Threshold- and information-based holding at multiple stops

www.ietdl.org Published in IET Intelligent Transport Systems Received on 28th August 2008 Revised on 1st May 2009 doi: 10.1049/iet-its.2008.0072 Special Issue – selected papers from AATT 2008 ISSN 1751-956X Threshold- and information-based holding at multiple stops G. Bellei K. Gkoumas Dipartimento di Idraulica Trasporti e Strade, Sapienza Università di Roma, Via Eudossiana, 18-00184 Rome, Italy E-mail: [email protected] Abstract: The objective of this study is the improvement in speed and regularity of transit systems using real-time control strategies, in particular, threshold-based and information-based vehicle holding. This objective is attained by taking into account the inherent uncertainty of transit operation, because of the random travel times and passenger arrivals at stops. A Monte Carlo simulation model of a single transit line is presented, with explicit representation of traffic lights. Both the straightforward threshold-based holding strategy and the strategy based on the availability of real-time information at stops, to take holding decision at single or multiple stops, are represented and compared, taking as a reference the results obtained by a conditional priority strategy at intersections, assumed to be given only by green extension actuated by local sensors. The results are evaluated and compared using performance indicators, coincident with waiting and on board time of transit users and with road traffic delay on transversal roads. 1 Introduction The paper focuses on intermediate capacity transit systems, a key factor in establishing a viable alternative to private car in medium-sized towns and in complementing high-capacity rapid transit in larger urban areas. The strategies aiming at improving transit performance have to be tested within the framework of an operation model, where perturbation formation and diffusion phenomena are duly represented. The impact of such phenomena on waiting times results in an increase of the headway variance, thus the regularity of operation can be improved by headway control through vehicle holding. Transit speed can be increased, while helping to keep operation regular, by applying conditional transit priority strategies at traffic lights. Conditional priority plays a role similar to vehicle holding since vehicles to which priority is denied are delayed with respect to others, so it is therefore natural, when considering different holding strategies, to compare them also with conditional priority. The modelling approach adopted in this study involves Monte Carlo simulation, implemented by repeatedly drawing the outcome of the main random phenomena, such as the arrival of passengers at stops, the alighting from transit vehicles and the running time between stops, which are given as input to the operation 304 & The Institution of Engineering and Technology 2009 model, to obtain a random drawing of the whole operation pattern. Consequently, the average performance of different holding strategies can be evaluated. To this aim, the traffic delay because of transit priority is computed as well, on the basis of a deterministic representation of queuing at intersections. In addition to a standard threshold-based holding strategy, a strategy aiming at equalising the headway of the controlled vehicle to that of the next one is represented. This strategy is implemented by an intelligent transportation system, since it is based on real-time information to forecast future events and take holding decision. 2 Literature review This study principally utilises premises from the field of operation models. A review of transit-related issues at various planning levels can be found in the survey paper by Desaulniers and Hickman [1]. The same authors also give a general review of control methods applied in the presence of irregularity, even if, as they state, ‘in normal service with only minor perturbations from the schedule and small service disruptions, vehicle holding and transit signal priority are the most common techniques that are applied’. Several transit line operation models have been studied in order to apply such IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304– 313 doi: 10.1049/iet-its.2008.0072 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org control methods. Eberlein et al. [2] formulated a deterministic operation model, utilising it to optimise holding within a rolling horizon approach. In their paper, the decision to hold a vehicle is taken by minimising the departure headway variance on a vehicle set, on the basis of automatic vehicle location (AVL) data, and the operation model is formulated as a set of constraints of the minimisation problem to which a schedule constraint is added. Hickman [3], on the basis of a stochastic operation model, formulated an analytic model to determine the optimal holding time for a single transit vehicle, taking into account its impact on a number of upstream uncontrolled vehicles. The