsymbol{\phi}} \\
\sf{s.t.} & \gamma_k^{\star}(\boldsymbol{\phi})\geq e^{r_k}-1, \forall k\in\{1,\cdots,K\}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\{1,\cdots,N\}.
\end{alignat}
\end{subequations}

To further simplify the above SINR constraints, which have fractional nature, we consider a closely related metric---mean square error (MSE), which is defined as
\begin{align}
\varepsilon_k(\boldsymbol{\phi},\boldsymbol{w}_k) \triangleq \mathbb{E}\Big\{\big|s_k\!-\!\boldsymbol{w}_k^H\big(\sum_{j}\sqrt{q}_j\boldsymbol{h}_js_j\!+\!\boldsymbol{n}\big)\big|^2\Big\}\label{MSE_def}
\end{align}
where the vector $\boldsymbol{w}_k$ is the linear receiver utilized at AP to suppress the interference and noise. The expectation in (\ref{MSE_def}) is taken over the distribution of noise $\boldsymbol{n}$ and information symbols $\{s_j\}$. When linear receiver is deployed, the following identity connects the minimal MSE $\varepsilon^{\star}_k(\boldsymbol{\phi})$ and maximal SINR $\gamma^{\star}_k(\boldsymbol{\phi})$:
\begin{align}
\varepsilon^{\star}_k(\boldsymbol{\phi}) = \frac{1}{1+\gamma^{\star}_k(\boldsymbol{\phi})}. \label{mse_sinr}
\end{align}
In fact, it can be proved that optimal linear MSE receiver is given as
\begin{align}
\boldsymbol{w}_k^{\star} = \widetilde{\boldsymbol{W}}(\boldsymbol{\phi})^{-\!1}\sqrt{q_k}\boldsymbol{h}_k(\boldsymbol{\phi}), \label{eq:w_mmse_lin}
\end{align}
with $\widetilde{\boldsymbol{W}}(\boldsymbol{\phi})$ being
\begin{align}
\widetilde{\boldsymbol{W}}(\boldsymbol{\phi}) = \sigma^2\boldsymbol{I}_M+\sum_{j}q_{i}\boldsymbol{h}_j(\boldsymbol{\phi})\boldsymbol{h}_j(\boldsymbol{\phi})^H.\label{W_tilde}
\end{align}
and the minimal $\varepsilon^{\star}_k(\boldsymbol{\phi})$ can be readily obtained as $\varepsilon^{\star}_k(\boldsymbol{\phi})=1\!-\!q_k\boldsymbol{h}_k(\boldsymbol{\phi})^H\widetilde{\boldsymbol{W}}_k(\boldsymbol{\phi})^{-\!1}\boldsymbol{h}_k(\boldsymbol{\phi})$. Combined with (\ref{eq-gamma-k-optimal}), (\ref{mse_sinr}) can be verified.

Based on the above observation in (\ref{mse_sinr}), (P4) can equivalently transformed into
\begin{subequations}
\begin{alignat}{2}
\textup{(P5):} \quad \sf{Find}\ & {\boldsymbol{\phi}} \\
\sf{s.t.} &\ \varepsilon_k^{\star}(\boldsymbol{\phi})\leq \frac{1}{1+e^{r_k}-1}=e^{-r_k}, \forall k\in\mathcal{K}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}.
\end{alignat}
\end{subequations}

The above optimization problem can be further transformed as follows
\begin{subequations}
\begin{alignat}{2}
\textup{(P6):} \quad \sf{Find}\ & \boldsymbol{\phi},\{\boldsymbol{w}_k\}  \\
\sf{s.t.} &\ \varepsilon_k(\boldsymbol{\phi},\{\boldsymbol{w}_k\})\leq e^{-r_k}, \forall k\in\mathcal{K}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}.
\end{alignat}
\end{subequations}
The equivalence between (P5) and (P6) can be seen by noticing the fact that $\varepsilon_k^{\star}(\boldsymbol{\phi})=\min_{\boldsymbol{w}_k}\varepsilon_k(\boldsymbol{\phi},\boldsymbol{w}_k)$ and the optima is achieved when choosing $\boldsymbol{w}_k=\boldsymbol{w}_k^{\star}$ given in (\ref{eq:w_mmse_lin}). In fact, if $\varepsilon_k^{\star}(\boldsymbol{\phi})\leq e^{r_k}$, then $\varepsilon_k(\boldsymbol{\phi},\boldsymbol{w}_k^{\star})=\varepsilon_k^{\star}(\boldsymbol{\phi})\leq e^{r_k}$. Conversely, if there exists $(\boldsymbol{\phi},\boldsymbol{w}_k)$ such that $\varepsilon_k(\boldsymbol{\phi},\boldsymbol{w}_k)\leq e^{r_k}$, then $\varepsilon_k^{\star}(\boldsymbol{\phi})\leq\varepsilon_k(\boldsymbol{\phi},\boldsymbol{w}_k)\leq e^{-r_k}$.

Now we consider the following optimization problem
\begin{subequations}
\begin{alignat}{2}
\textup{(P7):} \quad \min_{\boldsymbol{\phi},\{\boldsymbol{w}_k\},\alpha} \ &\alpha \\
\sf{s.t.} &\ e^{r_k}\varepsilon_k(\boldsymbol{\phi},\boldsymbol{w}_k)\leq\alpha, \forall k\in\mathcal{K}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}.
\end{alignat}
\end{subequations}
If the optimal value of (P7) is no greater than $1$, then (P6), and consequently the original problem (P3), is feasible. Otherwise the problem is infeasible. The intention behind the transformation from (P4) to (P7) lies in the fact that $\varepsilon_k(\boldsymbol{\phi},\boldsymbol{w}_k)$ has a quadratic form and is bi-convex with respect to $\boldsymbol{\phi}$ and $\{\boldsymbol{w}_k\}$, which is much simpler to tackle compared to $\gamma_k^{\star}(\boldsymbol{\phi})$.

Since the problem is still nonconvex with respect to $\boldsymbol{\phi}$ and $\{\boldsymbol{w}_k\}$ jointly, we can use block coordinate descent (BCD) method to solve it. Specifically we can alternatively update $(\boldsymbol{\phi},\alpha)$ and $(\{\boldsymbol{w}_k\},\alpha)$. When $\boldsymbol{\phi}$ is fixed, the optimal $\{\boldsymbol{w}_k\}$ is given in (\ref{eq:w_mmse_lin}) as discussed previously and $\alpha$ can be accordingly determined. When $\{\boldsymbol{w}_k\}$ are fixed, $\boldsymbol{\phi}$ should be obtained via solving the following problem:
\begin{subequations}
\begin{alignat}{2}
\!\!\!\!\!\!\textup{(P8):} \min_{\boldsymbol{\phi},\alpha} \ &\alpha \\
\sf{s.t.} &\ \boldsymbol{\phi}^H\boldsymbol{Q}_k\boldsymbol{\phi}+2\mathrm{Re}\{\boldsymbol{q}_k^H\boldsymbol{\phi}\}+d_k\leq\alpha, \forall k\in\mathcal{K}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}.
\end{alignat}
\end{subequations}
with the parameters defined as follows:
\begin{subequations}
\begin{align}
\!\!\!\!\boldsymbol{F}_k\triangleq &\ \ \boldsymbol{G}\sf{Diag}(\boldsymbol{h}_{\rm{r},k}), \\
\!\!\!\!\boldsymbol{Q}_k\triangleq & e^{r_k}\sum_{j}q_j\boldsymbol{F}_j^H\boldsymbol{w}_k\boldsymbol{w}_k^H\boldsymbol{F}_j, \label{eq_cnstr_Q}\\
\boldsymbol{q}_k\triangleq & e^{r_k}\sum_{j}q_j\boldsymbol{w}_k^H\boldsymbol{h}_{\rm{d},j}\boldsymbol{F}_j^H\boldsymbol{w}_k\!-\!e^{r_k}\sqrt{q_k}\boldsymbol{F}_k^H\boldsymbol{w}_k \\
d_k \triangleq & e^{r_k}\big(\sum_{j}q_j\big|\boldsymbol{w}_k^H\boldsymbol{h}_{\rm{d},j}\big|^2\!\!-\!\!2\sqrt{q_k}\mathrm{Re}\{\boldsymbol{w}_k^H\boldsymbol{h}_{\rm{d},k}\} \\
& \quad\quad+\!\sigma^2\|\boldsymbol{w}_k\|_2^2\!+\!1\big).\nonumber
\end{align}
\end{subequations}

According to (\ref{eq_cnstr_Q}), $\boldsymbol{Q}_k$ is obvioulsy positive-semidefinite $\forall k$ and therefore (P8) is convex. In fact (P8) can be transformed into standard second-order-cone-programming (SOCP) problem and can be efficiently solved via standard numerical solver like CVX.

