This paper is a postprint of a paper submitted to and accepted for publication in IET Intelligent Transport Systems and is subject to Institution of Engineering and Technology Copyright. The copy of record is available at IET Digital Library Empty Vehicles Management as a Method for Reducing Passenger Waiting Time in PRT Networks 1 W.B. Daszczuk , W. Choromański 2, J. Mieścicki 3, W. Grabski 4 Abstract: Empty vehicles management may improve the average waiting time for vehicle delivery in the PRT network. In this paper, original heuristic algorithm of empty vehicle management is presented. The algorithm is tested in several benchmark PRT structures under Feniks simulation environment. The results show that significant improvements of average waiting time may be achieved just due to the multi-parameter analysis of present network state alone rather than by the predictive use of forecasted demand. The algorithm does not use any central database of demand and location of free vehicles. Instead, it assumes the local exchange of data between stations on the their state and expected vehicles. Therefore, it seems well tailored to a distributed implementation. Keywords: Personal Rapid Transit; Empty Vehicles Management; Transport Simulation; Transportation Management
1. Introduction Eco-Mobility is the project on Personal Rapid Transit (PRT), now under development at the Warsaw University of Technology [1,2]. PRT [3,4,5,6] is based on unmanned vehicles driving along one-way tracks. Each vehicle can carry a group of 1-4 passengers to a common destination. It is known that in PRT networks the average time of waiting for the vehicle may be improved by reallocation of empty vehicles, at the expense of greater empty vehicles movement. In the literature, the reallocation algorithms addressing this issue are usually based on past demand estimates and future forecast. In [12,13,14,15] the demand forecasts are based on statistics from previous corresponding periods corrected for weather and special events. Demand forecast modified by currently observed input rate is proposed in [16]. In [17] the algorithm is based on a repository of requests (including future requests if possible) and empty vehicles (for occupied vehicles, approximate time of freeing is calculated analytically). The SV algorithm (Sampling and Voting [18]) chooses reactive movements (in response to requests) using a simple nearestneighbour rule, and it chooses proactive movements by generating an ensemble of possible sequences of future passenger requests, solving a deterministic optimization problem for each sequence individually, and then finding the empty vehicle movements that are common among the sequences. The DTP algorithm (Dynamic Transportation Problem, [19]) moves empty vehicles in anticipation of future requests. The redistribution is based on a target number of free vehicles allocated to every station. 1
Institute of Computer Science, Warsaw University of Technology, Nowowiejska str. 15/19, 00-665 Warszawa, Poland, Email:
[email protected] 2 Faculty of Transportation, Warsaw University of Technology, Koszykowa str. 75, 00-662 Warszawa, Poland, Email:
[email protected] 3 The same as 1, Email:
[email protected] 4 The same as 1, Email:
[email protected]
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Instead, in our research we have tested algorithms based on the multi-parameter analysis of a current state of a PRT network. In our approach the explicit predicted demand is not included into a set of algorithm parameters. However, the results suggest that the decisions based upon the analysis of many important parameters reporting the present state of the network also can play the role of local, very effective predictions. All above-mentioned algorithms [12-19] are based on central repository of requests (possibly including future ones) and empty vehicle supply (currently empty and about to become empty). In contrast, our multi-parameter algorithm is based on inter-station exchange of data, with possibility of limiting the distance among communicating stations. This feature makes our algorithm feasible for large networks with distributed control. One of the research experiments was focused on the influence of empty vehicles management on the passenger’s satisfaction, understood as passenger waiting time. This paper describes this experiment. In next sections, a model of a PRT network and the Feniks simulator are briefly presented, the empty vehicle management is described and the research on management effectiveness is discussed. 