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Time-Distance Diagrams: a Powerful Tool for Service Planning and Control

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Time-Distance Diagrams: a Powerful Tool for Service Planning and Control Journal of Public Transportation Time-Distance Diagrams: A Powerful Tool for Service Planning and Control Eric C. Broun National Transit Institute, Rutgers University Vukan R. Vuchic Department ofSystems Engineering, University ofPennsylvania Yong-Eun Shin Dong-Eui University, Pusan, Korea Abstract Graphical scheduling is an old technique that has been neglected, or never acquired, in many NorthAmerican transit agencies. It retains its advantages in basic schedule design and analysis as it eases the solution to problems that are difficult to solve analytically. Even information about simple routes is enhanced by the detailed operating characteristics inherent in detailed vehicle trajectories and by the relative ease with which accelerated service and service recovery strategies can be investigated. It also can be used to confirm and refine solutions that are generated by analytic methods. The methodology is reviewed in the context ofsuch planning applications. Graphical scheduling has additional advantages in operational control with the advent ofmodern /'IS technologies. By movement ofthe cursor on a terminal screen, detailed information about all activity along a route becomes available. It is possible to link the altering oftrajectories through clicking and dragging to the automatic issuance ofcontrol commands and updates ofpassenger information. These and other possible uses ofthe technique in an operational context are pre­ sented. Vol. 2, No. 2, 1999 2 Journal of Public Transportation Introduction Developmentand use of graphicalschedules for planningand supervisionof transitsystems operations is by no meansa newsubject. It is a time-provenmethod usedfor bothdevelopment and analysis of schedules.Yet, its use is far fromuniver­ sal. Whilemany transit and railway systems use graphicalschedules in theirdaily operationsfor multiplepurposes, the entireconcept and technique is virtuallyun­ knownin mostNorth American transit systems, including the largest ones. The latest dispatching/controlsoftware packages in the UnitedStates that monitorbuses in real-timethrough Automatic Vehicle Location (AVL) do not use it either.However, thissoftware does display GIS maps showing vehicle location along streets, as well as checkpointdata in spreadsheetformat. Whilethe spreadsheetformat is useful,the datadisplay format that for many purposesreveals the most information-the time-distance diagram-should also be available.Actually, the graphicalmethod is superiorto thenumerical ones for many applications.The purpose of thispaper is to broadenthe knowledge about this tech­ niquein thoseparts of thetransportation community where it is not usedand, often, whereit is not evenknown. Thispaper is organizedas follows.The following section describes the concept of the time-distancediagram and explains how it is designedand interpreted.The thirddescribes several applications in schedulingand the advantages this technique provides.The fourthsection describes its advantagesin real-timeoperational con­ trol,an areaof vitalinterest with the adventof highlycapable Intelligent Transpor­ tationSystems (ITS) that provide precise vehicle locations and numerous communi­ cationsoptions. The final section is a concludingsummary. Definition of Time-DistanceDiagram A time-distancediagram, as thename implies, is usedto plot motionof ave­ hicleor train-henceforthreferred to asTransit Unit or TU-with timeon the hori­ zontaland distance on the vertical axis. The distance axis usually has a lengthequal to the routelength, with the terminals defining the end points. Each TU has an individual trajectorydescribing its positionand movement over time. The family of trajectories Vol.2, No. 2, 1999 Journal of Public Transportation 3 fonna time-distancediagram representing a complete schedule for one, sometimes for several,lines. Thedetail at whichthe trajectory of a TU is presentedshould reflect the purpose of theanalysis. In somecases, particularly when the distances between planned stops are very long, the trajectoryneed be no morethan a straightline connectingthe departingterminal at the dispatchtime to the other terminalat the arrival time, usuallyin minutes.In othercases it maybe necessaryto providevery detailed trajec­ toryshowing all accelerations,decelerations, and stationdwell time periods. These detailedTU trajectoriesare commonly plotted in seconds. Usinga rapid transitline as an exampleof a detailedtrajectory, the line is modeledas a seriesof interstationspacings where the train accelerates with constant ratea., cruises at speedv, and decelerates at rateb. Eachstation i has a dwelltime t .. I ~ Traveltime on any interstationspacing is composedof the time