Since the BCD procedure alternatively updates $\boldsymbol{\phi}$ and $\{\boldsymbol{w}_k\}$ minimizing $\alpha$, $\alpha$ monotonically decreases until convergence. Once the value of $\alpha$ is found to be smaller than $1$, then the original problem is feasible and a feasible $\boldsymbol{\phi}$ has been found and BCD procedure could be stop. Otherwise, $\alpha$ converges to a value larger than $1$, and we have to claim that the problem is infeasible. The above discussion is summarized in Alg.~\ref{Alg-feas-prob}.

It should be noted that (P7) is itself nonconvex and its global optimal solution can hardly be obtained. Alg.~\ref{Alg-feas-prob} finds a suboptimal solution of (P7). Therefore Alg.~\ref{Alg-feas-prob} finding an objective value no greater than $1$ is a sufficient condition for feasibility of the original problem.

\begin{algorithm}[!t]
\caption{Feasibility Check Algorithm} \label{Alg-feas-prob}
\begin{algorithmic}[1]
\STATE Randomly initialize $\boldsymbol{\phi}^{(0)}$ with $\boldsymbol{\phi}^{(0)}_n=\kappa_n e^{j\theta_n}$ with $\kappa_n^2$ and $\theta_n$ uniformly distributed among $[0,1]$ and $[0, 2\pi)$ respectively, $\forall n\in\mathcal{N}$;
\STATE Initialize $\{\boldsymbol{w}_k^{(0)}\}$ via (\ref{eq:w_mmse_lin});
\STATE Initialize $\alpha^{(0)}:=\max_{k}\varepsilon_k(\boldsymbol{\phi}^{(0)},\boldsymbol{w}_k^{(0)})$;
\REPEAT
\STATE update $\boldsymbol{w}_k^{(t+1)}$ via (\ref{eq:w_mmse_lin});
\STATE update $\alpha^{(t+1)}$ and $\boldsymbol{\phi}^{(t+1)}$ by solving (P8);
\STATE $t++$;
\UNTIL{\emph{convergence or} $\alpha^{(t)}\leq1$}
\IF{$\alpha^{(t)}<=1$}
\STATE claim {\sf{Feasible}}, output $\boldsymbol{\phi}^{(t)}$ as feasible solution;
\ELSE
\STATE claim {\sf{Infeasible}}
\ENDIF
\end{algorithmic}
\end{algorithm

\section{Solving the Optimization} \label{sec-Analysis-Algorithms}

In this section we develop an algorithm to solve the Problem~(P3), whose objective has a sum-of-ratios form and is difficult to solve. Based on the discussion in previous section, we assume that the problem (P3) is feasible and we can find a feasible solution of (P3).

To attack (P3), we firstly introduce the an equivalent transformation of the rate function $R_k(\boldsymbol{\phi})$ \cite{Cioffi_wmmse, QingjiangShi_wmmse}, which will significantly simplify the optimization procedure, as will be shown later. The following identities hold:
\begin{subequations}
\label{wmmse_transform}
\begin{align}
&\!\!\!\!\!\!R_k(\boldsymbol{\phi})=\log(1+\gamma_k^\star(\boldsymbol{\phi}))\\
=&-\log\Big[\big(q_k^{-1}+\boldsymbol{h}_k^H\boldsymbol{W}_k^{-1}\boldsymbol{h}_k\big)^{-\!1}\Big]+\log(q_k))\\
\stackrel{(a)}{=}&\max_{\varpi_k\geq0}\Big\{\log(\varpi_k)-\varpi_k\Big[\big(q_k^{-1}+\boldsymbol{h}_k^H\boldsymbol{W}_k^{-1}\boldsymbol{h}_k\big)^{-1}\Big]\nonumber\\
&\quad\qquad+1+\log(q_k)\Big\}\label{wmmse_1st}\\
\stackrel{(b)}{=}&\max_{\varpi_k\geq0,\boldsymbol{v}_k}\Big\{-\varpi_k\Big[q_k\big|1-\boldsymbol{v}_k^H\boldsymbol{h}_k\big|^2+\boldsymbol{v}_k^H\boldsymbol{W}_k\boldsymbol{v}_k\Big]\nonumber\\
&\qquad\qquad+\log(\varpi_k)+1+\log(q_k)\Big\}\label{wmmse_2nd}\\
\triangleq&\max_{\varpi_k\geq0,\boldsymbol{v}_k}\widetilde{R}_k\big(\boldsymbol{\phi},\boldsymbol{v}_k,\varpi_k\big).
\end{align}
\end{subequations}
In the above step (a) can be readily verified by realizing the function in the maximization is concave with respect to $\varpi_k$ and its optima is obtained via
\begin{align}
\varpi_k^\star= q_k^{-1}+\boldsymbol{h}_k(\boldsymbol{\phi})^H\boldsymbol{W}_k(\boldsymbol{\phi})^{-1}\boldsymbol{h}_k(\boldsymbol{\phi}).\label{w_opt}
\end{align}
The transformation (b) holds since the function in (\ref{wmmse_2nd}) is quadratic concave in $\boldsymbol{v}_k$ and its optima is achieved at
\begin{align}
\boldsymbol{v}_k^\star=\boldsymbol{W}_k(\boldsymbol{\phi})^{-1}q_k\boldsymbol{h}_k(\boldsymbol{\phi}).\label{v_opt}
\end{align}

Abbreviating the notations $\boldsymbol{\varpi}\triangleq[\varpi_1,\cdots,\varpi_K]$ and $\boldsymbol{V}\triangleq[\boldsymbol{v}_1,\cdots,\boldsymbol{v}_K]$, we transform the problem (P3) as follows
\begin{subequations}
\begin{alignat}{2}
\textup{(P9):} \quad \min_{\boldsymbol{\phi},\boldsymbol{V},\boldsymbol{\varpi}} & \qquad \sum_{k=1}^K  \frac{A_k}{\widetilde{R}_k(\boldsymbol{\phi},\boldsymbol{v}_k,\varpi_k)} \\
\textup{s.t.} & \ \widetilde{R}_k(\boldsymbol{\phi},\boldsymbol{v}_k,\varpi_k))\geq r_k, \forall k\in\mathcal{K}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}.
\end{alignat}
\end{subequations}

Introducing an intermediate variable $\boldsymbol{\mu}\triangleq[\mu_1, \ldots, \mu_K]$, (P9) can be equivalently written as
\begin{subequations}
\begin{alignat}{2}
\textup{(P10):}\qquad \min_{\boldsymbol{\phi},\boldsymbol{V},\boldsymbol{\varpi},\boldsymbol{\mu}}& \sum_{k=1}^K  \mu_k     \\
~\hspace{-37pt}\textup{s.t.}&\ \frac{A_k}{\widetilde{R}_k(\boldsymbol{\phi},\boldsymbol{v}_k,\varpi_k)} \leq \mu_k, \forall k\in\mathcal{K},  \label{p11_frac_cnstr}\\
& \ \widetilde{R}_k(\boldsymbol{\phi},\boldsymbol{v}_k,\varpi_k)\geq r_k, \forall k\in\mathcal{K}, \label{p11_rate_cnstr}\\
&\ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}, \label{phi_cnstr}.
\end{alignat}
\end{subequations}