2. PRT model There are three types of nodes in a PRT network [5]: stations, capacitors and intersections. In stations the passengers order their trips, board/alight the vehicles at several berths, or wait for a vehicle in a queue if there is no empty vehicle available. A station is characterized by the number of berths and by the sizes of entry buffer (where the vehicles may wait for an empty berth) and exit buffer (where the vehicles wait if they cannot join the traffic due to traffic conditions). For a capacitor (or garage) the only parameter is the number of parking berths. Intersections are of three types: ‘fork’ (1=>2), ‘join’(2=>1) and ‘junction’ (1=>1, the latter are for technical purposes only). The control units of vehicles and network nodes’ controllers communicate via the radio network. This way every vehicle can get information on the status and movement of other vehicles as well as on the status of neighbouring nodes. Segments of a track connect the nodes (capacitors, stations and intersections). There are two types of segments. Highway segments connect intersections only and the velocity of vehicles on a highway is relatively high (maximum typically 15 m/s). Road segments may connect all types of nodes, and the velocity is lower (maximum 10 m/s). The type, length and maximum velocity are parameters of a segment. Each vehicle can accommodate up to 4 persons. The vehicles perform trips between stations and/or capacitors. Trips are either full (i.e. with passengers) or empty. We assume that all the vehicles are of one type, characterized by common parameters. They include maximum velocity on a highway and on a road, maximum acceleration and deceleration, minimum inductive deceleration and maximum friction deceleration (emergency brake), minimum separation between vehicles. It is assumed that the passengers arrive at network stations in groups of 1 to 4 persons. Number of persons in a group is a random variable with a uniform distribution (i.e. mean is equal to 2.5). Each group performs a trip (i.e. travels together) to the common destination (no ride sharing). The input stream of passenger groups is random, with exponential distribution of the inter-arrival time between groups. The mean value of the distribution is specific to the station and may vary during the day. If a group arrives at a station, it takes one of the empty vehicles, if available. If there is no empty vehicle, the passenger group waits in a queue until a vehicle is available (either
because it concludes the trip at this station or is delivered by the empty vehicles management algorithm). Every passenger group chooses its own destination of the trip. An ODM (OriginDestination Matrix) defines a probability of taking a trip between every pair of stations. The matrix may vary during the day. 3. The Feniks simulator The Feniks simulator has been implemented for the purposes of the research on various versions of control algorithms rather than for the simulation of a specific case study (e.g. for planning and design of some the PRT network under construction). A model structure under Feniks reflects a typical structure of PRT network. Capacitors and stations are assumed to be zero-sized, although they have internal structures necessary to model behaviour: parking places, entry and exit buffers, passenger queues etc. Feniks is a discrete event simulator [20], based on microsimulation [21]. Therefore, the segments are divided into smaller, equal-length sectors. Decisions are made on sector connections. The dynamic objects are vehicles and passenger groups. The routing follows the Dijkstra’s algorithm [22], with segment costs that are a combination of segment length, segment free passage time and traffic density (dynamic component). The three cost components are normalized (respectively: by average distance, average free trip time between every pair of stations, and number of vehicles in the model) and they have user-defined factors. A more detailed description of Feniks simulator can be found in [1,2]. 4. Control 4.1. Two levels of control algorithms The control algorithms of the PRT network can be divided into lower (coordination) level, and upper (management) level. The former include the algorithms for controlling vehicles following each other down the track, for the coordination on ‘join’ intersections and for the control of the behaviour inside stations and capacitors [23]. The latter include empty vehicle management and dynamic routing algorithms. 4.2. Empty vehicles management Empty vehicles management involves four types of activity, referred to as ‘calling’, ‘expelling’, ‘balancing’ and ‘withdrawing’. • Calling is used when there is no empty vehicle for a passenger group. • Expelling refers to situations when a vehicle approaches its destination station and no berth is free there. The algorithm determines which empty vehicle is to be expelled (if any) and to which station it has to go. • Balancing refers to situations when an empty vehicle has to leave the station where it stays and to move to another station, because it is expected that it may be needed there, even if it is not actually called. The decision is based upon the current state of two stations and the values of balancing parameters, for instance on the number of free berths or on the ratios of free berths to occupied berths in both stations. The algorithm determines when the vehicle is departed and to which station it has to move. Notice that balancing is a form of predictive control, but it is based upon the analysis of the current state of the net (or of its part) rather than on the explicit prediction of future demand. • A vehicle is withdrawn to the nearest capacitor if it stays in a station longer than a specified timeout.
