text:
\section{Introduction} \label{sec:1}

Radio Resource Allocation (RRA) mechanisms are expected to play a key role in
emerging and future OFDMA-based multiuser wireless access networks. RRA aims at
simultaneously guaranteeing high utilization of the available system resources,
satisfying the individual Quality-of-Service (QoS) requirements of the
competing users, and maximizing overall system performance. In order to
accomplish these targets, an RRA or Frequency-Domain Packet Scheduling (under
the LTE terminology) technique exploits the differentiated
channel conditions experienced by the various users over the available
bandwidth. In particular, a complete allocation decision comprises the specific
set of OFDM subchannels assigned to each user as well as the transmission
format, namely the amount of power and the modulation mode, for each resource
block~\cite{Pi06}.

\subsection{Background and Related Work}

Adaptive RRA aims at positively exploiting the rate differentiation that occurs on two levels
closely related to respective physical phenomena: the single-user centric
variation of the achieved rate on a subchannel basis and the multi-user centric
differentiation of the rate achieved by each user on each subchannel. The
former phenomenon is related to the inherent frequency selectivity of the
wideband wireless medium, while the latter to the so-called multi-user
diversity caused by the statistical independence of the corresponding
subchannels. Therefore, by assigning to each user the subchannels
that experience favorable channel (and thus rate) conditions, we expect to significantly
improve system performance.

In \cite{ViTs02} the above concepts were thoroughly presented first for single- and then for multi-antenna wireless mobile systems. According to this concept each receiver monitors the experienced SNR levels, feeds them back to the BS, while the BS schedules transmissions and adapts users' bit rates depending on the particular channel quality reports. Similar arguments were raised in \cite{NaBa00} where it was demonstrated that for 2G/3G systems the cellular spectral efficiency may significantly improve, even double at certain conditions, when the BS utilizes the per-user channel/rate information. In \cite{BoGr07} the ideas were extended to multi-channel OFDM wireless systems like the one we examine in this paper. The widely used term "opportunistic" bears a strong relation with our work, since we tend to allocate subchannels with high channel conditions (near to their peak) to the corresponding users (frequency-domain opportunism), while in \cite{ViTs02} a similar policy is employed in the time-domain.

The works of \cite{WoCh99} and \cite{RhCi00} were the first to introduce an
optimization framework for handling multi-user OFDM resource allocation
problems, paving the way for an extensive utilization of concepts and methods
addressed in the engineering optimization field, towards efficiently assigning
system resources to active users. The original formulations of
the respective system power minimization~\cite{WoCh99} and sum-throughput
maximization~\cite{RhCi00} problems indicated the hard large-scale non-linear
integer nature of the underlying RRA decision problems, rendering the
straightforward discovery of the optimal allocation practically impossible.
Towards relaxing the complexity of the related problems, a plethora of
simplifying approaches has been proposed in the literature aiming at the
development of suboptimal, yet efficient and computationally tractable
allocation routines~\cite{KiLi03,KiLe06}. The works in \cite{ChHo09} and \cite{SaAn09} provide extensive surveys on the methodological and algorithmic aspects of the particular research area. A popular assumption which was first proposed in~\cite{JaLe03} and further elaborated in~\cite{ZhLe04} and
\cite{LeCh08} regarded the uniform power loading (UPL) of the system
subchannels. Under a known power distribution, the achieved bit-rates for all
the possible subchannel/user combinations can be precalculated, and thus the
original multidimensional RRA problems resort to exclusive subchannel
allocation problems. From a system perspective, OFDMA is considered the major transmission and multiple access technology for
modern wireless networks such as 3GPP-LTE~\cite{HoTo09}, Mobile
WiMAX~\cite{AnGh07} and also plays a key role in the new paradigm of Cognitive
Radio Networking~\cite{Ho09}. In particular, LTE-oriented works may be found in \cite{PoKo06,MoPe08},
WiMax-oriented ones in \cite{YaBe10,CaSa11,KhBo11} and cognitive networking related in \cite{ZhLe09,TaCh11} correspondingly.

\subsection{Contributions and Novelty}
In this paper we examine a single-cell downlink OFDMA system scenario supporting
multiple QoS profiles under the UPL assumption. Unlike single-profile studies,
which have attracted enormous interest in the related literature, the heterogeneous problem,
which is more interesting and realistic has not been given great attention. In \cite{HoTe98} a mixed CBR/VBR scenario was introduced for the single-user case, while in our work we consider multiple users demanding mixed services. The multi-user case was studied in \cite{WaHw04,YuZh07} where suboptimal allocation algorithms were proposed. However the UPL hypothesis have not been taken into account as in our work and the efficiency of the algorithms compared to the maximum achieved performance (that is, the optimal one) was not demonstrated. Finally, in \cite{ZhLe04} an efficient UPL-based resource allocation algorithm was proposed guaranteeing a set of minimum bit rates, while in our paper a more realistic QoS scenario is assumed, comprising mixed CBR and Best-Effort traffic profiles.

In particular, in this work we employ two optimization approaches to our problem. The former is based on Integer
Linear Programming (ILP) and allows us to find: (a) the exact optimal resource
allocation decision as well as the maximum sum-rate performance in reasonable,
yet high execution time, and (b) a performance upper bound in polynomial time
solving the LP-relaxed version of the IP formulation. The second approach is based on our observation that the specific heterogeneous
formulation bears similarities with a well-known combinatorial optimization
problem, the Generalized Assignment Problem (GAP) \cite{MaTo90}. In order to
efficiently solve the GAP formulation we devise two heuristic schemes, which
are executed with polynomially bounded computational cost. The development is
mainly inspired by the ideas utilized in the respective GAP heuristics found in
the related literature. The examined schemes are tested in an
extensive set of realistic computational-based network scenarios, in terms of
the achieved sum-rate performance. The simulation results show that the devised heuristic schemes: (i) outperform drastically a semi-random subchannel assignment approach, and (ii) perform close to the exact optimal allocation, while their complexity cost is significantly lower.

\subsection{Paper Structure}
The remaining of the paper is structured as follows. In Sec.~\ref{sec:2} we
present the adopted system model and the holding assumptions, state the
multiple-QoS profile RRA problem and formulate it as a Binary Integer Linear
Programming optimization problem. In Sec.~\ref{sec:3} we discuss how we can
express our problem as a variation of the GAP combinatorial format, and which
are the advantages of this approach, while in Sec.~\ref{sec:4} we devise two
heuristic allocation algorithms inspired by the GAP concept and discuss
computational complexity issues. In Sec.~\ref{sec:5} we present the results of
the computational experiments and comment on them, while in Sec.~\ref{sec:6} we
summarize our work and propose possible extensions of it.

\section{System Model and Mathematical Formulation}\label{sec:2}

\subsection{System Description and Assumptions}

The downlink (DL) of an OFDMA cell is considered in the context of this work,
where a Base Station (BS) located at the center of a 2D area is fully
responsible for allocating system subchannels (or equivalently physical
resource blocks according to the LTE terminology) to the competing users. The
available bandwidth comprises $N$ mutually-orthogonal subchannels (or resource
blocks) forming the set $\mathcal{S} = \left\{ {1,2, \ldots ,n, \ldots N}
\right\}$. The available transmission power $P_{bs}$ is uniformly spread over
the bandwidth, namely, each subchannel $n$ is loaded with an equal amount of
power given by ${P_n} = {{{P_{bs}}} \mathord{\left/
 {\vphantom {{{P_{bs}}} N}} \right.
 \kern-\nulldelimiterspace} N}$. We assume that $K$ users are present in the
 cell, forming the corresponding set $\mathcal{U} = \left\{ {1,2, \ldots ,k, \ldots K}
 \right\}$, and that each user is assigned one of the two available QoS profiles/classes,
 the Constant-Bit-Rate (CBR) or the Best-Effort (BE) one. Users belonging to
 the {CBR-class} subset (${{\cal U}_{{\rm{CBR}}}}$) demand a specific constant service rate
 denoted by $R_k^{\min }(\forall k \in {{\cal U}_{{\rm{CBR}}}})$, representing
 Voice/Video-like services. On the other hand, users belonging to the {BE-class} subset (${{\cal
 U}_{{\rm{BE}}}}$) have
 infinitely backlogged queues and no strict QoS guarantee, reflecting FTP-downloading
 services.

 In order to perform adaptive resource allocation in the above typical multi-user OFDMA system setup the following key operations/assumptions are supported:
 \begin{itemize}
   \item Each user estimates the BS-user link channel quality over the available subchannels and feeds back this information to the BS through an error-free and delay-less uplink signaling channel.
   \item Due to the wideband and multi-user nature of the transmission medium \cite{TsVi04} each user $k$ achieves different rate performance over each subchannel $n$, given by $r_{n,k}$ bits/symbol. Given that the BS exactly knows the wideband and multi-user channel response as well as the allocated power per subchannel $P_n$, the $r_{n,k}$ bit rates are pre-calculated (please refer to the discussion in Appedix~\ref{sec:app1}).
   \item Based on the above 2D rate matrix and the QoS targets, the BS decides the exact subchannel set assignments for each user. This information is also transmitted to each user in order for the useful data to be correctly decoded.
   \item Each subchannel should be allocated to a single user, thus avoiding intra-cell inter-user interference~\cite{Pi06}.
 \end{itemize}

\subsection{Mathematical Formulation and Optimal Solution}

We introduce $N\cdot K$ binary integer variables notated by $\rho_{n,k}$ where
$\rho_{n,k}=1$ if the $n^{th}$ arbitrary subchannel is allocated to the
$k^{th}$ arbitrary user, else equals zero. The data-rate $r_{n,k}$ supported on
each subchannel/user combination can be pre-calculated based on the abstraction
modeling function given in Appedix~\ref{sec:app1}. We are now able to formally define the RRA problem, which constitutes the
identification of the allocation decision that maximizes the cell sum-rate
($R_{tot}$), satisfies the individual data-rate constraints for the CBR-class
users and preserves subchannels orthogonality. Eq.\eqref{eq:ILPform_of} express
the system-wise objective function, Eq.\eqref{eq:ILPform_concbr} the $K_1
(\left| {{\mathcal{U}_{{\text{CBR}}}}} \right|)$ QoS constraints, and
Eq.\eqref{eq:ILPform_consys} the $N$ subchannel-sharing-avoidance system
constraints. Power availability and minimum BER constraints are handled
implicitly as explained in Appendix~\ref{sec:app1}. Although the generalized
bandwidth and power allocation problem is non-linear~\cite{WoCh99}, the UPL
hypothesis allows us to recast RRA in a linear form, as shown in
Eq.\eqref{eq:ILPform}. Actually, due to the exclusive assignment
of each subchannel to one user, the UPL variation of RRA is actually an Integer
Linear Programming (ILP) optimization problem. In fact we are dealing with an
ILP problem, including $N \cdot K$ binary variables and $\left|
{{\mathcal{U}_{{\text{CBR}}}}} \right| + N$ equality constraints, where $N\sim
100$ and $K\sim 10 -20$ under a typical network setup~\cite{HoTo09}. Despite
the fact that ILP problems are still hard to solve due to their NP-hard nature,
efficient methods and solvers like CPLEX, SCIP or GUROBI are available and may
be employed for problems of such scale.

