1.1 Holding back buses, stop-skipping, deadheading

There have been much efforts carried out in attempting to address bus bunching. A common idea is to hold back some buses when bunching is perceived to be imminent, with extensions to adaptive and dynamic controls according to real-time situations [7–19], although holding back buses generally slows down the entire system. A holding strategy could be executed based on the goal of sticking to a schedule or maintaining the buses’ headway.

In contrast to holding buses, some have considered the option of skipping bus stops to speed up a slow bus [14, 20–24], as well as deadheading where an empty bus is dispatched directly to a target bus stop [21, 24–27]. Deadheading may be theoretically viewed as a special case of stop-skipping [24]. In the stop-skipping strategy, a simple formulation would be to designate a stretch of bus stops to be skipped and passengers who wish to go there would alight at a bus stop just before that stretch. These passengers are forced to wait for another bus, of course, losing priority for seats to those already on board [21, 23]. The solution method usually involves numerically optimising some objective function, using nonlinear integer programming for instance. An interesting extension to stop-skipping was investigated by Ref. [23]: A bus would allow passengers who wish to alight at a bus stop (where it would otherwise skip) to do so, and if so it would then also allow boarding at that bus stop. The performance of this policy, however, is quite susceptible to the passenger distribution patterns especially if every bus stop generally has people who would like to visit. Incidentally, some of these studies aimed to optimise both performance improvement for passengers as well as cost reduction for the transit operators [15, 22, 24].

Alternatively, a study which combined holding and stop-skipping strategies found this to be undesirable as such tight controls may induce poorer performance [28]. Perhaps a worthy way forward is to have buses with wide doors or “no doors” [29–31], so that multiple passengers may simultaneously board and alight as compared to queuing through a single narrow entry. However, this would risk other issues like fare evasion. Another possible approach involves city planning where the locations of bus stops along the bus routes are engineered to facilitate efficiency of the bus system [32]. Some other work considered optimising some objective function, involving various algorithms to enhance the bus system [33–35], as well as being data driven [36].

1.2 Boarding limits: A no-boarding policy

Buses can be sped up by limiting the number of people who are allowed to board the bus [37–39], i.e. a no-boarding policy. An initial investigation on trying to speed up a bus by limiting the number of people allowed to board was carried out by Ref. [37], where they combined this with the holding strategy. The performance of the bus system was compared with only the holding strategy and without any form of control. It turns out that the inclusion of boarding limits is beneficial, as observed from their simulations. The setup there comprises buses serving a loop of bus stops, where they start from one terminal and end at that same place after completing the loop. The optimal actions to be taken by a bus when it is at a bus stop are determined by an objective function (solved numerically, subject to a number of constraints) that comprises waiting time of commuters at bus stops, extra time spent on buses due to holding, as well as additional time spent at bus stops when passengers are denied boarding or if capacity limit is reached. All buses are assumed to have identical speed, no overtaking is allowed, and they introduced some kind of a “dummy bus” as a boundary condition that ensures that all passengers are picked up, after all other buses have returned to the terminal.

This model which dealt with boarding limits was subsequently improved and expanded in Ref. [38], with a similar objective function but with many more constraints to satisfy. Once again, the objective function that determines the action of a bus when it arrives at a bus stop is solved numerically, this time without the inclusion of the “dummy bus”. With a more elaborate model, it allowed for different finite bus capacities and different bus speeds. As before in Ref. [37], they found that the inclusion of boarding limits would help improve the system. Furthermore, the boarding limit strategy led to better comfort for passengers, since the passenger load would be spread out more evenly compared to having no such implementation. However, they noted that such improvements would depend on the conditions of the demand. In particular, the most significant benefit derived from the boarding limit strategy occurs when the headways between buses are short and when passengers demand is high.

Whilst the studies in Refs. [37, 38] are based on numerical optimisation of some objective function, Ref. [39] attempted an analytical treatment on how boarding limits would help improve the bus system. However, their theoretical construction did not appear to consider alighting (i.e. commuters are continually packed into the buses). Unlike those in Refs. [37, 38], they only studied the boarding limit strategy without involving the holding strategy.

