Modification in 5.2.2. Energy consumption

[Sensornets15.git] / Example.tex

diff --git a/Example.tex b/Example.tex
index db1b822f8cff9e417263b35e013d5b4a9f34810f..28aa4d4c82edc77e8ce045b8ce935f1f94b79515 100644 (file)
--- a/Example.tex
+++ b/Example.tex
@@ -102,76 +102,82 @@ where a  review of some related  works is presented. The  next section describes
 the  DiLCO  protocol,  followed   in  Section~\ref{cp}  by  the  coverage  model
 formulation    which    is    used     to    schedule    the    activation    of
 sensors. Section~\ref{sec:Simulation Results  and Analysis} shows the simulation
 the  DiLCO  protocol,  followed   in  Section~\ref{cp}  by  the  coverage  model
 formulation    which    is    used     to    schedule    the    activation    of
 sensors. Section~\ref{sec:Simulation Results  and Analysis} shows the simulation

-results. The paper  ends with a conclusion and some  suggestions for further work
+results. The paper ends with a  conclusion and some suggestions for further work

 in Section~\ref{sec:Conclusion and Future Works}.
 
 \section{\uppercase{Literature Review}}
 \label{sec:Literature Review}
 
 in Section~\ref{sec:Conclusion and Future Works}.
 
 \section{\uppercase{Literature Review}}
 \label{sec:Literature Review}
 

-\noindent In  this section, we  summarize some related works  regarding the coverage
-problem  and distinguish  our DiLCO  protocol from  the works  presented  in the
-literature.
-
-The most discussed coverage  problems in literature
-can  be classified into  three types  \cite{li2013survey}: area  coverage \cite{Misra} where
-every point inside an area is  to be monitored, target coverage  \cite{yang2014novel} where the main
-objective is to  cover only a finite number of  discrete points called targets,
-and  barrier coverage \cite{Kumar:2005}\cite{kim2013maximum} to  prevent intruders  from entering  into the  region of interest. In \cite{Deng2012} authors transform the area coverage problem to the target coverage problem taking into account the intersection points among disks of sensors nodes or between disk of sensor nodes and boundaries. 
-{\it In DiLCO  protocol, the area coverage, i.e. the coverage  of every point in
-  the sensing  region, is transformed  to the coverage  of a fraction  of points
-  called primary points. }
-
+\noindent  In  this section,  we  summarize  some  related works  regarding  the
+coverage problem and distinguish our  DiLCO protocol from the works presented in
+the literature.
+
+The most discussed coverage problems  in literature can be classified into three
+types \cite{li2013survey}:  area coverage \cite{Misra} where  every point inside
+an area is to be  monitored, target coverage \cite{yang2014novel} where the main
+objective is  to cover only a  finite number of discrete  points called targets,
+and barrier coverage \cite{Kumar:2005}\cite{kim2013maximum} to prevent intruders
+from entering into the region  of interest. In \cite{Deng2012} authors transform
+the area coverage problem to the target coverage problem taking into account the
+intersection points among disks of sensors nodes or between disk of sensor nodes
+and boundaries.  {\it In DiLCO protocol, the area coverage, i.e. the coverage of
+  every  point in  the  sensing region,  is  transformed to  the  coverage of  a
+  fraction of points called primary points. }

 
 The major  approach to extend network  lifetime while preserving  coverage is to
 divide/organize the  sensors into a suitable  number of set  covers (disjoint or
 
 The major  approach to extend network  lifetime while preserving  coverage is to
 divide/organize the  sensors into a suitable  number of set  covers (disjoint or

-non-disjoint),  where each  set completely  covers a  region of  interest,  and to
+non-disjoint), where  each set  completely covers a  region of interest,  and to

 activate these set  covers successively. The network activity  can be planned in
 advance and scheduled  for the entire network lifetime  or organized in periods,
 activate these set  covers successively. The network activity  can be planned in
 advance and scheduled  for the entire network lifetime  or organized in periods,