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Accordingly, four following functions have been defined: ‘calling function’, ‘expelling function’, ‘balancing function’ and ‘withdrawing function’. The following static and dynamic model parameters are used in each of these functions: • N – number of stations in the model, • NG – number of capacitors (garages) in the model, • M – number of vehicles in the model, • Hi – number of berths at station si or capacitor gi. • S={s1..sN} – the set of stations, • G={gN+1..gN+NG} – the set of capacitors, • V={v1..vM} – the set of vehicles, • Qi – current number of passenger groups in a queue at station si. • Ki – current number of vehicles in berths of station si or capacitor gi, • Li – current number of empty vehicles in berths and in the entry buffer of station si or capacitor gi, • Zi – current number of vehicles on a trip to station si or capacitor gi, • Dij – shortest distance (in metres) from station si to station sj or capacitor gi, • Dav – average distance (in metres) between pairs of distinct stations, • NDij – normalized distance between stations si and sj (or capacitor gj); NDij=Dav/Dij; the shorter is the distance the greater is NDij; ND=1 for mean distance. For the calculation of the values of the four functions, a set of weighting factors and threshold values has been defined: • FQ – passenger queue factor, • FEB – empty berths factor, • FND – normalized distance factor, • TQ – passenger queue threshold, • TEB – empty berths threshold, • TEV – empty vehicles threshold, • TND – normalized distance threshold, • T – total function threshold. Each function has its separate set of the above weighting factors and thresholds. The factors and thresholds for the balancing function have B prefix (i.e. BFQ, BFEB etc.), for the calling function they have C prefix (i.e. CFQ, CFEB etc.), E for the expelling function and W for the withdrawing function. The evaluation rules are almost identical for every function. Below, evaluation rules of the balancing function are given as an example. The general idea is that the algorithm consists in the three following steps: • Identify a subset of stations among which the balancing would be possible and beneficiary. The decisions are based on the actual queue lengths, the number of free berths, the distances between the stations etc., • Within such a restricted subset identify the station which is the “best” candidate for balancing. The evaluation is based on the value of special balancing function B, which is a weighted sum of station state indices and their weights, • Perform an empty trip to the “best” station, provided that the just-computed “best” value of balancing function Bmax exceeds some general threshold BT.
More specifically, the procedure is as follows: for a given station sx, we select a subset of stations si (out of the set of all stations) for which the following conditions simultaneously hold: • Qi – Li – Zi ≥ BTQ, (queue length is greater than (or equal to) ‘passenger queue threshold’; the number of empty vehicles at the station sx and the number of vehicles on a trip to the station sx are subtracted from the queue length, because these vehicles are about to take passengers soon), • (Hi – Ki + Qi – Zi)/ Hi ≥ BTEB, (normalized number of empty berths is greater than (or equal to) ‘empty berths threshold’; the number of passenger groups in a queue is added to the number of free berths H-K – they will take vehicles soon; and the number of vehicles on a trip to the station sx are subtracted from the number of free berths H-K, as they will take empty berths), • NDxj ≥ BTND, (normalized distance is greater than (or equal to) ‘distance threshold’, i.e. stations closer than 1/BTND are considered) • (Li + Zi – Qi)/Hi – (Lx + Zx – Qx)/Hx ≥ BTEV, (the difference between parts of empty vehicles in both stations is greater than (or equal to) ‘empty vehicles threshold’). Then, for every selected station si, the value of the balancing function (Bi) is calculated: Bi = BFQ*(Qi – Li – Zi) + BFEB*(Hi – Ki + Qi – Zi) + BFND*NDxi Note that the parameter ND plays a special role: it defines a distance to the stations that are taken into account in the algorithm. All features other than ND (number of vehicles, number of free berths, number of passenger groups, etc.) can be obtained by means of communication of sx with other stations. No central database holding these data is needed. Therefore, in real world the management algorithms can be implemented in a distributed way, basing on inter-station communication restricted to distance 1/ND. A station smax with the highest value of Bi (called Bmax) is chosen as a candidate for effective balancing. Then, if Bmax ≥ BT, an empty trip of the vehicle from station sx to smax is executed. In the calling function (C) is that the empty trip is executed from station smax to sx rather than from station sx to smax as in (B) For withdrawing (W) – capacitors are considered instead of other stations. For expelling (E) – stations as well as capacitors are considered. The described procedure is performed: • for balancing – periodically, • for calling – when a passenger group arrives and there is no empty vehicle, or when a vehicles finishes its trip, • for withdrawing – when an empty vehicle stays in a station longer than a specified timeout, • for expelling – each time a vehicle approaches a station where all berths are occupied and at least one empty vehicle is at the station. 