holding problem is thus formulated as a convex quadratic program on a single variable, having as objective the weighted sum of waiting time and holding delay. Sun and Hickman [4] extended the problem formulation to consider holding vehicles at a given subset of stations along the line, in the context of a deterministic operation model. Fu and Yang [5] implemented a holding strategy in which holding times are determined on the basis of both the preceding and the following headway, the latter predicted considering average values for the travel and dwell time, and compared the results with a threshold-based technique. Conditional transit priority strategies at traffic lights, that is to allow only selected vehicles to get priority, can increase transit speed while keeping operation regular. Sophisticated priority systems, like the one implemented within the UTOPIA project [6], allow for conditional priority, and take into account the effect of transit priority on arterial progression by adopting a rolling horizon approach. Nevertheless, no explicit reference to operation models is made in [6] in order to forecast the transit vehicle arrival times at traffic lights. Furth and Muller [7] tested a conditional priority strategy at the busiest intersection along a mixed traffic bus route, compared the results with an absolute priority strategy (where every vehicle, and not only those behind schedule, gets priority) and formulated additional considerations regarding the impact to the general traffic. In a more recent study by Kim et al. [8], the best conditional priority strategy for bus routes is determined on the basis of empirical forecasts, based on the ratio of headway delay, making reference to most widespread and simplest strategies and utilising the PARAMICS traffic micro simulation software. The application of conditional priority requires the location of the vehicles along the route. Hounsell and Shrestha [9] reviewed various architectures for AVL-based bus priority currently implemented in Europe, with explicit reference to the adopted criteria for conditional priority and to the priority request methods. A global positioning system (GPS) for a bus location has been adopted recently by transit authorities, for having the advantage of flexibility and reduced costs over fixed on-street hardware. Its accuracy has been tested by Hounsell et al. [10]. The operation model utilised in this study includes a dwell time model linear in the number of alighting and boarding passengers, as in [2, 3]. Numerical results have been obtained by assuming the same dwell time model coefficients as in [2], taken from the study by Lin and Wilson [11], where different dwell time models are IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304 – 313 doi: 10.1049/iet-its.2008.0072 identified on a grade-separated section common to several light rail lines, such that, although the right of way and operation framework are clearly different from the one we focus on here, the vehicles size and layout are similar. 3 Transit line operation model The operation model used to investigate different holding methods and to represent conditional priority results from extending a previous model developed by the authors in [12, 13]. It is a simulation model for a one-way transit line, defined by recursive relationships, similar to the equations of the analytic model in [3], and including random number drawings corresponding to the line operation simulation within a certain time horizon. With respect to such analytic model, some additional features of transit operation are considered. In fact, it is assumed that transit vehicles have a passenger capacity CV and boarding of passengers arriving during dwell time is taken into account. In addition, traffic lights are explicitly represented, although in a rather simplified form. Since the resulting formulation is somewhat cumbersome, a core operation model is presented first, including vehicle capacity, whereas arrivals during dwell time and traffic lights representation are dealt successively as extensions to such models. Finally, the main features of transit line operation utilised in numerical simulations are specified. 3.1 Core operation model Defining as TAmn and TDmn , respectively, the arrival and departure time of the vehicle m at the stop n, and denoting with Smn the dwell time of the vehicle m at the stop n, it is TDmn ¼ TAmn þ Smn (1) The dwell time may be expressed as a generic function S(Lmn , Bmn , Amn) of the on-board, the boarding and the alighting passengers but, at this stage, a simpler linear model dependent only on Amn and Bmn is utilised Smn ¼ a0 þ a1 Bmn þ a2 Amn (2) Denoting by Tn the running time between stops n 2 1 and n, and by d min the minimum interval between the arrival of a vehicle and the departure of the preceding one, it is TAmn ¼ max{TDm,n1 þ Tn ; TDm1 , n þ d min } (3) The probability density function of the random variable Tn is assumed to be triangular, defined by such exogenous and parameters