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

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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 during reconstruction of the Market-Frankford

Distance

-1' I I I I I I I I I I I I I I r---J C: 0

t5Q) u, .!Iii:: 0 ...~ Q) C> C: c:i5

I 8 I 1r-----'------I I I I I I I I I I I I

Time

Figure4. Useof time-distancediagram to scheduletrain meets outsideof a singletrack section.

Vol.2, No. 2, 1999 8 Journal of Public Transportation

MARl(ET .., ll'RANl(FORDSUBWAY - ELEVATEDLINE 2- HOURTIME-·DISTANCE OPERA'rlONS CHART ._ -cs__. (ii. llOPM - 6. 3Ul'M) PROPOSEDSINGLE TIIACKINC SCHEDIII.£

Figure5. Applicationof time-distancediagram for scheduling highfrequency service over a single-tracksection. rapidtransit line in Philadelphia,as described by Bruun and Sal peas ( 1991). Thefull peakperiod schedule was plotted using the incrementaltime-lost format to ensure thatno trajectoriescrossed each other on the single-track section. For clarity and size, onlypart of the scheduleis reproducedas Figure5; the twosolid horizontal lines showthe single-tracksection. Applicationsin Scheduling Themost common application of graphicalschedules is forsingle lines. How­ ever,there are a numberof caseswhere several lines that merge, diverge, intersect, or form a trianglecan also be scheduledgraphically, i.e., with applicationof time­ distancediagrams. Several typical cases of differenttypes of graphicalschedules are describedhere. SingleLine Operation Onceaverage operating speed from terminal to terminalis knownand desired headwaysare specified,development of thebasic schedule for single routes is alge-

Vol.2, No. 2, 1999 Journal of Public Transportation 9 braicallyand graphicallysimple. Its developmentinvolves only basic preliminary calculations,followed by adjustmentsto terminaltimes to maintainthe integercon­ strainton fleetsize. Even in this simplecase, a time-distancediagram showing the detailedtrajectory of eachindividual TU is helpful,because it revealsinformation on each individualterminal time, dwell times at differentstations, average running speeds,etc. It is a simplemethod to clearlydisplay deadheading, short turns, indi­ vidualruns and fleet size on theline. All pull-ins and pull-outs, short-term storage on sidings,and otherdetails can be shown.The diagramalso is helpfulfor planning transitionsbetween service plans during the day. SpedalOperations Oncea basicpattern for the schedule has been established, variants can be easily plottedand analyzed.These include short turn, skip-stop,and partial express (also calledzonal) operation. The modeling of eachwill be explainedbriefly. Shortturns can be treatedas if they simplywere intermediateterminals, al­ though,for trains,the minimumterminal time may have to includeallowances for maneuveringtime shorter or longerthan at the outerterminals, depending upon the lengthofline headways,short-tum cycle time and track layout. The simplestopera­ tionis whenline headways are longand timing of a short-turningtrain such that the short-turningtrain can reverse at a stationwith center platform without conflict with regulartrains passing in eitherdirection, as shownin Figure6a. In anothercase, the trainhas stoppedand had sufficient dwell time to dischargepassengers, it mustwait beforereversing. In this case,there is enoughtime for the operatorto changeends andaccept passengers, departing in reversedirection and crossing over to the oppo­ site track prior to arrivalof a followingtrain in eitherdirection, so that no delays occur. It canhappen, however, that the short-turningtrain cannot travel back immedi­ atelybecause the othertrack is occupiedby a trainpassing in the oppositedirection, or immediateturning would create irregular headways in the oppositedirection. Thenterminal time must be extendeduntil a trainin thereverse direction has passed beforethe short-turningtrain can travel back, as shownin Figure6b.

Vol.2, No. 2, 1999 10 Journal of Public Transportation

Distance

Time

Figure6a. Simpleshort tum.

As headwaysbecome shorter, it is likelythat the TU cannot wait long enough to reversewithout blocking traffic in its initialdirection. In thiscase, either a different reversinglocation must be used,or an extratrack for reversal must be provided.This situationis shownin Figure6c. The short-turning train must be movedpromptly into thecenter track for reversing, so as to allowat leasthmm for passing of thetrain that is followingit in its initialdirection. The center track then allows the reversingtrain to waitfor the desiredtime to departin thereturning direction. Thus, any possibility of delaysis eliminatedand regular headway can be easilymaintained.