5. Population and transit requests From available Polish statistical data we assumed that the population density is 3000 people/km2 for city and 2000 people/km2 for suburbs. As the data for PRT are not available, we have used the analogy to city tram transport. We roughly estimated that 1000 inhabitants of a typical neighbourhood generate about 15-20 tram transit requests (passengers) per hour in peak hours. For experiments with the simulated PRT networks we assumed the same value. As passengers arrive in groups of the mean size equal to 2.5 - every 1000 inhabitants generate the average input stream of 8 groups per hour. Then,
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by analysis of the model network topography, we determine the average nominal intensity of the input stream for individual stations. For instance, if in a given model the city centre is a circle of 3 km diameter, then its population is approximately 3.14*1.52 [km2] * 3000 [persons/km2] ≈ 21000 inhabitants, which generates a total of about 170 transit requests per hour. Now, if in this area there are four PRT stations then (and assuming that the requests are equally distributed among them) each station receives the input stream of 21.25 requests per hour. Similar way we can roughly estimate the total demand for transit requests for the whole network (below referred to as dem). 6. Simulated models To generalize the results, three following different network structures have been tested: • City (Fig. 1) resembling a typical city with suburbs, total segments length 33370m, • SeaShore (Fig. 2) – a linear model with side branches, total segments length 22200m, • TwinCity (Fig. 3) – a model of two connected cities or trade centres, total segments length 21380m. In the figures, double lines are two-way highways while single lines are one-way road segments. Circles represent stations and dashed boxes represent capacitors. Little circles without letters are roundabouts. A structure of a roundabout is shown in Fig. 4. The estimated values of total (nominal) request rate dem for each network are the following: for City, dem= 270, for SeaShore dem = 100 and for TwinCity dem = 74 [requests (groups) per hour]. The simulation parameters common for all models are: • 2 berths at every station, • 5-positions entry buffer at every station, • 1-4 passengers in every trip (with equal probability), • maximum highway velocity 15m/s, • maximum road velocity 10m/s, • maximum acceleration and deceleration 2m/s2, • boarding and alighting times: random, with triangular distribution (10;20;30 [s]) • minimum separation 10m, • sector size 5m, • ODM filled with equal values (trip destination chosen randomly), Empty vehicles management acts as follows: • Calling parameters cause the choice of nearest empty vehicle (CFND=5, other parameters have ‘neutral’ values), this makes calling from nearest stations preferable when vehicles are available. • Expelling parameters allow an empty vehicle to drive to a station where there are empty berths and these berths are not the targets of moving vehicles, stations with waiting passengers are preferred and nearest stations are preferred (ETEB=1, ETQ= –EH+1, EFEB=1, EFQ=1, EFND=1), • Balancing is turned on and off in pairs of experiments. When balancing is on, waiting passengers, empty berths and inter-station distance are considered and the distance is limited to half of mean distance: BTEB=1 (there is at least one empty berth), BTQ= –BH+1 (there may be waiting passengers, but if there are vehicles travelling to the station, the number of them should be less than the number of passenger groups waiting
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plus the number of free berths; otherwise a vehicle sent to this station may find all berths occupied), BTND=1 (balancing is limited to the scope of mean inter-station distance), BFEB=1, BFQ=1, BFND=1 (the factors of free space, passenger groups waiting and normalized distance are equal) BT=1 (at least one of the factors should give positive value to ‘fire’ the balancing move). Withdrawing is off (WT is very high).