as the mode tn , the minimum value tnkmin n the maximum value tnkmax n , so that the values taken within a simulation result from the random number drawings T max Tn ¼ T (tn , kmin n , kn ) (4) The passengers on-board Lmn arriving at a stop are the balance between on-board, alighting and boarding 305 & The Institution of Engineering and Technology 2009 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org passengers at the previous stop 3.2.1 Potential arrivals: To calculate the potential (5) Lmn ¼ Lm,n1 þ Bm,n1  Am,n1 The residual capacity after alightings CRmn is defined as CR mn ¼ CV  Lmn þ Amn (6) The passengers Bmn actually boarding vehicle m will be equal to passengers Wmn willing to board (in the core model they coincide with passengers present at the arrival of vehicle m), or to those able to board (the residual capacity), what is the less, thus enforcing a capacity constraint Lmn  CV Bmn ¼ min{Wmn ; CR mn } (7) The difference between passengers willing to board and able to board is the residual queue after departure of vehicle m Rmn ¼ Wmn  Bmn (8) Passengers in the residual queue after departure of vehicle m 2 1 are present at the arrival of vehicle m, together with the passengers Pmn arriving between the departure of vehicle m 2 1 and the arrival of vehicle m Wmn ¼ Rm1,n þ Pmn (9) The distributions of the random variables Amn are assumed to be binomial, whereas the Pmn are assumed to be Poisson. The parameters of the binomial distributions are the number Lmn of on-board passengers arriving at stop n on vehicle m (number of Bernoulli trials) and the average fraction an of alighting passengers (trial success probability), whereas the only parameter of Poisson distribution is the average number of passengers arriving at stop n within the given interval (expected number of events occurring during a fixed time period), that is, the product of the average flow bn times (TAmn 2 TDm21,n). The values taken within a simulation result from the random number drawings A and P (a) Amn ¼ A(an Lmn ); (b) Pmn ¼ P[bn (TAmn  TDm1,n )] (10) Equations (1)–(3) and (5)–(9) and the random numbers drawings (4) and (10) define a set of recursive calculations. Performing these calculations for m ¼ 2, . . . , M and n ¼ 1, . . . , N, and setting appropriate boundary conditions for m ¼ 1 and n ¼ 0, a simulation is obtained for the operation of M trips on a transit line with N þ 1 stops, terminals included. 3.2 Arrivals during dwell time Defining passengers willing to board as in (9) does not take into account passengers arriving during dwell time. Taking into account such passengers requires calculating their potential values in order to determine actual values consistent with vehicles residual capacity. 306 & The Institution of Engineering and Technology 2009 arrivals, the model formulation has to be modified to express passengers willing to board a vehicle m at stop n as the sum of those present at stop n at its arrival, denoted by Qmn , and those arriving during the dwell time, denoted by Dmn Wmn ¼ Qmn þ Dmn (11) Equally, Smn is the sum of components Umn (dwell time for the alighting of passengers Amn and the boarding of as many passengers Qmn as allowed by the capacity constraint) and Vmn (additional dwell time for the boarding of as many passengers Dmn arriving during the dwell time as allowed by the capacity constraint) Smn ¼ Umn þ Vmn (12) The interdependence between the number Dmn of passengers arriving during dwell time and its duration Smn complicates somehow the evaluation of passenger arrivals during dwell time. This problem is addressed calculating passenger and dwell time components by generalising the deterministic approach adopted by Vuchic [14], who assumes that dwell times are merely proportional to boardings. The basis for calculation is given by Qmn , which is independent of capacity and takes the place of Wmn in (9), which is thus substituted by Qmn ¼ Rm1,n þ Pmn (13) The arrivals during dwell time and the dwell time components, instead, are calculated by first determining their potential values, attained when boardings are not constrained by capacity, to obtain actual values from Qmn and the capacity constraint. The linear dwell time model (2) is applied to calculate the P potential dwell time Umn , because of alighting passengers and to passengers present at stop n at the arrival of vehicle m, assuming that they all succeed to board P Umn ¼ a0 þ a1 Qmn þ a2 Amn (14) Assuming an average passenger arrival rate at the stop, the P P would be bn Umn . Additional dwell time arrivals during Umn P because of these arrivals is, consistently with (14), a1 bn Umn . On the other hand, arrivals during this additional dwell P , additional dwell time because of these time are a1 b2n Umn 2 2 P arrivals is a1 bn Umn and so on. If a1bn , 1 the estimate of P the potential dwell time Vmn , because of arrivals during dwell time, is P Vmn ¼ P a1 bn Umn 1  a1 bn (15) IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304– 313 doi: 10.1049/iet-its.2008.0072 