To solve (P10), we have the following lemmas, which are proved in the Appendix of the online full version \cite{MEC_IRS_fullpaper} (i.e., this paper):
\begin{lem} \label{lem-part1}
If $(\boldsymbol{\phi}^\star,\boldsymbol{V}^\star,\boldsymbol{\varpi}^\star,\boldsymbol{\mu}^\star)$ is a solution to (P10), then there exists $\boldsymbol{\lambda}^\star=[\lambda_1^\star, \ldots, \lambda_K^\star]$ such that $(\boldsymbol{\phi}^\star,\boldsymbol{V}^\star,\boldsymbol{\varpi}^\star)$ satisfies the Karush-Kuhn-Tucker (KKT) condition of the following problem (P11) when $\boldsymbol{\lambda}=\boldsymbol{\lambda}^\star$ and $\boldsymbol{\mu}=\boldsymbol{\mu}^\star$:
\begin{subequations}
\begin{alignat}{2}
\textup{(P11):}\min_{\boldsymbol{\phi},\boldsymbol{V},\boldsymbol{\varpi}}&\ \quad \sum_{k=1}^K\lambda_k\big(A_k\!-\!\mu_k \widetilde{R}_k(\boldsymbol{\phi},\boldsymbol{v}_k,\varpi_k)\big)\label{opt-p5-obj} \\
\rm{s.t.} & \ \widetilde{R}_k(\boldsymbol{\phi},\boldsymbol{v}_k,\varpi_k)\geq r_k, \forall k\in\mathcal{K}, \label{reduced_func_rate_cnstr}\\
& |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N} \label{reduced_func_model_cnstr}.
\end{alignat}
\end{subequations}
Also, we have
\begin{align}
\lambda_k^\star \widetilde{R}_k(\boldsymbol{\phi}^\star,\boldsymbol{v}_k^\star,\varpi_k^\star)&=1,  \forall k\in\mathcal{K}, \label{lem-part1-lambda}\\
\mu_k^\star \widetilde{R}_k(\boldsymbol{\phi}^\star,\boldsymbol{v}_k^\star,\varpi_k^\star)&=A_k ,  \forall k\in\mathcal{K}. \label{lem-part1-gamma}
\end{align}
\end{lem}

\begin{lem} \label{lem-part2}
If both \ding{172} and \ding{173} below hold:
\begin{itemize}
\item[\ding{172}]  $(\boldsymbol{\phi}^\star,\boldsymbol{V}^\star,\boldsymbol{\varpi}^\star)$ solves (P11) when $\boldsymbol{\lambda}=\boldsymbol{\lambda}^\star$ and $\boldsymbol{\mu}^\star=\boldsymbol{\mu}^\star$;
\item[\ding{173}] $(\boldsymbol{\phi}^\star,\boldsymbol{V}^\star,\boldsymbol{\varpi}^\star)$, $\boldsymbol{\lambda}^\star$, and $\boldsymbol{\mu}^\star$ satisfy (\ref{lem-part1-lambda}) and (\ref{lem-part1-gamma}),
\end{itemize}
then
$(\boldsymbol{\phi}^\star,\boldsymbol{V}^\star,\boldsymbol{\varpi}^\star,\boldsymbol{\mu}^\star)$ satisfies the KKT condition of (P10).
\end{lem}

According to Lemmas~\ref{lem-part1} and~\ref{lem-part2}, we can solve (P10) via tackling (P11). Specifically, we adopt the approach of~\cite{Jonga-Sum-of-Ratios} to find $(\boldsymbol{\phi}^\star,\boldsymbol{V}^\star,\boldsymbol{\varpi}^\star)$, $\boldsymbol{\lambda}^*$, and $\boldsymbol{\mu}^*$ satisfying \ding{172} and \ding{173} of Lemma~\ref{lem-part2} via alternatively performing the following two steps:

\begin{itemize}
  \item \textbf{Step-1: Solve (P11)} Given $\boldsymbol{\lambda}^{(t-1)}$ and $\boldsymbol{\mu}^{(t-1)}$, we solve (P10) and denote the solution by  $\big(\boldsymbol{\phi}^{(t)},\boldsymbol{V}^{(t)},\boldsymbol{\varpi}^{(t)}\big)$.

  \item \textbf{Step-2: Update $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$} Given $\big(\boldsymbol{\phi}^{(t)},\boldsymbol{V}^{(t)},\boldsymbol{\varpi}^{(t)}\big)$, we utilize the modified Newton (MN) method of~\cite{Jonga-Sum-of-Ratios} to obtain $\boldsymbol{\lambda}^{(t)}$ and $\boldsymbol{\mu}^{(t)}$.

\end{itemize}

For the update in \textbf{Step-1}, (P11) is nonconvex in $(\boldsymbol{\phi},\boldsymbol{V},\boldsymbol{\varpi})$ jointly. Therefore we still adopt the BCD method to solve it. Recall the discussion explaining (\ref{wmmse_transform}), the optimal $\varpi$ is obtained in (\ref{w_opt}) when  $(\boldsymbol{\phi},\boldsymbol{V})$ is fixed, and the optimal $\boldsymbol{V}$ can be determined in (\ref{v_opt}) when $(\boldsymbol{\phi},\boldsymbol{\varpi})$ is fixed. Therefore we need to investigate the optimization of $\boldsymbol{\phi}$. When $(\boldsymbol{V},\boldsymbol{\varpi})$, the optimiztion of (P11) with respect to $\boldsymbol{\phi}$ can be written as

\begin{subequations}
\begin{alignat}{2}
\!\!\!\!\!\!\textup{(P12):} \max_{\boldsymbol{\phi}} \ & \boldsymbol{\phi}^H\boldsymbol{T}\boldsymbol{\phi}+2\mathrm{Re}\{\boldsymbol{t}^H\boldsymbol{\phi}\}+c\\
\sf{s.t.} &\ \boldsymbol{\phi}^H\boldsymbol{T}_k\boldsymbol{\phi}+2\mathrm{Re}\{\boldsymbol{t}_k^H\boldsymbol{\phi}\}+c_k\geq r_k, \forall k\in\mathcal{K}, \\
&\ \ |\boldsymbol{\phi}_n|\leq1, \forall n\in\mathcal{N}
\end{alignat}
\end{subequations}
with the parameters defined as follows:
\begin{subequations}
\begin{align}
\!\!\!\!\boldsymbol{T}_k\triangleq &-\varpi_k\Big[\sum_{j}q_j\boldsymbol{F}_j^H\boldsymbol{v}_k\boldsymbol{v}_k^H\boldsymbol{F}_j\Big], \label{eq_cnstr_T}\\
\boldsymbol{t}_k\triangleq & -\!\varpi_k\Big[q_k\big(\boldsymbol{v}_k^H\boldsymbol{h}_{{\rm{d}},k}\!-\!1\big)\boldsymbol{F}_k^H\boldsymbol{v}_k\!+\!\sum_{j\neq k}q_j\boldsymbol{v}_k^H\boldsymbol{h}_{{\rm{d}},j}\boldsymbol{F}_j^H\boldsymbol{v}_k\Big] \nonumber\\
c_k \triangleq & \log{\varpi_k}-\varpi_kq_k\!+\!1\!+\!\log{q_k}-\sigma^2\varpi_k\|\boldsymbol{v}_k\|_2^2 \\
&\quad -\varpi_k\sum_{j }q_j\big|\boldsymbol{h}_{{\rm{d}},j}^H\boldsymbol{v}_k\big|^2+2\varpi_kq_k\mathrm{Re}\big\{\boldsymbol{h}_{{\rm{d}},k}^H\boldsymbol{v}_k\big\} \nonumber \\
\boldsymbol{T} \triangleq & \sum_{k}\lambda_k\mu_k\boldsymbol{T}_k,
\boldsymbol{t} \triangleq \sum_{k}\lambda_k\mu_k\boldsymbol{t}_k,
c \triangleq \sum_{k}\lambda_k\mu_kc_k.
\end{align}
\end{subequations}
The above problem is convex and can be conveniently solved numerically.

For the update in \textbf{Step-2}, we introduce the following notations to simplify the subsequent expositions. According to (\ref{lem-part1-lambda}) and~(\ref{lem-part1-gamma}) define
\begin{align}
\Lambda_k^{(t)}(\lambda_k) & \triangleq \lambda_k \widetilde{R}_k(\boldsymbol{\phi}^{(t)},\boldsymbol{v}_k^{(t)},\varpi_k^{(t)}) - 1,  \forall k\in\mathcal{K}, \label{eq-Lambda-k-t} \\
\Gamma_k^{(t)}(\mu_k) & \triangleq \mu_k\widetilde{R}_k(\boldsymbol{\phi}^{(t)},\boldsymbol{v}_k^{(t)},\varpi_k^{(t)}) - A_k,  \forall k\in\mathcal{K}. \label{eq-Gamma-k-t}
\end{align}
Then according to the modified Newton's (MN) method in \cite{Jonga-Sum-of-Ratios}, we update $\boldsymbol{\lambda}$ and $\boldsymbol{\mu}$ in the  following way
\begin{align}
\lambda_{k,i}^{(t)} & : = \lambda_k^{(t-1)} - \xi^i \frac{\Lambda_k^{(t)}(\lambda_k^{(t-1)})}{ \frac{\textup{d} \Lambda_k^{(t)}(\lambda_k) }{\textup{d}\lambda_k }\big|_{\lambda_k = \lambda_k^{(t-1)}} }\nonumber\\
&= \lambda_k^{(t-1)} - \xi^i \frac{\Lambda_k^{(t)}(\lambda_k^{(t-1)})}{\widetilde{R}_k(\boldsymbol{\phi}^{(t)},\boldsymbol{v}_k^{(t)},\varpi_k^{(t)})}, \forall k\in\mathcal{K} \label{def-lambda-k-i} \\
\mu_{k,i}^{(t)} & := \mu_k^{(t-1)} -\xi^i \frac{\Gamma_k^{(t)}(\mu_k^{(t-1)})}{ \frac{\textup{d} \Gamma_k^{(t)}(\mu_k) }{\textup{d}\mu_k }\big|_{\mu_k=\mu_k^{(t-1)}}} \nonumber \\
&= \mu_k^{(t-1)} -\xi^i  \frac{\Gamma_k^{(t)}(\mu_k^{(t-1)})}{\widetilde{R}_k(\boldsymbol{\phi}^{(t)},\boldsymbol{v}_k^{(t)},\varpi_k^{(t)})},\forall k\in\mathcal{K}. \label{def-gamma-k-i}
\end{align}