Finally, an upper bound on the system performance could be estimated by
relaxing the integrality constraints or equivalently allowing for one or more
subchannels to be shared among users. For this case an LP solver could solve
the specific problem with 3rd order polynomial complexity cost. Note however
that the LP relaxation approach is able to provide us only with the system
performance (in fact the theoretical achieved upper bound) and not with an
implementable allocation decision due to the violation of the
subchannel-sharing constraints set.  We have to emphasize the implementation
limitations of both IP and LP approaches, underlying that their value is mostly
theoretical. For this reason we proceed with developing simpler allocation
schemes in the following two sections.

\begin{subequations}
\begin{gather}
  \mathop {\max }\limits_\text{\boldmath{$\rho$}}  {R_{tot}} = \sum\limits_{n \in \mathcal{S}} {\sum\limits_{k \in \mathcal{U}} {{r_{n,k}} \cdot {\rho _{n,k}}} }  = \sum\limits_{k \in {\mathcal{U}_{{\text{CBR}}}}} {R_{\min }^k}  + \sum\limits_{n \in \mathcal{S}} {\sum\limits_{k \in {\mathcal{U}_{{\text{BE}}}}} {{r_{n,k}} \cdot {\rho _{n,k}}} }  \hfill \label{eq:ILPform_of}\\
  {\text{subject to}} \hfill \nonumber\\
  \sum\limits_{n \in \mathcal{S}} {{r_{n,k}} \cdot {\rho _{n,k}}}  = R_{\min }^k,\forall k \in {\mathcal{U}_{{\text{CBR}}}} \hfill \label{eq:ILPform_concbr}\\
  \sum\limits_{n \in \mathcal{S}} {{\rho _{n,k}}}  = 1,\forall n \in \mathcal{S},\text{\boldmath{$\rho$}} \in {\mathbb{I}^{N \times K}},\mathbb{I} = \left\{ {0,1} \right\}
  \hfill \label{eq:ILPform_consys}
\end{gather}
\label{eq:ILPform}
\end{subequations}

\section{Radio Resource Allocation as a Variation of the Generalized Assignment Problem}\label{sec:3}

The Generalized Assignment Problem (GAP) along with its variations is a
well-studied special formulation of Combinatorial Optimization. Although GAP is
still an NP-hard problem, several efficient approximate heuristic algorithms
have been proposed for solving it~\cite{MaTo90,MoRo04} due to its wide
application in real-world problems. A Knapsack-based definition of GAP proposed
in~\cite{MaTo90} constitutes the assignment of items to knapsacks in order to
maximize the overall profit while not exceeding the capacity constraints of
each knapsack. The assignment of each item to a knapsack incurs different
profit and induces different cost. Note that an item can be obviously assigned
to a single knapsack. A slightly different definition is given in
\cite{MoRo04}, where one searches for the best scheduling of tasks or jobs to
agents, in order to minimize the overall processing cost and do not violate the
available resource budget of each agent. Again a task cannot be split into
multiple agents. Finally, a variation of the GAP, known as the Covering
Assignment Problem (CAP) is given in~\cite{FoWi97} which applies to a dairy
industry distribution scenario. Specifically, CAP regards the optimal
distribution procedure of the milk produced by several farms to a set of
factories, in order to minimize the cost of processing and simultaneously
satisfy the minimum resource demand of each factory.

Towards expressing our RRA problem as a GAP, we first define the correspondence
of an item/task/farm to a subchannel and a knapsack/agent/factory to a user of
the system. RRA may then be seen as the optimal exclusive allocation of each
subchannel to a single user in order to maximize the sum-rate (profit) function
and provide minimum rate (resource) assignments for a user subset.We also have
to mention that our formulation possesses several distinctive features compared
to the aforementioned classical GAP/CAP approaches:
\begin{itemize}
  \item The constraint expressions are tight whereas in GAP and CAP they correspond
  to left-hand-side or right-hand-side inequalities
  \item The constraint expressions are imposed on a subset of users and not
  over the complete users' set
  \item the profit/weight/cost factors are equal ($r_{n,k}$) and depend on both
  subchannel and user indexes through the experienced rate/channel conditions.
\end{itemize}
As far as GAP and its variations, several efficient heuristic algorithms could
be found in the literature. In~\cite{MaTo90} the authors proposed a scheme
which was based on the construction of an initial effective feasible solution
and its subsequent improvement through item reallocations. In~\cite{Wi97} the
author proposed a ``dual" algorithm, according to which the optimal solution
was approached from the exterior of the feasible region. Finally,
in~\cite{FoWi97} the authors devised similar schemes for the CAP variation. In
the following section we devise two algorithms, inspired by the previous
GAP/CAP works. We remark that due to the differences between our RRA problem
and the classical formulations, none of the existing algorithms could be
employed as-is. Moreover, although other solution approaches may be found in
the literature, such as the Lagrangean Relaxation, Branch-and-Bound,
Metaheuristics, etc. (see for example a survey in~\cite{CaWa92}) we focus on
the heuristic solution approach since it is appealing from an implementation
point of view. One should bear in mind that in a practical network scenario the
complete resource assignment must be updated every transmission frame. For
example, regarding contemporary wireless access systems like LTE and WiMAX, the
update rate should be of the order of 0.5--1 msec~\cite{HoTo09}.

\section{Heuristics for Solving the Resource Allocation Problem}\label{sec:4}

\subsection{Heuristic I: Approaching the Optimal Allocation from the Interior of the Feasible
Space} \label{sec:4a}

\paragraph{Description} The first heuristic is mainly inspired by the Knapsack
solution approach given in~\cite{MaTo90}. Resource Allocation is performed in 2
phases, that is the construction of an initial feasible solution and the search
for a gradually improved allocation. Hereafter, when we refer to a ``feasible"
allocation decision, we will mean that the $\left|
{{\mathcal{U}_{{\text{CBR}}}}} \right|$ data rate constraints are satisfied. As
far as the remaining $N$ orthogonality constraints, these will be implicitly
met, since at each subchannel allocation step, the subchannel which is assigned
to a user will be removed from the available bandwidth ``pool". Hence, we
should not confuse the use of the feasibility term with the one regarding the
LP relaxation given in Sec.\ref{sec:2}. The complete algorithm including all
the intermediate steps for each phase is given after the description of its key
features.

During the \text{1st phase}, our primary objective is to construct a feasible
RRA solution, that is, satisfy the strict data-rate constraints of all CBR
users, while secondarily the respective allocation has to be as efficient as
possible in terms of the overall system performance. Towards the 1st point we
prioritize the CBR-class subchannel allocation procedure over BE-class, thus
decoupling the inter-class problem to two consecutive intra-class sub-problems.
Concerning the secondary target, we state that if the above QoS constraints are
satisfied by utilizing the minimum amount of the available Tx power as well as
the ``best" subchannels in terms of the achieved channel/rate conditions, then
plenty of resources will be available for BE-class users. Such a policy would
enhance system performance, since BE users are the ones that contribute to the
objective function, due to the predetermined static nature of the CBR-class
users' data-rates. Hence, after the necessary initializations and declarations
(\textbf{Step 1}), an iterative joint subchannel/user allocation is applied
over the CBR users subset ignoring BE users (\textbf{Step 2}). The selection
criterion for picking the best CBR subchannel-user pair at each iteration is
dual: on the one hand the user experiencing the minimum averaged achieved
data-rate over the remaining available subchannels set is prioritized and on
the other hand the ``best" subchannel in terms of rate performance is promoted.
The user-selection criterion prioritizes the users with the worse rate
conditions, which are expected to consume the largest portion of resources, and
by allocating to them their most efficient subchannels we succeed into
constraining the overall amount of CBR-class resources consumption.

Subsequently, we finalize the initial feasible RRA solution by assigning the
unallocated resources to the BE users (\textbf{Step 3}). Since no QoS
requirements are imposed for the particular class, the simple policy of
best-rate user allocation is also the optimal one (see also~\cite{JaLe03}). Up
to now, we have provided a feasible allocation decision, however we argue that
we could further improve it since the previous steps may induce suboptimality
to the sum-rate performance. Note that this is caused by the ``greedy" (local)
nature of the heuristic allocations during Step 2. Towards the above purpose we
employ a series of subchannels swaps (\textbf{Step 4}). Specifically, we
perform a subchannel interchange between 2 users if and only if the system
performance is enhanced and simultaneously the feasibility for any CBR user is
not violated. Due to the presence of multiple QoS classes, we have to
discriminate between $2^2=4$ possible swap scenarios (2 classes are considered
in this work). Finally in order to limit the number of interchanges (and thus
the execution complexity) we perform only one round of comparisons and possible
reallocations by a single sweep of the set of system users. We remark that by increasing the comparison rounds, a marginal additional performance gain is expected, since already at each user iteration the owner's subchannels are compared with the subchannels of all the other users.

The last subprocedure given in \textbf{Step 5} comprises the release of
possible redundant subchannels of CBR-class users due to the active nature of
the related constraints and the reallocation of them to a BE user. We put an
emphasis on the fact that for CBR users surplus allocated data-rates are
ignored.

\paragraph{Complete Algorithm}

\paragraph{\textbf{Step 1}}  \emph{Initializations-Declarations}\\[3pt]
Let ${\left\{ {{r_{n,k}}} \right\}_{n \in \mathcal{S},k \in \mathcal{U}}}$ the
known achieved data-rates for all subchannel/user combinations and ${\left\{
{{\rho _{n,k}}} \right\}_{n \in \mathcal{S},k \in \mathcal{U}}}$ the set of the
allocation variables.\\[2pt]
Define: $lh{s_k} = \sum\limits_{n = 1}^N {{r_{n,k}} \cdot {\rho _{n,k}}} \text{
and } {b_k} = R_{\min }^k, \forall k \in {\mathcal{U}_{{\text{CBR}}}}$.

\paragraph{\textbf{Step 2}}  \emph{Subchannel Assignment to the CBR-class users -- Feasible Allocation}\\[3pt]
Define the available subchannels pool $\mathcal{N}$ and the subset of the
unsatisfied CBR users $\mathcal{K}$ as:\\[2pt]
$\mathcal{N} = \left\{ {n \in \mathcal{S}:\sum\limits_{k = 1}^K {{\rho _{n,k}}}
= 0} \right\},\mathcal{K} = \left\{ {k \in {\mathcal{U}_{{\text{CBR}}}}:lh{s_k}
- {b_k} < 0} \right\}$.\\[2pt]
Update $lh{s_k} = \sum\limits_{n = 1}^N {{r_{n,k}} \cdot {\rho _{n,k}}} \text{
and } {b_k} = R_{\min }^k, \forall k \in {\mathcal{U}_{{\text{CBR}}}}$.\\[2pt]
Pick the subchannel/user combination according to:\\
${k^*} = \arg \mathop {\min }\limits_{k \in \mathcal{K}} \left\{ {\left( {{1
\mathord{\left/
 {\vphantom {1 {\left| \mathcal{N} \right|}}} \right.
 \kern-\nulldelimiterspace} {\left| \mathcal{N} \right|}}} \right) \cdot \sum\limits_{n \in \mathcal{N}} {{r_{n,k}}} } \right\},{n^*} = \arg \mathop {\max }\limits_{n \in \mathcal{N}} \left\{ {{r_{n,{k^*}}}}
 \right\}$ and\\[2pt]
Perform the subchannel allocation: ${\rho _{{n^*},{k^*}}} = 1$\\[2pt]
If $\mathcal{K} \ne \emptyset$ Repeat Step 2 else Break the Loop and Go To Step
3

\paragraph{\textbf{Step 3}}\emph{Subchannel Allocation to BE-class users -- Finalization of the Feasible Solution}\\[3pt]
Assign each unallocated subchannel to the ``best" users in terms of the
achieved rate: ${\text{For each }}n \in \mathcal{N},{\text{ set }}{\rho
_{n,{k^*}}} = 1{\text{, where }}{k^*} = \arg \mathop {\max }\limits_{k \in
{\mathcal{U}_{{\text{BE}}}}} \left\{ {{r_{n,k}}} \right\}$\\[2pt]

