Crossover transitions in a bus-car mixed-traffic cellular automata model
TThis work is licensed under a Creative Commons “Attribution-NonCommercial-NoDerivs 3.0 Unported” license.
Crossover transitions in a bus–car mixed-traffic cellular automata model
Damian N. Dailisan ∗ , May T. Lim National Institute of Physics, University of the Philippines Diliman, 1101 Quezon City, Philippines
Abstract
We modify the Nagel-Schreckenberg (NaSch) cellular automata model to study mixed-traffic dynamics. Wefocus on the interplay between passenger availability and bus-stopping constraints. Buses stop next tooccupied cells of a discretized sidewalk model. By parametrizing the spacing distance between designatedstops, our simulation covers the range of load-anywhere behavior to that of well-spaced stops. The interplayof passenger arrival rates and bus densities drives crossover transitions from platooning to non-platooned(free-flow and congested) states. We show that platoons can be dissolved by either decreasing the passengerarrival rate or increasing the bus density. The critical passenger arrival rate at which platoons are dissolvedis an exponential function of vehicle density. We also find that at low densities, spacing stops close togetherinduces platooned states, which reduces system speeds and increases waiting times of passengers.
Keywords:
Computer modeling and simulation, Transportation, Monte Carlo methods statistical physicsand nonlinear dynamics
1. Introduction
Public transportation is universally acknowledged as a fundamental component in solving traffic con-gestion. Together with rail systems, buses form the backbone of medium- to long-haul modes of peopletransport. Since the interaction of buses and passengers introduces complex behavior in transportationsystems, more so in cities that do not have designated stops, several models have been proposed to studybus traffic [1–5]. In these models, a delay in the arrival time of a bus leads to more passengers waiting forthe bus, which then leads to further delays. Succeeding buses find fewer waiting passengers leading them tocatch up to the delayed bus. Thus, buses form platoons (or bunches). These models also show a transitionfrom the platooning state to a non-platooned state with increasing bus density.The tendency of buses to form platoons is problematic for public transport. In an ideal scenario, anefficient transport system would try to maintain equal time intervals between vehicle arrivals. However,an equal headway configuration of vehicles is unstable [6], more so with buses [1]. Since the instability isinherent to the interaction between public transport vehicles and passengers, approaches to maintain equalheadways should consider both traffic and passenger behavior [7].Bus route models [1–3] omit interactions between buses and other vehicle types. Yet we expect thatthese vehicular interactions play a key role in the dynamics of traffic flow — buses making curbside stopsimpede traffic flow outright [8], while small perturbations in vehicle speeds can induce congestion [9, 10].Even a single bus in two-lane mixed traffic alters traffic states and jam transition densities [11]. As such, it ∗ Corresponding author
Email addresses: [email protected] (Damian N. Dailisan), [email protected] (May T. Lim)
Preprint submitted to Physica A June 25, 2020 a r X i v : . [ n li n . C G ] J un igure 1: Diagram of road R and sidewalk S models. Pedestrians can spawn on an empty cell of S with probability α . Inmulti-lane scenarios, buses are limited to the lane adjacent to the sidewalk. is also critical to decision makers when curbside stops have to be replaced by bus bays to alleviate congestion[12]. Using a modified comfortable driving model, Yuan et al. [13] focused on system performance to showa dependence of the system capacity on the number of bus stops. They also found a gradual transition fromplatooned to non-platooned states. However, they did not explore the interplay of passenger arrival ratesand vehicle densities extensively nor considered passenger waiting times.It is important to understand transitions in traffic models with public transportation in mind. Slowermoving buses and the associated platoon formation can result in the perception that buses cause congestion.Even a few undisciplined bus drivers can often cause traffic jams when their large vehicles straddle multiplelanes [14, 15]. Such situations, as well as a lack of understanding of these systems, can lead to proposalsby policymakers to ban buses to alleviate congestion. An example of which would be the proposal to banprovincial buses (which account for less than 3% of traffic volume [16]) from Metro Manila’s highways toalleviate congestion in one of the city’s major thoroughfares [17, 18]. A Melbourne study found that theremoval of such bus services would see up to 30% of bus users shifting to cars, which defeats the purpose ofeliminating buses to alleviate congestion [8]. Knowledge of the factors that influence transitions can informdecisions, avoiding policies that can cause more harm to the current state of traffic congestion.This work studies the transitions induced by vehicle density and passenger arrival rate on a Nagel-Schreckenberg model modified to include buses. We show how the interplay between vehicle density andpassenger arrival rates affect the crossover transition from non-platooned to platooning states. Both single-and multi-lane cases, as well as mixed traffic scenarios, are studied. Lastly, we look at the effects of theplacement of bus stops on traffic flow, as well as passenger waiting times.
