From 2882625fcb41bd0ddb95800ac27a559e0bd19dee Mon Sep 17 00:00:00 2001 From: andetaille Date: Sun, 8 Dec 2024 22:03:30 +0100 Subject: [PATCH] =?UTF-8?q?Premi=C3=A8re=20version=20probl=C3=A8me=20des?= =?UTF-8?q?=20bus?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- fiches/descente-bus-fiche.tex | 20 +++++++- index.pdf | Bin 278028 -> 307827 bytes index_avec_fiches.pdf | Bin 261944 -> 314156 bytes src/descente-bus.tex | 86 ++++++++++++++++++++++++++++++++-- 4 files changed, 100 insertions(+), 6 deletions(-) diff --git a/fiches/descente-bus-fiche.tex b/fiches/descente-bus-fiche.tex index df50ad4..57f2fa0 100644 --- a/fiches/descente-bus-fiche.tex +++ b/fiches/descente-bus-fiche.tex @@ -1,5 +1,21 @@ \section*{Eléments de réponse} -\q (Facile) Première réponse +Temporaire, à vérifier et approfondir. +(Inclus uniquement à titre de discussion à ce stade.) -\q (Moyen) Deuxieme réponse +\q (Facile) Les bus se rattrapent toujours, quelle que soit la formule (sauf si la vitesse est constante). + +\q (Relativement facile) Les bus ne se rattrapent jamais, par un argument de symétrie temporelle. +Enfin, ça c'est si chaque bus démarre quand le précédent atteint le premier arrêt. +À voir si on veut imposer ça. + +\q (Relativement facile) La perturbation fait toujours se rattraper les bus, si le premier en est affecté. + +\q (Moyen, peut-être difficile à faire entièrement et proprement) Une telle stratégie existe, cela peut se faire de façon brutale. +Il semble clair qu'on ne peut pas avoir de gain de temps de parcours voyageur, mais il faut le prouver proprement. + +\q (Difficile) Pour deux bus, la stratégie existe. Quand on commence à avoir beaucoup de bus, c'est moins clair à prouver. +De nouveau, il semble raisonnable qu'on n'a pas de gain de temps de parcours, mais il faut le prouver proprement. + +\q (Moyen à difficile) On s'attend à un gain à la marge, mais il faut tenir compte du temps d'attente. +À la limite, dans le cas des arrêts qui se multiplient, on s'attend à trouver une EDO. diff --git a/index.pdf b/index.pdf index 4b772bb675fe19a4e5c4aee56e018c9e807b7b0a..ad333a4a9891d5ba76f789e57bc31e9798fe729d 100644 GIT binary patch delta 271255 zcmZsiQ*bV9ux4Xx$F^-}$F^dBX$+qhaP`+Uq~A@ zs$ZiH7IcS;2ng=`-v0LaJv}LfGfSZ@g`h7ye^Sm!m3Bloqsxe|x;|<9_`Y9Qz207` z`%gtlSfobE5ZTpcEP9f10+cP!9hdymMTP2_r<6owf~7KMBPkOC9Yx#u5x;G{NmzQ@ z4(Sy||E#k3xzds*Bydw}WSqw4>;+cPD#=I{?lIUmf~831icf%idZ|~o=Y`eU^L9yt z!IgF;NM)Ica1&Bv=1+sslG3Q|*ZmV5RCl}i(i&4Y`@kbb@@8RE1sHeH6Run03^;N= z4lYS6z<8Wk;)AVnOD!g%fqU+)Mz!7Q^}buCNl+ON9MMm3#?|TdcHQOUOmNkzd|k{i 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zkn@~KK=>JOD=flqL2eK+HWE1xrtgeP*_CFg##>r=_v+}Lilsw+=z%+01Ox5e*p|Z3kD^I z@S%|6!{z|EWpKL`kf&tH(EVBvEMz;K9pgu&sFJjLM=dIcgWxIBQz zBi~s}s|TPDgpUHG09AnL3l2v>#D5qZ0bz?Fw+ed~^g)DjXWD*2w&m&+F z5j@2KR>5TntTRFnAT@>SB!EGyjC3^Pz-L(CFaofXa6KiUz@&k_OC+M=0S3a`A^iuSP7FTxfjtF( zVejI>M+RW?5~wK&ZX+(NE_jP225X&u`aZ8v^TaqV`|SRA;BlA zW$6Y!$p8wiqM9}~V9Ob0sTX`i+1kto{0sC5o!LAf$9F`6gp&c{mj=NGN&$SvSWFBn shet^R3rxbx;zI#hi3v+5IyZ`_I diff --git a/src/descente-bus.tex b/src/descente-bus.tex index 77df04a..e67d5a5 100644 --- a/src/descente-bus.tex +++ b/src/descente-bus.tex @@ -1,9 +1,87 @@ -\section{Titre} +\section{Le cauchemar de la ligne 91-06} -Énoncé +Dans une contrée lointaine, les conditions de trafic se sont considérablement dégradées sur la ligne de bus 91-06, qui relie une importante gare ferroviaire à la cité universitaire, au grand dam des étudiants. +On assiste notamment à des scènes spectaculaires où un bus est tellement bondé qu'il ralentit d'arrêt en arrêt, pour finir par se faire rattraper par celui qui le suit, avec parfois trois bus consécutifs se suivant à la queue leu leu. +Face à ce problème, la société des transports TRAP a mandaté Antoine pour effectuer une analyse d'efficacité et proposer des pistes d'amélioration. +La situation étant plutôt complexe (c'est le moins qu'on puisse dire !), Antoine décide de travailler sur un modèle simplifié. +Il considère la ligne 91-06 comme une ligne droite, avec le dépôt situé en \( 0 \), puis un arrêt en chaque entier \( n \geq 1 \) (le dépôt n'est donc pas considéré comme un arrêt). +Après quelques observations, il lui apparaît que les bus se déplacent à une vitesse moyenne maximale \( V_{0} \), mais que leur vitesse moyenne diminue à mesure qu'ils chargent des passagers. +Antoine conjecture que la vitesse moyenne d'un bus contenant \( k \) passagers (le chauffeur ne compte pas comme un passager) est donnée par +\begin{equation} + \label{eq:VitesseBus} + V_{k} = \frac{V_{0}}{1+\ln{(k+1)}}\text{.