\paragraph{\textbf{Step 4}}\emph{Subchannel Swapping -- Improved Solution}\\[3pt]
For every user $k \in \mathcal{U}$ (either {CBR} or {BE}) repeat the following
procedure serially:\\[2pt]
(a) Find his/her allocated subchannels set:
${\mathcal{I}_k} = \left\{ {n \in \mathcal{S}:{\rho _{n,k}} = 1} \right\}$\\[2pt]
(b) Find the complementary users/subchan. sets : $\mathcal{U}'{_k}  =
\mathcal{U}\backslash \left\{ k \right\},{{\mathcal{I}'}_k} =
\mathcal{S}\backslash {\mathcal{I}_k}$\\[2pt]
(c) For all possible combinations $\left\langle {\left( {n,k}
\right){\text{,}}\left( {n',k'} \right)} \right\rangle$, where $n \in
{\mathcal{I}_k},n' \in {{\mathcal{I}'}_k},k \in \mathcal{U}'{_k}$ check the
inner conditional expression depending on the active scenario:\\[2pt]
(c.1) If ${k \in {\mathcal{U}_{{\text{CBR}}}}{\text{ and  }}k' \in
{\mathcal{U}_{{\text{CBR}}}}}$: ${r_{n',k}} > {r_{n,k}}{\text{ \&\&
}}lh{s_{k'}} + {r_{n,k'}} - {r_{n',k'}} \geqslant {b_{k'}}{\text{ }}$ \\ \text{
} \hspace{125pt} or ${r_{n,k'}}
> {r_{n',k'}}{\text{ \&\& }}lh{s_k} + {r_{n',k}} - {r_{n,k}} \geqslant
{b_k}$\\[2pt]
(c.2) If ${k \in {\mathcal{U}_{{\text{BE}}}}{\text{ and  }}k' \in
{\mathcal{U}_{{\text{BE}}}}}$: \hspace{8pt} ${r_{n',k}} - {r_{n,k}} +
{r_{n,k'}} - {r_{n',k'}}
> 0$\\[2pt]
(c.3) If $k \in {\mathcal{U}_{{\text{CBR}}}}{\text{ and }}k' \in
{\mathcal{U}_{{\text{BE}}}}$: \hspace{3pt} ${r_{n,k'}} > {r_{n',k'}}{\text{
\&\&
}}lh{s_k} + {r_{n',k}} - {r_{n,k}} \geqslant {b_k}$\\[2pt]
(c.4) If ${k \in {\mathcal{U}_{{\text{BE}}}}{\text{ and  }}k' \in
{\mathcal{U}_{{\text{CBR}}}}}$: \hspace{0.5pt} ${r_{n',k}} > {r_{n,k}}{\text{
\&\& }}lh{s_{k'}} + {r_{n,k'}} - {r_{n',k'}} \geqslant {b_{k'}}$\\[2pt]
If any of the conditions is TRUE employ the corresponding resource swapping:\\[2pt]
${\rho _{n,k}} = 0,{\rho _{n',k'}} = 0,{\rho _{n',k}} = 1,{\rho _{n,k'}} = 1$.

\paragraph{\textbf{Step 5}}\emph{Release of CBR-class Subchannels for BE-class}\\[3pt]
For every CBR user $k \in {\mathcal{U}_{{\text{CBR}}}}$:\\[2pt]
(a) Find his/her allocated subchannels set:
${\mathcal{I}_k} = \left\{ {n \in \mathcal{S}:{\rho _{n,k}} = 1} \right\}$\\[2pt]
(b) For each $n \in {\mathcal{I}_k}{\text{ if }}lh{s_k} - {r_{n,k}} \geqslant
{b_k}$ release it from this user and allocated it to the best BE-user: ${\rho
_{n,k}} = 0,{\rho _{n,{k^*}}} = 1,{\text{ where }}{k^*} = \arg \mathop {\max
}\limits_{k \in {\mathcal{U}_{{\text{BE}}}}} \left\{ {{r_{n,k}}} \right\}$

\subsection{Heuristic II: A Dual Approach of the Optimal Allocation}  \label{sec:4b}

\paragraph{Description} An alternative heuristic algorithm inspired by the works of \cite{Wi97} and
\cite{FoWi97}, where the optimal allocation decision is approached from the
exterior of the feasible space, is proposed in the present subsection. The
``dual" approach shares also several similarities with the Lagrangean
Relaxation technique, which dualizes the hard constraints. Specifically, the
solution is approached on 2 phases: the construction of the optimal
unconstrained subchannel allocation and the feasibility transformation of the
initial decision through a series of subchannel reallocations. All the
intermediate allocations generated by the dual approach are infeasible except
for the last one. Note, that a similar algorithm for a single-QoS profile RRA
problem has been proposed in~\cite{ZhLe04}. We now provide a summary of the
algorithm as well as a complete step-by-step description of the dual approach
in the context of our RRA problem.

During the \text{1st phase}, the necessary initializations and declarations
(\textbf{Step 1}) as well as the optimal unconstrained subchannel allocation
are performed (\textbf{Step~2}). The optimal solution to the corresponding
unconstrained problem is extracted by simply allocating each subchannel to the
user that experiences the maximum data-rate on it. Obviously, the sum-rate
performance of the best-user allocation policy is an upper bound on any
QoS-constrained scenario. The \text{2nd phase} on the other hand comprises a
series of subchannel reallocations aiming at satisfying the minimum QoS targets
or equivalently rendering the solution feasible (\textbf{Step
3})\footnote{During the reallocation subprocedure followed here, subchannels
are removed from one user and given to another, whereas in the 1st heuristic
subchannels interchanges occur}. The selection of user/subchannel pairs
participating in each reallocation cycle is dictated by the efficiency metric
shown in Eq.\eqref{eq:Realloc_crit}, where $n$ stands for a subchannel
candidate for reassignment, $k^*(n)$ for the current owner of the subchannel
(as determined by Step 2), and $k$ the index of the candidate reassigned user.
\begin{equation}\label{eq:Realloc_crit}
{a_{n,k}} = {\left. {\dfrac{{{r_{n,{k^*}\left( n \right)}} - {r_{n,k}}}}
{{{r_{n,k}}}}} \right|_{n \in \mathcal{N},k \in \mathcal{J}'}}
\end{equation}
The nominator of the metric expresses the decrease in sum-rate performance due
to the reallocation, whereas the denominator the increase of the the infeasible
user data-rate or equivalently the decrease of the distance from feasibility.
Thus, by minimizing this metric we simultaneously harm as less as possible the
original objective function value and approach as fast as possible the
feasibility region. Notice, however that a specific resource-block reallocation
is possible if the minimum QoS target for the owner-user is not violated. The
particular iterative procedure ends when all CBR users' demands are satisfied.
Finally, the last step of Phase 2 (\textbf{Step 4}) constitutes the release of
redundant subchannels (if any) from the CBR-class users, similarly to the
procedure followed in the 1st Heuristic.

\paragraph{Complete Algorithm}

\paragraph{\textbf{Step 1}}  \emph{Initializations -- Declarations}\\[3pt]
Let ${\left\{ {{r_{n,k}}} \right\}_{n \in \mathcal{S},k \in
\mathcal{U}}},{\left\{ {{\rho _{n,k}}} \right\}_{n \in \mathcal{S},k \in
\mathcal{U}}},{\left\{ {lh{s_k}} \right\}_{k \in
{\mathcal{U}_{{\text{CBR}}}}}},{\left\{ {{b_k}} \right\}_{k \in
{\mathcal{U}_{{\text{CBR}}}}}}$ as in Heuristic I.\\[2pt]

\paragraph{\textbf{Step 2}}  \emph{Optimum Unconstrained (QoS-unaware) Allocation}\\[3pt]
Best-Rate Allocation: For each $n \in \mathcal{S},{\text{ set }}{\rho
_{n,{k^*}}} = 1,{\text{ where }}{k^*} = \arg \mathop {\max }\limits_{k \in
\mathcal{U}} \left\{ {{r_{n,k}}} \right\}$\\[2pt]
Minimum QoS satisfaction check: If for all $k \in
{\mathcal{U}_{{\text{CBR}}}}{\text{ : }}lh{s_k} \geqslant {b_k}$ Then Go To
Step 4 Else Go To Step 3.

\paragraph{\textbf{Step 3}}  \emph{Subchannels Reallocation -- QoS Satisfaction}\\[3pt]
Define the following subchannels and users subsets:\\[2pt]
${\mathcal{J}_{{\text{CBR}}}} = \left\{ {k \in
{\mathcal{U}_{{\text{CBR}}}}:lh{s_k} \geqslant {b_k}} \right\} \subseteq
{\mathcal{U}_{{\text{CBR}}}}$: the subset of CBR users for which minimum QoS is
met.\\[2pt]
${{\mathcal{J}'}_{{\text{CBR}}}} = {\mathcal{U}_{{\text{CBR}}}}\backslash
{\mathcal{J}_{{\text{CBR}}}}$: the complementary subset of undersatisfied CBR
users.\\[2pt]
${{\mathcal{J}}_{{\text{BE}}}} = \left\{ {k \in
{\mathcal{U}_{{\text{BE}}}}:lh{s_k} > 0} \right\} \subseteq
{\mathcal{U}_{{\text{BE}}}}$: the subset of BE users allocated at least one
subchannel or equivalently posses non-zero data rate.\\[2pt]
$\mathcal{J} = {{\mathcal{J}}_{{\text{CBR}}}} \cup
{{\mathcal{J}}_{{\text{BE}}}}{\text{ }}$: the subset of users from which we can
remove subchannels\\[2pt]
$\mathcal{J}' = {{\mathcal{J}'}_{{\text{CBR}}}}$: the subset of users to which
we must add subchannels in order to meet the minimum QoS requirements.
$\mathcal{N} = \left\{ {n \in \mathcal{S},k \in \mathcal{J}:{\rho _{n,k}} = 1}
\right\}$: the subchannels pool from which we can extract subchannels and
reallocate to the users subset $\mathcal{J}'$.\\[4pt]
Repeat the following procedure while there exist unsatisfied CBR users, namely
as long as $\mathcal{J}' \ne \emptyset$:\\[2pt]
(a) For each subchannel $n \in \mathcal{N}$ identify the owner ${k^*}\left( n
\right) = \left\{ {k \in \mathcal{J}:{\rho _{n,k}} = 1} \right\}$.\\[2pt]
(b) Compose the 2D Reallocations Array ${{\mathbf{A}}_{\left| \mathcal{N}
\right| \times \left| {\mathcal{J}'} \right|}} = \left[ {{a_{n,k}}} \right]$,
where:\\[2pt]
${a_{n,k}} = \left\{ \begin{gathered}
  \dfrac{{{r_{n,{k^*}\left( n \right)}} - {r_{n,k}}}}
{{{r_{n,k}}}},{\text{ if }}lh{s_{{k^*}\left( n \right)}} - {r_{n,{k^*}\left( n \right)}} \geqslant {b_{{k^*}\left( n \right)}}{\text{ and }}{r_{n,k}} > 0 \hfill \\
   + \infty \hspace{45pt} {\text{,  elsewhere}} \hfill \\
\end{gathered}  \right.$\\[2pt]
(c) Check all the elements of the array:\\[2pt]
If for all elements ${a_{n,k}} =  + \infty$, Break the Loop and Go To Step
4.\\[2pt]
Else pick the subch./user combination according to $\left\{ {n',k'} \right\} =
\arg \mathop {\min }\limits_{n \in \left| \mathcal{N} \right|,k \in
\mathcal{J}'} {\mathbf{A}}$ and perform the reallocation: ${\rho
_{n',{k^*}\left( {n'} \right)}} = 0,{\rho _{n',k'}} = 1$. \\[3pt]
(d) Update the related quantities/subsets:
$\mathcal{J},\mathcal{J}',\mathcal{N},lh{s_k},k \in
{\mathcal{U}_{{\text{CBR}}}}$.