2. Modeling bus–car traffic
Our model combines elements from the Nagel–Schreckenberg (NaSch) [14, 19] model and Bus RouteModel (BRM) [1]. The two components of this hybrid model are the road and sidewalk (Fig. 1). Theroad model R has N lanes of length L sites, with periodic boundary conditions. A vehicle can interactwith passengers (buses), or ignore passengers (cars). A sidewalk model S with one lane of length L sites isupdated synchronously with the road model. States for the i th vehicle are lane l i , position x i , and speed0 ≤ v i ≤ v max . We denote sites occupied by cars as R ( x, l ) = 1 and those occupied by buses as R ( x, l ) = 2.The road lane adjacent to the sidewalk is denoted as l = 1. Passengers occupy the sidewalk if S ( x ) = 1.Similar to the BRM, we impose that R ( x, l = 1) = 2 and S ( x ) = 1 cannot occur simultaneously, i.e.passengers cannot spawn when a bus occupies the adjacent road cell.For simplicity, both vehicle types have similar parameters v max and p slow , with the only differencebetween the two being the additional interaction with pedestrians for buses. Additionally, buses have For the sake of brevity and since a passenger waiting for the bus to arrive occupies sidewalk space, we use the termspedestrian and waiting passenger interchangeably. ρ , passenger arrival rate α , and bus to vehicleratio f B . Vehicle states are updated in random sequential order at each timestep t ( δt = 1 .
65 sec / step)following these rules: R1: Acceleration : v it +1 = min (cid:0) v it + 1 , v max (cid:1) R2: Bus Loading : v it +1 = 0 if S ( x it ) = 1; S ( x it ) = 0 R3: Lane Change : l i = l i ± p l if v it +1 > ∆ x i R4: Deceleration : For cars, v it +1 = min (cid:16) v it +1 , ∆ x i ∆ t (cid:17) .For buses, v it +1 = min (cid:16) v it +1 , ∆ x i ∆ t (cid:17) , if (cid:98) v max a dec (cid:99) (cid:80) k =1 ( v t − ka dec ) > ∆ x pass ∆ t min (cid:16) v it +1 , ∆ x i ∆ t , max( v t − a dec , a dec ) (cid:17) , if v t + a dec ≥ x ∆ t ≥ a dec min (cid:16) v it +1 , ∆ x i ∆ t , ∆ x pass ∆ t (cid:17) , otherwise. R5: Random Slowdown : v it +1 = max (cid:0) v it +1 − , (cid:1) with probability p slow R6: Forward Movement : x it +1 = x it + v it +1 ∆ t Table 1: Parameters and their values used in the simulation. In this model, each cell is 5.5 meters long and a timestep is 1.65seconds.