intervalsrequired for acceleratingto cruisingspeed, running at cruisingspeed, decelerating into the station,and the stationdwell time. This kind of trajectory,shown in FigureI, maybe constructedfor individualinterstation spacings, or for entirelines. For certainanalyses, it is necessaryto computethe incrementaltime lost by stoppingat stationi, or T,.This is the timethat TU needsfor travelwith stopping at one station,as comparedto the traveltime on the same sectionwhile movingat cruisingspeed, without stopping at the station.This incrementaltime consistsof additionaltime due to decelerationin enteringstation i, ~' dwelltime at stationi, ts, andincremental time for accelerationwhile departing station i, t • Figure2 showsa 3 straight-lineapproximation of time lost for eachstop, referred to as Te,instead of exactacceleration and deceleration paths. This straight line simplification is conve­ nientfor plotting,and yet sufficientlyaccurate for mostscheduling purposes. Each incrementaltime lostis connectedto thenext by a straight-linewith slopeequal to thecruising speed between them. Therefore, if thecruising speeds on differentspac­ ingsare equal,the slopedlines are parallel.On linesections where cruising speeds vary,the slopesalso change among interstation spacings. Figure2 alsoshows trajectory of anotherTU, following the first one.The hori­ zontalseparation between trajectories represents the headwaybetween TUs, h. The Vol.2, No. 2, 1999 4 Journal of Public Transportation Distance Station i Time Figure1. Time-distance diagram showing elements of TU motion. minimumallowable headway depends on vehicle performance, route alignment, and controlsystem characteristics (e.g., manually driven on street,discrete block, mov­ ingblock, etc.). The vertical separation at anypoint in timerepresents the distance separationbetween corresponding points on successiveTUs, or theirspacing, s. The minimumspacing, which corresponds to the minimumheadway, consists of three components:the TU length, the distance passed during the driver's reaction time, and actualbraking distance. The minimum safe spacing is oftenalso presented as a con­ tinuousline in frontof theTU trajectory, and it is knownas theTU shadow.There are Vol. 2, No. 2, 1999 Journal of Public Transportation 5 Distance St2 ----------------------- h Time Figure2. Graphicalschedule of two successivetrains. actuallydifferent regimes or degreesof safety,and the readerinterested in this de­ tailedanalysis is referredto Vuchic (1981 ). ForTUsthat travel on thesame path, such as a railtrack or buslane, once any minimum is violated,the schedule is infeasible.In practice,there should be separationwell beyond the minimumto ensureschedule reliability.When trajectories reflect actual operations instead of the intendedsched­ ule, as soon as any revisedtrajectory can be projectedto violatethe minimums, delayscan be anticipated,unless corrective action is takento re-separatethem. Vol.2. No. 2, 1999 6 Journal of Public Transportation Graphicalschedules have a visualclarity that numerical schedules can never provide.Tables with numerical values ofTU departures at differentpoints along a linemay contain errors that are not immediatelynoticeable. When plotted graphi­ cally,every incorrect time is immediatelyconspicuous. Uneven travel speeds, TUs travelingat lessthan the minimumheadway, or anyconflicts in TUtravel paths are alsoeasy to detecton graphicalschedules. Graphicalschedules can also be useful to presentactual running of trainswhere someskip different stations along the line.An exampleis shownin Figure3 for the CalTrain commuter railroad serving communities south of SanFrancisco. The plot is basedon scheduleddepartures of everytrain at everystation, and showsdifferent headwaysduring the a.m.peak and midday periods, as wellas the relationshipof localand partial express peak period trains. In additionto schedulesfor TUs traveling on the sametrack in the samedirec­ tion,graphical analysis can also be usedto findmeetings of TUsmoving in opposite directionson the sametrack or path.Figure 4 showstrain scheduling for a single :;- u j!. l':llo Alla +-T--..-"""T'""1~--r-r-,--.--r--a ,..__....._,--:-.::.:; __ _ lteth\l>odCily 2~ SanCa,los llclmc1111 Soulh SI' ilafsliorc 22nd Succt o IS JO 4S GO 7S 90 IDS no US ISO 16S IIO 19S 210 22S 240 2SS 270 21S 300 6·00 7:00 1:00 9:00 10;00 11:00 TlmclminJ Figure3. Graphicalpresentation of a.m. peak and midday schedule for CalTrain,San Francisco Bay Area. Vol.2, No. 2, 1999 Journal of Public Transportation 7 tracksection A-B of a doubletrack line. If thetrajectories cross each other along the sharedsection, as shownby the dashedline in the figure,the operationis infeasible. A feasibleschedule is shownby the solidlines, which intersect on a doubletrack section. As anotherexample, this graphicalmethod was used to verify a temporary scheduleinvolving single-tracking
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