diff --git a/GPS_pw_modeling.mo b/GPS_pw_modeling.mo
index a93104a4c9ded12e26d236fa3d757d92c842fb2a..f89b0141dc73b6a591448f7366edef86fb9b8160 100644
--- a/GPS_pw_modeling.mo
+++ b/GPS_pw_modeling.mo
@@ -1,5 +1,7 @@
 package GPS_pw_modeling
   model GPS_pw_simulator_old
+  //prima versione - obsoleta
+  //
     //parameters
     parameter Integer required_satellites = 4;
     //number of satellites required for sufficient accuracy
@@ -222,6 +224,7 @@ package GPS_pw_modeling
   end GPS_pw_simulator_old;
 
 
+
   model test_GPS
     GPS_pw_modeling.GPS_pw_simulator GPS_pw_simulator1 annotation(
       Placement(visible = true, transformation(origin = {90, 50}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
@@ -442,4 +445,4 @@ package GPS_pw_modeling
 
   annotation(
     uses(Modelica(version = "3.2.1")));
-end GPS_pw_modeling;
\ No newline at end of file
+end GPS_pw_modeling;
diff --git a/report/main.tex b/report/main.tex
index d778060ddc4cb8440915dde8fa66ff51aaa0a89a..876e554bcef7cac0850f828048b687836d378fb5 100644
--- a/report/main.tex
+++ b/report/main.tex
@@ -34,10 +34,10 @@
 
 \begin{document}
 
-\title{The ``GPS'' project} 
+\title{Dynamical modeling of power consumpion in GPS sensors} 
 \author{
-  \IEEEauthorblockN{Your names}
-  \IEEEauthorblockA{Your affiliations}
+  \IEEEauthorblockN{Claudio Mandrioli}
+  \IEEEauthorblockA{Feedback control in computing systems - phd course, november 2017}
 }
 
 \maketitle
@@ -45,7 +45,7 @@
 \begin{abstract}
 ~This is a stub paper on the ``GPS'' subject as discussed in the last lecture of the PhD course in Lund, November 2017. Your task is to carry out the activities sketched out in the introduction, applying the notions and methods treated in the course wherever applicable, and to document your work by turning this stub into a 6--8 pages paper (indicative length). Needless to say, the title is just a placeholder, and you can change everything---including the introduction, that is a stub as well. For any question, just drop me a line.
 
-\vspace{5mm}\textit{Keywords---} You choose.
+\vspace{5mm}\textit{Keywords---} GPS, cyber-physical systems, energy consumption.
 \end{abstract}
 
 \section{Introduction}
diff --git a/report/sections/03-Modeling.tex b/report/sections/03-Modeling.tex
index 39582abd148554627deb2d3753691c2c63f75c4b..57a7b9c30b8580cc8f8c98540ba58a2cabe00c27 100644
--- a/report/sections/03-Modeling.tex
+++ b/report/sections/03-Modeling.tex
@@ -112,7 +112,7 @@ Along with the ephemeris data comes the necessity of representing the expiration
 %\end{array}
 %\end{equation}
 
-A possible representation of the states and the transitions are connected is given in figure~\ref{fig:cyberDynamics}.
+%A possible representation of the states and the transitions are connected is given in figure~\ref{fig:cyberDynamics}.
 %\footnote{In this representation it is assumed that after a \texttt{turn\_ON} signal no \texttt{turn\_OFF} signal will be received before the system is able to provide the position. This looks like a non desirable situation. The model could still easily be extended to include this case defining the following transitions (all triggered by the event \texttt{turn\_OFF}):
 
 %\begin{itemize}
@@ -142,15 +142,21 @@ font=\footnotesize]
 \node[thick] (wsa)  at ( 5,  -8) [p_off, label={Warm Start Available}] {8};
 \node[thick] (zero) at (-1.5, 1) [circle, draw=black!60, fill=black!60, inner sep=0.75mm] {};
 
-\draw [arr, bend right] (zero) to (ni); % arrows from beginning node
-\draw [arr] (ni) to node [above] {turn ON} (cs); % arrows from 1
+% arrows from beginning node
+\draw [arr, bend right] (zero) to (ni); 
+
+% arrows from 1
+\draw [arr] (ni) to node [above] {turn ON} (cs); 
+
+% arrows from 2
 \draw [arr] (cs) to node [above] {fetch freq and phase} (re);
 \draw [arr, in=60, out=120] (cs) to node [above] {turn OFF} (ni);
+\draw [arr, in=200, out=260] (cs) edge[loop] node [xshift=-10mm] {lose visibility} (cs);
 