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org P whereas potential arrivals during dwell time Dmn , consistent with (14) and (15) are P ¼ Dmn P bn Umn VP ¼ mn 1  a1 b n a1 (16) 3.2.2 Actual arrivals: With regard to actual arrivals, only the conclusions of the analysis are reported for the sake of brevity. The actual components of arrivals and dwell times are derived from potential ones, considering three cases for the residual capacity. P the actual arrivals during Case 1: If CR mn  Qmn þ Dmn dwell time and the dwell time components are equal to the potential ones P ; (a) Dmn ¼ Dmn P (b) Umn ¼ Umn ; P (c) Vmn ¼ Vmn (17) P . CR mn  Qmn all passengers present Case 2: If Qmn þ Dmn at the vehicle arrival will board, and so will do the passengers arriving during dwell time, until vehicle capacity is reached and the vehicle departs, thus the number of passengers arriving during dwell time who board is determined by passengers present at the vehicle arrival and residual capacity. The dwell time because of the alighting passengers and to the boarding of passengers present at the vehicle arrival is equal to the potential one, whereas the additional dwell time, because of the passengers arriving during dwell time is as much as needed to board by these passengers consistently with dwell time model (2) P ; (c) Vmn ¼ a1 Dmn (a) Dmn ¼ CR mn  Qmn ; (b) Umn ¼ Umn (18) Case 3: If CRmn , Qmn , the number of passengers present at the vehicle arrival who actually board is equal to residual capacity CRmn . The corresponding dwell time Umn is determined by model (2) with such boardings as argument. No additional dwell time is determined by passengers arriving during dwell time, since none of them can board, whereas their number is determined from the dwell time and the average passenger arrival rate (a) Dmn ¼ bn Umn ; (b) Umn ¼ a0 þ a1 CR mn þ a2 Amn ; (c) Vmn ¼ 0 (19) The operation model formulation with boarding of passengers arriving during dwell time implies substituting (13) into (9), and adding to (1), (8) and (10) the equations: † (11) and (12) representing segmentation of passengers willing to board and dwell times into arrival and dwell time components; † (13), (14) and (15) defining potential values of such components; IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304 – 313 doi: 10.1049/iet-its.2008.0072 † (17), (18) and (19) allowing the calculation of correspondent actual values; having also excluded (2) since redundant. 3.3 Traffic lights Traffic lights are represented within the operation model by assuming that there is a one-to-one correspondence among stops, traffic lights and intersections, being each stop and traffic light located immediately upstream the intersection. Moreover, the same fixed cycle C and green g are defined at each traffic light, the offsets are all zero and lost times are neglected. The departure time TDmn is equal to the sum of the arrival and the dwell time as calculated in Section 3.2.2, only if this sum, which in this context has the meaning of time TRmn when the vehicle is ready to depart TR mn ¼ TAmn þ Smn (20) falls within green time, being otherwise delayed to the beginning of the next green. The cycle when vehicle m is ready to depart from stop n is identified by kmn such that (kmn 2 1)C  TRmn , kmnC, and can be expressed as a function of TRmn and C, by kmn  TR mn ¼ C  (21) where [x] denotes integer part of x, whereas the departure time is given by TDmn ¼ TR mn TDmn ¼ kmn C if (kmn  1)C  TR mn , kmn C þ g if (kmn  1)C þ g  TR mn , kmn C (22) It is assumed that boarding continues while transit vehicle m is waiting for green at traffic light located at stop n, because of passengers arriving at rate bn as long as it is allowed by the capacity constraint. Such passengers give rise to a third component of passengers willing to board Tmn , those arriving in the interval from the time the vehicle is ready to depart to actual departure Tmn ¼ bn (TDmn  TR mn ) (23) so that the model formulation with traffic lights implies addition of (20) and substitution of (1) by (21) and (22), whereas (11) is substituted by Wmn ¼ Qmn þ Dmn þ Tmn (24) with Tmn given by (23). 307 & The Institution of Engineering and Technology 2009 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org 3.4 Transit line operation simulated by the model Reference is made to a long, high-frequency virtual light rail line, whose passenger flow in the most loaded section is close to the capacity. The supply parameters are the same for all the sections between stops and the average boardings and alightings are the same for all vehicles. From stop 0, the initial terminal, M ¼ 21 trips are dispatched at a regular headway h8 ¼ 180 s. The passenger boarding and alighting rates are defined in such a way to have, taking a vehicle capacity CV ¼ 300 passengers, a maximum load factor g ¼ 0.9 on the maximum load section. These boarding and alighting rates at stops and the line loading pattern (onboard passengers at the arrival) are shown in Fig. 1. The