Clearly, (\ref{lem-part1-lambda}) and (\ref{lem-part1-gamma}) will reduce to the standard Newton's method update when $\xi^i$ above is $1$. In the modified Newton's method, a suitable $i$ should be found. Let $i^{(t)}$ denote the smallest integer among $i\in\{0,1,\ldots\}$ satisfying that
\begin{align}
&\sum_{k=1}^{K}  \Bigg\{\left(\Lambda_k^{(t)}( \lambda_{k,i}^{(t)}) \right)^2 + \sum_{k=1}^{K}  \left(\Gamma_k^{(t)}( \mu_{k,i}^{(t)}) \right)^2\Bigg\} \label{eq-update-rule} \\
&\leq \left(1-\xi^i\epsilon \right)^2  \left[\sum_{k=1}^{K}  \left(\Lambda_k^{(t)}( \lambda_{k}^{(t-1)}) \right)^2 + \sum_{k=1}^{K}  \left(\Gamma_k^{(t)}( \mu_{k}^{(t-1)}) \right)^2 \right], \nonumber
\end{align}
where $\epsilon$ is a predefined precision parameter and $\xi$ is a positive constant smaller than $1$. Once $i^{(t)}$ is determined, $\boldsymbol{\lambda}^{(t)}$ and $\boldsymbol{\mu}^{(t)}$ should be updated as
\begin{align}
\lambda_{k}^{(t)} & \leftarrow \lambda_{k,i^{(t)}}^{(t)} \textup{in  (\ref{def-lambda-k-i})}\textup{\ with $i$ being $i^{(t)}$} , \label{set-lambda-k-t-plus-1} \\
\mu_{k}^{(t)}& \leftarrow \mu_{k,i^{(t)}}^{(t)}  \textup{in (\ref{def-gamma-k-i})}\ \textup{with $i$ being $i^{(t)}$}, \label{set-gamma-k-t-plus-1}
\end{align}

According to the analysis in \cite{Jonga-Sum-of-Ratios} the value
\begin{align}
\delta^{(t)}\triangleq\sum_{k=1}^{K}  \left(\Lambda_k^{(t)}( \lambda_{k}^{(t)}) \right)^2 + \sum_{k=1}^{K}  \left(\Gamma_k^{(t)}( \gamma_{k}^{(t)}) \right)^2 \label{eq-two-sum}
\end{align}
converges to zero as $t\rightarrow\infty$, and the rate of convergence is global linear and local quadratic. When $\delta^{(t)}$ in (\ref{eq-two-sum}) converges to zero, according to (\ref{eq-Lambda-k-t}) and (\ref{eq-Gamma-k-t})
\begin{align}
&\Lambda_k^{(t)}( \lambda_{k}^{(t)})\!=\!0 \Rightarrow  \lambda_k^{(t)} \widetilde{R}_k(\boldsymbol{\phi}^{(t)},\boldsymbol{v}_k^{(t)},\varpi_k^{(t)})\!\!=\!\!1,  \forall k\in\mathcal{K}, \label{eq-Lambda-k-t-terminate} \\
&\Gamma_k^{(t)}( \mu_{k}^{(t)})\!=\!0 \Rightarrow \mu_k^{(t)} \widetilde{R}_k(\boldsymbol{\phi}^{(t)},\boldsymbol{v}_k^{(t)},\varpi_k^{(t)})\!\!=\!\!A_k,  \forall k\in\mathcal{K}. \label{eq-Gamma-k-t-terminate}
\end{align}
Hence we will arrive at (\ref{lem-part1-lambda}) and (\ref{lem-part1-gamma}) and  consequently, by Lemmas~\ref{lem-part1} and~\ref{lem-part2}, the KKT condition of (P10). Therefore (\ref{eq-two-sum}), i.e. (\ref{eq-Lambda-k-t-terminate}) and (\ref{eq-Gamma-k-t-terminate}), performs as a termination condition for the iterative process.

\begin{algorithm}[!t]
\caption{Optimizing (P10)} \label{Alg-opt-overall}
\begin{algorithmic}[1]
\STATE Set $\xi\in(0,1)$; set the positive $\epsilon$, $\rho$ sufficiently small;
\STATE Invoke Alg.~\ref{Alg-feas-prob} to obtain a feasible $\boldsymbol{\phi}^{(0)}$; $t=0$;
\STATE Initialize $\boldsymbol{\varpi}^{(0)}$ and $\boldsymbol{V}^{(0)}$ via (\ref{w_opt}) and (\ref{v_opt}) respectively;
\STATE Set $\lambda_{k}^{(0)}=R_k(\boldsymbol{\phi}^{(0)})^{\!-\!1}$ and $\mu_{k}^{(0)}\!=\!A_k/R_k(\boldsymbol{\phi}^{(0)})$;
\REPEAT
\REPEAT
\STATE update $\boldsymbol{\varpi}^{(t)}$ by (\ref{w_opt});
\STATE update $\boldsymbol{V}^{(t)}$ by (\ref{v_opt});
\STATE update $\boldsymbol{\phi}^{(t)}$ by solving (P12);
\UNTIL{\textit{convergence}}
\STATE update $\lambda_k^{(t\!+\!1)}$ and $\mu_k^{(t\!+\!1)}$ by (\ref{set-lambda-k-t-plus-1}) and (\ref{set-gamma-k-t-plus-1}) respectively;
\STATE Evaluate $\delta^{(t)}$ by (\ref{eq-update-rule});
\STATE  $t\leftarrow t+1$;
\UNTIL{$\delta^{(0)}<\rho$}
\end{algorithmic}
\end{algorithm

\section{Simulation} \label{sec-Simulation}

\begin{figure}[ht]
\centering
  \includegraphics[width=0.5\textwidth]{images/Sim_Setting.eps} \vspace{-2pt}\caption{Simulation Setting-Up}
 \label{fig:system}
\end{figure}

In this section, we present numerical results to validate our proposed algorithms. In the experiment, we adopt a similar system setting-up reported in \cite{wu2018intelligent}. As shown in Fig.~\ref{fig:system}, the system comprises one AP with $M=4$ antennas, 4 single-antenna mobile users and one IRS device. The number of reflecting elements $N$ of the IRS will take different values from $\{30,60,90,120\}$. The distance between the AP and IRS is $50$m and it is assumed that the signal propagation environment between the AP and the IRS is dominated by the line-of-sight (LoS) link. Considering the mobility of the MDs will deteriorate the wireless propagation environment, we assume that the signals sent by MDs to both the AP and the IRS experience $10$dB penetration loss, independent Rayleigh fading and the pathloss exponent of $3$. The noise has -$170$dBm/Hz and the channel bandwidth is $100$KHz, so $\sigma_k^2 = 10^{-12}$mW. We set the transmission power of each MD as $10$dBm. Besides we assumed the antenna gain of both the AP and user is $0$dBi and that of each reflecting element at the IRS is $5$ dBi \cite{antennaRFID2009}. The distance between the AP and the  IRS is $50$m and the 4 MDs are located in a row parallel to the AP-IRS line and with a $5$m interval between the adjacent peers.

\begin{figure}[ht]
\centering
  \includegraphics[width=0.5\textwidth]{images/itinery_feas_alg.eps} \vspace{-2pt}\caption{Convergence of Feasibility Check Alg.~\ref{Alg-feas-prob}}
 \label{fig:feas_check_convg}
\end{figure}

Fig.~\ref{fig:feas_check_convg} shows us the behavior of the Alg.~\ref{Alg-feas-prob} for checking the feasibility associated with a given set of information rate requirements. In our experiment, we set $N=30$. To ensure fairness between different MDs, we just assume that all MDs will be guaranteed one identical information rate. We vary this common information rate requirement from $2.1$Nats/s/channel-use to $2.9$Nats/s/channel-use and perform Alg.~\ref{Alg-feas-prob}. Note that in our experiment one random channel realization is generated according to system setting explained above and then fixed. For each specific information rate requirement, $\boldsymbol{\phi}$ is started from a common initial point, which is randomly chosen. According to Fig.~\ref{fig:feas_check_convg}, Alg.~\ref{Alg-feas-prob} generally converges within 10 iterations. The convergent $\alpha$ value increases when the information rate requirement inflates. It can be inferred from Fig.~\ref{fig:feas_check_convg} that maximal feasible common information rate requirement should lie between $2.3$ to $2.5$ Nats/s/channel-use.