Symbol Description Typical Values Model Value L Length of the road 100 m (city), 10 km (highway) 500, 5120 v max Maximum speed 20 - 120 km/h (residential, highway) 5 (60 km/h) f B Fraction of buses 0.03 [16] 0 - 1 p slow Slowdown probability 0.01 - 0.5 [19–22] 0.1 p l Lane change probability – 0 (bus), 1 (car) a dec maximum deceleration 4 m/s ρ Vehicle density 0 - 180 vehicles/km 0.02 - 0.98 T τ Transient time – 3000 T Measurement time – 3000
The headway ∆ x i is the number of empty cells ahead of vehicle i while the passenger headway ∆ x pass isthe distance from the bus to the closest passenger. The implementation of Deceleration for buses ensuresdeceleration rates are limited to a dec when picking up passengers [23, 24]. The first criterion lets a bus ignorepassengers that appear when the bus is passing too fast to safely decelerate. The second criterion allows abus to anticipate stopping, and decelerate safely. For cars, Bus Loading is skipped. Vehicles may changelanes with probability p l ∈ { , } with equal chances of moving left or right. Vehicle i attempts to move intoadjacent lanes to avoid decelerating, but it may only change lanes if the target lane is empty and it satisfiesthe safety criteria. It is safe to change lanes when the trailing vehicle in the target lane can safely deceleratewithout crashing into vehicle i . The safety criteria can be written as v i + ( x i − x j ) ≥ v j − a dec , where j denotes the trailing vehicle. In the context of overtaking vehicles, Random Slowdown is skipped whena vehicle successfully changes lanes. Otherwise, the original NaSch rules (
R1, R4, R5, R6 ) are followed.Through these rules, interactions of vehicles and pedestrians give rise to complex dynamics of traffic flow.We measure the time averaged flow q of the system, defined as q = 1 T (cid:80) Tt =1 (cid:80) i v it ∆ tL , where v it is thespeed of the i th vehicle at time t . The average speed of vehicles in the system is then ¯ v = qρ . We allow for a transient simulation time of T τ = 3000 timesteps to remove transient behavior in the data[21]. For all realizations, measurements span T = 3000 timesteps with fifty trials for each set of parameters3 .0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.60.7 q f B = 0.0, = 0 f B = 0.2, = 10 f B = 0.5, = 10 f B = 1.0, = 10 f B = 1.0, = 10 f B = 1.0, = 10 f B = 1.0, = 10 Figure 2: Fundamental diagram for the single lane case. As density increases, jumps in q are observed. These jumps occur atdifferent densities, which appear to have a dependence on α . ρ , α , and f B . A complete list of parameters used in our simulations, with their corresponding calibrationfactor and real-world values, is given in Table 1.
3. Results and Discussion
In Sec. 3.1, we focus on the interplay of changing passenger arrival rates, fraction of buses, and transitionsfor the single lane case. Section 3.2 extends the model to multiple lanes, with buses being limited to theoutermost lane. Section 3.3 discusses placement of bus stops, and the effect of different bus fractions on thewaiting time of passengers.
If all vehicles on the road are buses ( f B = 1), we expect different behaviors at the extreme values of α . For the case α →
0, passengers do not arrive. With no passengers to pick up, buses do not slow down,and we recover the original NaSch model. In the NaSch model, we expect a phase transition to occurat ρ crit = v max +1 , which is the maximum allowable density where vehicles have enough headway to avoidslowing down [19, 25]. For the case of v max = 5, ρ crit ≈ . Thus, increasing the density beyond ρ crit inducesa phase transition from free-flowing traffic to congestion. On the other extreme, as α →
1, buses stop afterevery other timestep. High values of alpha guarantee that there will be a passenger waiting for a bus at thenext cell. Thus, at low densities, we expect buses in this system to move at an average speed of ¯ v = 0 . ρ ≈ , which corresponds to the critical density for v max = 0 . v → .
5; and (2) the dynamics of the NaSch model without buses ( α → f B = 0). For the values of f B = 1 . , α = [10 − , − , − ], we observe that for low densities, busespicking up passengers is the dominant behavior. However, increasing the density of buses results in jumpsin throughput in the fundamental diagram. These jumps indicate that the dominant behavior of the systemhas shifted towards the dynamics of the fundamental NaSch model (Fig. 2, f B = 0). We also observe thatthe density values at which these jumps occur shift to higher densities, as the passenger arrival rates areincreased.This interplay between α and ρ suggests that low values of density allow for passengers to accumulateon different sites of the sidewalk, which forces buses to make more frequent stops. On the other hand,4 v a = 0.2= 0.3= 0.4= 0.5= 0.6= 0.7= 0.8 * b f B = 0.1 f B = 0.4 f B = 0.7 f B = 1.0 Figure 3: (a) Comparisons of speed as a function of the passenger arrival rate α for different densities ( f B = 1). Sudden dropsin vehicle speeds as α is increased indicate a crossover from non-platooned to platooned flow. However, no clear transition isobserved for ρ ≥ .