1.3 This paper: An analytical treatment on the no-boarding policy, with numerical simulations and validation based on a real university bus loop service

Recently, work by Ref. [1] modelled a system of buses serving a loop of bus stops as a system of coupled oscillators [40], where each bus is represented by an autonomous oscillator. By doing so, bunching is derived as the ramification of phase synchronisation of these individual oscillators due to coupling at the bus stops, when demand exceeds a critical threshold. In fact, that framework revealed an insight on real buses where different human drivers have different natural speeds (frequency detuning): A system of buses having a greater degree of frequency detuning can help bunched buses to undo bunching since these differences in natural speeds allow a faster bus to break away from a slower one. As a result, such a system of buses do not exhibit sustained bunching during lull times when demand for service is low (below the critical threshold), but would contain clusters of sustained bunching or even be completely bunched during busy periods (above the critical threshold).

With the framework of bus bunching being described as a synchronisation phenomenon, Ref. [1] pointed out a common property of such systems of coupled oscillators, viz. they can be periodically entrained by an external force to preserve a desired configuration, just like an ordinary clock can be periodically perturbed by a centralised high-quality time-keeping source to maintain its accuracy [40]. Here for this paper, we focus and expand on that property by studying this mechanism for preventing bus bunching: The implementation of a no-boarding policy to achieve synchronisation of buses that maintains a reasonably staggered configuration. According to this policy, a bus would always allow alighting but disallow boarding of new passengers at bus stops if certain criteria are met. The goal of the policy is to speed up a “slow” bus so that the bus immediately behind would not end up bunching with it. The rationale for no-boarding is justifiable to the passengers waiting at the bus stop, since the following bus is nearby and approaching soon.

This idea is in contrast to the holding back strategy which slows down the system instead and lengthens the time spent by passengers on the bus [7–11, 13–15], since faster buses are held back in favour of slower ones in order to maintain their staggered headway. Moreover, if buses are held down at some point along the route, then designated waiting bays must have been planned and allocated beforehand otherwise they would obstruct traffic—something unfeasible in dense city centres which are already short of sought after prime land. A no-boarding policy does not require any specific engineering, making it readily applicable in any existing city or landspace. Whilst this idea is similar to skipping bus stops [14, 20–24], it always allows passengers to alight at their desired destinations. Plus, the instruction to the passengers is arguably clearer: “Please do not board.” This is direct, compared to the potentially confusing, “We are skipping this, ⋯, and that bus stops.”.

In this paper, we derive a comprehensive analytical theory of the no-boarding policy which improves upon the work in Ref. [39], as we account for passengers alighting the bus both in our theory and simulations. The aim of the no-boarding policy here is to maintain a reasonably staggered headway between all buses serving a loop of bus stops. Our theory allows us to understand the mechanism of no-boarding leading to stable headway between buses, as well as deriving when such a no-boarding policy would fail or backfire. As was noted in Ref. [38], this would work well for buses with short headway and high demand from their simulations. In fact, we will find this to be true—corroborating with Ref. [38], and are actually able to work out analytically that the no-boarding policy backfires in the lull period defined in Ref. [1], due to buses having different natural speeds. The full mathematical derivation for this critical transition where no-boarding fails is given in Ref. [41]. This paper then concludes with extensive simulations based on parameters measured from a real university campus loop shuttle service, where buses serve a loop of bus stops continually with no start/stop terminal. Live data for those buses can be found on a website that is listed as Additional Information at the end of this paper. In contrast to Refs. [37, 38], we allow overtaking in our theoretical analysis and simulations as this is commonly observed in the university shuttle bus loop service [1].

1.4 Outline and organisation of this paper

After some preliminaries in the next section, we proceed to investigate how such a no-boarding policy can significantly improve passenger service. To do so, we carry out exact analytical calculations in the case where buses move with identical natural frequency (which is the way forward, when self-driving buses become the norm in the near future [42]) to show that the average waiting time of passengers can be reduced dramatically compared to a system with no such interference policy where all buses end up bunching. This is because the no-boarding policy ensures that the buses are reasonably spaced out and not bunch. In fact, two possible no-boarding policies are explored, by looking at the phase difference of a bus from the bus immediately ahead (Section 3) or immediately behind (Section 4). Our analytical results provide clear evidence on the efficacy of this mechanism and how exactly improvement is achieved, even though it may appear a bit simplified under our assumptions.

To illustrate and test our analytical results, we carry out extensive simulations which turn out to agree very well with the analytical results. These simulations also provide further insights into the actual behaviour of the bus system under the no-boarding policy. Apart from that, we apply this no-boarding policy to a simulation of a bus system having M = 12 staggered bus stops in a loop, served by N buses moving with different natural frequencies, in Section 5. This setup is modelled after a university shuttle bus service [1] to see how the no-boarding policy would pan out in a real human-driven bus system, instead of programmable self-driving buses with identical natural frequency. In addition, we describe a simple adaptive algorithm which dynamically determines what angle (phase difference) to implement the no-boarding policy, in a bus system where the rates of people arriving at the various bus stops differ and are stochastic in Section 6. Those real parameters used in our simulations are measured from the university shuttle bus service [1], thereby providing support that our analytical theory’s results are indeed applicable to realistic systems with varying demand for service. The simplicity of this adaptive algorithm makes it general enough to be directly implemented in generic bus systems, which can be further improved and fine-tuned towards specific systems especially given prior knowledge and empirical data of the real systems.