Vol. 2, No. 2, 1999 Journal of Public Transportation 11

Distance

r------, _____ Terminal time I l-~-=~~~-:

Time

Figure6b. Shortturn scheduledbetween opposingtrain runs.

Thisexample shows that a situationthat is verycomplex to modelalgebraically dueto severalconstraints and solution possibilities can be quicklyanalyzed graphi­ callyto see whattype of solution(s) are actuallyfeasible for a givenbasic headway, cycletime and track layout. Skip-stopoperations, described in Vuchic(1976), are such that at somestations, onlyalternating trains stop (typically called A andB stops).This is donegenerally onlywhere the basicheadway is short,as the stationswhere only alternating trains

Vol.2, No. 2, 1999 12 Journal of Public Transportation

Distance

Track 1 Tracks 2

3

Time

Figure6c. Short tum with reversingtrack. stophave doubled headways. Skip-stop service reduces operating time, so thateither tenninaltimes or dispatchingtimes must be changed.Either possibility can be readily plottedand compared to regularoperation. The horizontal separation between tra­ jectoriesof trainsthat stop at a givenstation gives the headwaypassengers experi­ enceat thatstation. The diagram in thatcase shows that joint stations retain the same averageheadway as in regular,all-stop operation, while the A andB stations,being servedby alternatetrains, have twice longer headways. Since the joint stationshave shorterheadways, they remain critical for the linecapacity. The diagramfor skip-

Vol. 2, No. 2, 1999 Journal of Public Transportation 13

stopoperation can also illustrate the increase in operatingspeed on theline achieved by skippingseveral stations. Expressoperations on two-tracklines require careful scheduling oflocal and expresstrains. Again, graphical scheduling is greatlysuperior to numericalmethods, becauseinserting the trajectory of anexpress train between regularly scheduled train

Distance E E' 1 I

Express overtakes in a station with sidi~

I I I I ,' ~ If no overtaking ,' possible I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Time

Figure7. Synchronizationof localand expresstrains.

Vol.2, No. 2, 1999 14 Journal of Public Transportation runsis visuallyvery simple. As shown in Figure7, theexpress, shown by trajectoryE, shouldbe scheduledso thatit overtakesa localtrain in a stationwith siding; the initial headwaybetween the local and express trains h 1 whichis neededfor this operation is easyto obtaingraphically. lfthere areno stationswith sidings, so that overtakingis not possible,trajectory of the expresstrain is "slid"to the rightuntil it reachesthe dashed-lineposition on theright, E', whereit "catchesup" with the minimum head­ waybehind the localtrain on the lastinterstation spacing. Ifit is importantto utilize maximumline capacity achievable with this service regime, the expresscan be fol­ lowedat thebeginning of the lineby another local train after a minimumheadway, as shownon the diagram. In general,express running is practicalonly when average headway on the line is considerablylonger than the minimumone, so that the expresstrain can skip severalstations. The limitson expressruns are imposedeither by line capacityre­ quirementsor by the maximumacceptable headway. Trunkand BranchOperation Schedulingfor a trunkline that divides into two or morebranches is muchmore complicatedthan schedulingfor a singleline becauseit involvesdivergence and convergenceof trains.The sequencingof TUsarriving from the variousbranches andthe resultingregularity of the headwayalong the trunk section become impor­ tantconsiderations. Algebraical coordination of schedulesbecomes quite complex, involvingconstraints on terminaltimes and mergingsequences with multiple feasibilities,but not equallyefficient solutions. Even when solved analytically, it maybe difficultto visualizethese solutions. Bycontrast, using a graphicalapproach, feasible solutions can quickly be iden­ tified and compared.The methodis to use the pointof branchdivergence X as a reference.The trajectoriesfor all routes( each consisting of a trunk and a branch section)are showntogether on the trunksection, while the trajectoriesalong each branchsection are shown on individual,vertically separated but synchronizedtime­ distancediagrams. A verticaldashed line is drawnbetween corresponding diverging andmerging points for eachTU trajectory,as shownin Figure8.

Vol. 2. No. 2. 1999 Journal of Public Transportation 15

Toexplain in moredetail, a graphicaldiagram is startedby plottingtrajectories of successiveTUs at givenheadways on the trunk line.At point X, branchA is continuedwithout interruption, while the following TUs, going to branchesB andC, are transferredvertically to the tworespective diagrams. Terminal times on each

B

C

X A ------!-----______., _ _,...... ______---..,....--!-- __ _ A

X

Time

Figure8. Time-distancediagram of a trunk with three branches.