-and the set of  active sensor nodes is decided at the  beginning of each period \cite{ling2009energy}.
-Active node selection is determined based on the problem requirements (e.g. area
-monitoring,  connectivity,  power   efficiency). For instance, Jaggi et al. \cite{jaggi2006}
-address the problem of maximizing network lifetime by dividing sensors into the maximum number of disjoint subsets such that each subset can ensure both coverage and connectivity. A greedy algorithm is applied once to solve this problem and the computed sets are activated in succession to achieve the desired network lifetime. 
-Vu \cite{chin2007}, Padmatvathy et al. \cite{pc10}, propose algorithms working in a periodic fashion where a cover set is computed at the beginning of each period.
-{\it Motivated by these works, DiLCO protocol  works in periods, where each  period contains a preliminary
-  phase  for information  exchange and  decisions, followed  by a  sensing phase
-  where one cover set is in charge of the sensing task.}
-
-Various   approaches,   including   centralized,  or distributed
-algorithms, have been proposed to extend the network lifetime.
-%For instance, in order to hide the occurrence of faults, or the sudden unavailability of
-%sensor nodes, some distributed algorithms have been developed in~\cite{Gallais06,Tian02,Ye03,Zhang05,HeinzelmanCB02}. 
-In       distributed      algorithms~\cite{yangnovel,ChinhVu,qu2013distributed},
-information  is   disseminated  throughout   the  network  and   sensors  decide
-cooperatively by communicating with their neighbors which of them will remain in
-sleep    mode   for    a   certain    period   of    time.     The   centralized
+and the set  of active sensor nodes  is decided at the beginning  of each period
+\cite{ling2009energy}.  Active node selection is determined based on the problem
+requirements  (e.g.  area   monitoring,  connectivity,  power  efficiency).  For
+instance,  Jaggi  et al.  \cite{jaggi2006}  address  the  problem of  maximizing
+network lifetime by dividing sensors into the maximum number of disjoint subsets
+such  that each  subset  can ensure  both  coverage and  connectivity. A  greedy
+algorithm  is applied  once to  solve  this problem  and the  computed sets  are
+activated  in   succession  to  achieve   the  desired  network   lifetime.   Vu
+\cite{chin2007}, Padmatvathy et al. \cite{pc10}, propose algorithms working in a
+periodic fashion where a cover set  is computed at the beginning of each period.
+{\it  Motivated by  these works,  DiLCO protocol  works in  periods,  where each
+  period contains  a preliminary phase  for information exchange  and decisions,
+  followed by a  sensing phase where one  cover set is in charge  of the sensing
+  task.}
+
+Various approaches, including centralized,  or distributed algorithms, have been
+proposed     to    extend    the     network    lifetime.      In    distributed
+algorithms~\cite{yangnovel,ChinhVu,qu2013distributed},       information      is
+disseminated  throughout  the  network   and  sensors  decide  cooperatively  by
+communicating with their neighbors which of them will remain in sleep mode for a
+certain         period         of         time.          The         centralized

 algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high}     always
 provide nearly or close to optimal  solution since the algorithm has global view
 of the whole  network. But such a method has the  disadvantage of requiring high
 communication costs,  since the  node (located at  the base station)  making the
 algorithms~\cite{cardei2005improving,zorbas2010solving,pujari2011high}     always
 provide nearly or close to optimal  solution since the algorithm has global view
 of the whole  network. But such a method has the  disadvantage of requiring high
 communication costs,  since the  node (located at  the base station)  making the

-decision needs information from all the sensor nodes in the area and the amount of information can be huge.
-{\it  In order to be suitable for large-scale network,  in the DiLCO  protocol,  the area  coverage  is divided  into several  smaller
-  subregions, and in  each  one, a  node called the leader is  in charge for
+decision needs information from all the  sensor nodes in the area and the amount
+of  information can  be huge.   {\it  In order  to be  suitable for  large-scale
+  network,  in the DiLCO  protocol, the  area coverage  is divided  into several
+  smaller subregions, and in each one, a node called the leader is in charge for

   selecting the active sensors for the current period.}
 
 A large  variety of coverage scheduling  algorithms has been  developed. Many of
 the existing  algorithms, dealing with the  maximization of the  number of cover
 sets, are heuristics.  These heuristics  involve the construction of a cover set
 by including in priority the sensor  nodes which cover critical targets, that is
   selecting the active sensors for the current period.}
 
 A large  variety of coverage scheduling  algorithms has been  developed. Many of
 the existing  algorithms, dealing with the  maximization of the  number of cover
 sets, are heuristics.  These heuristics  involve the construction of a cover set
 by including in priority the sensor  nodes which cover critical targets, that is