\begin{figure}[ht]
\centering
  \includegraphics[width=0.5\textwidth]{images/irs_vs_feas.eps} \vspace{-2pt}\caption{The Impact of IRS on Feasibility of Rate Requirements}
 \label{fig:irs_vs_feas}
\end{figure}

Fig.~\ref{fig:irs_vs_feas} illustrates the impact of IRS on enhancing the guaranteed information rate to MDs. In our experiment, we still assume that all MDs require one common information rate requirement. For each specific $N$ and a predefined rate requirement, $600$ channel realizations are randomly generated according to the above settings. For each channel realization, we generate a batch of random IRS elements and then invoke Alg.~\ref{Alg-feas-prob} to optimize the IRS elements trying to reach feasibility. We compare the probability of getting feasibility for the case of no-IRS, random IRS and optimized IRS in Fig.~\ref{fig:irs_vs_feas}. The result in Fig.~\ref{fig:irs_vs_feas} undoubtedly convince us the significance of IRS. Among the range of rate requirement in the figure, the optimized IRS can improve the feasibility probability by $40\%$ to $50\%$. Besides, more improvement could be obtained with more number of IRS elements being deployed.

\begin{figure}[ht]
\centering
  \includegraphics[width=0.5\textwidth]{images/itinery_sor_obj.eps} \vspace{-2pt}\caption{Convergence of Obj. of (P3) by Alg.~\ref{Alg-opt-overall}}
 \label{fig:itinery_sor_obj}
\end{figure}

\begin{figure}[ht]
\centering
  \includegraphics[width=0.5\textwidth]{images/itinery_sor_delta.eps} \vspace{-2pt}\caption{Convergence of $\delta^{(t)}$ by Alg.~\ref{Alg-opt-overall}}
 \label{fig:itinery_sor_delta}
\end{figure}

Fig.~\ref{fig:itinery_sor_obj} and Fig.~\ref{fig:itinery_sor_delta} show the convergence behavior of Alg.~\ref{Alg-opt-overall}. In the experiment we generate a group (5 in the figure) of random channel realizations, for which a feasible point can be found (e.g. by Alg.~\ref{Alg-feas-prob}). For each specific channel realization, we perform Alg.~\ref{Alg-opt-overall} starting from the feasible point. The objective value and the $\delta^{(t)}$ defined in (\ref{eq-two-sum}) are presented in Fig.~\ref{fig:itinery_sor_obj} and Fig.~\ref{fig:itinery_sor_delta} respectively. As shown in Fig.~\ref{fig:itinery_sor_obj}, Alg.~\ref{Alg-opt-overall} usually converges in several outer loops. Fig.~\ref{fig:itinery_sor_delta} convince us that Alg.~\ref{Alg-opt-overall} exhibits superlinear convergence rate. The results in Fig.~\ref{fig:itinery_sor_obj} and Fig.~\ref{fig:itinery_sor_delta} suggest that Alg.~\ref{Alg-opt-overall} has fast converge and is therefore very efficient.

\begin{figure}[ht]
\centering
  \includegraphics[width=0.5\textwidth]{images/irs_vs_obj.eps} \vspace{-2pt}\caption{Optimizing (P3)}
 \label{fig:irs_vs_obj}
\end{figure}

Finally Fig.~\ref{fig:irs_vs_obj} illustrates the effect of Alg.~\ref{Alg-opt-overall} in optimizing (P3). Taking into account of fairness of different MDs, we just assume that $A_k$ are equal to each MDs and consequently set $A_k=1$ $\forall k\in \mathcal{K}$. For each specific $N$, we generate a bunch (10 in the figure) of random channel realizations and their feasible solutions which are found via randomization. We compare the objective values of (P3) obtained via Alg.~\ref{Alg-opt-overall} with those associated with those random IRS elements. As presented in Fig.~\ref{fig:irs_vs_obj}, for various values of $N$, Alg.~\ref{Alg-opt-overall} can effectively decrease the objective value of (P3) and therefore equivalently increase the earnings of the edge server.

\section{Conclusion} \label{sec-Conclusion}

In this paper we study the mobile edge computing problem with the aid of the novel technology of IRS. In details, we formulate the earning maximization problem with a group of information rate constraints in the presence of IRS device. We develop an iterative algorithm which can efficiently identify the feasibility of the information rate requirements and find a feasible solution. Besides we also develop an algorithm to optimize the earning of the edge server for loading computing. Substantial numerical results have shown that our proposed IRS aided scheme can significantly improve the guaranteed information rate to MDs by the AP and also effectively enlarge the earning of the edge server.

redpajama_set_name: RedPajamaArXiv


I was flying from SFO to Delhi (India) during February on a quarterly business trip. From SFO to Newark (EWR) it was United, and then Continental to Delhi (approx 14 hours non-stop). My flight is fully confirmed all the way and I am looking forward to the ordeal armed with a laptop and a book. The whole trip started on a pretty bad note; they tell me that I will get the EWR/Delhi boarding pass at the Continental "gate." I should have just gone home; Bangalore could wait.
We all got in the plane and the pilot announced that a video monitor is not working. Then they went looking for a part somewhere in the airport. Replacement was done and things still didn't work. It was already getting late and my four hour margin at Newark (EWR) was quickly shrinking. In reality I shouldn't have bothered given what was coming next, but I did. We all got out of the plane looking for an alternative route to Newark (there were none). But then suddenly the problem was fixed and soon we are in the air headed to EWR from SFO.
We land in Newark. By this time I have found a wonderful Verizon lady headed to Delhi on the same flight. We all make our way to the gate where the Continental flight will leave with about 45 minutes to spare. I ask for my boarding pass and there is none to be had. The flight is overbooked. Lots of people are milling around and it is nearly 10 pm. Apparently some other international flights were canceled and everyone is booked to India on this single plane. It is as complete pandemonium as I have seen in a US airport.
An hour goes by and the plane is full. Now they look at overbooked people. There is no logic to who they ask to board; there is no list or priority order at all. All of a sudden the key "guy" (I know his name but will omit here) asks me to board. I am half the way in, and then he changed his mind to let someone else go. Then someone else was a family with a kid, so I didn't really get upset but the whole process was frustrating.
Finally, the "key guy" said you come back tomorrow and we will have a place for you. There were several people in the same situation. We all said fine, and were given hotel/food vouchers and about $1000 (or lower) which was OK. Frustrating, but these things happen. Next flight is after 24 hours.
An hour later and we are in a hotel with cake for dinner (very late for food, everything is closed). At least I had made a couple of friends by then.
Next day we all visit Times Square courtesy of Continental. We are dutifully back at Continental's counter at 6 pm for our "boarding passes." Guess what – there are none to be had. The "key guy" is missing; customer service can not find him. There are no "reserved boarding passes." We are back with the crowd just like the day before.
Again, the same drama plays but it is only $400 this time. I decline; going to Times Square again is not fun. And my yesterday's clothes are in a bag and I now have brand new clothes thanks to Continental. They are now feeding, clothing and providing me shelter with some spend money. Perhaps some aggravation is worth it.
I see many passengers being visibly upset; maybe that is why the "key guy" does not appear. Life is too hard for him. I wonder how Continental expects to be in international business given the high standards set by multiple Asian airlines, including carriers like Jet Airways from India, a country which is not exactly known for its high standards of customer service.
Meanwhile I am deciding whether to head back to SFO some $600 richer or take the $400 and wait another day. I did get a seat on the plane but… never again.
Are you saying they gave you flight vouchers worth a $1000.00 then $400.00?
Surely, you are not meaning they gave you cash?
When I have been bumped or delayed, I have only been given food and hotel vouchers. I have never been given "spending" money. Nor, have they ever given me money for clothing.
The joys of Newark Liberty….my rule of thumb is never fly through New York/Newark. Bad things always happen because of air traffic congestion.


Johannesburg- Former Muvhango actress Simphiwe Ngema is set to host e.tv's new cultural show, Isiko which debuts on 10 January.
Simphiwe, who was popularly known for her character Thuli on the soapie, left fans saddened when she announced her exit.
This week e.tv announced that the channel will be bringing viewers a new show, which will showcase the country's rich cultural diversity.
And, each week Simphiwe will be the one to take viewers on this journey.
The actress will travel the country to meet with traditional leaders and community members of the Va-Tsonga, Zulu, Xhosa, Ndebele, Swati, Vha-Wenda, Basotho, Batswana, Bapedi, Jewish, Indians and Afrikaans groups.
The show debuts on 10 January at 7,30pm on eKasi + (Channel 105).