Vol.2, No. 2, I 999 16 Journal of Public Transportation branchmust be determinedso thatTUs merge at pointX with equalheadways. Graphically,these terminal times are obtainedeasily, and they can be visualized muchmore clearly than in numericalschedule tables. This solution, which provides evenheadways along the trunk, can be comparedto alternateones that would mini­ mizeterminal times and fleet size for each route, but, instead, have uneven headways alongthe trunk.

Richmond Concord

+-- Mac Arthur

San FranciscoTrunk

Oakland "Y" Colma

Fremont Figure9. SanFrandsco BART network and lines(1996).

Vol.2, No. 2, 1999 Journal of Public Transportation 17

In somecomplicated networks, analytic solutions can also become intractable. A goodexample is the BayArea Rapid Transit (BART) network, which, until re­ cently,consisted of threebranches plus a "cross-connection"between two of the branches.Thus, the operationrepresents a triangleplus an additionaltrunk-and­ branchline. The line pattern is shownin Figure9. Althoughvery difficult to model analytically,this networkwas successfullyanalyzed using a graphicalmethod to arriveat severalpractical candidate scheduling solutions. This work was presented in severalreports by Vuchic, Bruun and Krstanoski ( 1995-97). Althoughspace does not allow discussionof a complicatednetwork likeBART, application of graphical analysisto a networkconsisting of a trunkwith two branchesand cross­ linebetween the branches will be ex- X plainedhere. This networkalso can be interpretedas a three-legnetwork withlines between all three terminals, as shownin Figure10. Graphicalpresentation for this case,shown in Figure11, is thesame A as for a trunkline with two branches: Figure10. Simple two-branch network the maindiagram shows the A-X-B with cross-connectingline B-C. section, while the X-C branch is shownabove it. A-BandA-C trains are shownas in Figure8. If the B-Ctrains are shownas inboundtrains on the B-X section,then, when transferring to the X-C diagram,they are shownas outboundtrains. Once understood, this diagram is very helpfulin coordinatingthe schedules of thethree branches. Because of the interde­ pendency,such schedules usually cannot provide regular headways on all lines,and thegraphical schedule can be veryuseful in makingadjustments that make the best candidatesets of headwaysfor the three branches.

Vol.2. No. 2. 1999 18 Journal of Public Transportation

X

X

A Figure11. Time-distance diagram showing cross-connecting line betweenbranches Band C (see Fig. 10).

Useof Math ProgramGenerated Schedules Severalcomputer-based packages, using math-programming methods, have beenin use for morethan 20 yearsfor schedulingtransit services and crews.How­ ever,there are occasionally constraints that cannot be expressedwell mathematically that may affectthe practicalityof the generatedsolutions. Examples include the operationof differentrail serviceson sharedtrackage that do not fullycooperate in schedulingand dispatching, or transitlines that require large amounts of slacktime in orderto operatereliably. Interconnected lines, such as the above-discussedBART network,may also be moreconducive to graphicalthan numerical scheduling and analysis. Plottingthe resultsgenerated by theprogram as a time-distancediagram will allowvisualization of the latitudethat is availablefor deviations from the schedule

Vol. 2, No. 2, 1999 Journal of Public Transportation 19 beforea recoverystrategy must be usedto repositionTUs due to loss of slot in the schedule.This is doneby checking the horizontal separation between the non-agency­ controlledvehicles and its own. The history of scheduleadherence and delays along theshared section should provide some insight into the likelihoodof delaysserious enoughto losea slot as a functionof the timeslack available. Should the schedule appeartoo tightfor reliable operation, the programcan be rerunwith changed con­ straintsto includeadditional slack or "cushions"in the schedule. An exampleapplication where a math-program-basedschedule should be fur­ theranalyzed is the SoutheasternPennsylvania Transportation Authority (SEPTA) RegionalRail Division. This is a regionalrail network that includes not only SEPTA­ controlledtracks, but alsomajor portions where Amtrak controls all traindispatch­ ing.By usinga time-distancediagram, it is possibleto see howtight the gaps are betweenthe Amtrak and SEPTA trains at cruciallocations along the network. Crucial locationsare thosewhere SEPTA trains are likelyto be heldwhen there are delays. Adjustmentscan be madeto the scheduleto providerecovery time or, if possible,to avoidtight scheduling in thefirst place. Such a diagramalso allows quick visualiza­ tionof the qualityof connectionsbetween the variousinterconnected lines at key transferpoints. Applicationsin OperationsControl So far, the discussionhas been in a planningcontext. Graphical analysis is usefulin an operationalcontext as well. Real-limeOversight and Reporting Real-timeoversight of trainoperations with elaborate control centers has been aroundfor many decades, although, in somecases, the positionalresolution is poor whensignal blocks are long. Oversight of busesthrough Automatic Vehicle Location and otherIntelligent Transportation Systems (ITS) technologies is a muchmore recentcapability with generally high accuracy. In bothcases, numerical schedules in spreadsheetformat provide information with which to monitoradherence in real­ timeby displayingthe numericaldeviation, perhaps highlighted by a blinkingor color-changingdisplay when deviations become sufficiently large. For trains, a sys-