-to  say targets  that  are covered  by  the smallest  number  of sensors \cite{berman04,zorbas2010solving}.  Other
-approaches  are based  on  mathematical programming  formulations~\cite{cardei2005energy,5714480,pujari2011high,Yang2014} and  dedicated
-techniques (solving with a branch-and-bound algorithms available in optimization
-solver).  The problem is formulated  as an optimization problem (maximization of
-the  lifetime  or  number  of  cover  sets) under  target  coverage  and  energy
-constraints.   Column  generation techniques,  well-known  and widely  practiced
-techniques for solving  linear programs with too many  variables, have also been 
-used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it  In DiLCO  protocol,  each leader,  in  each subregion,  solves an  integer
-  program with a double objective  consisting in minimizing the overcoverage and
-  limiting  the  undercoverage.  This  program  is inspired  from  the  work  of
-  \cite{pedraza2006}  where the  objective is  to maximize  the number  of cover
-  sets.}
-
-
-
+to  say   targets  that   are  covered  by   the  smallest  number   of  sensors
+\cite{berman04,zorbas2010solving}.  Other  approaches are based  on mathematical
+programming formulations~\cite{cardei2005energy,5714480,pujari2011high,Yang2014}
+and dedicated  techniques (solving with a  branch-and-bound algorithms available
+in optimization solver).   The problem is formulated as  an optimization problem
+(maximization of the lifetime or number of cover sets) under target coverage and
+energy  constraints.   Column   generation  techniques,  well-known  and  widely
+practiced techniques for  solving linear programs with too  many variables, have
+also                                                                        been
+used~\cite{castano2013column,rossi2012exact,deschinkel2012column}. {\it In DiLCO
+  protocol, each  leader, in  each subregion, solves  an integer program  with a
+  double objective  consisting in minimizing  the overcoverage and  limiting the
+  undercoverage.  This  program is inspired from the  work of \cite{pedraza2006}
+  where the objective is to maximize the number of cover sets.}

 
 \section{\uppercase{Description of the DiLCO protocol}}
 \label{sec:The DiLCO Protocol Description}
 
 \section{\uppercase{Description of the DiLCO protocol}}
 \label{sec:The DiLCO Protocol Description}


@@ -182,7 +188,6 @@ techniques: network leader election  and sensor activity scheduling for coverage
 preservation  and  energy  conservation,  applied  periodically  to  efficiently
 maximize the lifetime in the network.
 
 preservation  and  energy  conservation,  applied  periodically  to  efficiently
 maximize the lifetime in the network.
 

-

 \subsection{Assumptions and models}
 
 \noindent  We consider  a sensor  network composed  of static  nodes distributed
 \subsection{Assumptions and models}
 
 \noindent  We consider  a sensor  network composed  of static  nodes distributed


@@ -198,7 +203,7 @@ sensor coverage  model in the  literature. Thus, since  a sensor has  a constant
 sensing range $R_s$, every space points  within a disk centered at a sensor with
 the radius of  the sensing range is said  to be covered by this  sensor. We also
 assume  that  the communication  range  $R_c \geq  2R_s$.   In  fact, Zhang  and
 sensing range $R_s$, every space points  within a disk centered at a sensor with
 the radius of  the sensing range is said  to be covered by this  sensor. We also
 assume  that  the communication  range  $R_c \geq  2R_s$.   In  fact, Zhang  and

-Zhou~\cite{Zhang05} proved that if  the transmission range fulfills the previous
+Hou~\cite{Zhang05} proved  that if the transmission range  fulfills the previous

 hypothesis, a complete coverage of  a convex area implies connectivity among the
 working nodes in the active mode.
 
 hypothesis, a complete coverage of  a convex area implies connectivity among the
 working nodes in the active mode.
 


@@ -212,7 +217,6 @@ less accurate according to the number of primary points.
 
 \subsection{Main idea}
 \label{main_idea}
 
 \subsection{Main idea}
 \label{main_idea}

-

 \noindent We start  by applying a divide-and-conquer algorithm  to partition the
 area of interest  into smaller areas called subregions and  then our protocol is
 executed   simultaneously  in   each   subregion.
 \noindent We start  by applying a divide-and-conquer algorithm  to partition the
 area of interest  into smaller areas called subregions and  then our protocol is
 executed   simultaneously  in   each   subregion.


@@ -278,9 +282,6 @@ not.  Alternately, if  the  sensor  is not  the  leader, it  will  wait for  the
 Active-Sleep packet to know its state for the coming sensing phase.
 
 
 Active-Sleep packet to know its state for the coming sensing phase.
 
 

-
-
-

 \begin{algorithm}[h!]                
 