demand varies as in an urban diametrical line, passing through and terminating at transfer nodes, where a large number of passengers board and alight. This demand resulting from g, CV and h8 values adopted, corresponds to an average flow of 3600gCV/h8 ¼ 5400 passengers/h on the maximum load section, to be compared with a 3600CV/h8 ¼ 6000 places/h line capacity. The number of stops is 31 including the dispatching terminal, that is to say N ¼ 30. A running time t ¼ 40 s is assumed between stops, representing the mode of the triangular probability density function for the running time Tn , with k min ¼ 0.9 and k max ¼ 1.2. Finally, the coefficients of the linear dwell time model implemented, as identified by a survey in [11], are a0 ¼ 11.73, a1 ¼ 0.42 and a2 ¼ 0.49. The random numbers drawing is performed by Zrandom (available online on 04/2009 at www.zrandom.com) software, whereas the operations needed to evaluate transit line performance indexes are implemented in an Excel worksheet. 4 Definition of the real-time control strategies The real-time control methods considered are holding at one or more stops along the line, both adopting a threshold-based and an information-based strategy. Conditional priority is also represented. 4.1 Vehicle-holding strategies The vehicle-holding strategies represented are proposed in two different ways: a first one considering, with respect to the vehicle candidate to be held, only the preceding vehicle, and a second, taking also into account the following vehicle. In the second case, a prediction model is proposed for the estimation of the headway of the following vehicle at the control stop. 4.1.1 Threshold-based holding strategy: A simple holding strategy is formulated by redefining the ready time TRmn in order to restore the threshold value hH min of the headway with respect to such time, when it would not be respected by simply taking TRmn equal to the sum of arrival and dwell time, as in (20) H TRmn ¼ TR m1,n þ hH min if TAmn þ Smn  TR m1,n , hmin TRmn ¼ TAmn þ Smn if TAmn þ Smn  TR m1,n hH min (25) It is worth noting that implementing such control strategy requires the control system to be informed that vehicle is ready to depart. The operation model with threshold-based holding strategy is obtained by substituting (25) into (21) and (22) of the standard traffic light model. 4.1.2 Holding strategy with real-time information: The holding strategy with the use of real-time information consists in modifying the ready time (TRmn) of the vehicle candidate to be held in such a way that headway variance is reduced. The headways with respect to the ready time of the vehicle candidate to be held and of the next one are thus equalised. The first headway, HR N mn ¼ TAmn þ Smn  TR m1,n , is observed if the driver requests permission to depart at time TAmn þ Smn , whereas the latter, HR Emþ1,n , is estimated on the basis of operation data made available by an AVL system. Figure 1 Boarding and alighting passengers and loading pattern at stops 308 & The Institution of Engineering and Technology 2009 IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304– 313 doi: 10.1049/iet-its.2008.0072 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org Naturally enough, only holding is possible in order to equalise headways, thus the ready time TRmn can only be increased with respect to TAmn þ Smn . an estimated ready time, what green should be eventually extended to reduce the delay caused to vehicle m by the traffic light in correspondence to stop n. If we denote by h a threshold value, in the present analysis equal to 5 s, introduced in order to avoid holding a vehicle for just a few seconds, it is The first potentially useful green extension for a vehicle m arriving at stop n at time TAmn , calculated from estimated ready time, belongs to cycle k8mn satisfying E TR mn ¼ TR m1,n þ 0:5(HR N mn þ HR mþ1,n ) if (k8mn  2)C þ g þ Dmax  TAmn þ SnE , (k8mn  1)C þ g þ Dmax (28) E TRm1,n þ 0:5(HR N mn þ HR mþ1,n )  TAmn þ Smn þ h TR ¼ TAmn þ Smn if whereas the potentially useful green extension for such a vehicle, according to actual ready time TRmn , belongs to cycle kmn satisfying E TRm1,n þ 0:5(HR N mn þ HR mþ1,n ) , TAmn þ Smn þ h (26) The prediction of the headway HR Emþ1,n is carried out implementing a linear regression, on simulated data, considering four independent variables, namely: HRN mn 1. the headway with respect to ready time, in case of no holding, of the vehicle m candidate to holding at the control stop n; 2. the headway with respect to ready time HRmþ1;n ¼ TRmþ1,n 2 TRmn of the following vehicle m þ 1 at the stop n where its last ready time is known; (kmn  2)C þ g þ Dmax  TR mn , (kmn  1)C þ g þ Dmax (29) Considering the cycle where a potential useful green extension can occur leads to a traffic light model formulation, which is a generalisation of (22), although it is formally different since the cycle it refers to is a different one. To derive such formulation, we observe that (28) is equivalent to k8mn  2  TAmn þ SnE  g  Dmax C , k8mn  1 (30) S ¼ n  n between the control 3. the number of stops Dnn  stop n and the stop n ; thus k8mn is given taking the integer part of the ratio in (30) T ¼ TAmn þ Smn  TR mþ1,n between 4. the difference Dmnn the control time and the ready time of the following vehicle at stop n . " # TAmn þ SnE  g  Dmax þ2 k8mn ¼ C With a, b, c, d and e denoting coefficients, the function estimated by the regression is S T HR Emþ1,n ¼ a þ bHR N mn þ cHR mþ1,n þ dDnn þ eDmnn  and an analogous expression can be derived for kmn kmn ¼ (27) The operation model with information-based holding strategy is obtained by substituting (26), with HR Emþ1,n given by (27), into (21) and (22) of the standard traffic light operation model.   