Vol.2, No. 2, 1999 20 Journal of Public Transportation ternschematic with lights indicating block occupancy is common.For buses, a GIS mapshowing vehicles of differentcolors (yellow-early, green-on time,red-late, forexample) can also be usedto highlightcurrent status. Spreadsheet-typedisplays have, however, some definite limitations. The bigger picturethat might reveal how any deviations might be interrelatedand where devia­ tionsare likely to propagateis notreadily apparent with spreadsheets, schematics, or maps.A time-distancediagram where trajectories, or portionsof trajectorieschange colors,used, for example,in the Italianltaltel bus dispatching/monitoring system, canreadily reveal not onlythe currentsituation but impending deviations and con­ flicts.Even more importantly,as discussedin the next section,it providessome insightinto when and where recovery strategies must be employed. Moreover,when a real-timetime-distance diagram is viewedon a high-resolu­ tiongraphics terminal with a cursorball, detailed information is instantlyavailable. The horizontalcursor shows time informationfor any positionalong a route or commonsection. Every point where a trajectorycrosses this cursorcan be labeled withthe timea vehiclepassed or is projectedto pass,which run number,vehicle, crew,etc. The vertical cursor provides position information at anyparticular time. Everypoint where the cursorcrosses a trajectorycan be labeledwith the position coordinates(the precisionof displaydepends upon the locatingtechnology and updaterate), run number,vehicle, distance separation from leading and following vehicles,etc. If thereis morethan one vehicle at a terminal,this is alsoreadily shown automaticallywith more than one label. Dependingupon the ITS technologiesin use, additionalinformation about vehicleloading on a particularrun, vehicle condition and other system attributes also can be instantlyavailable. Thus, a graphicalschedule is a meansnot onlyto track vehicles,but to monitorthe entireoperational status along a lineor in a network,if desired.The data also can be stored,either temporarily or permanently.In the caseof an incidenta recordis available,analogous to air-trafficcontrol tapes or blackboxes on airliners.Interesting records can be usedfor training pwposes by replayingthem to investigatecontrol decisions that were taken.

Vol.2, No. 2. I 999 Journal of Public Transportation 21

ScheduleRecovery Strategy Design A numericalformat provideslittle assistancein designinga delay recovery strategy.A GIS displaymap on a computerterminal provides some visualization of the situation,but it doesnot provide an accuratepicture of headwaybetween vehicles as it showsdistance between vehicles. Distance is not necessarilyproportional to headwayin urbansituations where speeds vary widely over the courseof a route.By _ comparison,an experienceddispatcher can quickly assess the statusof an entirefleet operatingon a route,or assessan entiresector in the casewhere dispatching respon­ sibilityis dividedby regioninstead ofby route. Severalrecovery techniques can be testedon the time-distancediagram, alone or in combination.Ideally, likely delay scenarios should be studiedbeforehand so dispatcherscan be trainedin effectivestrategies, but it is also possibleto investigate alternativesin real-time,at thecost of a fewminutes of elapsedtime ( and the riskthat the situationmay deterioratefurther). The basic recoverytechnique is to simplydelay the dispatchingtime of a fol­ lowingTU, which is done by "sliding"an entiretrajectory. Another technique is to extendthe dwelltime at a station.By extendinga dwelltime, part of a trajectorycan be shiftedto representholding a TU at a stationto restore its headwayseparation from the first delayedone. In turn, the next TU can be held at another station to restoreits headwayseparation. This process is repeateduntil scheduled headways are re-established.An exampleof a recoverystrategy involving holding two TUs follow­ ing delayedones is shownin Figure12. Other techniques include improvised short­ turns,express running, and insertionof extraTVs. These concepts were already dis­ cussed,so it suffices to say that these techniquescan also be readilytested on the diagramas recoverystrategies as wellas basicscheduling strategies. Real-TimeControl Once the capabilityfor oversightand for design of recoverystrategies is in place,the capabilityis almostin handto activelycontrol operations. Instructions can, of course,be sentmanually through verbal, either oral or written,messages. But it is alsopossible to automatethe instructions,thereby shortening the responsetime and reducingthe layersof supervisionrequired. An examplefollows.