6. (b) The crossover transition involves a critical value α ∗ for different densities and bus fractions. Thevalue of α ∗ increases monotonically with ρ . increasing the density of buses not only accounts for the pick-ups of these passengers, but also preventsthe accumulation of passengers in the system. An interesting consequence is that the average speed of thesystem actually increases.We can look at the interplay between ρ and α in another way, by varying α for fixed values of ρ . Weobserve that the crossover from non-platooned to platooned states depends on α (Fig. 3a). For a fixeddensity, once α becomes sufficiently large, passengers accumulate in the system, which results in morefrequent stops for buses. This sets the formation of a platoon of vehicles, while at the same time allowingfor more passengers to accumulate ahead of the platoon.Figure 4 illustrates the crossover from either a free flow or congested phase to a platooned phase for twodifferent values of α . A platoon develops from an initial non-platooned configuration as arrival rates areincreased above some critical value α ∗ . For a system with a density of ρ = 0 .
4, we find that α ∗ ≈ − . (Fig 3a). In the case of α = 10 − , platoons do not form in the system at all. However, when we look at thecase of α = 10 − . , a platoon develops after 600 timesteps, despite having the same initial configuration forthe case α = 10 − . As passengers accumulate in the gaps between buses, the decrease in the speed of busescreates larger gaps. This feedback loop drives the formation of platoons and also results in longer passengerwaiting times.For the case of mixed traffic (Fig. 2, α = 10 − ), the transition from a platooned to a non-platoonedphase occurs at lower densities with increasing f B . This observation is still consistent with the notion of thetransition from platooned to a non-platooned phase due to increasing density of buses. As the effective busdensity scales with f B , higher densities are needed to compensate for lower f B . Thus, reducing f B shiftsthe crossover transition to higher densities.We can make two observations of our single lane model. First, we see that there exists a ρ ∗ ( α ) responsiblefor a crossover behavior from a platooned phase to either a free-flow or congested phase. When ρ < ρ ∗ , alow f B gives more time for passengers to accumulate in the gaps between buses. At the same time, buseswill tend to clump together, creating larger gaps. Both processes aid in the formation of platoons. When ρ > ρ ∗ , the gaps between buses are sufficiently small to substantially reduce passenger accumulation thatcause a cascade of slowdowns. Thus, smaller or no platoons are formed, and we obtain a non-platoonedphase.Secondly, the crossover from a platooned to a non-platooned phase is also determined by α ∗ ( ρ ). Forarrival rates α < α ∗ , the system would be found in the non-platooned phase, while for α > α ∗ the systemwill exhibit platooning. The dependence on ρ is only up to ρ ≈ , beyond which the density-dependentdynamics of the NaSch dominates the system. Figure 3b highlights the interplay between α ∗ and ρ . The5
200 4000100020003000 = 10 = 10 position t i m e Figure 4: Spatio-temporal diagrams of the system of buses ( f B = 1) near the transition point of ρ = 0 .
4, for low passengerarrival rate ( α = 10 − , left) and high passenger arrival rate ( α = 10 − . , right). Increasing α induces a transition fromnon-platooned to platooned flow. crossover transition value appears to scale as α ∗ ∼ exp( ρ + c ). We observe the scaling only until ρ = 0 . ρ crit . While the phase transition from free flow to congestion is undoubtedly an important aspect in the studyof transport and vehicular traffic, the densities involved are typically low densities. Since the problem ofcongestion in city traffic involves densities above ρ crit , this particular phase transition is rather insignificantwhen it comes to policy intervention since controlling vehicle volume is difficult. However, the crossovertransition of passenger–bus interactions is present for a larger range of densities. The key to managing trafficflow at densities above ρ crit must then lie in avoiding the formation of platoons.The behavior of buses in our simplified model has underlying assumptions that are absent in real traffic.Buses in our model have infinite capacities, do not wait for passengers, and our model does not have passengerdrop-offs. What happens when we relax these assumptions? From infinite to finite bus capacities.
With each bus having a passenger capacity c , the “leading bus”gets filled up and eventually stops picking up passengers. Said bus breaks away from the platoon that itleads, leaving the next bus as the new “leading bus”. The notion of a “leading bus” in this case is notspecific to a single bus, as any unfilled bus can be the “leading bus” and cause a platoon. A spatio-temporaldiagram of our model with different capacities shows this phenomenon (Fig. 5).If the passenger volume eventually exceeds aggregate bus capacity, all buses will be full after some time T cap . In the case of Fig. 5, T cap ≈ T cap scales linearly with the passenger capacity c , for all densities and passenger arrival rates (Fig. 6). From no-waiting to waiting for passengers.