TR mn  g  Dmax þ2 C Priority at each traffic light is defined by the maximum green extension Dmax. If the first cycle begins at a conventional 0 time, the kth cycle begins at time (k 2 1)C and the kth green, ending at (k 2 1)C þ g, can be extended up to (k 2 1)C þ g þ Dmax if a transit vehicle may take advantage of such extension. The arrival time TAmn of vehicle m at stop/traffic light n is assumed to be known by means of a local presence sensor and an estimated dwell time SnE at stop n is taken into account to determine, on the basis of IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304 – 313 doi: 10.1049/iet-its.2008.0072 (32) Taking into account that green extension can be given only if the corresponding green is not already finished at vehicle’s arrival time, that is if TAmn , (k8mn  1)C þ g 4.2 Conditional priority at traffic lights (31) (33) the departure time with green extension is given, if condition (33) is satisfied, arrival headway TAmn 2 TAm21,n is higher than a given threshold hPmin , and it is k8mn ¼ kmn , by TDmn ¼ (k8mn  1)C if (k8mn  2)C þ g þ Dmax  TR mn , (k8mn  1)C TDmn ¼ TR mn if (k8mn  1)C  TRmn , (k8mn  1)C þ g þ Dmax (34) 309 & The Institution of Engineering and Technology 2009 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org otherwise green extension is not given, or it is given within a cycle where it is not needed, and departure time is given by sum of TT and WT, and total system time (ST) as the sum of PT and TD. TDmn ¼ (kmn  1)C It is assumed that only two kinds of road traffic flow exist: one on the road along the line and the other traversing the line at intersections. Delay of road traffic flow along the line axis is not taken into account, since, without green share adjustment, green for this flow may only increase. Neglecting such delay implies, in general, underestimation of priority benefits, while it is consistent with a compact platoon assumption for road traffic flow along the line axis. (kmn TDmn  2)C þ g þ D ¼ TR mn if if max  TR mn , (kmn  1)C (kmn  1)C  TR mn , (kmn  1)C þ g TDmn ¼ kmn C if (kmn  1)C þ g  TR mn , (kmn  1)C þ g þ Dmax (35) The estimated dwell time SnE is computed by applying average boardings and alightings at stop n to the linear dwell time model (2). The green extensions determined by the priority strategy, each time a traffic light controller is alerted to extend the green at cycle k8mn until the transit vehicle present at stop departs, can be calculated as follows: TT is calculated with considerations on the traffic load and the travel times and is given by X TT ¼ † for kmn . k8mn , the transit vehicle gets no priority, maximum green extension is given, but not utilised since the vehicle is not yet ready to depart when the extension expires; † for kmn ¼ k8mn , a green extension is given and utilised if (k8mn  1)C þ g  TR mn , (k8mn  1)C þ g þ Dmax ; the extension’s length depends on TRmn ¼ TDmn . þ kmn , k8mn if Dkn ¼ 0; k ¼ k8mn kmn . k8mn if Dkn ¼ Dmax ; k ¼ k8mn kmn ¼ k8mn if Dkn ¼ max{0; TPmn  (k  1)C þ g}; k ¼ k8mn (36) Although (31) – (35) are part of the operation model representing priority, substituting (21) and (22) of the standard traffic light model, (36) is not part of the operation model and it is utilised only to compute the road traffic delay consequent to the implementation of transit priority. The whole process is explained in a more comprehensive manner in [15]. 5 Performance indicators The performance indicators considered are the passenger travel time (TT), passenger waiting time (WT) and the delay caused by the traffic lights to the road traffic flow (TD). Then, total passenger time (PT) is defined as the 310 & The Institution of Engineering and Technology 2009 Lmn (TAmn  TPm,n1 ) X (Lmn  Amn )(TPmn  TAmn ) n¼1,N 1 # (37) WT is calculated by measuring the area between cumulate arrival and cumulate departure curves, and is given by WT ¼ X X {P[bn (TAmn  TPm1,n )] m¼2,M n¼1,N 1  (TPmn  TAmn ) þ Rm1,n (TPmn  TPm1,n ) þ P[bn (TAmn  TPm1,n )] þ Denoting by Dkn the green extension determined by priority given to vehicle m at stop n within cycle k ¼ k8mn , it is, summarising the three cases above X m¼2,M n¼1,N kmn , k8mn , the transit vehicle gets no priority, no green † for extension is actually given because the