   \BlankLine
 \begin{algorithm}[h!]                
 
   \BlankLine


@@ -516,15 +517,15 @@ receive such  packets, we  use the equation  giving the  energy spent to  send a
 1-bit-content   message  defined   in~\cite{raghunathan2002energy}   (we  assume
 symmetric  communication costs), and  we set  their respective  size to  112 and
 24~bits. The energy required to send  or receive a 1-bit-content message is thus
 1-bit-content   message  defined   in~\cite{raghunathan2002energy}   (we  assume
 symmetric  communication costs), and  we set  their respective  size to  112 and
 24~bits. The energy required to send  or receive a 1-bit-content message is thus

- equal to 0.2575 mW.
-
-Each node has an initial energy level, in Joules, which is randomly drawn in the
-interval  $[500-700]$.  If  its  energy  provision reaches  a  value below  the
-threshold  $E_{th}=36$~Joules, the  minimum energy  needed  for a  node to  stay
-active during one period, it will no longer take part in the coverage task. This
-value  corresponds  to the  energy  needed by  the  sensing  phase, obtained  by
-multiplying the energy consumed in active  state (9.72 mW) by the time in seconds
-for one period (3,600 seconds), and  adding the energy for the pre-sensing phases.
+ equal to 0.2575~mW.
+
+Each node  has an initial  energy level, in  Joules, which is randomly  drawn in
+$[500-700]$.   If its  energy  provision  reaches a  value  below the  threshold
+$E_{th}=36$~Joules, the minimum  energy needed for a node  to stay active during
+one  period, it  will  no longer  take part  in  the coverage  task. This  value
+corresponds to the  energy needed by the sensing  phase, obtained by multiplying
+the energy  consumed in active state  (9.72 mW) by  the time in seconds  for one
+period  (3,600 seconds),  and  adding  the energy  for  the pre-sensing  phases.

 According to  the interval of initial energy,  a sensor may be  active during at
 most 20 periods.
 
 According to  the interval of initial energy,  a sensor may be  active during at
 most 20 periods.
 


@@ -542,8 +543,7 @@ the efficiency of our approach:
   connectivity  is crucial because  an active  sensor node  without connectivity
   towards a base  station cannot transmit any information  regarding an observed
   event in the area that it monitors.
   connectivity  is crucial because  an active  sensor node  without connectivity
   towards a base  station cannot transmit any information  regarding an observed
   event in the area that it monitors.

-  
-    
+     

 \item {{\bf Coverage Ratio (CR)}:} it measures how well the WSN is able to 
   observe the area of interest. In our case, we discretized the sensor field
   as a regular grid, which yields the following equation to compute the
 \item {{\bf Coverage Ratio (CR)}:} it measures how well the WSN is able to 
   observe the area of interest. In our case, we discretized the sensor field
   as a regular grid, which yields the following equation to compute the


@@ -556,16 +556,10 @@ where  $n$ is  the number  of covered  grid points  by active  sensors  of every
 subregions during  the current  sensing phase  and $N$ is the total number  of grid
 points in  the sensing field. In  our simulations, we have  a layout of  $N = 51
 \times 26 = 1326$ grid points.
 subregions during  the current  sensing phase  and $N$ is the total number  of grid
 points in  the sensing field. In  our simulations, we have  a layout of  $N = 51
 \times 26 = 1326$ grid points.

-%The accuracy of this method depends on the distance between grids. In our
-%simulations, the sensing field has been divided into 50 by 25 grid points, which means
-%there are $51 \times 26~ = ~ 1326$ points in total.
-% Therefore, for our simulations, the error in the coverage calculation is less than ~ 1 $\% $.
-
-

 
 \item {{\bf  Energy Consumption}:}  energy consumption (EC)  can be seen  as the
 
 \item {{\bf  Energy Consumption}:}  energy consumption (EC)  can be seen  as the

-  total amount of  energy   consumed   by   the   sensors   during   $Lifetime_{95}$   or
-  $Lifetime_{50}$, divided  by the number of periods.  Formally, the computation
+  total amount of  energy   consumed   by   the   sensors   during   $Lifetime_{95}$   
+  or $Lifetime_{50}$, divided  by the number of periods.  Formally, the computation

   of EC can be expressed as follows:
   \begin{equation*}
     \scriptsize
   of EC can be expressed as follows:
   \begin{equation*}
     \scriptsize