Allowing buses in our model to stop for a duration T wait isanalogous to waiting for passengers or making scheduled stops. In effect, this slows down the movement ofthe platoon (Fig. 7a). In the limit that α is sufficiently large, we can calculate the average speed of theplatoon as ¯ v = T wait +1 . In the case of our simplified model, T wait = 1 and we have ¯ v = 0 . α (Fig. 7b). This ties into the mechanism of platoon formation, as when buseswait longer at stops, the average speed of the platoon decreases. For the same pedestrian spawning rate α ,this decrease in platoon speeds results in more pedestrians accumulating ahead of the platoon. Adding drop-offs.
Our description of the model does not explicitly let passengers alight. While the6
200 4000100020003000 c = c = 40 position t i m e Figure 5: Spatio-temporal diagrams of the system of buses ( f B = 1 , α = 10 − . , ρ = 0 .
4) with different passenger capacities c . The system with finite capacity ( c = 40) platoons like our simplified model ( c = ∞ ) for some time T cap ≈ term picking up passengers is useful in creating a mental picture of what goes on in the model, a betterinterpretation is that these are stopping events as a result of passengers boarding and alighting the bus.Instead of explicitly modeling multiple passengers boarding (and alighting) buses at the sidewalk, buses canbe filled (and emptied) by multiple passengers at a single stop. This works in the regime where the net fluxof passengers do not fill up buses.When buses are close to capacity, explicitly modeling drop offs influences the dynamics of buses. In Fig.5, we show that as buses reach their capacities, they break away from the platoon, and can end up at thetail of a different platoon. If no passengers alight, the bus catches up with the tail of the next platoon.On the other hand, if a passenger alights within the route, the bus once again becomes a potential platoonstarter. In this way, a bus can alternate between a platoon starter, and a platoon trailer, depending onthe distribution of drop-off points of passengers. If passenger drop-offs are spread uniformly throughout theroute, we expect to see several smaller platoons instead of a single large platoon. Note that some commuteroutes would have a concentration of pickups at the start of the route, and drop-offs at the end of the route.This situation would resemble the dynamics of buses at capacity, with faster transit speeds as stops midroute are minimized. We now extend the single-lane model to two lanes. In this model, we restrict buses to drive along thesecond lane, while cars can move freely on any lane. As buses stop to pick up passengers, they block themovement of vehicles on the bus lane. Lane changing allows cars to overtake stopped or slow-moving buses,though the maneuver can cause congestion on the adjacent lane. Buses cause bottlenecks, which greatlyreduce the flow of vehicles in the system. These bottlenecks are similar to work zone scenarios [26], withthe key difference being that the flow reduction is temporary and that the bottleneck is non-stationary.In the case of multiple lanes, we see three regions in the fundamental diagram for the case of α ≥ − (Fig. 8). The first two regions are from 0 < ρ < .
12 and 0 . < ρ < .
66. In the single lane case, when α is sufficiently high, a bus moves slowly as it picks up passengers, which obstructs all other vehicles behindit. In the multiple lane case, cars can overtake the slow-moving buses and prevent the platoon formation(0 < ρ < . . < ρ < . T c a p × = 10
20 40 60 c × = 10
20 40 600.00.51.01.52.02.5 × = 10
1= 0.1= 0.4= 0.8
Figure 6: The time it buses to reach capacity T cap as a function of the passenger capacity c for different combinations of α, ρ . T cap scales linearly with the passenger capacity for all combinations of density and passenger arrival rate. q a T wait = 1 T wait = 2 T wait = 3 T wait = 5 T wait = 9 * b Figure 7: (a) Fundamental diagram for different bus waiting times T wait . Waiting for passengers for a longer duration reducesthe average speed of buses in the system, and shifts the crossover transitions to higher densities. (b) The critical value α ∗ vs.densities for different bus waiting times. Increasing T wait changes the slope of log α ∗ vs. ρ . multi-lane model do not appear as a single clump of closely spaced vehicles followed by a large gap of space.Instead, many smaller platoons form, with overtaking cars utilizing the gaps between platoons. However,this region does not exist for the case of α ≤ − , as the arrival rates of passengers is not high enough tocause the platoon formation. Thus, in the absence of platoons, the speeds of buses and cars are similar, andwe recover dynamics similar to the NaSch model.The remaining region ( ρ > .