The rule for balancing (all factors equal) is very simple, yet even this set of factors gives significant result (as will be shown in section 8). Numerical values of these parameters were determined heuristically, based on several dozen preliminary experiments that examined how changes in individual parameters affect the results. These simulations show that the algorithm is stable in the sense that ±30% value changes did not result in dramatic changes of the number of empty courses. However, the more detailed description of this research is beyond the scope of the paper. 7. Determining the number of vehicles and maximum ridership Even from the early simulations it has been observed that the maximum ridership RS(M)max, i.e. the maximum number of full trips per unit time that can be done in the given network with M vehicles – grows linearly with M. In our three models this remains valid for number of vehicles ranging from just a few up to several hundred (250 – 300) vehicles. For greater values of M the network would become saturated with vehicles: they begin interfere with each other slowing down the traffic, causing traffic congestions, etc. For every version of the model, a preliminary simulation has been performed in order to obtain the maximum ridership RS(M)max for an example M=48 vehicles (arbitrarily chosen but well below the saturation range). To do this, we set the initial number of passenger groups in all input queues large enough so that they could not be served during the simulated time. Under this condition, as the passenger queues are never empty, every vehicle arriving at a station immediately takes the new passenger group and starts a new full trip. There are no empty trips, except for a short warm-up period when the vehicles move from capacitors to stations. The obtained RS(48)max is for a given model (with its structure and ODM) with 48 vehicles, but the principle of proportionality allows to easily estimate the ridership also for other numbers of vehicles in the same network, provided that the network remains safely far from saturation. Another approach to estimate the maximum ridership has been presented in [24], where the sophisticated analysis was based on static analysis of flow. In contrast to such approach, we prefer one simple experiment, organized as described above. 8. Input rate and safety margin In the simulations the input stream of passenger groups (for individual stations as well as for the whole network) is random, with the inter-arrival time generated from the exponential distribution with parameter λ (where λ is the input rate). The mean time between consecutive arrivals is 1/λ. Let for given network with M vehicles, the total input rate be λ [groups/hour]. The condition for the network to be in an equilibrium state is λ < RS(M)max . Knowing the total request rate dem (as in Section 5) and the results of RS(M)max estimation we could determine the number of vehicles M so that for λ = dem the equilibrium condition holds.
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However, when λ approaches RS(M)max, the queue lengths as well as the waiting times grow very rapidly. Therefore, the number of vehicles M should be set so that some safety margin is provided. This margin can be characterized as m=RS(M)max/dem. The recommended value of m is at least 2 (the value of maximum ridership being twice as big as actual input rate). A real-world network usually significantly differs from the simulated simplified model, for instance: • stations are not symmetrically located in the topography; • mean input rate varies during the day; • different stations may have various mean values of input at the same time; • ODM is not uniform, etc. Therefore, designer should choose a value of m even much greater than 2. For every model two “safe” (in above sense) numbers of vehicles have been arbitrarily determined. For every model and every number of vehicles six simulations have been performed. One simulation run has been done in order to confirm the exact value of RS(M)max for given M. Other five ones have been performed for random input streams with input rates λ covering evenly the range from 0 to RS(M)max. The basic parameters of the models are collected in Table 1. Table 1: Input rates and safety margins in individual models model vehicles dem [groups/h] RSmax [groups/h] 48 638 City 270 76 986 24 259 SeaShore 100 50 532 50 452 TwinCity 74 76 681
safety margin m 2.3 3.6 2.6 5.3 6.1 9.2
9. The simulation experiments Below we discuss only one aspect of empty vehicle management, namely – the role of balancing. The results collected in Table 2, confirm that the waiting time dramatically grows as input rate approaches the equilibrium limit. Table 2: Passenger waiting time in individual experiments (semilogarithmic plots) w/o bal. with bal.
City
Smaller number of vehicles
Greater number of vehicles
10000
10000
1000
1000
100
100
10
10 1
1 0
200
400
600
800
10000
SeaShore
0
200
400
0
100
200
600
800
1000
1200
10000
1000
1000
100
100
10
10 1
1 0
50
100
150
200
250
300
300
400
500
600
10000
TwinCity
10000
1000
1000
100
100
10
10 1
1 0
100
200
300
400
0
500
200
400
600
800
Notes:
• Horizontal axes – input rate [passenger groups/h] • Vertical axes – average passenger waiting time [s] – logarithmic scale • Numbers of vehicles are taken from Table 1 (smaller, greater): City (48, 76), SeaShore (24, 50), TwinCity (50, 76)
For higher input rates the balancing is not so important, because a vehicle finishing its trip finds immediately a passenger for the next trip and empty trips hardly occur. However, for low input rate balancing is very effective: average waiting time can be shortened several dozen times! The relative values (the ratio of average waiting time with and without balancing) are presented in Table 3. Table 3: The ratio of average waiting time without balancing to average waiting time with balancing Model Smaller number of vehicles Greater number of vehicles
City
100
100
10
10
1
1
0
SeaShore
200
400
600
800
100
100
10
10
1
200
400
600
800
1000
200
300
400
500
1200
1
0
TwinCity
0
50
100
150
200
250
0
300
100
100
10
10
1
100
600
1
0
100
200
300
400
500
0
100
200
300
400
500
600
700
800
Notes:
• Horizontal axes – input rate [passenger groups/h] • Vertical axes – average passenger waiting time without balancing divided by average waiting time with balancing – logarithmic scale • Numbers of vehicles are taken from Table 1 (smaller, greater): City (48, 76), SeaShore (24, 50), TwinCity (50, 76)
The shortening of passenger waiting time is achieved at the cost of increased number of empty trips. Some example values of the waiting time shortening compared with empty trips growth are collected in Table 4.