Spanish Dance is a fiery and passionate art form made up by sound from castanets, intricate footwork and flamboyant costumes. We welcome dancers age 3 to 18, to join our Spanish Dance classes.
Start by learning the basics and progressing through the levels by taking part in annual exams and medal tests.
At the Love To Dance Academy we currently offer Spanish Exams for Spanish Stars, Spanish Sparkles, Beginner Junior, Beginner Senior, Level 1, 2, 3 and 4. We welcome students age 4+ to our Spanish Exam program.
The Medal tests are similar to examinations and a way for a young dancer to mark his or her progress in the art of dance. They have been very popular for many years in various countries, sometimes also referred to as "Badge Tests". After successfully completing the test sequence and dance, candidates will receive a medal and certificate. This is particularly good for the students because of the "exam" environment practise they will get. We recommend that all students partake in these Medal Tests.


Lynn Phillips, president of the Southlake Residential Association, speaks during a public meeting about potentail Hoover sewer rate increases at the Jefferson State Community College Shelby-Hoover campus on Tuesday, July 31, 2018.
Tim Westhoven, the city of Hoover's chief operations officer, speaks during a public meeting about potential Hoover sewer rate increases at the Jefferson State Community College Shelby-Hoover campus on Tuesday, July 31, 2018. Next to him is the city's chief financial officer, Melinda Lopez.
Dan Murchison, a resident of The Sanctuary at Caldwell Crossings, speaks during a public meeting about potential Hoover sewer rate increases at the Jefferson State Community College Shelby-Hoover campus on Tuesday, July 31, 2018.
Hoover Mayor Frank Brocato speaks during a public meeting about potential Hoover sewer rate increases at the Jefferson State Community College Shelby-Hoover campus on Tuesday, July 31, 2018.
About 15 people showed up for a public meeting about potential rate increases for the Hoover sewer system at the Jefferson State Community College Shelby-Hoover campus on Tuesday, July 31, 2018.
Melinda Lopez, the chief financial officer for the city of Hoover, Alabama, speaks during a public meeting about potential rate increases for the Hoover sewer system at the Jefferson State Community College Shelby-Hoover campus on Tuesday, July 31, 2018.
Hoover City Administrator Allan Rice speaks during a public meeting about potential rate increases for the Hoover sewer system at the Jefferson State Community College Shelby-Hoover campus on Tuesday, July 31, 2018.
Only about 15 people showed up tonight for a second public meeting about a potential sewer rate increase in Hoover — much different than the 150 or so people who packed the Riverchase Country Club last month.
The low turnout at Jefferson State Community College's Shelby-Hoover campus could have been due to the rainy weather, but one resident said people may now simply understand the necessity of a sewer rate increase after 14 years without one.
"When you understand what they're saying, it's a no-brainer," said Sarah Moseley, who lives in The Sanctuary at Caldwell Crossings off Caldwell Mill Road.
Hoover's sewer system, which serves about 4,500 residential customers and 200 commercial customers in and around the Riverchase, Inverness and Southlake communities, has not been taking in enough revenue to provide long-term support for the system's future, city officials say.
Sewer rates have not increased since 2004, but more revenue is needed to keep the system functioning properly, Chief Financial Officer Melinda Lopez told those in attendance tonight.
Revenues are strong enough to pay for current operations, pay off debt obligations and provide positive cash flows, but the sewer system has had negative retained earnings since 2008, Lopez said.
An engineering study by Goodwyn, Mills & Cawood indicates the Hoover sewer system has $7.23 million in capital projects needed in the next five years. A separate rate study done by utility consultant Jackson Thornton indicated Hoover's sewer system needs about $6.5 million in revenue per year to get back on track, and is currently $2.3 million short of that.
The current average residential customer on the Hoover sewer system produces about 6,000 gallons of wastewater per month, which would cost a homeowner $36.66 per month, Lopez said.
To make up the $2.3 million shortfall, the average residential sewer bill would need to rise by $30.53 a month, Lopez said. However, City Administrator Allan Rice said that doesn't mean everybody's sewer bill would go up that much.
Most city officials are not in favor of a flat fee per residence, but rather a sewer bill that factors in the amount of wastewater produced, Rice said. That way, a single person would likely pay much less than a household with five or seven people in it, he said.
The Hoover sewer system currently doesn't charge a base fee. It's all based on the amount of water each homeowner uses, Lopez said. One option city officials are considering would add a base fee for everyone to cover fixed costs such as pipes, and then base the rest of the sewer bill on usage, Rice said.
Mayor Frank Brocato tonight emphasized that no decisions have been made. City leaders are simply explaining the situation to residents and getting feedback, he said.
Dan Murchison, another resident of that community, said nobody wants a rate increase, but it looks like they have to have it. He said he's just glad to see Hoover doing something about it before it gets into a worse situation like Jefferson County had.
Lynn Phillips, president of the Southlake Residential Association and a civil engineer and licensed wastewater operator who has been in the utility business 35 years, said he believes some level of rate increase is needed, but he recommended the city consider using some property tax money to help supplement the money paid directly by sewer customers.
When the Riverchase and Inverness communities were annexed, former city leaders made commitments for the city to help pay for sewer operations, he said. The thinking was that adding the Riverchase and Inverness communities into Hoover would serve as a economic catalyst for the city and help increase the city's property and sales taxes, he said.
Rice said there probably are many opinions about whether the city should subsidize the sewer system with other city revenues, but he would guess that a majority of Hoover residents who now are paying much higher sewer rates on the Jefferson County system would not be in favor of the city giving a subsidy to only those 4,500 people on the city sewer system. That's especially considering that some of the people on the Hoover sewer system live in unincorporated Shelby County, he said.
Phillips said it may be a hard sell, but using property taxes to help pay for sewer systems is common in some parts of the United States.
Phillips also recommended that city leaders develop a strategic plan for the sewer system and long-range financial plan that goes beyond five years.
Brocato said the final decision lies with the Hoover City Council, and he expects the council to have a public hearing and address the matter later this year.
One key piece of feedback he has heard so far is that people want the city to take action and not linger on this decision too long.


The Museum was inaugurated on the 27th May 1973, date of the 32nd anniversary of the Battle of Crete. It includes models of ships, various nautical instruments and objects such as range finders, barometers, wind-gauges, course and speed abaci, torpedo firing computer, magnetic boat compasses, meteorological instruments, sextants, a planisphere, torpedo-boat steering gear, etc.
There are also paintings, portraits, authentic historic photographs, such as the raising of the flag of the Union of Crete with Greece in the Firka fortress at Chania, on the 1st December 1913, etc. Also exhibited are relics such as uniforms and other memorabilia which belonged to the heroic dead of the Greek Navy, guns which had been used in the war of 1912-13, etc.
All the exhibits from the naval tradition of Greece are classified according to their historical periods, that is: Bronze age-Classical and Hellenistic (2800 BC-6th century AD). Venetian period (1200 AD to the 17th century), Byzantine and post-Byzantine period (650 AD to the 18th century), the period of the Ottoman rule-the struggles for Independence (1650-1909), the Balkan Wars (1912-1913), World War II (1940-1944), the post-War period (after 1944).