A flow diagram in Fig 1 summarises the layout and organisation of this paper.

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Fig 1. Outline and organisation of this paper.

https://doi.org/10.1371/journal.pone.0230377.g001

3.1 Analytical theory for N = 2 buses serving M = 1 bus stop

3.1.2 Lower bound to θ₀: θ_min.

Note that there is a lower bound to θ₀ where no-boarding is implemented if Δθ > θ₀. This lower bound arises, because if θ₀ is too small, then the buses would too frequently disallow boarding whenever Δθ > θ₀ with the repercussion that they are not picking up people faster than people arriving at the bus stop—leading to an unbounded growth of people at the bus stop waiting for service. We can calculate the lower bound to θ₀, denoted as θ_min as follows. In Fig 2(a) (top panel), bus A implements the no-boarding policy since it sees Δθ = θ_min, and leaves the bus stop. Fig 2(b) (top panel) is the moment when bus B just arrives at the bus stop, and the phase difference remains as θ_min—strictly speaking, since bus B is now at the bus stop, it is bus B that measures its phase difference with respect to bus A. Therefore, the phase difference is Δθ = 2π − θ_min < π. As bus B remains at the bus stop, bus A advances until in Fig 2(c) (top panel) where Δθ grows to θ_min. If the time taken by bus A to go from (b) to (c) is less than τ, then bus B has to implement no-boarding and stop for a duration less than τ given in Eq (3). Consequently, bus B picks up less than L people given in Eq (5)—monotonically creating a growing list of disgruntled people waiting for service. This θ_min is the critical lower bound such that bus B is able to spend exactly τ to be able to pick up the required L number of people. Since the time interval from (b) to (c) is (2θ_min − 2π)/ω (the time taken by bus A to travel) and this equals to τ (the stoppage duration of bus B), we get (6) (7) where we have defined x_min ≔ θ_min/2π being a normalisation unit. We find that this minimum angle depends on k. If demand is higher, then the lower bound to the angle for implementing no-boarding is larger, i.e. the buses should be given greater leeway and duration to actually let people board since there are more people demanding service.

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Fig 2. N = 2 buses serving M = 1 bus stop in a loop.

The buses travel clockwise, and implement the no-boarding policy if Δθ > θ₀. Top panel: Shown here is the situation where θ₀ is the lower bound θ_min, before the system is picking up people slower than people arriving at the bus stops. Bottom panel: In steady state, the phase difference Δθ(t) fluctuates around the effective angle θ_eff, which is bounded by θ₀ (i.e. θ_eff < θ₀). On average, bus A and bus B would stop over the same duration τ and pick up the same number of people L. Since bus B is lagging due to its large phase difference with respect to bus A, a portion of (1 − π/θ_eff) × 100% of the people waiting for bus B would have to wait longer to board bus A instead, because bus B would have implemented the no-boarding policy when Δθ > θ₀. Hence if θ_eff is closer to π, then fewer people would have to be denied boarding by bus B. A closer-to-antipodal configuration would improve the overall average waiting time, given by Eq (15).

https://doi.org/10.1371/journal.pone.0230377.g002

3.2 Simulations based on real bus loop service data

A recent study on Singapore’s Nanyang Technological University’s (NTU) campus shuttle bus system comprising N buses serving M = 12 reasonably staggered bus stops in a loop was carried out to investigate the mechanisms of bus bunching [1]. The study reported that these buses typically have different natural speeds, due to different human drivers’ driving styles. There are distinctively slow drivers and fast drivers. Their average periods around the loop without stopping are within a range of 12 to 18 minutes, i.e. f_i ∈ [0.93, 1.39] mHz. Further data analysis on this shuttle bus system found that the ratio of the rate of people arriving at each bus stop to the rate of people loading up the bus, k ≔ s/l, averaged over all M = 12 bus stops, is of the order of k ∼ 0.020 during lull periods and k ∼ 0.065 during busy periods. This information implies that with the people taking about a second to get up the bus so that l ∼ 1 person per second, then the arrival rate of people at a bus stop is s ∼ 0.020 person per second (1.2 person per minute) during lull periods and s ∼ 0.065 person per second (almost 4 people per minute) during busy periods.

