Fig. 3: Penetration rate estimation and reconstructed PTS diagram of an example movement.
From: Traffic light optimization with low penetration rate vehicle trajectory data
All sub-figures are generated based on the mid-day period (10:00–15:00) of three consecutive weeks’ (only weekdays from Monday to Friday) data from 03/07/2022 to 03/25/2022. a A short period (3 cycles) of the time-space (TS) diagram for a specific movement. b Aggregated TS diagram that aggregates trajectories of the same time of day (TOD) into one cycle. c Arrival and departure histograms (\({a}^{{{{{{\rm{obs}}}}}}}\left(t\right),\, {b}^{{{{{{\rm{obs}}}}}}}(t)\)) generated from the aggregated TS diagram. All arrivals are within the first cycle (0–90 sec) while the departures might extend to the following cycle. d Arrival and departure profiles under a given penetration rate \(\phi\) (\(6\%\) in the displayed case). Blue and red bars show the scaled arrival and departure profile, \({a}^{{{{{{\rm{sc}}}}}}}\left(t\right)\) and\(\,{b}^{{{{{{\rm{sc}}}}}}}(t)\). The scaled arrival \({a}^{{{{{{\rm{sc}}}}}}}\left(t\right)\) is used as the input arrival \(\hat{a}(t)\), denoted by the red dashed line. The blue dashed line \(\hat{b}(t)\) is the resulting departure profile derived from the queueing model. e Penetration rate estimation. The purple dashed line shows the average delay \({\bar{d}}^{{{{{{\rm{obs}}}}}}}\) directly calculated from the observed trajectories while the dashed blue curve shows the model-estimated average delay \(\hat{d}(\phi )\) under different penetration rates. The red vertical line shows the estimated penetration rate \({\phi }^{*}\) such that \(\hat{d}\left({\phi }^{*}\right)={\bar{d}}^{{{{{{\rm{obs}}}}}}}\). f, g Aggregated TS diagram (a complete cycle) and the reconstructed PTS diagram of the example movement.