\paragraph{\textbf{Step 4}}  \emph{Redundant CBR-class Subchannels Reallocation}\\[3pt]
Define the following subchannels and users subsets:\\[2pt]
$\mathcal{J} = \left\{ {k \in {\mathcal{U}_{{\text{CBR}}}}:lh{s_k} > {b_k}}
\right\}$: the subset of CBR users assigned redundant data-rate (over-satisfied
users).\\[2pt]
$\mathcal{N} = \left\{ {n \in \mathcal{S},k \in \mathcal{J}:{\rho _{n,k}} = 1}
\right\}$: possible redundant subchannels pool.\\[2pt]
Repeat the following procedure while there exist over-satisfied users, namely
as long as $\mathcal{J} \ne \emptyset$:\\[2pt]
For each $n \in \mathcal{N}$:\\[2pt]
(a) find the owner ${k^*}\left( n \right) = \left\{ {k \in \mathcal{J}:{\rho
_{n,k}} = 1} \right\}$\\[2pt]
(b) If $lh{s_{{k^*}}} - {r_{n,{k^*}}} \geqslant {b_{{k^*}}}$ remove the
subchannel $n$ from the owner ${k^*}\left( n \right)$ and reallocate it to the
``best" BE user in terms of achieved data-rate $k'$:\\[2pt]
${\rho _{n',{k^*}\left( {n'} \right)}} = 0,{\rho _{n',k'}} = 1$, where
$k'\left( n \right) = \arg \mathop {\max }\limits_{k \in
{\mathcal{U}_{{\text{BE}}}}} \left\{ {{r_{n,k}}} \right\}$, Else Go To the next
subchannel.

\subsection{Computational Complexity Estimation}
An estimation of the computational complexity of the proposed optimal and
suboptimal schemes follows. With respect to the heuristic algorithms we take
into account the involved searching, sorting and comparison operations. We
assume that the order of CBR and BE users is the same that is, $\left|
{{\mathcal{U}_{{\text{CBR}}}}} \right| \sim \left|{{\mathcal{U}_{{\text{BE}}}}}
\right| \sim K$ and that the number of subchannels is an order greater than the
number of active users, namely $N \gg K$.

\paragraph{Exhaustive and ILP/LP models} We first consider an exhaustive
search approach, where all the possible combinations of subchannels-to-users
assignments are examined. The particular procedure has an exponential
complexity order of $O(K^N)$, which is obviously computationally intractable.
As far as the exact optimal ILP model there is no guarantee for the execution
complexity, due to its NP-hard nature, however it is expected to be
significantly lower than the complete enumeration. The relaxed LP model on the
other hand has a provable third-order polynomial solution complexity with
respect to the number of the involved variables. Hence, in order to find the
performance upper bound of the RRA problem we have to spend a set of
computational operations of order $O\left( {{N^3}{K^3}} \right)$.

\paragraph{Heuristic I} Since the introductory Step 1 induces no complexity, we proceed directly with the next step.
We first employ a necessary pre-sorting operation regarding the achieved
data-rate values over all users, which costs $KN\log N$ operations (this is an
implementation issue, and this is why it was omitted from the previous
algorithmic description). Regarding Step 2 we perform at most $N$ iterations
for satisfying the CBR-class QoS constraints, and at each iteration we have to
select the worst user in terms of the average achieved data-rate with
complexity cost $K$ and locate its best subchannel with cost $logN$ (due to
pre-sorting), leading to a total cost of $NK + N\log N$ operations. Step 3
involves $N\log K$ operations for identifying the best-rate user for each
remaining subchannel. Step 4 is the most complex one, since it entails a series
of comparisons: without loss of accuracy we assume that each user is
preallocated on average ${N \mathord{\left/{\vphantom {N K}}
\right.\kern-\nulldelimiterspace} K}$ subchannels, and then for each user the
assigned ${N \mathord{\left/
 {\vphantom {N K}} \right.
 \kern-\nulldelimiterspace} K}$ subchannels are compared with the ${N \mathord{\left/
 {\vphantom {N K}} \right.
 \kern-\nulldelimiterspace} K}$ subchannels of the complementary $\left( {K - 1} \right)$
 users. All users are scanned for a full cycle, hence the whole process costs
 $\dfrac{N} {K} \cdot \dfrac{N} {K} \cdot \left( {K - 1} \right) \cdot K \approx
{N^2}$ operations. Finally, Step 5 involves at most $N$ subchannel releases,
and for each one the best-rate BE user must be located with cost $\log K$,
hence $N\log K$ operations are needed. Combining all the above algorithmic
steps we get $KN\log N + NK + N\log N + {N^2} + N\log K$ operations and after
performing several manipulations and approximations we resort to an estimated
computational complexity order of $O\left( {{N^2} + NK\log N} \right)$.

\paragraph{Heuristic II} Similarly to the 1st heuristic we first employ a pre-sorting operation for the
2D rates array which costs $KN\log N$ operations. Step 2 needs $N\log K$
operations in order to find the optimal unconstrained allocation. Step 3
contributes significantly to the overall complexity: at most $N$ subchannel
reallocations are performed and for each one a reallocation cost matrix
containing $NK$ elements must be devised and then searched for the combination
experiencing the minimal cost. Thus $N \cdot NK = {N^2}K$ operations are needed
to fulfil the particular step. Step 4 induces an additional cost of $N \log K$
operations as in the 1st heuristic. Accounting for all the algorithmic steps we
result to an overall approximate complexity order of $O\left( {{N^2} + NK\log
N} \right)$.

\section{Simulation Results and Discussion}\label{sec:5}

\paragraph{Simulation Setup}

The DL of a single-cell OFDMA-based packet data network is modeled and
simulated in the context of this work, while the selection of system parameters
reflects an LTE scenario~\cite{HoTo09}. A system bandwidth of 20~MHz is
assumed, consisting of $N$~=~100 orthogonal data subchannels. DL transmissions
occur on frame bursts of $T_f$~=~1~msec. A realistic pedestrian NLOS macro-cell
urban channel from the WinnerII models family is adopted~\cite{WINII07a}. The
PHY-abstraction function of Cioffi is utilized for associating each
channel-to-noise ratio level with an achieved data rate (see
Appendix~\ref{sec:app1}). Channel conditions are assumed perfectly known at the
BS, allowing for an opportunistic subchannel allocation. At the receiver side
transmissions are harmed due to thermal noise with power density of
$N_0$~=~$-174$ dBm/Hz.

As far as the traffic/QoS models, a CBR/BE dual-class scenario is formed. Each
CBR user requires $R_k^{\min}$= 36 {bits/OFDMA symbol}\footnote{For simplicity
reasons we may assume that an OFDMA frame carries one data-symbol. Thus all the
data-rate/throughput quantities may be identically expressed in bits/OFDMA
symbol/frame or simply in bits.}; the overall required CBR load is determined
by varying the number of users, where $\left| {{\mathcal{U}_{{\text{CBR}}}}}
\right|$~=~6--12. The number of BE users is held constant at $\left|
{{\mathcal{U}_{{\text{BE}}}}} \right|$~=~5. System capacity may be calculated
as $C_{max}=N \cdot c_{max}$~=~600~bits, where $c_{max}$ is the maximum
supported data-rate per subchannel, however the true achieved cell-rate is
expected to be lower due to the hard CBR rate constraints and BE-class users'
power shortage. The simulation parameters are summarized in
Table~\ref{ta:SimParam}.

\begin{table}[t]
\footnotesize  \caption{Simulation Parameters} \label{ta:SimParam} \centering
\vspace{6pt}
\begin{tabular}{lll}
 \hline\noalign{\smallskip}
\textbf{Quantity} & \textbf{Symbol} & \textbf{Value/Comment}  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Carrier Frequency & $f_c$ & 2.5 GHz \\
System Bandwidth & BW &  20 MHz \\
Subchannel Bandwidth & $\Delta f$ & 200 kHz \\
Number of Subchannels & $N$ & 100 \\
Transmission Error Rate & $P_e$ & $10^{-6}$ \\
Noise Power Density & $N_0$ & -174 dBm/Hz \\
Higher Tx Order per suchannel & $c_{max}$ & 6 bits/symbol \\
Cell Radius & $R_{cell}$ & 2 km \\
Channel Model & -- & WinnerII C3 BU-NLOS Macro\\
Users Distribution & -- & Uniform\\
Users Mobility Model & -- & Pedestrian (4 km/hr)\\
Number of OFDMA frames per drop & -- & 100\\
CBR QoS model & ${\left\{ {R_k^{\min }} \right\}_{k \in
{\mathcal{U}_{{\text{CBR}}}}}}$ & 36 bits/OFDMA symbol \\
CBR users & $K_1=\left| {{\mathcal{U}_{{\text{CBR}}}}} \right|$ & 6--12 \\
BE users & $K_2=\left| {{\mathcal{U}_{{\text{BE}}}}} \right|$ & 5 \\
Max Number of drops & $N_{drops}^{ulim}$ & 1000 \\
Min Number of drops & $N_{drops}^{llim}$ & 25 \\
Statistics Convergence Threshold & $\sigma_{norm}$ & 0.02 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

We consider five schemes which are compared below in terms of the achieved
sum-rate performance while we guarantee the CBR-class rates:
\begin{itemize}
  \item The exact optimal scheme extracted by solving the ILP optimization problem
  given in in Eq.~\eqref{eq:ILPform} (\textbf{IP})
  \item A scheme that provides us with a performance upper bound, obtained by
  solving the continuous relaxed version of the previous scheme (\textbf{LP})
  \item The 1st heuristic proposed in Sec.\ref{sec:4a} (\textbf{HEUR1}), for which we also run a version without using the swapping sub-procedure (\text{HEUR1 no swap})
  \item The 2nd heuristic proposed in Sec.\ref{sec:4b} (\textbf{HEUR2})
  \item A semi-random allocation algorithm  (\textbf{RANDOM}) which: (a) satisfies the target rates for the CBR-class users, by picking them one by one and assigning their best available channel until all requirements are met and, (b) assigns the remaining subchannels to the BE users randomly
\end{itemize}

All the algorithms are implemented in MATLAB and for the ILP/LP problems we
utilize the CPLEX solver~\cite{CPLEX}, calling it through the TOMLAB
interface~\cite{Tomlab99}. In order to capture the effect of different system
parameters to the achieved performance, we simulate 20 realistic system
scenarios by varying:
\begin{itemize}
   \item [(i)] the CBR-loading/QoS levels in the service area, by
   considering a different number of active CBR users ($K_1 = \left| {{\mathcal{U}_{{\text{CBR}}}}}
   \right|$~=~6,8,10,12)
   \item [(ii)] the average experienced SNR conditions in the cell, by
   tuning the ratio of the BS transmission power ($P_{bs}$) to the minimum required Tx power for guaranteeing feasibility regarding the $K_1$
   data-rate constraints ($P_{CBR}^{feas}(K_1)$) in each scenario ($P_{bs}/P_{CBR}^{feas}$~=~$2.0,2.5,\ldots,4.0$
   \footnote{Note that lower values for the power availability metric (e.g. 1.0 or 1.5) are not examined. This is justified
   by the fact that at lower SNR conditions, one or more CBR rate constraints can not be
   satisfied (``outage" conditions), rendering the QoS constrained problem infeasible. In order to cope with such situations, an adaptive power
   allocation strategy must be employed, like the one we proposed in our recently published work~\cite{GoKo10}.
   })
 \end{itemize}

We define an arbitrary realization or ``drop" as a system setup consisting of a
set of users randomly placed in the cell, for which their large-scale channel
conditions are held constant, whilst the small-scale conditions vary in time.
Each drop consists of 100 consecutive OFDMA frames, spanning $100 \cdot
T_f$~=~100~msec. For each scenario we simulate multiple statistically
independent drops and record the average sum-rate performance of each scheme.
The simulation is terminated if the normalized (to the mean) variance of the
performance statistics drops below the convergence target threshold (which is
set to 0.02) or if the maximum number of simulated drops (which is set to 1000)
has been reached~\cite{JeBa00}. Moreover, a minimum number of 25 drops are
executed in any scenario, in order to avoid transient effects. Thus, at least
$25 \cdot 100$ = 2,500 and at most $1000 \cdot 100$ = 100,000 optimization
problems are formulated and solved for each scenario.