text:
The Karmapa
News from around the Net
Karmapa Foundation
Helpers and Givers
His Holiness the 17th Karmapa in the U.S.A. Public Events Schedule for 2015
Karmapa Foundation organized for His Holiness a shirt-time trip to the United States. During two months, they were regulating all the planned events along with some invited headliners. Below you can see full information about Karmapa's lectures in different cities of the US.
PUBLIC TEACHINGS
Where: Stanford, California
Name of speech: "Interconnection between technological progress and nature"
Partners of the occasion:
CCARE (Stanford);
MIT (Massachusetts).
Follow this link to see the full description of the speech and pictures. The online translation is recorded here.
Where: San Francisco, California
Time: 9:00 a.m.; 11:00 a.m.
Follow this link to watch the broadcast.
NORTHERN CALIFORNIA TIBETAN ASSOCIATION
Where: South Richmond, California
For pictures click here, and the broadcast is here.
GYUTO FOUNDATION
Where: East Richmond Heights, California
All reports and recorded materials are here.
Where: Redlands, California
Name of speech: "Interrelationship".
You can see the pictures of the event by clicking hereand video here.
Where: Cambridge, Massachusetts
Topic: "Ways of caring for the planet in the 21st century".
Watch the recorded video here. View pictures here.
Where: Princeton, New Jersey
Lecture about Buddhism and social problems (environment, gender, etc.)
For detailed report go here. Visual materials are here.
KARMA THEGSUM CHÖLING
Where: Mount Laurel, New Jersey
Time: from10:00 a.m. to 12:00 p.m.; from 2:00 p.m. to 4:00 p.m.
Teaching: "Meditation step by step".
The report is available on this page. For online translation go here.
UNIVERSITY OF YALE
Where: New Haven, Connecticut
Chubb Fellowship Discourse
What is Chubb Fellowship here. Written report is here, visuals – here.
NEW YORK and NEW JERSEY TIBETAN COMMUNITY
Where: Hunter College, NY
Time: 6:30 p.m.; 8:30 p.m.
Dialogue with public
Follow this link to watch records.
DANANG FOUNDATION
Where: New York City
When: 11/04/2015 – 12/04/2015
Time: from 9:30 a.m. to 12:00 p.m. on Saturday; from 9:30 a.m. to 12:00 p.m. on Sunday;from 3:00 p.m. to 5:30 p.m. on Sunday
First topic: "Four Teachings of Gampopa".
Watch the first a href="https://www.youtube.com/watch?v=VCDCxHRgw3A" target="_blank" rel="noopener noreferrer">topic video here, webcast from the 2nd lecture here, and the third record here.
Where: NYCity
Expanding opportunities for followers from Tibet and Himalayi
Visual materials of the lecture for KSS members and pictured description are here.
Time: from 7:00 p.m. to 9:00 p.m.
Social dialogue
Written reporting is viewed here. The broadcast is here.
KARMA TRIYANA DHARMACHAKRA
Where: Kingston, NY
Time: from 10:00 a.m. to 12:00 p.m.; from 3 p.m. to 5:00 p.m.
Lecture: "Developing of true compassion"
Video materials: morning event – here, evening event – here. Reports: morning event – here, evening event – here.
Where: Woodstock, NY
When: 19/04/2015-20/04/2015
Time: from 10:00 a.m. to 12:00 p.m.; from 2:30 p.m. to 4:30 p.m. on Sunday; from 10:00 a.m. to 12 p.m.; from 2:30 p.m. to 4:30 p.m. on Monday
Lectures: The Four Dharmas of Gampopa
The organizers provided translations of all lectures during 19 and 20 April in Chinese, Spanish, English. Please watch video from the first day: morning lecture is here, afternoon lecture is here. Observe 2nd-day sessions: 10-12 – here, 2:30-4:30 p.m. – here.
Where: Red Hook, NY
Time: 10 a.m.; 12:00
Learning Buddhist Medicine
Find the pictures and written description of the teaching here. Find video records here.
KAGYU THUBTEN CHÖLING MONASTERY
Where: Wappingers Falls, NY
Time: after 12:00
For visual materials go here. Reporting and pictures from KTC are here.
The Adventist Development and Relief Agency
Where: Manhattan, NY
Watch the record here. All reports are here.
TERGAR MEDITATION COMMUNITY
Where: Madison, Wisconsin
The essence and basic concepts of Kagyü prayer
The full description is placed here. Online translation is viewed here.
Speech by R. Davidson and S. Dimidzhyan on "How Buddhism and Science Affect Human Welfare."
Description of the occasion is here, the live session recorded here.
TAFM
Where: Minneapolis, Minnesota
How to make the world better
All the written reports are placed here, and the webcasts are here.
TIBETAN ASSOCIATION OF WASHINGTON
Where: Lynnwood
Durable expansion of opportunities
View the broadcast here.
SAKYA MONASTERY
Where: Seattle, Washington
Time: 10:30 a.m.; 11:30 a.m.
Green Tara mantra of Tibetan Buddhism
For video records go here.
NALANDABODHI
Time: from 2:00 p.m. to 4:00 p.m.; from 7:00 p.m. to 9:00 p.m.
Debate with students heads on justice in the modern world: teaching and call to action.
Description of the first day is placed here, an overview of the 2nd day is placed here.
© 2023 Karmapa America 2015

redpajama_set_name: RedPajamaCommonCrawl


text:
UConn Law In the Media
Hillary Greene in New York Times: What Will Each Senator Say About Trump? We Deserve to Know (Jan 16, 2020)
Alexandra D. Lahav in Washington Post: Major drugstore chains sue doctors in sprawling federal opioid case (Jan 8, 2020)
Sara C. Bronin in Huffington Post: Trump Walks Back Threat To Commit War Crimes Against Iran's Cultural Sites (Jan 7, 2020)
Sara C. Bronin in Cheddar: Attacking Iranian Cultural Sites Would Be War Crime, Pentagon Says After Trump Threat (Jan 7, 2020)
Sara C. Bronin in Sirius POTUS 124: Cultural Sites and War Crimes (Jan 6, 2020)
Sara C. Bronin in CBC News Network: Trump threatens Iran's cultural sites (Jan 6, 2020)
Sara C. Bronin in Fox 61: Elected officials and local experts react to President Trump's tweets threatening the middle east (Jan 6, 2020)
Sara C. Bronin in Los Angeles Times: Opinion: Destroying cultural heritage sites is a war crime (Jan 5, 2020)
Richard Pomp in Law360: Photo Retailer B&H Knocks NY AG's Tax Savvy In Dismiss Bid (Jan 2, 2020)
Leslie C. Levin in Hartford Courant: Hartford police officer sues blogger Kevin Brookman seeking to unmask anonymous commenters (Jan 2, 2020)
View more Faculty In the Media »
The awards, accomplishments, speaking engagements and news of the distinguished faculty at UConn School of Law:
Professor Peter Kochenburger Named to New York Consumer Protection Task Force
Professor Peter Kochenburger, Executive Director of the Insurance LLM Program and Deputy Director of the Insurance Law Center, has been named to the newly created Consumer Protection Task Force, a group within New York's Department of Financial Services that will further it's mission to protect consumers by providing formal input on the department's consumer engagement, policy development and research. One of the task force's first priorities will be to help the Department of Financial Services to build support for and implement consumer protections, including measures to strengthen regulatory oversight of debt collectors, cracking down on elder financial abuse, increased access to affordable banking services, and strengthening the state's consumer protection laws. Members were selected based on their experience and expertise in the areas of economic justice, housing, health and debt collection, and advocacy on behalf of communities throughout New York.
Professor Sachin Pandya Serves as Working-Paper Discussant at Association of American Law Schools Annual Meeting
Professor Sachin Pandya served as a working-paper discussant at the annual meeting of the Association of American Law Schools on Jan. 4, 2020, at a panel titled, "New and Emerging Voices in Workplace Law." It was jointly sponsored by the Association of American Law Schools sections on employment discrimination law and on labor relations and employment law.
Jessica de Perio Wittman Joins SALLD Board
Professor Jessica de Perio Wittman, director of the law library at UConn School of Law, has been appointed a member-at-large of the Executive Committee of the Society of American Law Library Directors. She was introduced as a member on Jan. 5, 2020, at the annual meeting of the American Association of Law Schools in Washington, D.C.
Professor Jessica de Perio Wittman named to board of Special Education Equity for Kids of Connecticut
Professor Jessica de Perio Wittman has been named to the board of Special Education Equity for Kids of Connecticut. The organization's mission is "break down barriers in special education and increase equity and opportunity for children with disabilities." De Perio Wittman will be formally welcomed at the next board meeting on Jan. 14, 2020, in New Haven.
Professor Peter Kochenburger Gives Presentation to National Association of Insurance Commissioners
Dec 19, 2019 -
Professor Peter Kochenburger gave a presentation on pricing practices in auto and homeowners insurance in the United Kingdom at the 2019 Fall National Meeting of the National Association of Insurance Commissioners on Dec. 7, 2019, in Austin, Texas.
Joseph MacDougald '96 Honored for Energy Work
Professor Joseph MacDougald '96, executive director of the Center for Energy and Environmental Law, received a 2019 Excellence in Energy Award from the Connecticut Power & Energy Society on Dec. 11, 2019. The award, which recognized MacDougald for his influence in shaping the Connecticut energy landscape, was presented at the society's annual holiday party at the Connecticut Science Center on Dec. 11, 2019.
Mathilde Cohen to Moderate Panels at AALS Annual Meeting
Professor Mathilde Cohen will moderate three panel discussions; a session on European Law; the AALS Discussion Group; and a session of the Africa and European Law joint program, at the 2020 Annual Meeting of the American Association of Law Schools in Washington, D.C., on Jan. 4, 2020.
Timothy Fisher to Speak at AALS Annual Meeting
Dean Timothy Fisher will facilitate a breakout session at the 2020 Annual Meeting of the American Association of Law Schools in Washington, D.C., on Jan. 4, 2020. The topic will be "Naming Places, Moving Statues, Portraiture, Representation in the Physical Space."
Alexandra D. Lahav to Speak at AALS Annual Meeting
Professor Alexandra D. Lahav will speak at the Litigation and Remedies Joint Program at the 2020 Annual Meeting of the American Association of Law Schools in Washington, D.C., on Jan. 4, 2020. She will moderate a panel discussion on how voting rights remedies are administered, their timing and their limitations.
Leslie C. Levin to Speak at AALS Annual Meeting
Professor Leslie C. Levin will speak at the Professional Responsibility program at the 2020 Annual Meeting of the American Association of Law Schools in Washington, D.C., on Jan. 4, 2020. She will participate in a panel discussion on changes to the legal profession, and discuss the possibility of fundamental reconsideration of how societies regulate the legal profession, the practice of law, and the delivery of legal services.


