\nBut when we refine network representations<\/strong> (e.g., explicit dwell links and transfer-station modeling) and let effective service frequency depend on passenger flow<\/strong> (to capture congestion), two practical problems emerge:<\/p>\n
- \n
- Re-boarding artifacts at transfer stations:<\/strong> expanded representations may allow passengers to alight and re-board the same line<\/em> at the same stop, creating unrealistic movements. <\/li>\n
- Algorithmic instability:<\/strong> with flow-dependent effective frequencies, standard iterative solvers can exhibit period-two oscillations<\/strong> or very slow convergence.<\/li>\n<\/ol>\n
This paper addresses both issues with a refined representation and a lightweight stabilizer that improves convergence while preserving the main assignment workflow.<\/p>\n
\n2. Core Contribution 1: PLM Representation to Eliminate Re-boarding<\/h2>\n
What is PLM?<\/h3>\n
We introduce a Pre-Line Marked (PLM)<\/strong> representation that conditions each station decision on (stop, destination)<\/em> and the preceding line<\/strong> (\u201cpre-line\u201d). This preserves transfer context and yields clearer station-level flow decomposition.<\/p>\n
A key reinterpretation is that \u201csame-line transfer\u201d becomes an on-board dwell-window decision<\/strong>:<\/p>\n
- \n
- If a passenger already arrives on line m<\/em>, the \u201cwaiting time\u201d for m<\/em> is the remaining dwell time<\/strong> (not a headway-based wait).<\/li>\n
- Other candidate lines still use effective frequency-based waiting.<\/li>\n<\/ul>\n
This removes incentives for irrational re-boarding and produces more realistic transfer behavior.<\/p>\n
\n
- \n
- \nFigure 2:<\/b> Expanded network vs. PLM representation. PLM conditions decisions on the preceding line, preventing re-boarding artifacts and refining station-level flow allocation.\n<\/li>\n<\/ul>\n
\n3. Core Contribution 2: A Tractable UE Formulation via Variational Inequality (VI)<\/h2>\n
PLM defines user equilibrium (UE) over strategy proportions<\/strong> at each decision state (stop i, destination s, pre-line m)<\/em>.<\/p>\n
Because strategy choices affect flows, flows affect effective frequencies, and effective frequencies affect costs, the equilibrium is reformulated as a finite-dimensional Variational Inequality (VI)<\/strong>. This provides a principled basis for existence analysis and solver design.<\/p>\n
- \n
- \nFigure 3:<\/b> Reformulation of the PLM-based equilibrium into a solvable VI problem.\n<\/li>\n<\/ul>\n
\n4. Core Contribution 3: ODA \u2014 A Plug-in Stabilizer for Oscillation Control<\/h2>\n
Why oscillations happen<\/h3>\n
With flow-dependent effective frequencies, the loading step becomes an inner fixed-point problem<\/strong>. Lagged-frequency updates inside standard MSA-style loops can be non-contractive, producing period-two flip oscillations<\/strong>.<\/p>\n
ODA: detect + damp<\/h3>\n
We propose an Oscillation Detection Algorithm (ODA)<\/strong> that:<\/p>\n
- \n
- Monitors<\/strong> normalized share updates of effective frequencies,<\/li>\n
- Detects<\/strong> a period-two signature,<\/li>\n
- Applies damping<\/strong> only when instability is detected (via short-window averaging).<\/li>\n<\/ul>\n
ODA can be inserted modularly without redesigning the outer assignment loop.<\/p>\n
- \n
- \nFigure 4:<\/b> MSA workflow with an inner stabilization slot. ODA is inserted modularly without altering the overall structure.\n<\/li>\n<\/ul>\n
\n5. Numerical Results: From Small Networks to Winnipeg<\/h2>\n
5.1 Small network: realism + stability<\/h3>\n
On a benchmark small network, PLM removes re-boarding artifacts while maintaining sensible line-level outcomes.
\nWhen congestion sensitivity increases, baseline MSA exhibits oscillations; MSA-ODA<\/strong> suppresses oscillations and improves convergence.<\/p>\n- \n
- \nFigure 7:<\/b> Convergence performance. ODA removes saw-tooth oscillations while remaining lightweight.\n<\/li>\n<\/ul>\n
5.2 Station-level interpretability<\/h3>\n
Even when line flows are similar, station-level movements can differ. PLM separates flows by provenance (arriving line vs. new entrants) and yields behaviorally meaningful transfers.<\/p>\n
- \n
- \nFigure 9:<\/b> Station-level flows at a transfer node: PLM provides provenance-aware movements; classical representations may imply re-boarding artifacts.\n<\/li>\n<\/ul>\n
5.3 Winnipeg network: scalability<\/h3>\n
On the Winnipeg bus network, ODA remains practical and robust across demand regimes, improving convergence compared with baseline methods while avoiding heavy inner solves.<\/p>\n
- \n
- \nFigure 10:<\/b> Winnipeg benchmark: ODA improves robustness across demand regimes and scales efficiently.\n<\/li>\n<\/ul>\n
\n6. Takeaways<\/h2>\n
- \n
- PLM representation<\/strong> eliminates re-boarding artifacts by conditioning decisions on the preceding line and reinterpreting same-line behavior via dwell-window choice.<\/li>\n
- VI reformulation<\/strong> makes the refined equilibrium analyzable and solvable in a principled way.<\/li>\n
- ODA stabilizer<\/strong> targets inner-loop instability and improves convergence while preserving standard assignment workflows.<\/li>\n<\/ul>","protected":false},"excerpt":{"rendered":"
- \n
- The Challenge: Why Transit Assignment Oscillates on […]<\/li>\n<\/ol>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_import_markdown_pro_load_document_selector":0,"_import_markdown_pro_submit_text_textarea":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-69","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=\/wp\/v2\/posts\/69","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=69"}],"version-history":[{"count":1,"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=\/wp\/v2\/posts\/69\/revisions"}],"predecessor-version":[{"id":70,"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=\/wp\/v2\/posts\/69\/revisions\/70"}],"wp:attachment":[{"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=69"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=69"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/deigel.me\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=69"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
- VI reformulation<\/strong> makes the refined equilibrium analyzable and solvable in a principled way.<\/li>\n
- PLM representation<\/strong> eliminates re-boarding artifacts by conditioning decisions on the preceding line and reinterpreting same-line behavior via dwell-window choice.<\/li>\n
- \nFigure 10:<\/b> Winnipeg benchmark: ODA improves robustness across demand regimes and scales efficiently.\n<\/li>\n<\/ul>\n
- \nFigure 9:<\/b> Station-level flows at a transfer node: PLM provides provenance-aware movements; classical representations may imply re-boarding artifacts.\n<\/li>\n<\/ul>\n
- \nFigure 7:<\/b> Convergence performance. ODA removes saw-tooth oscillations while remaining lightweight.\n<\/li>\n<\/ul>\n
- Detects<\/strong> a period-two signature,<\/li>\n
- Monitors<\/strong> normalized share updates of effective frequencies,<\/li>\n
- \nFigure 3:<\/b> Reformulation of the PLM-based equilibrium into a solvable VI problem.\n<\/li>\n<\/ul>\n
- Other candidate lines still use effective frequency-based waiting.<\/li>\n<\/ul>\n
- Algorithmic instability:<\/strong> with flow-dependent effective frequencies, standard iterative solvers can exhibit period-two oscillations<\/strong> or very slow convergence.<\/li>\n<\/ol>\n