Vol.2, No. 2, 1999 22 Journal of Public Transportation

Distance

Time Figure12. Schedule recovery by delayingtwo followingtrains.

Assumethat a tentativeschedule recovery strategy has been tried on the com­ puterscreen by clickingand dragging the dwell time of a particulartrain to prolong its stayat a station.The result is a tentativerevised trajectory. A verificationprompt couldask if the dispatcherwould like to implementthis plan. If answeredin the affirmative,the command can automatically be issuedto thein-vehicle control panel as instructionsto the operatorin the caseof manuallyoperated vehicles, or to the AutomaticTrain Operation (ATO) unit in thecase of automaticoperation. Further-

Vol. 2, No. 2, 1999 Journal of Public Transportation 23 more,the delayinfonnation can be automaticallyforwarded to the PassengerInfor­ mationSystem and to the controlcenters for connectingmodes. In turn, decisions canbe automaticallymade with preset criteria whether to holdor dispatchconnect­ ing services,if desired.

Conclusions Graphicalscheduling is an oldtechnique that has beenneglected, or neverac­ quired,in manyNorth American transit agencies. Introduction of diversifiedand complextransit network operations in manycities creates potential for increasing applicationsof graphicalmethods in developingschedules, as well as in operations control.Actually, with modern computer graphics and ITStechnologies it is more powerfulthan ever and deserves consideration where it is not currentlybeing used. However,the latestdispatching/control software offered by transitITS vendorsin the U.S.does not takeadvantage of thisapproach. Graphicalscheduling retains its advantages in basicschedule design and analy­ sis. It allowsthe solutionof problemsthat are quitedifficult to solveanalytically. Eveninformation about simple routes is enhancedby the detailedoperating charac­ teristicsinherent in detailedvehicle trajectories and by the relativeease with which acceleratedservice and service recovery strategies can be investigated.It alsocan be usedto confirmand refine solutions that are generatedby analyticmethods. Graphicalscheduling has additional advantages in operationalcontrol with the adventof modernITS technologies. By movement of thecursor on a terminalscreen, detailedinfonnation about any position along a route,or aboutall activityalong a routeat anyparticular moment, becomes available. An experienceddispatcher can gaina perspectiveon an entireroute or geographicsector and anticipate problems at an earlystage. In addition,it is possibleto linkthe alteringof trajectoriesthrough clickingand dragging to theautomatic issuance of controlcommands and updates of passengerinfonnation.

References Bruun,Eric C., andP. Takis Salpeas. 1991. Train operations computer simulation case study: Single-trackingoperations for Philadelphia sMruket-Frankford Subway Elevated Rail Rapid

Vol.2, No. 2, 1999 24 Journal of Public Transportation

TransitLine. Transportation Research Record 1308, Transportation Research Board, Wash­ ington,D.C. Vuchic,Vukan R 1976.Skip-stop operation: High speed with good area coverage. UITP Revue No. 2, InternationalUnion ofPublic Transport, Brussels, Belgium. Vuchic,Vukan R 1981.Urban public transportation systems and technology. Englewood Cliffs, NJ:Prentice Hall. Vuchic,V. R, E.C. BnlWl, andN. Krstanoski. 1995-97. Schedule analysis of the BART system with SFIA-MillbraeExtensions, and other reports to BayArea Rapid Transit District. Oakland, CA: unpublished.

Aboutthe Authors ErucC. BRUUN isAssistant Director for Advanced Technologies and Innovative Practicesat theNational Transit Institute, Rutgers University. He received his Ph.D. in SystemsEngineering (Transportation focus) from the University of Pennsylvania in 1992. VUKANR. Vucmcis UPS Foundation Professor off ransportationEngineering at theUniversity of Pennsylvania.He received his Ph.D. in TransportationEngineering fromthe Universityof Californiaat Berkeleyin 1966. YoNG-EUNSHIN is Assistant Professor at Dong-EuiUniversity in Pusan,Korea. Hereceived his Ph.D.in Cityand Regional Planning from the University of Pennsyl­ vaniain 1997.

Vol.2, No. 2, 1999