MACRO DESIGN MODELS FOR A SINGLE ROUTEIntroduction to Analysis ApproachBus Frequency Model: the Square Root ModelSquare Root Model (cont’d) Square Root Model (cont’d) Square Root Model (cont’d) Bus Frequency ExampleStop/Station Spacing ModelStop/Station Spacing Model (cont’d)Bus Stop SpacingStop Spacing TradeoffsWalk Access: Block-Level ModelingResults: MBTA Route 39*Nigel H.M. Wilson 1.201, Fall 2006 1 Lecture 12MACRO DESIGN MODELSFOR A SINGLE ROUTEOutline1. Introduction to analysis approach2. Bus frequency model3. Stop/station spacing modelNigel H.M. Wilson 1.201, Fall 2006 2 Lecture 12Introduction to Analysis Approach• Basic approach is to establish an aggregate total cost function including:• operator cost as f(design parameters)• user cost as g(design parameters)• Minimize total cost function to determine optimal design parameter (s.t. constraints)Variants include:• Maximize service quality s.t. budget constraint• Maximize consumer surplus s.t. budget constraintTotal CostUser CosthOPTHeadwayOperator CostCostNigel H.M. Wilson 1.201, Fall 2006 3Lecture 12Bus Frequency Model:the Square Root ModelProblem: define bus service frequency on a route as a function of ridershipTotal Cost = operator cost + user costNigel H.M. Wilson 1.201, Fall 2006 4Lecture 12Square Root Model (cont’d) This is the Square Rule with the followingimplications:• high frequency is appropriate where (cost of wait time/cost of operations time) is high• frequency is proportional to the square root of ridership per unit time for routes of similar lengthRidershipFrequencyCapacityConstant Load FactorFrequency-Ridership RelationshipNigel H.M. Wilson 1.201, Fall 2006 5Lecture 12Square Root Model (cont’d)• load factor is proportional to the square root of the product of ridership and route length.RidershipPassengers/ busBus Capacity-Ridership RelationshipBus Capacityt1t2t1>t2Load Factor >1Nigel H.M. Wilson 1.201, Fall 2006 6Lecture 12Square Root Model (cont’d) Critical Assumptions:• bus capacity is never binding• wait time savings are only frequency benefits• ridership ≠ f (frequency)• simple wait time model• budget constraint is not bindingPossible Remedies:• introduce bus capacity constraint• modify objective function• introduce r=f(h) and re-define objective function• modify objective function• introduce budget constraintNigel H.M. Wilson 1.201, Fall 2006 7Lecture 12Bus Frequency ExampleIf: c = $90/bus hour, b = $10/passenger hour.t = 90 mins, r = 1000 passengers/hour,Then: hOPT= 11 minsNigel H.M. Wilson 1.201, Fall 2006 8Lecture 12Stop/Station Spacing ModelProblem: determine optimal stop or station spacingTrade-off is between walk access time (which increases with station spacing), and in-vehicle time (which decreases as station spacing increases) for the user, and operating cost (which decreases as station spacing increases)Define Z = total cost per unit distance along route and per headwayand Tst= time lost by vehicle making a stopc = vehicle operating cost per unit times = station/stop spacing - the decision variable to be determinedN = number of passengers on board vehiclev = value of passenger in-vehicle timeD = demand density in passenger per unit route length per headwayvacc= value of passenger access timew = walk speedcs= station/stop cost per headwayNigel H.M. Wilson 1.201, Fall 2006 9Lecture 12Stop/Station Spacing Model (cont’d)Yet another square root relationship, implying that station/stop spacing increases with:• walk speed• station/stop cost• time lost per stop• vehicle operating cost• number of passengers on board vehicle• value of in-vehicle timeand decreases with:• demand density• value of access timeNigel H.M. Wilson 1.201, Fall 2006 10Lecture 12Bus Stop SpacingU.S. Practice• 200 m between stops (8 per mile)• shelters are rare• little or no schedule informationEuropean Practice• 320 m between stops (5 per mile)• named & sheltered• up to date schedule information• scheduled time for every stopNigel H.M. Wilson 1.201, Fall 2006 11Lecture 12Stop Spacing Tradeoffs• Walking time• Riding time• Operating cost• Ride qualityStop spacing (m)Operator + User Costextra walk timeextra riding timeextra operating costtotal extra costNigel H.M. Wilson 1.201, Fall 2006 12Lecture 12Walk Access: Block-Level ModelingShed LineMain Street with Existing StopsabFigure by MIT OCW.Nigel H.M. Wilson 1.201, Fall 2006 13Lecture 12Results: MBTA Route 39*AM Peak Inbound results•Avg walking time up 40 s•Avg riding time down 110 s•Running time down 4.2 min•Save 1, maybe 2 busesSource: Furth, P.G. and A. B. Rahbee, “Optimal Bus Stop Spacing Using Dynamic Programming and Geographic Modeling." Transportation Research Record 1731, pp. 15-22, 2000.10IM8/SP6OT4S2SHUYLTNALAIBEH BHFSCALE0 0.5 1MIExisting stopDiscrete model optimal stopExisting routeContinuous model optimumMBTA guidelineDiscrete model optimumFigure by MIT
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