 % arrows from 3
 \draw [arr] (re) to node [above] {get ephemeris} (pa); 
 \draw [arr, in=60, out=120] (re) to node [above] {turn OFF} (ni);
-\draw [arr, in=60, out=120] (re) to node [above] {lose visibility} (cs);
+\draw [arr, in=60, out=120] (re) to node [left] {lose visibility} (cs);
 
 % arrows from 4 - position available
 \draw [arr, in=60, out=120] (pa) to node [above]
@@ -158,15 +164,13 @@ font=\footnotesize]
 \draw [arr, in=0, out=-45]  (pa) to node
   [right, at start, yshift=-5mm, xshift=4mm] {turn OFF} (hsa);
 \draw [arr] (pa) edge[loop right] node {get ephemeris data} (pa);
+\draw [arr, in=90, out=-90]  (pa) to node  [right, at start, yshift=-7mm, xshift=-19mm] {lose visibility} (hs);
 
 % arrows from 5 - hot start
-\draw [arr, in=-90, out=45] (hs) to node [right, align=left,
-  yshift=-8mm, xshift=-1mm] {fetch freq and phase quick\\
-  $\land \neg$ ephemeris data expired} (pa);
-\draw [arr, in=-90, out=135] (hs) to node [right, align=left,
-  yshift=11mm, xshift=-8mm] {fetch freq and phase quick\\
-  $\land$ ephemeris data expired} (re);
+\draw [arr, in=-75, out=45] (hs) to node [right, align=left, yshift=-0mm, xshift=-0mm] {fetch freq and phase quick} (pa);
+\draw [arr, in=-45, out=135] (hs) to node [right, near end, yshift=0mm, xshift=0mm] {fetch freq and phase quick} (cs);
 \draw [arr, in=135, out=225] (hs) to node [right, near start] {turn OFF} (hsa);
+\draw [arr] (hs) to node [below] {lose visibility} (ws);
 
 % arrows from 6 - hot start available
 \draw [arr] (hsa) to node [below] {lose hot start} (wsa);
@@ -176,12 +180,10 @@ font=\footnotesize]
 
 % arrows from 7 - warm start
 \draw [arr, in=225, out=45] (ws) to node [right, align=left, at start,
-  xshift=5mm, yshift=2mm] {fetch freq and phase\\
-  $\land \neg$ ephemeris data expired} (pa);
-\draw [arr, in=225, out=90] (ws) to node [left, near end, align=right,
-  xshift=-4mm, yshift=-1mm] {fetch freq and phase\\
-  $\land$ ephemeris data expired} (re);
+  xshift=5mm, yshift=2mm] {fetch freq and phase} (pa);
+\draw [arr, in=285, out=110] (ws) to node [left, near start, align=right] {ephemeris data expired} (cs);
 \draw [arr, in=135, out=225] (ws) to node [right, near start] {turn OFF} (wsa);
+\draw [arr, in=200, out=160] (ws) edge[loop left] node [xshift=0mm] {lose visibility} (ws);
 
 % arrows from 8 - warm start available
 \draw [arr, in=-45, out=45] (wsa) to node [right, near end] {turn ON} (ws);
@@ -193,11 +195,14 @@ font=\footnotesize]
 }
 \end{figure*}
 
-This state machine models how the signal acquisition and the ephemeris data acquisition are related to each other and how they can happen in different sequences. For instance we can see that at the first start up only a cold start is available and therefore the sensor is forced first to fetch the signal and then to read the ephemeris data before actually providing the position. i.e. path 1-2-3-4. Instead after the first position fix the system is able to go back in the state \emph{Position Available} without aquiring again the position data of the satellites. This is represented by path 6-5-4 in case of hot start and 8-7-4 for the warm start.
+The way the states and the transitions are connected is shown in figure~\ref{fig:cyberDynamics}. This state machine models how the signal acquisition and the ephemeris data acquisition are related to each other and how they can happen in different sequences. For instance we can see that at the first start up only a cold start is available and therefore the sensor is forced first to fetch the signal and then to read the ephemeris data before actually providing the position. i.e. path 1-2-3-4. Instead after the first position fix the system is able to go back in the state \emph{Position Available} without aquiring again the position data of the satellites. This is represented by path 6-5-4 in case of hot start and 8-7-4 for the warm start.
+
+When ephemeris data stop being available (condition \emph{ephemeris data expired}) the sensor can behave in two ways according to whether the antenna is turned on or off. If it is off then the sensor doesn't have either of the required information for positioning itself any more and therefore it goes back to the state \emph{No Info}, transitions 6-1 and 8-1. If instead the antenna is turned on and frequency and phase are fetched the sensor will just go back from state \emph{Position Available} (in the described scenario in fact both the information are available and therefore also the positioning) to the state in which it is reading the ephemeris data, \emph{Read Ephemeris} with the transition 4-3. 
+
 