\paragraph{Results and Discussion}
In Figs.\ref{fig:Thr-Load} and \ref{fig:Thr-Power} we depict the sum-rate
performance for all schemes as a function of the required CBR loading and the
average experienced SNR conditions. The effect of the subchannel swapping procedure to the performance of the HEUR1 algorithm is illustrated in Fig.\ref{fig:HEUR1SwapEffect} (HEUR1 with/no swap). In Figs.\ref{fig:Thr-Load-Bars-LowerSNR} and
\ref{fig:Thr-Load-Bars-HigherSNR} we provide the optimality gaps of our heuristic schemes (HEUR1,HEUR2) for two representative power availability
scenarios. Finally in Table~\ref{ta:AllScenPerf} we present the achieved performance of the
heuristic schemes  compared to the exact optimal (IP) as well as
the performance difference between the IP and the relaxed LP approach. The main
observations/comments regarding the behavior of the various schemes/algorithms
are the following.

\begin{itemize}
  \item [(i)] Both heuristic approaches perform close to the optimal allocation scheme
  and follow its performance trend for different system conditions (Figs.\ref{fig:Thr-Load},\ref{fig:Thr-Power}) while
  their complexity cost is significantly lower.
  \vspace{3pt}

  \item [(ii)] The proposed heuristics outperform the RANDOM assignment scheme significantly. This means that exploiting the inherent 2D rate selectivity/diversity of the system leads to a dramatic increase on the cell performance. The performance gain of the 1st heuristic is 60.6\% on average. In particular in low SNR conditions the gain is approximately 80\% and at higher SNR conditions drops to 48.64\%, which is still high. This is reasonable, since at deteriorating channel conditions, the selection of the ``best" subchannels-set for each user becomes more critical. On the other hand the 2nd heuristic provides us with an average gain of 52.8\%, which is still remarkable.
      \vspace{3pt}

  \item [(iii)] The optimality gap (defined as the percentage sum-rate loss from the optimal scheme)
  for each algorithm depends on the
  CBR-loading and the received SNR conditions experienced in the cell. The gap
  seems to narrow as SNR conditions are improved through
  the increase in the BS power. This is justified by the fact that for higher
  amounts of BS power, larger data-rates per subchannel are supported (close to the upper bound of 6 bits), leading to the vast majority of available bandwidth resources experiencing high performance, and thus the selection
  of the optimal subchannel set for each user is not so critical anymore.
  \vspace{3pt}

  \item [(iv)] The 1st heuristic has an average optimality gap of 6.7\% for
  the lowest BS-power scenario which decreases to 2.3\% for the highest BS-power scenario.
  On the other hand the alternative dual heuristic performs worse than the first
  one. Its recorded optimality gap is 15.4\% for the lowest SNR scenario and
  4.6\% for the highest on average.
  \vspace{3pt}

  \item [(v)] The worst-case performance for the 2nd heuristic is observed at
  the lowest SNR and highest CBR-loading scenario, where the performance loss compared to
  the optimal scheme is 23.87\% (Fig.\ref{fig:Thr-Load-Bars-LowerSNR}--rightmost group of bars).
  Recall that the particular algorithm first
  allocates the available resources ignoring the demanded QoS levels. Under
  such conditions, the CBR-class sum-rate heavily dominates the overall dual-class sum-rate, and thus the subsequent resources reallocation phase finds great
  difficulty in leading to a feasible allocation. In other words when the propagation conditions are harsh and the QoS targets demanding, it is better to first guarantee the strict CBR rate constraints and then look for an improved allocation. On the contrary at higher
  SNR/lower CBR-loading conditions (Fig.\ref{fig:Thr-Load-Bars-HigherSNR}--leftmost group of bars) the achieved performance is quite high (or
  equivalently the associated optimality gap is quite low) since the distance between the original solution and the feasibility is
  significantly smaller.
  \vspace{3pt}

  \item [(vi)] The performance of the 1st heuristic is not significantly
  affected by the CBR-loading conditions, contrary to the 2nd heuristic
  behavior. For the lowest power scenario the optimality gap of the 1st heuristic increases by
  2.3\% (from 5.9\% goes to 8.2\%)
  as the number of CBR users increases from 6 to 12, whereas for the 2nd
  heuristic the corresponding increase is 14.3\% (from 9.6\% goes to
  23.9\%). This is justified by the fact that the 1st heuristic focuses on
  finding a feasible solution by prioritizing CBR-class users at the
  initial allocation phase. Similar conclusions may be drawn for higher SNR scenarios as well.
  \vspace{3pt}

  \item[(vii)] As far as the 1st heuristic the importance of the subchannel swapping step is demonstrated through Fig.\ref{fig:HEUR1SwapEffect}, where one may observe the improvement level of the sum-rate performance for all the simulation experiments. The performance gain varies from 1.2--7.7\% and it is more noticeable in lower CBR-loading scenarios. In such cases a lower number of subchannels is required for the CBR-class and thus a larger number of re-asssignments is expected to occur.
    \vspace{3pt}

  \item [(viii)] Averaged over all conditions, the 1st
  heuristic achieves 96.21\% of the optimal sum-rate performance while the 2nd 91.63\% of
  it.
  \vspace{3pt}

  \item [(ix)] The performance gap between the actual optimal solution (IP) and
  the upper bound (LP) is not negligible. For the lowest power scenario (Fig.\ref{fig:Thr-Load-Bars-LowerSNR}) this gap
  ranges between 4.4\% and 17.9\% whereas for higher BS power (Fig.\ref{fig:Thr-Load-Bars-HigherSNR}) varies between 2.8\%
  and
  11.9\%. Therefore, although the LP solution is extracted very efficiently
  compared to the IP, it often fails on providing a tight bound on the exact optimal
  performance. Consequently, if the LP bound is used as a performance benchmark
  for the evaluation of a suboptimal scheme, then the actual efficiency of the
  latter would be underestimated.
\vspace{9pt}

\end{itemize}

\section{Conclusions - Future Work}\label{sec:6}
The resource allocation problem of maximizing the sum-rate performance for a
downlink OFDMA single-cell network (like LTE) assuming heterogeneous traffic
requests and uniform power loading over the system subchannels was studied in
this work. The problem was first mathematically modeled as an ILP optimization
problem, allowing us to extract the actual optimal allocation decision and the
associated maximum achieved sum-rate performance in reasonable execution time.
By relaxing the integrality constraints on the above model, a performance upper
bound may also be extracted with  3rd order polynomial complexity cost. In the
second part of this work, motivated by the resemblance of the specific problem
with a well-known combinatorial optimization problem that is, the Generalized
Assignment, we developed two heuristic algorithms for efficiently allocating
system subchannels to the competing classes and users. We finally demonstrated
through extensive simulation experiments that the performance loss of the
heuristic schemes compared to the optimal is rather low, especially for the 1st
heuristic, and that our schemes heavily outperform semi-random subchannel assignments. Possible
extensions of this work may involve the cooperation of the proposed schemes
with time-domain scheduling algorithms handling packet delay and fairness QoS objectives, the employment of practical channel
state reporting schemes and the consideration of interference in multi-cell
deployments.

\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Thr-Load-Revised}
\caption{System Rate performance vs CBR-Loading conditions}
\label{fig:Thr-Load}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.75\textwidth]{Thr-Power-Revised.eps}
\caption{System Rate performance vs average received SNR conditions}
\label{fig:Thr-Power}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=0.78\textwidth]{HEUR1SwapEffect.eps}
\caption{The effect of the subchannels swap procedure to the HEUR1 performance}
\label{fig:HEUR1SwapEffect}
\end{figure}

\begin{figure}
\subfigure[Lower SNR-conditions]{
\includegraphics[scale=0.52]{Thr-Load-Bars-Plvl2-grayscale.eps}
\label{fig:Thr-Load-Bars-LowerSNR} } \subfigure[Higher SNR-conditions]{
\includegraphics[scale=0.52]{Thr-Load-Bars-Plvl35-grayscale.eps}
\label{fig:Thr-Load-Bars-HigherSNR} } \caption[]{Sum-Rate and Optimality Gap
Performance}
\end{figure}

\begin{table}[!h]
\centering \footnotesize \caption{Achieved Optimality Gap values for all
Scenarios} \vspace{6pt} \label{ta:AllScenPerf}
\begin{tabular}{llll}
\hline\noalign{\smallskip}
\textbf{Scenario} & \textbf{IP/LP} & \textbf{HEUR1/IP}   & \textbf{HEUR2/IP}\\
\text{\{${K_1,P_{bs}/P_{\text{CBR}}^{\text{feas}}}$\}} & \% & \% & \%\\[2pt]
\hline\\[1pt]
\{6, 2.0\}  & 95.59 & 94.11 & 90.36 \\
\{6, 2.5\}  & 95.99 & 96.94 & 93.41 \\
\{6, 3.0\}  & 97.18 & 97.46 & 94.58 \\
\{6, 3.5\}  & 96.03 & 98.21 & 95.89 \\
\{6, 4.0\}  & 97.29 & 96.92 & 96.32 \\
\{8, 2.0\}  & 93.50 & 94.45 & 86.24 \\
\{8, 2.5\}  & 93.60 & 93.77 & 92.16 \\
\{8, 3.0\}  & 94.11 & 96.74 & 94.31 \\
\{8, 3.5\}  & 94.62 & 97.39 & 95.48 \\
\{8, 4.0\}  & 94.91 & 97.80 & 96.23 \\
\{10, 2.0\} & 88.55 & 92.90 & 85.56 \\
\{10, 2.5\} & 90.12 & 95.61 & 90.83 \\
\{10, 3.0\} & 91.24 & 96.93 & 93.28 \\
\{10, 3.5\} & 91.94 & 97.63 & 94.65 \\
\{10, 4.0\} & 92.62 & 98.11 & 95.59 \\
\{12, 2.0\} & 82.12 & 91.84 & 76.12 \\
\{12, 2.5\} & 85.54 & 94.51 & 85.27 \\
\{12, 3.0\} & 88.09 & 96.44 & 90.05 \\
\{12, 3.5\} & 88.62 & 98.60 & 92.78 \\
\{12, 4.0\} & 91.11 & 97.97 & 95.40 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