} +\end{equation} +(Antoine considère que les bus se déplacent constamment à leur vitesse moyenne, et le temps passé aux arrêts est compris dans cette vitesse moyenne. +Autrement dit, il fait comme si les bus se déplaçaient constamment à vitesse \( V_{k} \), et lorsqu'ils atteignent un arrêt, ils embarquent tous les passagers qui s'y trouvent et changent de vitesse instantanément.) +Les bus ne peuvent pas se dépasser : lorsqu'un bus rattrape son prédécesseur, il le suit à la même vitesse que lui (y compris si cela implique de rouler à une vitesse inférieure à sa vitesse moyenne), et en arrivant à un arrêt, les passagers sont répartis équitablement entre les bus (s'il devait y avoir plus de bus que de passagers, en supposant qu'il y a \( N \) passagers, chacun des \( N \) bus les moins remplis recevrait un passager). -\q Première question +\q +Dans un premier temps, Antoine s'intéresse à ce qui se produit aux heures de pointe. +Il considère qu'il y a \( N \) passagers à chaque arrêt, et que ceux-ci ne se remplissent pas par la suite (ce qui correspond à dire qu'il y a tant de passagers aux arrêts que le remplissage sur la période étudiée est négligeable). -\q Deuxième question +\begin{enumerate} + \item\label{item:DeuxBusRattrapent} Deux bus quittent le dépôt, le premier au temps \( t = 0 \), le second au temps \( t = 1 \). + Finissent-ils par se rattraper ? + \item Antoine se demande si cette réponse dépend de sa modélisation de la vitesse. + Reprendre le point~\ref{item:DeuxBusRattrapent} si \( V_{k} \) peut être donnée par une expression quelconque, et non nécessairement celle en~\eqref{eq:VitesseBus}. + (On supposera juste que \( V_{k} \) est décroissante en \( k \), car il est raisonnable de penser que les bus ralentissent s'ils sont davantage chargés.) +\end{enumerate} + +\q +Antoine se demande à présent ce qui se passe lorsque la journée commence, et que les arrêts se remplissent progressivement. +Il considère donc toujours deux bus, un démarrant au temps \( t = 0 \), et un démarrant au temps \( t = 1 \). +Au temps \( t = 0 \), les arrêts sont tous vides, et se remplissent à un taux constant de \( \rho \) passagers par unité de temps. +Lorsqu'un bus arrive à un arrêt, il récupère tous les passagers qui y sont arrivés depuis le passage du bus précédent, soit \( \rho \) multiplié par le temps écoulé (on permet un nombre non entier de passagers, pour simplifier la modélisation). +Reprendre la question précédente dans ce cadre, et considérer également le cas où on fait démarrer \( m \) bus, le premier au temps \( t = 0 \), et ensuite un par unité de temps. + +\q +Antoine étant bien conscient que la modélisation qui précède ne tient pas compte des petits aléas de la circulation ou des afflux imprévus de passagers, il souhaite désormais les inclure. +Il considère donc les variantes suivantes. + +\begin{enumerate} + \item Au temps \( t = T \), le bus de tête est immobilisé pendant un intervalle de temps de \( \frac{1}{10} \) en raison d'un embouteillage, avant de reprendre à vitesse normale. + \item Au temps \( t = T \), une quantité \( q \) de passagers arrive à l'arrêt situé en \( n = 10 \) en plus du remplissage normal (par exemple, suite à l'arrivée d'un TGV, dont les passagers souhaitent rallier la cité universitaire en bus\dots). +\end{enumerate} + +Toujours dans le cas de deux bus séparés d'une unité de temps, évaluer l'impact des deux perturbations ci-dessus sur la suite de leur parcours. +(On discutera les résultats en fonction de \( T \) et \( q \).) + +\q +Antoine souhaite à présent concevoir une stratégie pour retenir les bus aux heures