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| Notation | Description |
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| Indices and variables |
| p | Phase index, of range |
| m | Cycle index, of range |
| i | Index of vehicles detected by mobile data source during the phase of cycle, of range [1, 2, …… ] |
| Number of the first vehicle at the downstream location that arrives at the start of the red phase |
| Number of the last vehicle at the downstream location that is affected by red phase |
| Time taken to dissipate the queue formed during red of phase of cycle in hours |
| Time stamp at which vehicle of phase during cycle passes the upstream location |
| Time stamp at which vehicle of phase during cycle crosses the downstream location |
| Delay experienced by vehicle of phase of cycle in hours |
| Jam spacing between and probe vehicles of phase of cycle in km |
| Maximum queue length from the stop line during queue formation of phase of cycle in km |
| Number of vehicle on arrival at the downstream location in phase of cycle |
| Actual backward queue forming shock wave speed of phase of cycle in kmph |
| Estimated backward queue forming shock wave speed between and vehicles of phase of cycle in kmph |
| Estimated backward queue forming shock wave speed using probe data of phase of cycle in kmph |
| Arrival flow rate of phase during cycle in veh/hr |
| Capacity of phase during cycle in veh/hr |
| Effective green duration of phase during cycle in veh/hr |
| Upper bound of effective red duration of phase during cycle in hours |
| Lower bound of effective red duration of phase during cycle in hours |
| Upper bound of effective green duration of phase during cycle in hours |
| Lower bound of effective green duration of phase during cycle in hours |
| Parameters |
| Free-flow speed of vehicles in kmph |
| Critical density in veh/km |
| Wave speed in kmph |
| Jam density in veh/km |
| Saturation flow rate in veh/hr |
| Cycle length in hours |
| Distance between upstream and downstream detector locations corresponding to phase, in km |
| Average pedestrian crossing time for phase, in hours |
| Decision variables |
| Optimal effective red duration of phase of cycle, in hours |
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