vehicle departs, cancelling the request for extension before it begins; " (TAmn  TPm1,n ) 2 bn (TPmn  TAmn )2 2 (38) The delay of passengers in the residual queues Rmn is taken into account by the sum of the products of such queues and the average departure headway at each stop, which is added to (38). Finally, assuming a constant arrival rate a and a saturation flow s, and representing a medium-high road congestion (degree of saturation 0.85), TD is given by the area between cumulate vehicle arrivals and departures at the intersection, evaluated as follows TD ¼ P n¼1,N 1 þ min ( P k¼1, K gkn (Qkn þ QRkn ) 2 QRkn (s  a), C  gkn ) (Qkþ1,n þ QRkn ) 2 (39) as a function of the extended green times gkn ¼ g þ Dkn determined by priority, Dkn supplied by (36), and of the queues on transversal roads Qkn and QRkn , in correspondence to the beginning of the green and of the red, respectively, with reference to the line axis traffic light, for each cycle k IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304– 313 doi: 10.1049/iet-its.2008.0072 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org and stop n. The queues are recursively calculated from s, a, C and gkn . The formulation of recursive calculation and the derivation of (39) from geometrical considerations can be found in [15]. of determination) increases from an initial value of 0.25 at stops 4 and 5 (making prediction rather tricky) to 0.55 at stop 6, and it reaches a value of 0.81 at stop 9, of 0.90 at stop 11 and of 0.95 or more at stops 16 and beyond (where predicted times are very close to the simulated times). 6 6.2 Multiple-stop holding control Numerical results Initially, the effect of holding control at a single stop is inquired, implementing the threshold-based and the information-based techniques. Then holding control at multiple stops is considered to decide on whether the addition of a second, or third, control stop has a positive effect. Finally, in order to evaluate the variation among the results obtained in different instances with the best threshold and information-based holding strategies, these results are compared with those obtained by implementing the conditional priority strategy. 6.1 Single-stop holding control In the single-stop holding control, threshold-based and information-based control is performed at all stops along the line, one at a time, as explained in Section 4.1. The information-based technique is applied only from the fourth stop on, since the prediction model cannot be applied to the first three stops. Regarding PT, the information-based technique gives better results at stops 11– 20, whereas the global best is obtained with the threshold-based technique at the 10th stop, with a value only slightly better than the one obtained with the information-based technique at stop 15 (Fig. 2). It is worth noting that the threshold-based holding strategy gives better results for WT at almost every stop, whereas the information-based one is better, at least for the central stops, on the front of TT. The implementation of the multiple-stop holding control is similar to the one of single-stop control; however, in this case, threshold-based and information-based controls are performed at selected combinations of stops along the line. In Fig. 3, with reference to the reduction of PT, results are presented implementing threshold-based and informationbased holding at selected combinations of two stops (stops 10 and 15, 10 and 19, 15 and 19) and three stops (stops 10, 15 and 19). These are compared with the results of the single holding control (at stops 10, 15 and 19). In all cases of multiple-stop holding control, the information-based technique gives slightly better results. Adding a second holding stop, the improvement is more evident for the information-based technique. Although the best solution found is the one with three stops, the improvement of adding a third stop is less marked. 6.3 Representation of conditional priority In order to obtain the optimal parameters for the conditional priority strategy, five different values for the green extension (Dmax) are considered, namely 0 (no extension), 8, 12, 16 and 20 s, the latter in order to check for a limit case, being unrealistic for a 60 s cycle. Non-zero Dmax cases are combined with seven hPmin values, which vary from 60 to 180 s in 20 s intervals. The results for the ST reduction are reported in Fig. 4. The minimum is achieved with hPmin ¼ 120 s and Dmax ¼ 16 s values. The consistency of the information-based technique is strongly influenced by the prediction model, which is founded on the linear regression. The results of the regression vary at stops along the line: the R 2 (coefficient 6.4 Compared results Figure 2 Total passenger time reduction – single-stop holding control Figure 3 Total passenger time reduction – multiple-stop holding control IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304 – 313 doi: 10.1049/iet-its.2008.0072 The multiple-stop holding control strategies that give the best