-When ephemeris data stop being available (condition \emph{ephemeris data expired}) the sensor can behave in two ways according to whether the antenna is turned on or off. If it is off then the sensor doesn't have either of the required information for positioning itself any more and therefore it goes back to the state \emph{No Info}, transitions 6-1 and 8-1. If instead the antenna is turned on and frequency and phase are fetched the sensor will just go back from state \emph{Position Available} (in the described sccenario in fact both the information are available and therefore also the positioning) to the state in which it is reading the ephemeris data, \emph{Read Ephemeris} with the transition 4-3. Different consideration must be done is the antenna is on but has not yet fetched the signal, this is the case in states 5 and 7. In fact in those states moving directly to state 2 would overestimate the time required for fetching the signal since the search for frequency and phase shift would be ``reset'', and instead moving to state 3 would underestimate it skipping directly to the moment when the devivce is able to read the ephemeris data. Therefore what is defined in the model is that the sensor remains in state 5 and 7 until it has fetched the signal and then move to state 4 or 3 depending on whether ephemeris data are respectively available or not.
+%Different consideration must be done if the antenna is on but has not yet fetched the signal, this is the case in states 5 and 7. In fact in those states moving directly to state 2 would overestimate the time required for fetching the signal since the search for frequency and phase shift would be ``reset'', and instead moving to state 3 would underestimate it skipping directly to the moment when the devivce is able to read the ephemeris data. Therefore what is defined in the model is that the sensor remains in state 5 and 7 until it has fetched the signal and then move to state 4 or 3 depending on whether ephemeris data are respectively available or not.
 
-\textcolor{red}{fai venire fuori che conoscenza otteniamo dal modello (puo' venire dagli esempi)}
+%\textcolor{red}{fai venire fuori che conoscenza otteniamo dal modello (puo' venire dagli esempi)}
 
 
 
diff --git a/report/sections/04-Examples.tex b/report/sections/04-Examples.tex
index 00ca14e685c604c9cb4be7c52541d55caaf1ba03..fbdec2486d085765d9898608059de6f5f877eff2 100644
--- a/report/sections/04-Examples.tex
+++ b/report/sections/04-Examples.tex
@@ -2,7 +2,7 @@
 
 %couple of examples of ways you could use this model
 
-\textcolor{red}{fai vedere come strategie del controllo possono sfruttare le dinamiche descritte dal modello}
+%\textcolor{red}{fai vedere come strategie del controllo possono sfruttare le dinamiche descritte dal modello}
 
 In this section is discussed an object-oriented implementation of the described model with the modeling language Modelica\footnote{https://www.modelica.org/}. 
 
@@ -10,26 +10,30 @@ In this section is discussed an object-oriented implementation of the described
 What is now to be discussed is how to define the firing of the different transitions defined in the model.
 
 \subsubsection{turn ON/OFF} 
-Those transitions as defined in section~\ref{sec:input-def} are the controlled inputs of the defined system. Since apparently to have coherency the two must be one the negation of the other  they can be defined as a boolean being \emph{true} whe the input is turn ON and \emph{false} when the input is turn OFF.
+Those transitions as defined in section~\ref{sec:input-def} are the controlled inputs of the defined system. Since apparently to have coherency the two must be one the negation of the other  they can be defined as a boolean being \emph{true} whe the input is turn ON and \emph{false} when the input is turn OFF. This signal can also be used for defining the output power consumption, since it is equivalent to distinguishing the states in which the antenna is on and off.
 
 \subsubsection{fetch frequency and phase}
-This transition represents when the devices matches the phase shift and frequency distortion of the signal. This procedure takes some time after the antenna has been turned on, usually of the order of milliseconds~\ref{bib:gps-book} from 1mS to 10mS depending on signal strength. Under some assumptions (discussed in section~\ref{sec:info-dynamics}) it has been hypotized that this process can be sped up using the information from previous samplings~\cite{bib:microsoft-leap}. Anyhow this has not been implementes so there is no actual estimation of how much of improvement could be achieved.
+This transition represents when the devices matches the phase shift and frequency distortion of the signal. This procedure takes some time after the antenna has been turned on, usually of the order of milliseconds~\cite{bib:gps-book} from 1mS to 10mS depending on signal strength. 
 