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Table 4: Waiting time shortening (with/without balancing) compared with empty trips growth (without/with balancing), with uniform ODM Model
Smaller number of vehicles b Waiting time Empty trips shortening growth [times] [times]
Input [groups/h]a
small (100, 150) medium (320, 500) small (40, 80) SeaShore medium (130, 260) small (60, 100) TwinCity medium (220, 340) EXAMPLE ABSOLUTE VALUES Balancing: w-with/wo-without small (100, 150) City medium (320, 500) City
a b
69.1 2.5 40.8 4.1 22.1 5.7
3.54 1.66 5.90 7.43 2.55 1.68
Greater number of vehicles b Waiting time Empty trips shortening growth [times] [times]
28.7 4.4 15.9 5.1 12.6 3.4
3.22 1.99 2.34 1.27 1.47 1.24
Waiting Empty Waiting Empty time [s] trips time [s] trips w wo w wo W wo w wo 0.68 46.91 1823 515 1.32 37.95 3027 940 18.79 47.33 2577 1550 18.28 80.92 4608 2319 left value is for lower number of vehicles, right value is for upper number numbers of vehicles are taken from Table 1 (smaller, greater): City (48, 76), SeaShore (24, 50), TwinCity (50, 76)
Although shortening of the waiting time is obvious, the average waiting time has dropped from about a minute to several seconds in absolute numbers. As a minute of waiting time is not very burdensome for a passenger, a larger model has been tested, also to learn about the scalability of the effect. A model called Manhattan (Fig. 5) has been prepared, with total length equal to 56km and 56 stations (now stations are little circles, without letters). Experiments with 40, 80 and 160 vehicles have been performed. The results are collected in Table 6. There is less relative waiting time shortening: 2.77 times to 5.47 times for the Manhattan model with various numbers of vehicles (see the upper row for every number of vehicles in Table 6) vs. up to 69 times for the City model. However, in absolute values, the passenger waiting time dropped from almost 17 minutes (without balancing) to 6 minutes (with balancing) in the Manhattan model (experiment with 40 vehicles). This is still a significant improvement of passenger satisfaction. As before, the result is achieved at the cost of increasing the number of empty trips (the lower row for every number of vehicles in Table 5). Table 5: Results for model Manhattan
Model
Vehicles
Input [groups/h]
Manhattan
40
33.6
Manhattan
80
84
Manhattan
160
336
Without balancing: Avg waiting time [s] Number of empty trips/h 1013.9 247 263.0 686 91.1 2161
With balancing: Avg waiting time [s] Number of empty trips/h 366.6 671 61.0 1168 16.7 3210
Ratios Improvement [times] Growth [times] 2.77 2.72 4.31 1.70 5.47 1.49
10. Results for non-uniform ODM All the experiments summarized in tables 4 and 5 have been performed for uniform OriginDestination Matrix (ODM). In this situation the average in- and outflow of vehicles at each station are equal. In order to see how the management algorithm performs with non-uniform ODM, all four models have been simulated again. This time ODM was first filled with random numbers from range [0,1], then normalized (to achieve sum of 1 in every row). This way a “randomly non-uniform” ODM was obtained which has been used in all simulations described in this section. While input streams are (statistically) identical in each station, because of non-uniform ODM the output streams are different. A sum of values in a column is proportional to the rate of the corresponding station output stream. For example, sums of columns in City model range from 0.58 to 1.44, the latter being almost 2.5 times greater than the former one. This makes the flow unbalanced. Table 6 contains results for four models with such non-uniform ODM. The shortening of waiting time in individual models is not equal to the one for uniform ODM (in terms of numerical values), but in every case the effect is clearly visible and significant. Table 6: Waiting time shortening (with/without balancing) compared with empty trips growth (without/with balancing), with non-uniform ODM Model
Input [groups/h]a