OF ALL THE financial tasks performed by small businesses, preparing accurate financial forecasts may be the most important. All major business decisions flow from these forecasts: decisions about hiring, borrowing, spending and investing in new equipment, facilities and product lines, for example.
Creating financial forecasts is both an art and a science: art in the sense that it involves some creativity in making future assumptions, and science in the sense that the exercise is based in part on historical data. "The science comes from recognizing that the future is uncertain, and then measuring and handling these uncertainties in a quantitative manner," says Gerald A. Bush, a Costco member and the chairman of Decision Strategies (www. decisionstrategies.com) in Atlanta.
FOR HELP IN preparing financial forecasts, talk to your banker, accountant or an outsourced CFO services provider. Other sources of assistance include Small Business Development Centers (SBDCs) and the Service Corps of Retired Executives (SCORE); go to www.sba.gov and click on the "Find Resources" box to locate an SBDC office or SCORE executive near you.
The best way to start the financial forecasting process is by looking backward at your company's recent financial history— specifically, the past six to 12 months. Then make pro forma revenue and expense projections on your financial statements.
ber and partner with CFO Edge ( www.cfoedge.
keting, and travel and entertainment.
Finally, there are the direct costs incurred in the manufacture and/or delivery of products or services; these are referred to as "cost of goods sold," or CGS. "For example, inventory and raw materials would be considered CGS for a typical manufacturing company," says Goldman.
With these pro formas in hand, you can now make profitability projections. Subtracting CGS from total sales will yield projected gross profit, while subtracting fixed and variable expenses and taxes from gross profit will yield net profit, or your projected bottom line.
Don Sadler, don@don sadlerwriter.com, is a freelance writer based in Atlanta who specializes in topics related to small business and finance.


This trail does not go to Watchman itself but ends at a bench near the base of the Watchman spire. A few Zion landmarks are visible while hiking the Watchman Trail. This hike starts across the street from the Visitor's Center. You will pass junipers, pinyon pines, cactus, and cross a perennial spring. At the end of the trail from the loop there are panoramic views of the lower canyon, the town of Springdale, Oak Creek Canyon, and the Watchman.
The classic Watchman Trail is a moderate hike of 2.7 miles taking about two hours. The elevation gain is a mild 368 feet, which makes for a good family hike. At the end there is an optional loop around the bench. I strongly suggest taking some time at the loop. There is enough room for the family to spread out, play, and picnic around the loop.
There are a few minor drop-offs, but that is common in Zion. The well-maintained dirt path runs along the Virgin River for a short bit before running along the foot hills beneath the massive Bridge Mountain.
This trail is in full sun and through a hot desert environment. Take lots of water, wear good shoes, pace yourself, and take periodic breaks especially with children. This hike is better when it is colder since it tends to be one of the sunnier strolls.
Jason's thoughts - A good, steady, fairly good, middle of the road hike, kinda sunny.
Kelly's thoughts - A good hike the first time. Better the second time, because I made this a trail run. This was a little taxing being it was all uphill, but definitely a great workout. I spent some extra time walking around the loop and relaxing and taking some pictures. I ran downhill when I could. There are some rocky areas to this trail. That mixed with some drop-offs is not so cool.
Ben's thoughts - It was hard kinda getting up. It was fun making the stacks of rocks. It was hotter than the Emerald Pool trail.


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If you were a partner in a partnership that made a loss, you cannot lodge a paper tax return. You must lodge your tax return using e-tax or a registered tax agent. Otherwise, read on.
If you were entitled to an amount of trust income at 30 June 2015, you need to include your share of the net income of the trust in your 2014–15 tax return even if you did not receive an amount from the trust until after 30 June 2015. If you have not been advised about all of your trust entitlements, contact your trustee.
Note: If the trust income which you have received or are entitled to includes an amount described as tax-free, tax deferred, tax exempted or as a capital gains tax (CGT) concession, then read the information on non-assessable payments in the Guide to capital gains tax 2015 (NAT 4151).
You should not receive a distribution of a net capital gain or a net capital loss from a partnership. For information about how a partner shows their share of a capital gain or capital loss, see the Guide to capital gains tax 2015.
If you conducted a business activity as a partner in a partnership that resulted in a loss, or resulted in a loss after deducting your expenses, you must complete items P3 and P9 in the Business and professional items section in Business and professional items 2015 in addition to item 13. If this applies to you, then you must lodge your tax return using e-tax or a registered tax agent.
If you are a foreign resident who has received a fund payment from a managed investment trust on which an amount was withheld, see Withholding tax arrangements for managed investment trust fund payments (Nat 72719).
If you answered Yes to one or more of these questions, you will need to provide additional information. Print 'SCHEDULE OF ADDITIONAL INFORMATION - ITEM 13' at the top of a separate sheet of paper. Print your name, address, tax file number, the name of the trust, your share of income from the trust and any credits you are entitled to claim for that income, and explain your situation. Print X in the Yes box at Taxpayer's declaration on your tax return. Attach your schedule to page 3 of your tax return. Read on.
If the partnership made a loss, you cannot lodge a paper tax return. You must lodge your tax return using e-tax or a registered tax agent.
If the trust made an overall loss for tax law purposes in 2014–15, the loss is retained in the trust. You will have no share of the net income of the trust.
For more information see the Guide to capital gains tax 2015.
If you have incurred debt deductions, such as interest and borrowing costs, in deriving assessable income from a trust or partnership, of more than $2 million (alone or combined with those of your associate entities) for 2014–15, the amount that you can deduct at X and Y may be affected by the thin capitalisation rules. For more information see Thin capitalisation.
For more information on deductions for expenditure on landcare operations and water facilities, see the Guide to depreciating assets 2015 (NAT 1996).
If you were a partner in a partnership and you can claim a deduction in relation to your share of eligible expenditure incurred by the partnership on horticultural plants, grapevines, electricity connections or phone lines, include any such deduction that relates to primary production income or loss from a partnership at X. For information about deductions for expenditure on horticultural plants, grapevines, electricity connections and phone lines, see the Guide to depreciating assets 2015.
If a trustee incurred eligible expenditure on landcare operations, only the trustee, not a beneficiary of the trust, can claim deductions for that expenditure. For more information on deductions for expenditure on landcare operations, see the Guide to depreciating assets 2015.
If you were a partner in a partnership and you can claim a deduction in relation to your share of eligible expenditure incurred by the partnership on electricity connections, include any such deduction that relates to non-primary production income or loss from partnerships at Y item 13. For information about deductions for expenditure on electricity connections, see the Guide to depreciating assets 2015.
The partnership did not carry on the computer consultancy business in 2014–15. Lisa cannot include her $1,000 loss from the computer consultancy business at Y. This amount does not relate to a business activity which is the same as, or similar to, her current year partnership business activity.
In 2014–15 Lisa's partnership distribution from the furniture restoration business was $2,000. Lisa includes the $2,000 at O, $5,000 as a deferred loss relating to the furniture restoration business at Y and prints D in the TYPE box. Therefore, her net distribution from this partnership business activity is a loss of $3,000.
If the partnership or trust income which you have received, or to which you are entitled, includes income from activities as an author of a literary, dramatic, musical or artistic work, or as an inventor, performing artist, production associate or active sportsperson, you must also write the amount of this taxable professional income at Z item 24. You will not be taxed twice on this income. For more information, see question 24 Other income.
other credits for tax paid by a trustee on trust income; write the total of your share of credits for tax paid by a trustee at S item 13 (show cents). However, if you are the principal beneficiary of a Special Disability Trust, do not include your share of credits for tax paid by the trustee here. Include your share of credits for tax paid by the trustee at T9.
If any of these measures are likely to affect you, see You and your shares 2015 (NAT 2632).