66) marks the dissolution of the platoons in the bus lane. In this region,the density has increased enough such that both lanes move at similar speeds. As a result, the effect of buspickups is indistinguishable from the congested dynamics of the NaSch model. We also observe this phasein the case of a crossover transition occurring when ρ < .
66 (such is the case when α = [10 − , − ]).In this multi-lane scenario, stop durations increase when buses are allowed to wait for passengers. Thisslow-down has a two-fold effect: it reduces overall platoon movement; and it impedes the adjacent lane asmore cars move out of the bus lane. 8 .0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.50.6 q =10 =10 =10 =10 =10 position t i m e lane 1 lane 2 Figure 8: (Left) Fundamental diagram of multi-lane traffic. Jumps in q are not as obvious as in the single lane case, but arestill observed for α = (cid:2) − , − (cid:3) . (Right) Spatio-temporal diagram for ρ = 0 . , α = 10 − , with cars (black) and buses (red).Coupling of both lanes occurs at the trailing end of platoons due to cars changing lanes. In the previous sections, passengers may board and alight on any segment of the road. Such a scenario iscommon in cities like Manila and Jakarta, whereas having designated bus stops is a more common occurrenceworldwide. As such, it is natural to take accessibility into account when designing stop locations. Closerspacing between stops can improve accessibility for commuters. But when spaced too closely, stops slowdown transit travel speeds [27].The decentralization of bus franchise operators in cities like Manila makes it challenging to plan fordifferent aspects of a transit system. In routes that do not have sufficient demand, buses will be infrequent.Uncertainty in arrival times of buses in such cases can lead to either long waiting times, as well as areduction of demand as commuters search for a more reliable alternative. Implementing bus timetables canhelp manage passenger expectations and establish the reliability of the transit system.In comparing realizations with different distances between designated stops d , we assumed the same totalnumber of passengers. Thus, we fixed the value of the pedestrian arrival rate for the entire system A = α (cid:0) Ld (cid:1) per timestep. With fewer stops (larger d ), the arrival of passengers at each stop, α , rises commensurately.We ran simulations for d ∈ [1 , , , , A = α max Ld max , such that when d = 256, α = 0 .
1. Amaximum value of d = 256 is specifically chosen for our cell size of 5.5 m, as this corresponds to a 1.408 kmdistance between stops. We choose this based on the assumption that at most, a pedestrian would be forcedto walk 704 m for a destination in between two stops. To account for our choice of d max , we use L = 5120and set A = 2 for simulations in this section. For comparison, we will look at the cases ranging from allbuses ( f B = 1) to a low bus volume, Philippines-inspired case ( f B = 0 .
03) [16].First, we look at the single-lane case of an all-bus system. Increasing the distances between stops improvesthe overall speed of the system for ρ < .
12 (Fig. 9a and 9b). The inflection point at ρ ≈ .
12 correspondsto the crossover transition from platooning to a non-platooned phase. Configurations with d ≤
64 exhibit aninflection at ρ ≈ .
12, a crossover transition which is absent in farther-spaced stops ( d = 256). We observesimilar behavior in the f B = 0 .
03 case (Fig. 9c), where system speeds increase with station separationdistances, but with two key differences from the all-bus ( f B = 1) case. Only d = 1 and d = 4 have crossovertransitions that occur at higher densities, and the case of d = 256 does not have slower system speeds than d ≤
64 for all densities.We attribute the slowdown of vehicles in the case of d = 256 , f B = 1 to the high arrival rates ofpassengers, which form slow-moving jams near the stops where buses have a high probability of stopping.Due to our model design, buses do not wait for passengers at stops. Since the distance (and time) headwaysbetween succeeding buses are short, the high arrival rates of passengers would make succeeding buses stopat the station. If a bus can wait for multiple passengers, buses behind it would not have to stop often, whichwould improve the overall system speeds.In general, we expect the average number of buses that stop to board passengers should be the samefor all realizations of d . Allocating larger separation distances between stops reduces the frequency of stopsmade by buses. This allows buses to maintain their top speeds for longer, and results in a higher system9 .0 0.2 0.4 0.6 0.8 1.0012345 v a f B = 1.00 f B = 1.00 f B = 1.00 f B = 1.00 f B = 1.00 b f B = 0.50 f B = 0.50 f B = 0.50 f B = 0.50 f B = 0.50 c f B = 0.03 f B = 0.03 f B = 0.03 f B = 0.03 f B = 0.03 d = 1 d = 4 d = 16 d = 64 d = 256 Figure 9: Speed vs. density plots for a single-lane model with varying station separation distances d ∈ [1 , , , , throughput.System speeds, however, only paint half of the picture. Travel begins when the pedestrian leaves hisorigin, not at the moment the pedestrian boards a vehicle. Thus, it is necessary to include the time spentwaiting for public transport. We find that the spacing of stops matters most in medium to low densityregimes ( ρ < . f ∈ [0 . , . , ρ < .