redpajama_set_name: RedPajamaArXiv


On your wedding day, you're already wearing the most gorgeous of your life, carrying the most beautiful flowers, and rocking the prettiest, glossiest hair your head's ever seen. The final step? Making sure you're not partaking in any common wedding makeup mistakes ahead of the big day. Even if you've had a skincare and beauty routine down pat since 2008, it's time to step up your game before you say I Do. Because chances are, you've been making some major makeup faux pas without even realizing it.
Before you walk down the aisle, take an extra minute in front of the bathroom mirror and see if you're guilty of any of these top eight wedding makeup mistakes. Reversing them is the easiest way to a flawless wedding look—even in the unretouched photos!
A great long wear lipstick can stay put for hours, but it's no match for champagne toasts and wedding cake—unless you give it a little help. Before you swipe on lip color, create a smooth surface with lip primer we love MAC's version ($17.50, available at Nordstrom); it makes for easier application and keeps lipstick on way longer.
2. You're using the wrong foundation shade.
If your base makeup makes you look chalky, it's time to switch out your shade for summer. Even if you're planning to go the drugstore, head to a department store first to find your shade. Swipe three options on your jawline, then go to a mirror near a window to see how it looks in natural light. The shade that blends most closely to your skin tone is the winner.
3. Your concealer's doing double duty.
Wondering why your concealer is settling into the creases under your eyes? It could be because you're using the same all-purpose pot you use to cover up breakouts. While blemishes usually call for thick, full-coverage concealers, under-eyes benefit more from smoother formulas with light-catching pigments. Try a liquid formula, like Maybelline Dream Lumi Touch Highlighting Concealer ($6.44, available on Amazon).
4. You're not applying primer.
5. You're using a dry makeup sponge.
The invention of the BeautyBlender has been a saving grace for makeup aficionados, but only when used properly. Using a dry makeup sponge to blend your makeup means the beauty tool will absorb more of the product, putting your beloved (and most likely expensive) foundation to waste. Splash a little water on your sponge for a streak-free, photo-ready result. Make sure you're also storing the damp makeup sponge in an open environment after use to prevent mold (the last thing you want on your face).
6. You're contouring too much.
While Kim K. can rock the contoured look like nobody's business, this beauty trend seems to be a thing of the past. Plus, too much contour can create a noticeably heavy contrast between your face and the rest of your body, according to makeup expert Sonia Kashuk. For a more natural sculpted look, she recommends applying sheer products under the cheekbones for extra depth.
7. You're depriving your lashes of maximum lift potential.
Goopy, clumpy eyelashes aren't the best look on your wedding day. But chances are, you're weighing your lashes down with every swipe of the mascara wand. If you usually swipe from underneath your lashes upward, try reversing it: Apply mascara from the top of your lashes, sweeping down and out, which makes for the most separation.
8. You're not refreshing your makeup bag every few months.
Just uncovered your favorite lipstick from 2015 at the bottom of your makeup bag? Toss it before your wedding day. Left in a makeup bag at room temperature, makeup grows bacteria fast—most after six months to a year, but some after just a few months. Check the expiration date at the bottom of each tube (it looks like a jar of face cream with a number in it; that number shows how many months until it's donezo) and stick to it.
Wishing the very best to our Ikonica Beauty Virginia and her new husband Ross.
We are extremely grateful to have been a part of this happy couples BIG Day!
Virginia was looking for a old Hollywood glam look that would match her sense of style!
We think she rocked it with a true petal pink lip colour from alistbeauty and snow frost and baby-boll pink eye shadow enhanced with a thick cat-eye eyeliner.
The day was amazing and truly Ikonica!


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Featured Designer: Sandra Boynton
You've probably read a couple of Sandra Boynton's books growing up, or have read them to your children. She is kind of a big deal in the literary world, recognized as an extraordinary talent. What you may not know about Boynton is that she has her very own Zazzle store! This amazing artist and entrepreneur has lead an interesting life, and her works speak to both children and adults. Read on and learn more about Boynton, her books, music, and her designs!
Hippopotenuse Poster
Since 1974 Boynton has written over fifty children's books. You may recognize some of the titles from your childhood, or from purchases you've made for children in your family: "The Going-To-Bed Book," "Doggies," "Happy Hippo, Angry Duck," "But Not The Hippopotamus," "Pajama Time, "Barnyard Dance." These books are accompanied by simple illustrations and language.
The Moon Is High Poster
The simple illustrations and language hide a deeper context according to Ian Bogost. Writing in The New Yorker, Bogost uses "The Going-To-Bed Book" as an example of Boynton's subliminal writing style. On one page of the book, once the animals have gone through their pre-sleep rituals, they do something unexpected: they head to the deck of the ship and exercise! Seems out of place, right? Bogost breaks it down thusly:
"…the boat isn't just a boat. It bears the unmistakable silhouette of Noah's ark, host to the animals saved from the flood in Genesis…[i]magine the terror of repetition without relief among the flood-bound Boynton hippos. This is also the terror of new parenting: the sameness of every day—the same rituals, the same books, the same diapers and tears and wails and suckles, joy and pride all bound up with the dread and exhaustion of infancy."
The success of a children's book is not just tied into how much kid's like them. Adults need to like them as well, since they will be reading them every night. Adding in elements that entertain the parents reading these books without changing the meaning for children is a high-wire act, and Boynton manages it beautifully. Books, however, aren't her only artistic pursuit.
Frog Trouble Postcard
Boynton has also released 5 albums of music with collaborator Michael Ford: Rhinoceros Tap (1996), Philadelphia Chickens (2002), Dog Train (2005), Blue Moo (2007), and Frog Trouble (2013). Each album is a send-up of a different musical genre. For instance, Philadelphia Chickens is an imaginary musical, while Frog Trouble is a country album. Boynton has teamed with some legendary musicians, including Brian Wilson of The Beach Boys, Blues Traveler, Neil Sedaka, B.B. King, Hootie & the Blowfish, Alison Krauss, Brad Paisley, and many more. Her first three albums were certified gold, and Philadelphia Chickens was nominated for a Grammy Award in 2003.
Famous Barnyard Composers Poster
As Boynton told the Harvard Gazzette her process for getting these huge artists to work on a track with her is pretty simple: "I tend to write each song with a particular performer in mind. With my music production partner, Mike Ford, I create a tight demo, and then I send it off to the artist's manager. When artists like a song, they say yes." Judging from the list of people she has worked with, more and more artists are saying yes!
Christmas Tree of Kitties Card
Working for Recycled Paper Greetings, Boynton's illustrated card have sold over 500 million copies from 1973 through 2003. One single card featuring a hippopotamus, a bird, and two ewes, and punning on the traditional "Happy Birthday to You" card (Hippo Birdie Two Ewes), has sold over 10 million copies on its own.
Happy Valentine's Day Card
A prolific career that has spanned decades, Boynton has extended her reach into producing designs for her own personal Zazzle store. You can check out all of her artistic endeavors and enjoy the talent, passion, and drive she brings to each.
Which are your kids' favorite Boynton books or songs? Do you have a favorite? Please share with us in the comments section below!
Tags: designer, Featured designer, sandra boynton
Jamar'S ARTICLES (92) • DESIGNERS
3 thoughts on "Featured Designer: Sandra Boynton"
I just bought the board book "Moo, Baa, Lalala" for my one-year-old niece because my own kids loved her books when they were little and it's a great "sound" book! Their favorite was the Going to Bed book. I still have a post-it note pad from 1985 that says "Don't Let the Turkeys Get You Down" which I love and which helped me get through some tough times in my college years when I would see it and repeat it. I've been a fan of Sandra Boynton's for decades!!!
Mary Sullivan says:
I have it all -cards, mugs, books, music, stuffed animals even baby quilt (as a wall quilt, since my guys were too big when it came out). When I'm having a bad day, I listen to Phily Chicken or Rhino Tap or break into verse "…it's quiet now, what do you say? …or what's wrong little Pooky?" But I am especially fond of Chloe & Maude" great moral plot!
Debby Phillips says:
My husband bought me the Hippo Birdie card for my birthday in college in 1976 and the started my love affair with all things Boynton. I bought my kids her board books, GreenHat, Blue Hat and Doggies were my son's favorites. He laughed every time I'd say oops. Recently I have bought the same books for my grandson. He loves them all but really loves Doggies when the howl at the moon, he howls too. I give the books to all new parents. Thank you for so much joy.

redpajama_set_name: RedPajamaCommonCrawl


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MADE IN PLANO
10° F Rolling Ice Cream
by Alex Rejaee
The Country's Newest Dessert Trend Has Rolled into Town
Tucked away in the back corner of The Shops at Willow Bend food court, the atmosphere of 10° F Rolling Ice Cream is bright and cheerful. Teenage employees take orders and bustle around like bees while the manager rolls out their customers' creations (no pun intended…well maybe a little). A solid line is formed in front of the sole register, and a throng of about twenty people are crowded around the glass panels watching their ice cream concoctions being created.
The rolled ice cream trend began in the street carts of Thailand and quickly spread across the globe due to its unique look and healthier nature (because of the liquid composition and instant freezing process, additives are not necessary). Not surprisingly, the frozen treat became an instant hit in Manhattan. The owners of 10° F, originally from New York, noticed its success and knew there was an open market in the Dallas area. The shop opened this past November and has quickly become a hit. There are already plans of extending the 10° F brand to five more locations in the DFW Metroplex, with one of the proposed spots potentially being in The Colony.
photos Nicole Forzano
According to an employee, one order is equal to about two scoops of traditionally made ice cream. The menu is fairly simple: pick a flavor of ice cream for the base – either original, green tea or chocolate – and then choose up to two mix-ins. These range from freshly cut fruit to Oreo cookies. Once the rolls have been made, customers have the option to select an unlimited amount of toppings as a finishing touch.
The process itself is mesmerizing. The ice cream is stored in liquid form and then poured onto large flat, metal surfaces. The surfaces look identical to what crepes are made on, except instead of being heated, they are kept frozen. The mix-ins are then placed in the middle of the ice cream "batter" and, using two flat spatulas, chopped up until finely blended. All the while the ice cream has become colder and stiffer due to the frozen surface.
Up until this point, the whole process is almost identical to that of Marble Slab or Cold Stone Creamery. But it differs when the semi-frozen mix is spread out into a thin square across the surface, allowing the ice cream to harden. Once spread, the rolling finally begins. Taking one of the spatulas, the flattened ice cream is scraped from bottom to top in strips, causing them to curl into small, individual rolls.
The employees are like machines. Pour, chop, smooth, roll. It seems like they could go on an entire day without stopping. Both children and adults take photos and Snapchats of the process, sending them to friends and family.
Compared to its competition across the country, the menu here at 10° F Rolling Ice Cream is fairly concise, having three options as base flavors and nine as mix-ins. There is only one register and two of the frozen surfaces, so orders can easily become backed up during peak times.
Hopefully, we will see that with the expansion of locations, the menu options and customizations will grow as well. Only time will tell whether or not the rolling trend sticks around as long as frozen yogurt. But for now, 10° F seems to be the perfect place for those looking to try a hip, new dessert spot or, if anything, to snap a couple of pictures for Instagram.
10F Rolling Ice Cream
Alex Rejaee
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Precision Resource Group feature in the Sunday Times Virgin Atlantic Fast Track 100 2020
December 7, 2020 /in Blog /by PRG
Precision Resource Group (PRG) are extremely excited to this weekend have featured in the 24th annual Sunday Times Virgin Atlantic Fast Track 100 league table.
We couldn't be more proud with this year's league table including only three companies based in the South West and just two companies from the recruitment sector UK wide. Ranked number 89 on the list, we are the first of the newest entrants for the South West and are proud to have tracked a three-year annual sales growth of 41% with £31.3m in April 2020.
The Sunday Times Virgin Atlantic Fast Track 100 league table ranks Britain's 100 private companies with the fastest-growing sales over the last three years. It is compiled by Fast Track and published in The Sunday Times each December, with an awards event the following September.
Even though it has been a difficult year for all businesses, we are so proud that PRG and our group companies iO Associates, Panoramic Associates, Panoramic Care and Finitas have come out of it stronger than ever. We couldn't have achieved any of this without our incredible employees who have worked so hard and put in so much effort this year!
A huge congratulations to all the other winners, you can view the full league here: https://www.fasttrack.co.uk/league-tables/fast-track-100/league-table/
https://precisionresourcegroup.co.uk/wp-content/uploads/2020/12/Fast100_2020_Press-release-graphic-PRG-1.png 365 938 PRG https://precisionresourcegroup.co.uk/wp-content/uploads/2018/07/logo.png PRG2020-12-07 09:33:442020-12-07 09:39:48Precision Resource Group feature in the Sunday Times Virgin Atlantic Fast Track 100 2020
PRECISION RESOURCE GROUP
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Bristol, BS1 2NH
E: info@precisionresourcegroup.co.uk
© Copyright - PRG
Jessie May and our Board of Trustee, Tom Selman We have joined the Employers Network for Equality & Inclusion (ENEI)