small (100, 150) medium (320, 500) small (40, 80) SeaShore medium (130, 260) small (60, 100) TwinCity medium (220, 340) small (33.6, 84) Manhattan medium (80, 336) EXAMPLE ABSOLUTE VALUES Balancing: w-with/wo-without small (100, 150) City medium (320, 500) small (33.6, 84) Manhattan medium (80, 336) City
a b
Smaller number of vehicles b Waiting time Empty trips shortening growth [times] [times]
15.2 2.1 10.8 1.4 86.1 2.5 5.9 3.6
3.24 1.86 0.78 1.28 1.13 1.57 3.31 3.74
Greater number of vehicles b Waiting time Empty trips shortening growth [times] [times]
19.4 3.0 72.5 3.9 49.8 3.6 13.3 3.1
3.22 1.99 2.34 1.27 1.47 1.24 10.20 6.41
Waiting Empty Waiting Empty time [s] trips time [s] trips w wo w wo w wo w wo 2.47 37.51 1984 612 1.66 32.18 1610 806 25.91 55.04 2440 1313 21.38 64.93 3022 1800 123.1 729.6 873 264 15.63 208.1 5245 514 10785 105.9 382.0 2186 584 48.69 149.7 1682 left value is for lower number of vehicles, right value is for upper number numbers of vehicles are taken from Table 1 (smaller, greater): City (48, 76), SeaShore (24, 50), TwinCity (50, 76) and from Table 6 (smaller, medium): Manhattan (40, 80)
11. Conclusions and further work The results show that the use of the described heuristic algorithm is beneficial. The important features of presented balancing algorithm are:
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it does not rely on forecasting (which may be inadequate and may fail to predict nontypical behaviour of people, for example before and after rally, community meeting, mass events etc.), • it is not based on any centrally collected data. Instead, it makes the use of information interchanged between stations and vehicles, while one of parameters allows to restrict the maximum distance for which the exchange of information takes place. This approach seems to be well suited to the distributed implementation of control. Moreover, the same, uniform idea of the algorithm is applied to other aspects of vehicle management (calling, expelling, withdrawing). The procedure is almost identical while the sets of parameters are specific (however their meaning and purpose are still analogous). Of course, prediction data may be used in addition to parameters describing the state of the network, and the effect may be better than described in the present paper. Yet the prediction has been discussed in a number of papers (e.g. [12-19]) therefore it has not been used here for clarity of results and of the approach itself. The complexity of the algorithm is not troublesome: every station or capacitor obtains relevant data (passenger queue length, number of empty berths, number of empty vehicles) from neighbouring stations/capacitors located no more than BTND , counts the value of function B for every node and finally compares these values. Nodes perform this procedure periodically, independently of one another. It is difficult to compare the efficiency of the described algorithm to other ones, known from the literature. These competitive solutions are also heuristic and any quantitative comparison should be based on the numerical results of multiple simulations of the same network model. This suggests that for the purposes of the research on control algorithms it would be very profitable to establish some set of reference models (or benchmarks) which could serve as a testbed for comparisons and evaluation. Finally we may note that the Feniks simulator (current version: 4.0) proved to be a very useful tool in the analysis of the behaviour of vehicles in a PRT network and testing of management algorithms. Future experiments are planned with other aspects of vehicle management, dynamic routing, fault-tolerant properties and optimization of network operation. •
Funding The research is carried out under the project ECO-mobility co-funded by European Regional Development Fund under the Operational Program Innovative Economy (the ECO-Mobility project [WND POIG.01.03.01-14-154/09, 2009-2013])