@@ -573,24 +567,21 @@ points in  the sensing field. In  our simulations, we have  a layout of  $N = 51
       + E^{a}_m+E^{s}_m \right)}{M},
   \end{equation*}
 
       + E^{a}_m+E^{s}_m \right)}{M},
   \end{equation*}
 

-where $M$  corresponds to the number  of periods.  The total amount of energy consumed by
-the  sensors (EC)  comes  through  taking into  consideration  four main  energy
-factors. The  first one, denoted $E^{\scriptsize  \mbox{com}}_m$, represents the
-energy consumption  spent by  all the nodes  for wireless  communications during
-period $m$.   $E^{\scriptsize \mbox{list}}_m$,  the next factor,  corresponds to
-the  energy consumed by  the sensors  in LISTENING  status before  receiving the
-decision to  go active or  sleep in period $m$.  $E^{\scriptsize \mbox{comp}}_m$
-refers to the energy needed by all the leader nodes to solve the integer program
-during a period.  Finally, $E^a_{m}$ and $E^s_{m}$ indicate  the energy consumed
-by the whole network in the sensing phase (active and sleeping nodes).
-
-
-
+where $M$  corresponds to  the number  of periods.  The  total amount  of energy
+consumed by the  sensors (EC) comes through taking  into consideration four main
+energy   factors.  The  first   one,  denoted   $E^{\scriptsize  \mbox{com}}_m$,
+represents  the  energy  consumption  spent   by  all  the  nodes  for  wireless
+communications  during period  $m$.  $E^{\scriptsize  \mbox{list}}_m$,  the next
+factor, corresponds  to the energy consumed  by the sensors  in LISTENING status
+before  receiving   the  decision  to  go   active  or  sleep   in  period  $m$.
+$E^{\scriptsize \mbox{comp}}_m$  refers to the  energy needed by all  the leader
+nodes  to solve the  integer program  during a  period.  Finally,  $E^a_{m}$ and
+$E^s_{m}$ indicate the energy consumed by the whole network in the sensing phase
+(active and sleeping nodes).

 
 \end{itemize}
 %\end{enumerate}
 
 
 \end{itemize}
 %\end{enumerate}
 

-

 %\subsection{Performance Analysis for different subregions}
 \subsection{Performance analysis}
 \label{sub1}
 %\subsection{Performance Analysis for different subregions}
 \subsection{Performance analysis}
 \label{sub1}


@@ -612,16 +603,16 @@ chosen to remain active during the sensing phase.
 \subsubsection{Coverage ratio} 
 
 Figure~\ref{fig3} shows  the average coverage  ratio for 150 deployed  nodes. It
 \subsubsection{Coverage ratio} 
 
 Figure~\ref{fig3} shows  the average coverage  ratio for 150 deployed  nodes. It

-can  be seen  that both  DESK and  GAF provide  a   coverage ratio which is slightly better
-compared to DiLCO  in the first thirty periods. This can  be easily explained by
-the number of  active nodes: the optimization process  of our protocol activates
-less nodes  than DESK  or GAF, resulting  in a  slight decrease of  the coverage
-ratio. In case of DiLCO-2  (respectively DiLCO-4), the coverage ratio exhibits a
-fast decrease  with the number  of periods and  reaches zero value  in period~18
-(respectively 46), whereas  the other versions of DiLCO, DESK,  and GAF ensure a
-coverage ratio above  50\% for subsequent periods.  We  believe that the results
-obtained with these two methods can be explained by a high consumption of energy
-and we will check this assumption in the next subsection.
+can be seen  that both DESK and  GAF provide a coverage ratio  which is slightly
+better  compared to  DiLCO  in the  first  thirty periods.  This  can be  easily
+explained  by  the number  of  active nodes:  the  optimization  process of  our
+protocol activates less  nodes than DESK or GAF, resulting  in a slight decrease
+of the coverage  ratio. In case of DiLCO-2  (respectively DiLCO-4), the coverage
+ratio exhibits a fast decrease with the number of periods and reaches zero value
+in period~18 (respectively  46), whereas the other versions  of DiLCO, DESK, and
+GAF ensure a coverage ratio above  50\% for subsequent periods.  We believe that
+the  results  obtained  with these  two  methods  can  be  explained by  a  high
+consumption of energy and we will check this assumption in the next subsection.