Ukulele Underground Forums > Ukulele Boards > Ukulele Tips, Tricks and Technique > Using a (guitar) pick?
View Full Version : Using a (guitar) pick?
Good for you! So do I. Let's break down those self-imposed barriers.
Why does everyone seem so opposed to picks?
Pick as you please, rock the free world! But beware lest you be ousted from the uke community, soft and cuddly on the outside but rock hard and clique-ified on the inside. Many radicalist groups such as the "1-Inch Thumbnails", the "Soprano Inferiority Complex Correction Squad", "People with Manner Play the Tenor" and others exist. Needless to mention the group found in dark alleys and behind trash containers, ready to strike on those who dare to jump out of the mold. Their name is not spoken on these forums, but I will leave you with their secret slogan: "If you use a pick, you are a ....".
I guess uke players just like to fondle their baby as much as possible. Nylon over skin, many fall for it. Just do your thing and rock that uke.
It comes down to personal preference and no one is gonna knock you if do use on or you don't use one. Here's a similar thread (http://www.ukuleleunderground.com/forum/showthread.php?t=3640) and this one (http://www.ukuleleunderground.com/forum/showthread.php?t=3340) as well.
Here's an article about Jake and how he used to always us a pick.
Uke Society Policy: "Don't Ask, Don't Tell." Opps, this policy is for a different subject. Never mind.
I think it's more about rythms and strums. When I use a pick I tend to use more of a guitar technique. When I use my fingers my strums are more uke-centric.


Previously the following code (3.7.1 and earlier) would remove all the dashboard panels and still works with the exception of Site Content and WordPress News.
Tried attaching it to other actions (wp, init, current_screen) without any change.
We're not against WordPress news, but clients get easily confused, and we like to keep things as minimal as possible.
The Site Content widget can be removed by adding it to your list using its ID, site-right-now. We should update the instructions in the Codex to reflect that change, and also to remove the widgets that were removed from the Codex list.
BUT - something different is going on with the WordPress News widget -- it uses the same ID, dashboard_primary, as the old version, but it isn't being removed, although I can still show and hide it from the checkboxes in Site Options.
Restore original properties of the right now (activity) and primary (news) dashboard widgets.
This avoids issues with those trying to remove/disable core meta boxes.
The right now widget was renamed. This renames it back to its original name.
The news widget was set to "low" priority. This means it loses its special "core" priority designation, which add_meta_box() specifically checks for to see if it needs to block the widget.
Sacrificing low priority is not that big of a deal. This restores 3.7 behavior.
Renaming the widget broke some CSS.
Restore CSS for the Site Content, née Right Now, Dashboard widget. Removes a nice large swath of unnecessary styles. fixes #26392, see #25824.


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Maplewood Middle School Home and School Association
MMS HSA
HSA News Submission Form
Request for HSA Funds
Staff and School Info
Youthnet
MMS HSA News - Week 22
*|MC_PREVIEW_TEXT|*
MAPLEWOOD MIDDLE SCHOOL HOME & SCHOOL ASSOCIATION
To all that are celebrating the Lunar New Year, the HSA wishes you luck in the Year of the Rat!
For 2020, the Lunar New Year officially begins on Feb. 21. The Lunar calendar follows the phases of the Moon and varies slightly from the Solar calendar every year.
The holiday is celebrated by a fifth (1/5) of the world's population and observed by over 20 Asian and Southeast Asian countries, including China, Vietnam, and Korea.
It is a time to tie up loose ends, make amends, reconnect with loved ones, and say goodbye to the past year (by literally sweeping the dust out the door). You will see lots of RED decorations everywhere. The color is a symbol of luck, happiness, and prosperity.
February is packed with Winter Band and Orchestra, School Assemblies and 13! The Musical - stay on top of the calendar by adding the HSA calendar.
Tuesday, Feb. 4: Half day for students (PD for staff)
Wednesday, Feb. 5: Black History Month Event . Stay tuned!
Feb. 13: Musical fundraiser winners will be drawn. See details below.
Feb. 14 and Feb. 17: School closed for President's Day
Friday, Feb. 21: Trivia Night. Tickets are on sale. See link below!
MMS 7th ANNUAL TRIVIA NIGHT!
Friday, February 21st @ The Woodland
Tickets: $70 AND includes more fun than you can imagine with a fierce trivia competition if you're up for it! Plenty of food!
1 drink ticket included / welcome to bring you own or buy wine & beer there.
Tickets sell out fast! Think about your team now. CLICK HERE TO ORDER TICKETS OR TO SPONSOR THE EVENT!
All proceeds go to support MMS HSA programs.
ATTN: 8th GRADE FAMILIES
Two events are fast approaching and we need your help. The annual MMS/SOMS Mixer is Tuesday, Feb. 11 from 7 p.m. to 9 p.m. at CHS. This is a free event but we could use some snacks and volunteers. Click here to sign up. More details are available on the flyer.
The 8th grade pretzel and water fundraiser at Table Tennis tournament will be held on Thursday, Feb. 13 during school hours (1:30 p.m. to 2:55 p.m.). If you can volunteer to help sell pretzels, please email mfitzp100@gmail.com.
13! Front Row Seats Fundraiser
This month's fundraiser is a Front Row Seat Raffle! Just $10 buys you a chance to win FOUR prime seats, a retail value of $80!One winner per performance will get four front-row seats, a 13! poster and a T-shirt.Click here to register. Winners will be drawn Feb. 13.
Nights of 100 Dinners - Achieve's Fundraiser
Do you enjoy entertaining? Want to support our schools? Host a party in March for the Achieve Foundation's Nights of 100 Dinners! Send your guest list to Achieve and we'll send the invitations. Guests make a donation to the organization. Learn more.
Achieve Foundation Direct Funding
Last month, the Achieve Foundation, awarded $34,000 in educator grants for projects across the district. Five worthy grants - including for MMS - are now open for Direct Funding from the community. Help us spread the word and fully fund these wonderful and varied programs that benefit our students.
HSA Meeting
The next HSA meeting will take place on Wednesday, Feb. 19 at 7:30 p.m. in the MMS Library. The meeting originally scheduled for Jan. 15 was postponed.
Yearbooks on Sale!
Yearbooks are on sale! Don't wait until it is too late. Order your copy today! Order and payment must be received by April 15, 2020.
Download the yearbook order form or order online. Our school code is 6667.
You can also order a Well Wisher online. Send a message to a special 8th grader! Your Well Wisher must be ordered by April 1, 2020.
Join The MMS HSA
No Frills Fundraiser
MMS Makerspace
Families - We are so proud of our MMS students and their participation in the MLK Peace Walk last week. MMS Parent Chad Hunt took amazing photos - take a look here.
A friendly reminder that school is closed on Monday, Jan. 20 in observance of MLK day. In addition, SEPAC will hold its next meeting on Wednesday, Jan. 22 at 7:30 p.m. at MMS (note the location change). This meeting is for families with students who are currently IN or ENTERING middle school or high school. Dr. Morana and supervisors have been invited to attend for the first half of the meeting.
Additional reminders:
The Winter Band and Orchestra dates have landed - mark your calendars for Feb. 4 (Orchestra) and Feb. 6 (Band). They each start at 7 p.m. and should last for about an hour.
Trivia Night is Friday, Feb. 22. Tickets will go on sale soon and will sell out FAST! If you are planning on attending, get your teams together ASAP. (Team sizes are about 10 people.)
13! The Musical runs Feb 28. to March 1 at MMS.
Tickets are on sale now!
Messages and ads can still be placed in the playbill.
Musical T-Shirts and Candygrams can also be pre-ordered.
Enjoy the week,
Veronica and Kelly
Tickets sell out fast! Think about your team now. STAY TUNED FOR AN EMAIL THIS WEEK WITH INFORMATION ON TICKET SALES!
This month's fundraiser is a Front Row Seat Raffle! One winner per performance will get four front-row seats, a 13! poster and a T-shirt. Note: There are no refunds so winners who have purchased tickets can share with friends or donate tickets back to the house. Winners will be drawn Feb. 13. Stay tuned for more details!
Schools Closed
All district schools will be closed Monday, Jan. 20 in honor of Dr. Martin Luther King Jr.
The next HSA meeting will take place on Tuesday, Jan. 21 at 7:30 p.m. in the MMS Library. The meeting originally scheduled for Jan. 15 was postponed.
SEPAC Meeting
The next SEPAC meeting will be Wednesday, Jan. 22 at 7:30 p.m. at MMS. Note that the meeting location has changed. The target audience is anyone with a student in or entering middle school or high school.
In Tune's Open House
Stop by on Saturday, Jan. 25 to find out more about our music school. Meet our teachers, try out an instrument and learn about our new lessons and classes.
In Tune is located above the Maplewood Movie theater (155 Maplewood Avenue), and we offer music lessons for all ages and levels!
Our lives begin to end the day we become silent about things that matter.
- Dr. Martin Luther King Jr.
Dr. King's powerful words continue to inspire, teach and shape individuals in our community and around the world.
Maplewood Middle School will honor the life and legacy of Dr. King when it holds its annual Silent Peace March this week. The silent march includes all students and staff from the middle school and represents a continued commitment to the ideals for which Dr. King and so many others peaceably fought. The sight of hundreds of middle schoolers walking silently through Maplewood Village is stunning and profoundly affecting. The march takes place Friday, Jan. 17 at approximately 9:15 a.m., beginning at the school and winding its way through the village.
Other events taking place this week include:
Tuesday, Jan. 14: Screenagers at South Mountain Elementary School @ 7:30 p.m.
Wednesday, Jan. 15: MMS HSA Meeting, MMS Library @ 7:30 p.m.
Reminders for next week:
Monday, Jan. 20: ALL SCHOOLS CLOSED in honor of Dr. King.
Thursday, Feb. 6: MMS Band Winter Concert, 7 p.m.
SCREENAGERS is back!
Come learn about what too much screen time does to your student's brain. Screenagers will be shown on Tuesday, Jan. 14 at 730 p.m. at South Mountain Elementary School, 444 West South Orange Ave., South Orange.
Donate to Classroom Libraries
As we promote a love of reading at MMS, we would like our kids to have visually pleasing libraries and cozy places to read! Our teachers are looking for small bookshelves, bean bag chairs, and other comfy seating options for our kids.
Pictured above is the library in 8th grade ELA teacher Ms. Herstatt's class. This is a sample of what we're looking for. Please reach out to Ms. Gronau if you have items you wish to donate.
The next HSA meeting will take place on Wednesday, Jan. 15 at 7:30 p.m. in the MMS Library.
The next SEPAC meeting will be Wednesday, Jan. 22 at 7:30 p.m. in the Central Office District Meeting Room (525 Academy St., Maplewood). The target audience is anyone with a student in or entering middle school or high school.
MMS HSA Families -
We have our first full week back at school this week! As always, stay connected on upcoming events by adding the MMS HSA google calendar to your own calendar.
A few things to keep a lookout for:
- Tickets for 13! The Musical will go on sale this week.
- This Wednesday, Jan. 8, the district is hosting an event on integration and equity at 7 p.m. at Columbia High School. Learn more here.
- Jan 14: Screenagers will be showing at South Mountain Elementary School at 7:30 p.m.
- Jan. 17: Silent Peace March for MLK Day.
- Jan. 20: School is closed for MLK Day.
- SEPAC will hold its monthly meeting on Jan. 22.
Stay tuned for details about the Winter Band and Orchestra Concerts later this month.
Welcome Back MMS Families!
Hoping you had a restful winter break and are ready for 2020. A few reminders to kick of the short week:
- Congratulations to the 6th Graders who won the Battle of the Classes!
- Thursday is a B Day and a regular full day of school.
- After school YouthNet Registration is up - be sure to sign up here.
- Next HSA Meeting is Wednesday, Jan. 15!
- The SOMSD 2020-21 School Year Calendar is out! You can find it here.
Your MMS HSA,
Sign Up for YouthNet Winter 2020
Permission slips will be collected through the end of the week for the YouthNet Winter 2020 Season. Check out our selection of clubs and afterschool activities. The session begins Jan. 6.
Maplewood Middle School Home and School Association  © 2021 973-378-7660