de pointe, afin d'éviter qu'ils ne se rattrapent. +On suppose donc à présent que, lorsqu'ils arrivent aux arrêts (et uniquement à ce moment), les bus peuvent s'arrêter et attendre un temps arbitraire avant de repartir à leur vitesse normale. +Une \emph{stratégie} est une façon pour les bus de décider, en connaissance de la position de tous les bus et du nombre de passagers qu'ils transportent, du temps à attendre lorsqu'ils arrivent à un arrêt. +Dans le cadre de la question 1a), existe-t-il une stratégie pour retenir les bus de façon à ce que le second ne rattrape jamais le premier ? +Le cas échéant, une telle stratégie peut-elle être choisie de façon à améliorer le temps de parcours voyageur (c'est-à-dire que certains passagers iraient de l'arrêt où ils sont montés à un arrêt ultérieur plus vite que si cette stratégie n'était pas mise en \oe uvre) ? + +\q +Antoine souhaite ajouter encore davantage de réalisme à son modèle, et tenir compte du fait que les bus font plusieurs rotations sur le même trajet. +Il considère donc que la ligne se termine à l'arrêt \( n = 20 \), considéré comme le dépôt de fin de ligne. +Il n'y a aucun passager à cet arrêt, lorsqu'un bus l'atteint, il se vide de tous ses passagers, et repart dans l'autre sens. +Il effectue ensuite des allers et retours sur le trajet, et à chaque fois qu'il atteint le dépôt en \( n = 0 \) ou en \( n = 20 \), il dépose tous ses passagers et repart dans l'autre sens. +Il considère également qu'au temps \( t = 0 \), il y a \( N \) passagers à chaque arrêt et dans chaque sens, et qu'ensuite, les arrêts continuent à se remplir à un taux de \( \rho \) passagers par unité de temps. + +On suppose que deux bus circulent sur la ligne, partant tous deux du dépôt en \( n = 0 \), le premier au temps \( t = 0 \), le second au temps \( t = 1 \). +Existe-il une stratégie pour retenir les bus aux arrêts de façon à éviter qu'ils ne se rattrapent ? +Que se passe-t-il dans le cas de \( m \) bus ? +Discuter de l'optimalité d'une éventuelle stratégie par rapport au temps de parcours voyageur. + +\q +Antoine cherche à explorer une dernière idée pour améliorer le temps de parcours voyageur. +On se place à nouveau dans le cadre de la question 2, avec deux bus circulant, et où les arrêts sont initialement vides et se remplissent progressivement. +Antoine propose que le premier bus desserve uniquement les arrêts impairs, et le second uniquement les arrêts pairs. + +\begin{enumerate} + \item Cette stratégie présente-t-elle un gain en termes de temps de parcours voyageur ? + Le quantifier aussi précisément que possible. + \item Pour pousser son idée encore plus loin son idée, Antoine suppose à présent qu'un bus démarre à chaque temps \( t \) entier, et que les bus parcourent les arrêts de \( k \) en \( k \). + Plus précisément, le premier bus dessert les arrêts multiples de \( k \), le suivant les arrêts multiples de \( k \) plus \( 1 \), et ainsi de suite. + Après \( k \) bus, le schéma se répète : le \( (k+1) \)-ème bus dessert les arrêts multiples de \( k \), le \( (k+2) \)-ème bus dessert les arrêts multiples de \( k \) plus \( 1 \), et ainsi de suite. + Quantifier le gain éventuel en termes de temps de parcours voyageur, et examiner ce qui se produit lorsque \( k \to +\infty \). + \item Antoine décide de pousser son idée encore plus loin, et d'ajouter des arrêts, pour répartir les voyageurs sur davantage d'arrêts. + Il suppose donc qu'il y a à présent un arrêt en \( \frac{n}{k} \) pour chaque entier \( n \geq 1 \), et que les voyageurs arrivent aux arrêts à un taux de \( \frac{\rho}{k} \) passagers par unité de temps. + On considère toujours qu'un bus quitte le dépôt à chaque unité de temps. + Quantifier l'impact de ce changement sur le temps de parcours voyageur, à la fois dans le cas où chaque bus dessert tous les arrêts ou dans le cas où les bus desservent les arrêts de \( k \) en \( k \). + On s'intéressera particulièrement à ce qui se produit à la limite lorsque \( k \to +\infty \). +\end{enumerate} \q Proposer et étudier d'autres pistes de recherche. \ No newline at end of file