 \subsubsection{get ephemeris data}
-The retrieval of the ephemeris data requires the device to listen to one full cycle of the GPS message of each of the visible satellites. One of those cycles lasts 30 seconds this means that in the best case -- i.e. if the device starts to listen exactly at the beginning of the cycle -- this is the amount of time required to fire the transition, in the worst case it will be 59 seconds. It must be also noted that it is the maximum over the all the satellites that matters. Different choices can be done according to the specific use-cases, whether for example the average or worst case must be considered.
+The retrieval of the ephemeris data requires the device to listen to one full cycle of the GPS message of each of the visible satellites. Each of those cycles lasts 30 seconds and this means that in the best case -- i.e. if the device starts to listen exactly at the beginning of the cycle -- this is the amount of time required to acquire the data and fire the transition, in the worst case it will instead be 59 seconds. It must be also noted that it is the maximum over the all the satellites that matters. Different choices can be done according to the specific use-cases, whether for example the average or worst case must be considered.
 
 \subsubsection{ephemeris data expire}
-This event represents the fact that the stored ephemeris data are no more available and the device is not listening to the satellites. Since those data have a prescribed time validity of 30 minutes it is always fireable from 30 minutes after the last update of those. 
+This event represents the fact that the stored ephemeris data are no more available and the device is not listening to the satellites. Since those data have a prescribed time validity of 30 minutes it is always fireable from 30 minutes after the last update of those. Formally:
 
-\subsubsection{lose hot start}
-This event is diffucult to be defined for the same reasons discussed for the event \texttt{fetch frequency and phase} in the case of hot start. For such reason here we only consider a deelay of some milliseconds. When modeling an actual device of the ones available nowadays since the idea of the hot start is not yet available this time should be set to zero.
+\begin{equation}
+	time>expiration\_time .
+\end{equation}
+
+\subsubsection{fetch frequency and phase quick - lose hot start}
+Under some assumptions -- discussed in section~\ref{sec:info-dynamics} -- it has been hypotized that the process of fetching the signal can be sped up using the information from previous samplings~\cite{bib:microsoft-leap}. This possibility is captured by the states \emph{hot start} and \emph{hot start available}. Anyhow this idea has not yet been implemented there is no actual estimation of how much of improvement could be achieved, for his reason what is implemented in modelica is a simplified version of the model that excludes the hot start procedure. The model is very similar to the presented one: the cited states are removed and the two transitions entering them, \texttt{lose\_visibility} and \texttt{turn\_off}, should instead go respectively to \emph{warm start} and \emph{warm start available}.
 
 \subsection{Model execution examples}
 
 In this section will be presented through the simulation of different scenarios which kind of considerations we can make on how GPS sensors work according to the retrieved model. Such considerations are related to (i) the presence of two different dynamcs in the retrieval of the position, one that is slow, the other that is fast, (ii) the different delays the system can present , (iii) the different ways we can duty-cycle the sensor and (iv) how variations on the number of visible satellites can affect the availability of the position.
 %first two examples use the boolead table called slow_and_fast with a constant and sufficient number of visible satellites
 \subsubsection{Start up of the sensor}
-In the first simulation we want to show the existance of two dynamics in the sensor: a slow one related to the ascquisition and validity of the ephemeris data, and a fast one related to the acquisition of the ranging data. This simulation points also out what is the difference between warm and cold start. To do so we turn on the antenna first for a long time in order to be sure to acuire the ephemeris data of the visible satellies and then start to duty cycle the sensor to acquire the position at different points in time. A example of a turn on signal doing so is given in figure~\ref{fig:control1}.In figure~\ref{fig:position1} instead we can see the availability of the position measure given the input above described. We can see how at the first turn on of the sensor it takes a minute before the position becomes actually available, while afterward the position is available after only milliseconds(recall that the antenna is turned on and consumes powerexactly completely cohordinated to the turn\_on signal).
+In the first simulation we want to show the existance of two dynamics in the sensor: a slow one related to the ascquisition and validity of the ephemeris data, and a fast one related to the acquisition of the ranging data. This simulation points also out what is the difference between warm and cold start. To do so we turn on the antenna first for a long time in order to be sure to acuire the ephemeris data of the visible satellies and then start to duty cycle the sensor to acquire the position at different points in time. A example of a turn on signal doing so is given in figure~\ref{fig:control1}.In figure~\ref{fig:position1} instead we can see the availability of the position measure given the input above described. We can see how at the first turn on of the sensor it takes a minute before the position becomes actually available, while afterward the position is available after only milliseconds(recall that the antenna is turned on and consumes powerexactly completely cohordinated to the turn\_ON signal).
 