References [1] Choromański, W., Daszczuk, W. B., Grabski, W., Dyduch, J., Maciejewski, M., Brach, P.: ‘PRT (Personal Rapid Transit) computer network simulation, analysis of flow capacity’, in Proc. 14th International Conference on Automated People Movers and Automated Transit Systems, April 2124, 2013, Phoenix, AZ, pp. 296-312 [2] Choromański, W., Daszczuk, W. B., Dyduch, J., Maciejewski, M., Brach, P., Grabski, W.: ‘PRT (Pesonal Rapid Transit) Network Simulation’ in Proc. WCTR - World Conference on Transport Research 2013, Rio de Janeiro; July 07-10, 2013, in print
[3] Irwing, J. et al.: ‘Fundamentals of Personal Rapid Transit’, Lexington books, Lexington MA, 1978, ISBN is 0-669-02520-8 [4] Anderson J.E.: ‘A review of the state of the art of personal rapid transit’. Journal of Advanced Transportation. 34(1), 2000, pp.3-29 [5] Andréasson, I. J.: ‘Staged introduction of PRT with Mass Transit’, Proc. PRT@LHR 2010 Conference, September 2010, London Heathrow Airport [6] MacDonald, R.: ‘The Future of High Capacity PRT’, Proc., 13th International Conference on Automated People Movers and Transit Systems, Paris, France, May 22-25, 2011, ISBN: 978-07844-1193-3, pp.250-262 [7] Hermes, http://archive.is/X3KX, accessed November 2013 [8] Beamways, http://www.beamways.com/, accessed November 2013 [9] RUF, http://www.ruf.dk/, accessed November 2013 [10] NETSIMMOD, http://prtconsulting.com/simulation.html, accessed November 2013 [11] Castangia, M., Guala, L.: ‘Modelling and simulation of a PRT network’, Proc. 17th International Conference on Urban Transport and the Environment, June 6-8, 2011, Pisa, Italy, pp.459-472 [12] Andréasson, I. J.: ‘Vehicle distribution in large personal rapid transit systems’, Transportation Research Record 1451, 1994, pp.95-99 [13] Andréasson I. J.: ‘Quasi-optimum redistribution of empty PRT vehicles’, Proc. 6th International Conference on Automated People Movers. Las Vegas, Nevada, April 9-12, 1997, pp.541-550 [14] Andréasson, I. J.: ‘Reallocation of Empty PRT vehicles en route’, TRB annual meeting, Washington DC January 2003 [15] Carnegie J. A., Hoffman P. S.: ‘Viability of Personal Rapid Transit In New Jersey (Final Report).’ Presented to Governor Jon S. Corzine and the New Jersey State Legislature. February 2007, http://policy.rutgers.edu/vtc/reports/REPORTS/PRTfinalreport.pdf, accessed November 2013 [16] Zheng, P., Jeffery, D. and McDonald, M.: ‘Development and evaluation of traffic management strategies for personal rapid transit’, in Proc. Industrial Simulation Conference 2009, Loughborough, UK, June 1-3, 2009, pp.191-195 [17] van der Heijden, M., Ebben, M., Gademann, N., van Harten, A.: ‘Scheduling vehicles in automated transportation systems Algorithms and case study, OR Spectrum, Vol. 24, No. 1. (15 February 2002), pp.31-58
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[18] Lees-Miller, J. D. and Wilson, R. E.: ‘Sampling for Personal Rapid Transit Empty Vehicle Redistribution’, Proc., 90th Annual Meeting of the Transportation Research Board, 2011 [19] Lees-Miller, J. D. and Wilson, R. E.: ‘Proactive empty vehicle redistribution for personal rapid transit and taxis’, Transportation Planning and Technology, 35(1), 2012, ISSN 0308-1060, pp.17– 30 [20] McHaney, R.: ‘Understanding Computer Simulation’, Roger McHaney & Ventus Publishing Aps, 2009, ISBN 978-87-7681-505-9 [21] Szillat M. T:’ A Low-level PRT Microsimulation’, PhD thesis, University of Bristol, April 2001 [22] Dijkstra, E. W.: ‘A note on two problems in connexion with graphs’. Numerische Mathematik 1 (1959), pp.269–271 [23] Daszczuk, W. B., Mieścicki, J., Grabski, W.: ‘Principles of building PRT models and simulation experiments in FeniksOnArena v.1.0 environment’, Eco-Mobility Project Research Report, 2011 (in Polish) [24] Lees-Miller, J. D., Hammersley, J. C., and Wilson, R. E.: ‘Theoretical maximum capacity as benchmark for empty vehicle redistribution in personal rapid transit’, Transportation Research Record: Journal of the Transportation Research Board, 2146, December 2010, pp.76–83
one-way road
two-way highway
G station
F
H
capacitor
B A city
I
E C D J
L
suburbs
K roundabout 3000m
Figure 1: The “City” model
15
6000m
E hotels
A
G
F
H
resorts
B
C
sea
Figure 2: The “SeaShore” model
D
A
B
E
F
D
C
G
H
3000m Figure 3: The “TwinCity” model
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Figure 4: The roundabout
Figure 5: The “Manhattan” model
19
