From 37eab9bdcda6c945b9f2251cc639206fe78f731c Mon Sep 17 00:00:00 2001 From: youainti Date: Fri, 9 Jul 2021 17:14:16 -0700 Subject: [PATCH] got computational to a good state, started cleaning up laws of motion and survival analysis. Added a 4th and 5th level of subsections (titled paragraphs) --- CurrentWriting/Main.tex | 2 +- .../assets/preambles/GeneralPreamble.tex | 2 + CurrentWriting/sections/01_LawsOfMotion.tex | 9 ++-- .../sections/03_SurvivalAnalysis.tex | 49 +++++++++++-------- .../sections/07_ComputationalApproach.tex | 10 ++-- 5 files changed, 43 insertions(+), 29 deletions(-) diff --git a/CurrentWriting/Main.tex b/CurrentWriting/Main.tex index 7bca9c8..36acc35 100644 --- a/CurrentWriting/Main.tex +++ b/CurrentWriting/Main.tex @@ -24,7 +24,7 @@ \section{Modeling the Environment} \subsection{Laws of Motion} -\subfile{sections/01_LawsOfMotion} %DONE +\subfile{sections/01_LawsOfMotion} %Needs headings fixed \subsubsection{Survival Analysis}\label{SEC:Survival} \subfile{sections/03_SurvivalAnalysis} %TODO diff --git a/CurrentWriting/assets/preambles/GeneralPreamble.tex b/CurrentWriting/assets/preambles/GeneralPreamble.tex index 669035c..a0444ff 100644 --- a/CurrentWriting/assets/preambles/GeneralPreamble.tex +++ b/CurrentWriting/assets/preambles/GeneralPreamble.tex @@ -12,3 +12,5 @@ \usepackage{graphicx} \graphicspath{assets/img/} +%setup paragraph level indexing +\setcounter{secnumdepth}{5} diff --git a/CurrentWriting/sections/01_LawsOfMotion.tex b/CurrentWriting/sections/01_LawsOfMotion.tex index 9138694..4465842 100644 --- a/CurrentWriting/sections/01_LawsOfMotion.tex +++ b/CurrentWriting/sections/01_LawsOfMotion.tex @@ -7,7 +7,7 @@ i.e. constellation-level satellite stocks and debris. These laws are the foundations to the results found in \cref{SEC:Kessler,SEC:Survival}, and are crucial elements of the models presented in sections \cref{SEC:Operator,SEC:Planner}. -\subsection{Satellite Stocks} +\subsubsection{Satellite Stocks} Each constellation consists of a number of satellites in orbit, controlled by the same operator and operated for the same purpose. Satellites can be destroyed by collisions with other satellites or debris. @@ -21,7 +21,7 @@ Where $l^i(\cdot)$ represents the rate at which satellites are destroyed by coll Note that it is reasonable to assume that the loss of satellites to collisions should be increasing in the level of debris: $\parder{l^i}{D_t}{} >0$. -\subsubsection{Collision Efficiencies} +\paragraph{Collision Efficiencies} %TODO: Explain bit about constellation collision efficiencies. As demonstrated by \cite{reiland2020}, there are constellation designs by which an operator can minimize the risk of intra-constellation collisions. @@ -34,9 +34,8 @@ While some of the steps could be taken, a fundamental issue arises in that const are operated for different purposes and require different orbital properties. %Maybe 2 operators can place themselves in low risk orbits, but adding a 3rd increases the risk to all of them. %This could be explained as Coordination across time (time travel doesn't exist yet) -This coordination is also complicated by the fact that many of the constellations that -will add to the overall risk have not been concieved by their designers yet. - +This coordination is also complicated by the fact that constellations are not +designed nor launched at the same time. Consequent to these reasons, I believe the loss function $l^i$ should have the following properties related to satellite stocks. diff --git a/CurrentWriting/sections/03_SurvivalAnalysis.tex b/CurrentWriting/sections/03_SurvivalAnalysis.tex index f74dde0..20d264b 100644 --- a/CurrentWriting/sections/03_SurvivalAnalysis.tex +++ b/CurrentWriting/sections/03_SurvivalAnalysis.tex @@ -4,8 +4,9 @@ \begin{document} In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation. I've applied this to my model and extended the results. -This approach allows us to construct a elasticity of survival and satellite additions, i.e. an elasticity -of risk. +%This approach allows us to construct a elasticity of survival and satellite additions, +%i.e. an elasticity of risk. + %I should probably look up how to analyze changes in risk level and quantitative representations etc. % Marginal survival. @@ -18,7 +19,8 @@ To extend this