 \begin{figure}[h]
  \begin{center}
@@ -51,7 +55,7 @@ In the first simulation we want to show the existance of two dynamics in the sen
 
 
 \subsubsection{Ephemeris data expiration}
-In this second simulation we show an outage of the position measure availability due to the expiration of the ephemeris data. In this scenario the device is duty the cycling sensor until at some point the ephemeris data expire and a prolonged turn\_ON signal is required in order to update the ephemeris data and make available again the position. In figure~\ref{fig:control2} we can see the described turn\_ON signal while in~\ref{fig:position2} we can see the availability of the position measure. In this example the ephemeris data expire around time=1880 , we can see how the sampling suddently becomes uneffective and the position becomes available again only after around a minute in which it is continuously turned on and reads the ephemeris data.
+In this second simulation we show an outage of the position measure availability due to the expiration of the ephemeris data. In this scenario the device is duty cycling the sensor until at some point the ephemeris data expire and a prolonged turn\_ON signal is required in order to update the ephemeris data and make available again the position. In figure~\ref{fig:control2} we can see the described turn\_ON signal while in~\ref{fig:position2} we can see the availability of the position measure. In this example the ephemeris data expire around time=1880 , we can see how the sampling suddently becomes uneffective and the position becomes available again only after around a minute in which it is continuously turned on and reads the ephemeris data.
 
 \begin{figure}[h]
  \begin{center}
@@ -72,7 +76,9 @@ In this second simulation we show an outage of the position measure availability
 \end{figure}
 
 \subsubsection{Visible satellites}
-In this third scenario it is shown a possible way the number of visible satellites can influence the availability of the position. In figure~\ref{fig:control3} are shown the number of visible satellites and the control signal that turns on the GPS antenna. We simulate an satellites outage at $time=100$ where the number of visible satellites becomes 3 which is not sufficient for providing any positioning. The device before that tracks the 5 visible satellites and starts samling the position every 10 seconds for one second. After the satellite outage it instead stays turned on trying to fetch more sateliites. This happens only at the time instant 200 where the number of visible satellites becomes 4. After having updated again the ephemeris data the device is able to start sampling again the position, as shown in figure~\ref{fig:position3}.
+In this third scenario it is shown a possible way the number of visible satellites can impact the availability of the position. In figure~\ref{fig:control3} are shown the number of visible satellites and the control signal that turns on the GPS antenna. We simulate a satellites outage at $time=100$ where the number of visible satellites becomes 3 which is not sufficient for providing any positioning. 
+
+The device before the outage tracks the 5 visible satellites and starts sampling the position every 10 seconds for one second. After the satellite outage it instead stays turned on trying to fetch more sateliites. This happens only at the time instant 200 where the number of visible satellites goes up to 4. After having updated again the ephemeris data the device is able to start again sampling the position, as shown in figure~\ref{fig:position3}.
 
 \begin{figure}[h]
  \begin{center}
diff --git a/report/sections/05-Related-Work.tex b/report/sections/05-Related-Work.tex
index 2dc6de8784ec2a74b3ec7417f347a7b38219b338..497a906bf8afa92c9043720665569cf63455fa40 100644
--- a/report/sections/05-Related-Work.tex
+++ b/report/sections/05-Related-Work.tex
@@ -1,6 +1,6 @@
-While many works discuss how to improve power consumption in devices that use the GPS positioning system, only few of those explicitly discuss the problem of modeling how such sensors consume power and even less discuss this modeling problem analyzing how the system actually works. Arguably this is related to the fact that almost all of those solutions are designed for smartphones which can access the internet and therefore run assisted-GPS~\cite{bib:desing-principles-for-energy-efficiency, bib:energy-accuracy-tradeoff, bib:traffic-deelay, bib:modeldriven-pw-consumption-smartphones}. This decouples the retrieval of the ephemeris and ranging data, also allowing a much faster acquisition of the former. This means that the deivce must retrieve only the ranging data using the GPS antenna making the sensor dynamics simpler. Anyway this approach presents some drawbacks like: (i) fetching the GPS data from the internet has as well an impact on the energy consumption, (ii) the use of internet connection might have an economic cost, (iii) this process depends on the underlying OS of the device as it uses funcitonalities external to the sensor and (iv) the device location privacy is compromised reaching the Assisted-GPS servers.
+While many works discuss how to improve power consumption in devices that use the GPS positioning system, only few of those explicitly discuss the problem of modeling how such sensors consume power and even less discuss this problem analyzing how the system actually works. Arguably this is related to the fact that almost all of those solutions are designed for smartphones which can access the internet and therefore run assisted-GPS~\cite{bib:desing-principles-for-energy-efficiency, bib:energy-accuracy-tradeoff, bib:traffic-deelay, bib:modeldriven-pw-consumption-smartphones}. This decouples the retrieval of the ephemeris and ranging data, also allowing a much faster acquisition of the former. This means that the deivce must retrieve only the ranging data using the GPS antenna making the sensor dynamics simpler. Anyway this approach presents some drawbacks like: (i) fetching the GPS data from the internet has as well an impact on the energy consumption, (ii) the use of internet connection might have an economic cost, (iii) this process depends on the underlying OS of the device as it uses funcitonalities external to the sensor and (iv) the device location privacy is compromised reaching the Assisted-GPS servers.
 