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Widow Programs
Orphan Programs
IDFWO Canada
Elad Blobstein
During the Second Lebanon War, in a bold operation, Sikursky CH-53 ('Yas'ur') pilot Colonel Dani Schiffenbauer rescued the body of Lieutenant Colonel Emanuel Moreno, who was killed in action, deep beyond enemy lines • Schiffenbauer himself was killed four years later in a training accident • Their two widows, Yael and Maya, have since come together, connected by their shared fate — and developed a unique friendship
Translated by Zoe Jordan
Photos by Yossi Aloni
"I always found airplanes exciting," says Yael Schiffenbauer, sitting in the garden outside of her home in Moshav Kidron, a few dozen meters from the runway at the nearby Air Force base. "When I hear the planes or helicopters coming through here, I go out right away; it's still exciting to me. In recent years, that feeling has become all mixed up with a lot of longing, too."
The longing is for her husband, the late Colonel Dani Schiffenbauer, a Sikursky CH-53 pilot who was killed in 2010 when his helicopter crashed during a joint training for both the Israeli and Romanian Air Forces in the Carpathian Mountains.
"We had just moved to a new apartment and the whole place was in chaos," Yael recalls. "We moved on a Thursday and on Sunday he flew to Romania. He had spent three nights in our new home. Through the window I saw a friend of ours pass by the house and I thought to myself, 'how sweet, he came to help', but then I saw that he wasn't smiling back at me. And then I saw the representatives from the local IDF bureau. I didn't fall apart but he held me like I was about to faint. It took me a few days to digest the news."
Sitting beside Yael is her close friend Maya Moreno-Ohana. Theirs is the kind of friendship that only the reality of Israel can forge. She listens to her friend's painful story, to which she can relate all too well. Maya was widowed in 2006 when her husband, the late Lieutenant Colonel Emanuel Moreno, a fighter in the special forces, was killed during the Second Lebanon War in a covert operation beyond enemy lines. The person who retrieved his body from the battlefield was none other than the late Col. Schiffenbauer, in an operation for which he later received a citation.
The pastoral stillness and the chirping of birds are occasionally interrupted by the sound of a fighter jet taking off. It seems like this reality is a reflection of the two widows' character. They radiate a gentle calm and quiet that is violated, every so often, by the noise of the disasters that befell their lives.
"It took me time to understand how extraordinary it is that there was a pilot here who is the reason I now have this grave," says Maya. "When Emanuel was killed I had to deal with my personal grief as well as the national reality that came with it: an extremely challenging situation. The first year, I was just busy with the kids. Afterwards, I began to think of myself again and by the third year after his death things started to open up within me. I began to understand that he had a grave and that I was able to visit it thanks to a very complex rescue operation."
Yael agrees with Maya that it was a particularly challenging rescue, which required quick decision-making in the field. She recalls that after the operation, Dani told her that Emanuel's death was a huge loss for the defense forces. Dani also attended Emanuel's funeral and a few days later took Yael to the squadron base and played her a recording of the voices from the rescue operation's two-way radio. According to Yael, the dramatic voices gave some indication of what was happening on the ground.
Three years after Emanuel's death, it occurred to Maya to arrange a meeting with the man who had retrieved her late husband's body. "A good friend of Emanuel's was on the team — today he is a member of parliament, Matan Kahana — who finished the special forces track and moved on to the Air Force," she says. "One day, a few years after Emanuel's death, Matan came back from an investigation they did about his rescue and asked me if I was interested in seeing the report. I said yes, but first I wanted to meet the pilot who had been responsible. He didn't really understand why, but I explained to him that I realized that it was no small thing that I had been granted this grave, it could have ended differently, and I wanted to thank him," she recalls.
"He arranged a meeting for me with Dani and added, 'he's just going to Romania for a training exercise and then coming back'. But then they informed me that Dani had died. I remember that moment vividly. I was in Yavne, I remember the precise spot where I was. The woman on the phone told me to stop the car and then she told me. I decided that I would come to his shiva to meet Yael, Dani's widow." And that is how the two women met.
"Shivas are intense, and the amount of people that come is sometimes overwhelming," says Yael. "But when Maya sat down across from me to say thank you — I was extremely touched. I thought to myself, what is she thanking me for? Emanuel was killed for all that he did out there for us — we should be thanking him."
Since then, Maya and Yael have been inseparable. While they don't meet on a daily basis, the two agree that their heartfelt connection goes way beyond words. "Maya is a special woman, she is such an inspiration to me," says Yael. Maya adds that "there's something that brings hearts together. We connect through our values and our world views."
Don't Wait, Act Now
Maya Moreno-Ohana, age 46, currently lives in Moshav Shokeda, alongside the Gaza strip. She is remarried to Eliran Ohana and together they are raising five kids, three of whom are Emanuel's children.
She tells her story all over the country. And although each lecture is different, at most talks she focuses on the reason she chose to expose her story. "I always say that Emanuel is my story's architect, but in fact the person who brought that to light for me was actually Dani, despite me not having met him. The central message for my audience is that if there is something you feel compelled to do — you must do it; you shouldn't wait. I waited too long and I missed my chance. If you have an inner desire to thank someone or to clarify something — you must act on it immediately," she says.
Yael Schiffenbauer, 52, has three children that Dani fathered. She has a doctorate in chemistry from the University of Tel Aviv, and manages a biomedical engineering project.
Do your kids know each other?
Maya: "Yes. From the 'Otzma' camps. My oldest daughter and Yael's middle child flew in a helicopter together during one of the activities. They are slightly different ages, but the name Schiffenbauer is well-known in our home."
Yael: "Every Israeli household knows the name Moreno. Including ours, of course."
Do you see aspects of Dani and Emanuel in your kids?
"Yes. The boys are grown now, and the way they move reminds me of him. Also the way they talk, look, and think," says Maya. Yael adds: "I don't see as much similarity. But someone always says how much one of the kids looks like Dani."
Did the tragedies that you experienced influence your kids' decisions about where to serve in the military?
Maya: "I want them to do whatever it is they want. I remember a moment during the shiva when I told myself, 'don't let that fear control you'. So, I decided then that I would relinquish control of that. My kids will go where they want and do what is right for them."
Yael: "I know that I won't stop my children from doing what they want. It's hard enough that their lives changed in this way. There's no need to punish them further."
Maya devotes a lot of her time to the fight for the return of prisoners and missing persons. According to her, most Israelis identify with the fight and get involved by simple means such as getting engaged on social media platforms or making other small gestures, but she is convinced that we must take a more active role in the struggle.
"I realized that from my very specific vantage point, I understand that it is not to be taken for granted that my husband has a grave, so I am obligated to rise up and fight this fight. I'm not related to those currently imprisoned or missing. I didn't know them or their families," she says.
She recalls a personal anecdote: "I remember a conversation that I had with Avia, my eldest daughter. When Gilad Shalit was kidnapped, Emanuel returned home, came inside, took his tefillin and left again for the army. I realized that if he was taking his tefillin with him he wouldn't be coming home. He told me that a soldier had been kidnapped so he had to go down south. I barely saw him after that. He would come home every few days and behave like a caged lion. He was really angry about the situation. One evening Avia told me, 'Enough with this soldier, already. I want Dad to come home.' She really missed her father.
"I sat down on the floor beside her and told her that Dad was searching for the soldier. I asked her what she thought the soldier was feeling. She said that he must be feeling very alone, in the dark, afraid and probably missing home.
"Then she got up, looked me in the eye and said: 'Okay Mom, I'll let go of Dad for the sake of that soldier.' She said it and she did it. And a little under a month later, Emanuel was killed. But that conversation really stood out to me: if a five-year-old girl can give up her deepest wish of having her dad around, I should be able to step out of my own private world and create a bit of noise on behalf of the cause, because we can't forget the true meaning of a mutual guarantee and of the IDF's values. Israel is founded on the principle of a mutual guarantee. If I can broadcast that message — it's my duty to do so," Maya concludes.
Grief, Lebanon War, Widows
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Paige Hill: Saturday Inspiration ... Bring Mary Englebreit's Home Companion Back!
Saturday Inspiration ... Bring Mary Englebreit's Home Companion Back!
By show of hands who used to stalk the mail box or the newsstand for this FABULOUS magazine? Does everyone miss it as much as I do?
Mary Engelbreit's Home Companion "Got me". I felt at home in the pages ... These artists felt like MY People ... MY FRIENDS! Ironically enough when I was fortunate enough to MEET many of these wonderful women, I was treated as if I actually was the old friend I felt I was. All because of the way I was introduced to them in ME's magazine. I was filled with creative projects, decorating ideas and inspiration. I have saved each issue and treasure each one. I consult them every time I need a creative boost. Just this week I was down in the decorating dumps. Luckily, I just pulled out one of my ME holiday issues and VOILA! Dilemma solved. The ideas that were published in Home Companion are AMAZING and timeless ... I miss it so much!
Were you a Home Companion fan?
Mary has released a special digital mini mag just for us!
It's a special peek into Mary's home for Christmas ... It's BEAUTIFUL!
I have a collection of old paper mache houses and bottle brush trees, so I especially love how she has added them to her holiday decor. Not only can you download the magazine, you can also get some DARLING gift tags for FREE!
I am going to print mine on heavy card stock, then attach them to my neighbor gifts!
As you can see, I love Mary Englbreit's Home Companion and I so wish they would bring it back! Unfortunately, the publishing industry is volatile and it's hard to get funding, but me? I'd like my voice heard. How about you? Can you help?
Click the link below to cast your vote and spread the word!
We creatives need this magazine!
You can share this post on Facebook or Twitter by using the buttons at the bottom of the post and I would so appreciate if you would. Pulling together is the only way we get things done, and our creative world is a better place because of Mary!
Let's show her we care and want our voices heard!
I loved ME too! It was one of my favorites!!!!! I'd love to see it back again!
Thank you for this link! My mother-in-law bought me a subscription when it first came out. After 5 moves, I still have several copies. Would love it to come back!
I loved loved loved home Companion. I still have all my favorites. I Was very sad to see it end and still am. One of my favorites.