definition to all fleets, we can measure the total number of satellites that survive. This can be calculated as the weighted sum of survival rates. \begin{align} - R =& \frac{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_t} + R =& \frac{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_t} \\ + %=& \frac{\text{Total Surviving Satellites}}\frac{\text{Total Starting Satellites}} \end{align} \subsubsection{Marginal Survival Rates} @@ -59,25 +61,32 @@ This can also be written in differential form as From \cref{EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation}, we can see that the fleetwide marginal survival rate is made up of two components. +We'll call these the direct and relative survival effects, +corresponding to the $dR^j$ and $ds^i_t$ terms respectively. \begin{itemize} - \item $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$ - represents the effect on each satellite constellation, and is always negative because - $\parder{R^j}{s^i_t}{} < 0$ by assumption. - Thus each constellations' survival rate will decrease as satellites are added to - any constellation. - \item $\frac{ R^i - R }{\sum_{j=1}^n s^j_t}$, - represents the effect of averaging out marginal survival rates. - Intuitively, when a constellation has a higher survival rate - than the fleet's survival rate, adding a satellite to that fleet contributes - less colision risk than if it were given to another - Note that it is positive but only when $R^i > R$. - Additionally, it disappears quickly as the total number of satellites increase. - Thus when there are a large number of satellites in orbit, regardless of who - owns them, it is almost certain that any increase in satellite stocks will - lead to a reduction in the survival rate. - \footnote{I believe Rao makes this an assumption, I show it is a result} + \item The direct survival effect, + $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$, + represents the effect of a new satellite on each constellation. + It is always negative because + $\parder{R^j}{s^i_t}{} < 0$ by assumption. + Thus each constellations' survival rate will decrease as satellites are added to + any constellation. + \item The relative survival effect, found in + $\frac{ R^i - R }{\sum_{j=1}^n s^j_t}$, + represents the effect of averaging out marginal survival rates. + Intuitively, when a constellation has a higher survival rate + than the general fleet's survival rate, adding a satellite to + that constellation contributes less colision risk than if it were given + to another constellation. + Thus when there are a large number of satellites in orbit, regardless of who + owns them, this effect is removed. \end{itemize} -Consequently, we can see that in many cases, the marginal survival rate will be negative. +Consequently, we can see that in most cases, the marginal survival rate will be negative. +In most models, this is either not examined or is assumed, but now we have the opportunity +to examine incentives in the case that it is not true. +One particular case where this may be important is when there is low utilization, +low internal risk, and near-monopolistic use of an orbital shell. + \end{document} diff --git a/CurrentWriting/sections/07_ComputationalApproach.tex b/CurrentWriting/sections/07_ComputationalApproach.tex index 3b79968..70c156a 100644 --- a/CurrentWriting/sections/07_ComputationalApproach.tex +++ b/CurrentWriting/sections/07_ComputationalApproach.tex @@ -87,12 +87,16 @@ from the state space. If I can data on how satellites are and have been distributed, I plan on selecting from that distribution. -\subsections{Extensions} +\subsections{Heterogeneous Agents} + One key question is how to handle the case of heterogeneous agents. -I believe I can address this in the constellation operator's case +When the laws of motion depend on other agents' decisions, as is the case +described in \ref{lawsOFMotion}, intertemporal iteration may +require knowing the other agents best response function. +I believe I can model this in the constellation operator's case by solving for the policy functions of each class of operator simultaneously. -I still have some questions about this approach and have not dived into +I would like to verify this approach as I have not dived into some of the mathemeatics that deeply.