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62 lines
3.6 KiB
TeX
62 lines
3.6 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation.
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I've applied this to my model and extended the results.
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% Marginal survival.
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The survival rate for a constellation $i$ is defined as $R_i = 1-l^i(\cdot)$, the proportion of satellites-
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that were not lost (degraded nor destroyed) between period $t$ and $t+1$.
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Thus the marginal survival rate represents the additional loss of
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satellites due to a slightly larger constellation or fleet stock.
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Let $S_t = \sum^n_{j=1} s^j_t$.
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Then the survival rates for a constellation and for society's fleet are respectively defined as:
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\begin{align}
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R_i =& \frac{s^i_{t+1}- x^i_t}{s^i_t} = 1- l^i(s^i_t,S_t,D_t) \\
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R =& \frac{S_{t+1}- X_t}{S_t} = \frac{\sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] }{S_t} \label{EQ:socsurv}
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\end{align}
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In this case, the fleet survival rate \cref{EQ:socsurv}, represents the proportion of satellites-
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in period $t+1$ that survived from period $t$.
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The marginal survival rates when a given constellation $i$ changes size are:
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\begin{align}
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\parder{R_i}{s^i_t}{} =& -\left(\parder{l^i}{s^i_t}{} + \parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{} \right)
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= - \parder{l^i}{s^i_t}{} - \parder{l^i}{S_t}{} \label{EQ:iii} \\
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\parder{R}{s^i_t}{} =& \frac{S_t \sum_{i=1}^N \left( [1-l^i(s^i_t,S_t,D_t)]
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+ s^i_t [ -\parder{l^i}{s^i_t}{} -\parder{l^i}{S_t}{}\parder{S_t}{s^i_t}{}] \right)
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- \left( \sum_{i=1}^N s^i_t[1-l^i(s^i_t,S_t,D_t)] \right)}
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{(S_t)^2} \notag{}\\
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=& \sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{} \label{EQ:i}
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\end{align}
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Note that $ \sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}$ is the weighted, average marginal survival rate-
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across constellation operators.
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The derivation of \cref{EQ:i} is in Appendix \ref{APX:Derivations:SurvivalRates}.
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Direct comparison between the marginal survival rates of an individual operator and the social planner's fleet
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cannot proceed further without specifying the functional loss forms $l^i(\cdot)$
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and specifying which firm will be compared to society.
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In spite of this, conditions on the average effects can be developted as follows.
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The marginal survival rate of the fleet is less than the weighted, arithmetic mean of marginal survival rates-
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of the constellations when:
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\begin{align}
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\sum_{i=1}^N \left[ \frac{R_i}{S_t} \right] - \frac{R}{S_t}
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+\sum_{i=1}^N \frac{ s^i_t}{ S_t} \parder{R_i}{s^i_t}{}
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\leq& \sum_{i=1}^N \frac{s^i_t}{S_t} \parder{R_i}{s^i_t}{} \\
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\sum_{i=1}^N R_i - R \leq& 0\\
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\sum_{i=1}^N [1- l^i(s^i_t,S_t,D_t)] - \sum_{i=1}^N s^i_t [1- l^i(s^i_t,S_t,D_t)] \leq& 0\\
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\sum_{i=1}^N (1 - s^i_t) [1- l^i(s^i_t,S_t,D_t)] \leq& 0 \label{EQ:ii}
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\end{align}
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which is true if every constellation has at least one satellite.
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As any constellation of interest has at least one satellite
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and $\parder{R_i}{s^i_t}{} < 0$ from the assumption on collision mechanics that $\der{l^i}{s_t^i}{}>0$,
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we conclude that the marginal survival rate of the entire satellite fleet is lower
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than the weighted arithmetic mean of marginal survival rates across constellations.
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Note that it is possible for some constellations to have a lower marginal survival rate than the fleet,
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but the survival rate for many operators must be higher than the societal rate.
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Consequently, we would expect many operators to underestimate the impact of their behaviors on others
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if they use their own observed or expected risk factors to estimate the risk they impose on others.
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\end{document}
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