32 for f B = 0 .
03, Fig. 10). Having stops close to each otherincreases waiting times of passengers, more so at low densities. Because of platooning, bus delays lead tomore waiting passengers, which in turn causes further delays. Thus, waiting times can reach more than 30minutes when buses are allowed to stop anywhere ( d ∈ [1 , ρ > . f B = 1).Furthermore, the wait is just one timestep: the pedestrian gets to ride the bus one timestep after getting tothe sidewalk. While this looks good from the point of view of passengers, it is quite inefficient since somebuses fail to pick up passengers. We set the net pedestrian arrival rate A = 2, but the number of busesin the system for ρ > .
18 is more than 921. Even if our model does not take into account bus passengercapacities, we can already see that reducing the number of buses may improve system speeds without lossof service reliability.In mixed traffic settings, waiting times do not remain constant with increasing vehicle density. Atsufficiently high road densities, waiting times also increase. Increased waiting times at higher road densitiesare an indication of insufficient supply of buses. The lack of buses, coupled with the slow speeds of vehicleson the road, lead to prolonged waiting times as congestion gets worse (Fig. 10, f B = 0 . habal-habal , which are prevalent in the Philippines. This mentality bleeds into other forms of public transportationsuch as buses, Jeepneys, and AUVs, which do not have designated stops. Our work supports the notion ofplanning out stops that are spaced further apart, but we must also take into account the added time andeffort to the commute of a passenger whose stops would lie in between two stops.The growing popularity of ride-sharing services also causes problems. Although these have infrequent10 f B = 1.00 f B = 0.50 d = 1 d = 4 d = 16 d = 64 d = 256 f B = 0.25 f B = 0.03 m e a n w a i t i n g t i m e ( m i n s ) Figure 10: Mean waiting times at various densities for different station separation distances d . Station distances affect waitingtimes at middle to low density values ( ρ < . f B ), and at highdensities for mixed traffic ( f B < stops, their point-to-point nature is similar to a system of buses that do not have well spaced stops. Entirefleets of these vehicles can end up clogging roads and side-streets while they wait for their next booking.If left unchecked, the sheer number of ride-sharing vehicles waiting for passengers can end up negating anybenefits from a well designed public transport system.
4. Conclusion
Platooned flow is characterized by bunched slow-moving buses that leave gaps on the road. Platoonsdisappear either by having fewer riders or more buses, and results to an equal headway configuration. Withmultiple lanes, platooned dynamics spill over to the non-bus lane, and persists until the platoon is dissolved.Platoon formation is driven by interactions at stops. Spacing out stops speeds up traffic by minimizingplatooning and queuing-induced congestion. However, spacing them too far would make walking to amidpoint destination impractical. Load-anywhere behavior, while convenient for an individual, comes at thecost of slower traffic flow and longer waiting times. To mitigate the formation of platoons, sticking to a busschedule helps. Cities may also incentivize staggered work schedules while ensuring adequate bus supply.Though our model did not take into account multiple passenger arrivals on sidewalk cells, finite buscapacities, and alighting scenarios; we can estimate their effects. If we account for multiple passengerarrivals, the waiting times belong to the first passenger in the queue, and thus a lower limit. Adding finitecapacities to buses dissolves platoons when buses are full, but the resulting increase in vehicle speeds comeswith longer passenger waiting times. Allowing buses to wait for passengers at stops, rather than the otherway around, slows down the overall movement of platoons resulting to congestion.Microscopic models, like ours, would benefit from the increasing use of real-time sensors that drivesmart city initiatives worldwide. With rapid parameter calibration, we can realize the idea of immediately11ctionable predictions.
Acknowledgment
The authors acknowledge the support provided by the Commission on Higher Education-PCARI [IIID-2016-006].
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