 
 Concerning  DiLCO-8, DiLCO-16,  and  DiLCO-32,  these methods  seem  to be  more
 efficient than DESK  and GAF, since they can provide the  same level of coverage
 
 Concerning  DiLCO-8, DiLCO-16,  and  DiLCO-32,  these methods  seem  to be  more
 efficient than DESK  and GAF, since they can provide the  same level of coverage


@@ -655,7 +646,7 @@ used for the different performance metrics.
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/EC.pdf} 
 \begin{figure}[h!]
 \centering
 \includegraphics[scale=0.45]{R/EC.pdf} 

-\caption{Energy consumption}
+\caption{Energy consumption per period}

 \label{fig95}
 \end{figure} 
 
 \label{fig95}
 \end{figure} 
 


@@ -663,11 +654,14 @@ The  results  depict the  good  performance of  the  different  versions of  our
 protocol.   Indeed,  the protocols  DiLCO-16/50,  DiLCO-32/50, DiLCO-16/95,  and
 DiLCO-32/95  consume less  energy than  their DESK  and GAF  counterparts  for a
 similar level of area coverage.   This observation reflects the larger number of
 protocol.   Indeed,  the protocols  DiLCO-16/50,  DiLCO-32/50, DiLCO-16/95,  and
 DiLCO-32/95  consume less  energy than  their DESK  and GAF  counterparts  for a
 similar level of area coverage.   This observation reflects the larger number of

-nodes set active by DESK and GAF.
-
-
-%In fact,  a distributed  method on the subregions greatly reduces the number of communications and the time of listening so thanks to the partitioning of the initial network into several independent subnetworks. 
-%As shown in Figures~\ref{fig95} and ~\ref{fig50} , DiLCO-2 consumes more energy than the other versions of DiLCO, especially for large sizes of network. This is easy to understand since the bigger the number of sensors involved in the integer program, the larger the time computation to solve the optimization problem as well as the higher energy consumed during the communication.  
+nodes set active  by DESK and GAF. Now,  if we consider a same  protocol, we can
+notice  that the  average  consumption  per period  increases  slightly for  our
+protocol  when increasing  the  level  of coverage,  whereas  it increases  more
+largely for  DESK and  GAF.  In case  of DiLCO  It means that  even if  a larger
+network allows  to improve the number  of periods with a  minimum coverage level
+value,  this   improvement  has  a  higher   energy  cost  per   period  due  to
+communications and  a more difficult optimization.  However,  in comparison with
+DSK and GAF, our approach has a reasonable energy overhead.

 
 \subsubsection{Execution time}
 
 
 \subsubsection{Execution time}
 


@@ -702,8 +696,6 @@ prevents it  to  ensure a  good  coverage   especially  on   the  borders   of
 subregions. Thus,  the optimal number of  subregions can be seen  as a trade-off
 between execution time and coverage performance.
 
 subregions. Thus,  the optimal number of  subregions can be seen  as a trade-off
 between execution time and coverage performance.
 

-%The DiLCO-32 has more suitable times in the same time it turn on redundent nodes more.  We think that in distributed fashion the solving of the  optimization problem in a subregion can be tackled by sensor nodes. Overall, to be able to deal  with very large networks,  a distributed method is clearly required.
-

 \subsubsection{Network lifetime}
 
 In the next figure, the network lifetime is illustrated. Obviously, the lifetime
 \subsubsection{Network lifetime}
 
 In the next figure, the network lifetime is illustrated. Obviously, the lifetime


@@ -728,7 +720,6 @@ the larger level  of coverage ($95\%$) in the case of  our protocol, the subdivi
 in $16$~subregions seems to be the most appropriate.
 
 
 in $16$~subregions seems to be the most appropriate.
 
 

-

 \section{\uppercase{Conclusion and future work}}
 \label{sec:Conclusion and Future Works} 
 
 \section{\uppercase{Conclusion and future work}}
 \label{sec:Conclusion and Future Works} 
 


@@ -755,8 +746,6 @@ there is an optimal number of  subregions.  Therefore, in case of our simulation
 context  a subdivision in  $16$~subregions seems  to be  the most  relevant. The
 optimal number of subregions will be investigated in the future.
 
 context  a subdivision in  $16$~subregions seems  to be  the most  relevant. The
 optimal number of subregions will be investigated in the future.
 

-
-

 \section*{\uppercase{Acknowledgements}}
 
 \noindent  As a  Ph.D.   student, Ali  Kadhum  IDREES would  like to  gratefully
 \section*{\uppercase{Acknowledgements}}
 
 \noindent  As a  Ph.D.   student, Ali  Kadhum  IDREES would  like to  gratefully