 
 Some works propose frameworks~\cite{bib:framework-for-energy-efficiency} and design principles~\cite{bib:desing-principles-for-energy-efficiency} for energy efficient implementation of smartphone applications. Those mainly focus on how different sensors can be joined in order to acieve more energy efficient positioning--- mainly just minimizing the usage of GPS which is the most enery consuming one\footnote{Other sensors involved are: inertial sensors, giros, wifi trilateration and other internet based methods.}.
 
-This work~\cite{bib:entracked-datadriven-modeling} explicitly addresses the problem of modeling how the different sensors consume power on mobile devices but presents mainly an high-level data driven approach instead of looking in detail at how the sensors work. The model includes constant power consumption and a deelay in the position availability that depends on for how long the sensor has been turned off. Those characteristics are correct but with a model that looks at how actually the system works it is possible to get better estimation of those.
+One work~\cite{bib:entracked-datadriven-modeling} explicitly addresses the problem of modeling how the different sensors consume power on mobile devices but presents mainly an high-level and data-driven approach instead of looking at how the sensors actually work. The model here introduced includes constant power consumption and a deelay in the position availability that depends on for how long the sensor has been turned off. Those characteristics are correct but with a model that looks at how actually the system works it is possible to get better and rigorous estimation of those.
diff --git a/report/sections/06-Conclusions.tex b/report/sections/06-Conclusions.tex
index 131ff8e8fb3f530bd61808c1215181677f428833..8236f8b40eac795cf17c0fd141e4ee5acc8cd5c6 100644
--- a/report/sections/06-Conclusions.tex
+++ b/report/sections/06-Conclusions.tex
@@ -6,6 +6,6 @@
 
 %in the end: two discrete dynamics, one slow (ephemeris) and one fast (ranging) but what makes the difference is how you use sensor
 
-In this work a dynamical modeling of how GPS sensors consume power has been presented with respect to how they are able to provide positioning of the device. Such devices are basically characterized by two coupled discrete dynamics: one slow of the ephemeris data and one faster of the ranging data\footnote{Where slow and fast are referred to both the acquisition and validity of the information.}.
+In this work has been presented a dynamical modeling of how GPS sensors consume power with respect to how they are able to provide positioning of the device. Such devices are basically characterized by two coupled discrete dynamics: one slow of the ephemeris data and one faster of the ranging data\footnote{Where slow and fast are referred to both the acquisition and validity of the information.}.
 
-Another aspect of the modeling problem is how each satellite is tracked and eventually lost as some point: while the former can be done in parallel the latter is decoupled. This decoupling is not captured by the presented model and it results in a underestimation of the sensor performances when a satellite disappears for a short time. The model could be extended to overcome this limitaiton but it would require distinguishing the different satellites which would introduce more complexity in the model: discussion of this is left to specific use-cases for which such level of detail could be necessary.
\ No newline at end of file
+Another aspect of the modeling problem is how each satellite is tracked and eventually lost as some point: while the former can be done in parallel the latter is decoupled. This decoupling is not captured by the presented model and it results in a underestimation of the sensor performances when a satellite disappears for a short time. It also limits the detail at which the time for the acquisition of the ephemeris datacan be modeled. The model could be extended to overcome this limitaiton but it would require distinguishing the different satellites and therefore introduce more complexity: discussion of this is left to specific use-cases for which such level of detail could be required.