results are compared with the optimal conditional 311 & The Institution of Engineering and Technology 2009 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org comparable to the one determined by conditional priority, whereas the total system time reduction determined by threshold-based holding is significantly less. Figure 4 Total system time reduction – conditional priority strategy A further operative consideration has to be made about the investment cost of control hardware, which is higher for the more efficient strategies, that is, in this case, informationbased holding and conditional priority. In addition, threshold-based holding, although less efficient, allows non-negligible time reductions. Finally, priority determines management and public acceptance issues. Therefore all the considered control strategies may be suitable for specific applications, and their implementation in different cases (e.g. different transit modes), or in integrated forms, are worth inquiring. 8 [1] References DESAULNIERS G., HICKMAN M.D.: ‘Public transit’, in BARNHART C. , ‘Handbooks in operations research & management science: transportation’ (Elsevier, 2007, 1st edn.), vol. 14, pp. 69 – 127 LAPORTE G. (EDS.): [2] EBERLEIN X.J. , WILSON N.H.M. , BERNSTEIN D.: ‘The holding problem with real-time information available’, Transp. Sci., 2001, 35, pp. 1 – 18 [3] HICKMAN M.D.: ‘An analytic stochastic model for the transit vehicle holding problem’, Transp. Sci., 2001, 35, pp. 215– 237 Figure 5 Time reduction – multiple-stop holding control and conditional priority strategy priority strategy case described in Section 6.3, and the results are shown in Fig. 5. In terms of ST, the information-based and thresholdbased holding strategies give only slightly worse results when compared with conditional priority (respectively a reduction of 3.8, 3.0 and 4.3%). The information-based holding strategy is slightly better for all performance indicators compared to the threshold-based one. 7 Conclusions The transit operational model presented in this study, even though based on Monte Carlo simulation, includes, with respect to widely known models, explicit representation of traffic lights and considers the vehicle capacity and the passenger arrivals during dwell time. The results obtained indicate that the best informationbased holding strategy appears to be competitive with the best threshold-based strategy, since the total system time reduction determined by information-based holding is 312 & The Institution of Engineering and Technology 2009 [4] SUN A., HICKMAN M.: ‘The holding problem at multiple holding stations’, in HICKMAN M. , MIRCHANDANI P. , VOSS S. (EDS.): ‘Computer-aided systems in public transport’ (Springer, 2008), pp. 339–362 [5] FU L., YANG X. : ‘Design and implementation of busholding control strategies with real-time information’, Transp. Res. Rec., 2002, 1791, pp. 6 – 12 [6] MAURO V., DI TARANTO C.: ‘UTOPIA’. Proc. CCCT 89 – AFCET, Paris, France, September 1989 [7] FURTH P.G., MULLER T.H.J.: ‘Conditional bus priority at signalized intersections: better service with less traffic disruption’, Transp. Res. Rec., 2000, 1731, pp. 23– 30 [8] KIM S., PARK M., CHON K.: ‘A bus priority signal strategy for regulating headways of buses’, J. Eastern Asia Soc. Transp. Stud., 2005, 6, pp. 435– 448 [9] HOUNSELL N.B., SHRESTHA B.P.: ‘AVL based bus priority at traffic signals: a review and case study of architectures’, Eur. J. Transp. Infrastruct. Res., 2005, 5, (1), pp. 13 – 29 [10] HOUNSELL N.B., SHRESTHA B.P., MCLEOD F.N., PALMER S., BOWEN T., HEAD J.R.: ‘Using global positioning system for bus priority in IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304– 313 doi: 10.1049/iet-its.2008.0072 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply. www.ietdl.org London: traffic signals close to bus stops’, IET Intell. Transp. Syst., 2007, 1, (2), pp. 131 – 137 [11] LIN T., WILSON N.H.M. : ‘Dwell time relationships for light rail systems’, Transp. Res. Rec., 1992, 1361, pp. 287– 295 [12] BELLEI G., GKOUMAS K.: ‘Una distribuzione degli intertempi che tiene conto del pairing’ (in Italian), in FERRARI P. , CEPOLINA E.M. (EDS.): ‘Didattica e ricerca nell’ingegneria dei trasporti’ (Franco Angeli editori, 2005), pp. 92– 99 IET Intell. Transp. Syst., 2009, Vol. 3, Iss. 3, pp. 304 – 313 doi: 10.1049/iet-its.2008.0072 [13] BELLEI G., GKOUMAS K.: ‘Pairing, headway distributions and dwell time’. Proc. 10th Int Conf. Computer-Aided Scheduling of Public Transport, Leeds, UK, June 2006 [14] VUCHIC V.R.: ‘Propagation of schedule disturbances in line-haul passenger transportation’, Revue de l’UITP, 1969, 18, (4), pp. 281– 285 [15] BELLEI G., GKOUMAS K.: ‘Priority and holding strategies to improve transit performance’. Proc. 11th World Conf. Transport Research, Berkeley, California, June 2007 313 & The Institution of Engineering and Technology 2009 Authorized licensed use limited to: Universitaet Goettingen. Downloaded on November 2, 2009 at 09:55 from IEEE Xplore. Restrictions apply.