Based on these real data for M = 12 bus stops, we can test our analytical theory of the no-boarding policy for a realistic simulated system having N = 2 buses serving M = 1 bus stop in a loop. Let the two buses have identical natural period of T = 12 minutes to complete one loop without stopping. Identical natural period would be applicable to programmable self-driving buses which would be rolled out in the near future [42]. The case of human-driven buses with different natural periods is investigated in Section 5. Normally, there are three or four buses serving the route, with up to seven or even eight buses during busy times. Since we are only dealing with two buses here, we consider a lull period with k = 0.010. This would then produce reasonable number of people demanding service and they are able to fit within the buses’ usual capacity of an order of 50 passengers. Nevertheless, we emphasise here that these results hold for any value of k even in busy periods, since there is nothing in the theory that specifically restricts to the lull period or any value of k. Our use of a lull k = 0.010 in this subsection is simply because we are here considering N = 2 buses and pushing up k would make the number of people on each bus higher—which is fine, unless a maximum bus capacity is additionally imposed. Subsequently when we generalise to N buses (last part of this section, below), then more people (higher k, busy period) can be served without unrealistically carrying “hundreds of people per bus”.

Anyway, suppose each person takes l = 1 second to board or alight, which translates to a people arrival rate of s = 0.010 person arriving per second or one person arrives every 100 seconds. Since k = 0.010 is a value applicable to M = 12 bus stops in the NTU bus system, for our simulations with only M = 1 bus stop, we should multiply this arrival rate by 12 to generate a comparable proportion of people using the service, i.e. a person arriving every 100/12 seconds. Note that typically people would travel from one part of the loop to another, travelling say an average of about half a loop (or less, since people can take the bus service going in the opposite direction for a shorter path), rather than making one full loop. As our simulations with M = 1 bus stop force people to travel one full loop, we halve the arrival rate in order to more realistically reflect the number of people on the two buses. Therefore, we have one person arriving every 100/6 seconds or rounded down to 16 seconds (because the simulations run on discrete time steps of 1 second).

In the usual situation with no intervention which corresponds to setting θ₀ = 360° = 2π radians, the two buses bunch and would remain so since their natural speeds are identical. In steady state, our simulation results are as follows: The average waiting time for passengers is 0.515±0.299 units of T = 12 minutes (0.299 is one standard deviation, since people arrive uniformly over time with some having to wait longer than others for the bus to arrive), the time spent on the bus is 1.032 units (there is no uncertainty here, since every passenger spends this same amount of time on the bus), giving a total travel time of 1.546±0.299 units. Both buses spend units stopping at the bus stop (there is no uncertainty here, since the bunched pair of buses always spend this same amount of time stopping at the bus stop) to allow alighting followed by boarding (bunched buses share the load). Each bus load carries L = 24 people. These value of and L are exactly the values predicted by Eqs (4) and (5). In addition, Eq (15) gives for θ_eff = 2π radians (x = 1) and k = 1/16, an average waiting time of units which is in agreement with our simulation value.

Next, we show results for this lull period where a person arrives every 16 seconds at this single bus stop, with the no-boarding policy in action. We carry out separate simulations for various (but fixed) θ₀ ∈ {180°, 181°, …, 360°} (360° means no implementation). Fig 3(a) indicates excellent agreement between the analytical theory and the simulation results (which are averaged over many rounds during steady state). The stoppage duration is typically averaged around units for both buses [with standard deviation of the order of 0 to 0.025 units, depending on the actual θ₀ being implemented, since Δθ(t) fluctuates around θ_eff and depends on θ₀], and the number of people that they pick up is about L = 24 each (with standard deviation of the order of 0 to 20 people, depending on θ₀). These graphs also affirm that when the implemented angle is below the lower bound θ₀ < θ_min, then τ and consequently L are less than the demand level [see also Fig 5, which shows the lower bound θ_min for any k].

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Fig 3.

(a) Graphs of and L versus θ₀ for each of the two buses. These simulation results agree with Eqs (4) and (5), as well as the lower bound θ_min from Eq (7). Shown here [as well as in (b)] is for k = 1/16, and these features similarly hold for any value of (fixed) k. (b) Graphs of Δθ(t) versus time for each of the two buses. The no-boarding policy kicks in if Δθ > 225°.

https://doi.org/10.1371/journal.pone.0230377.g003

Fig 3(b) shows the phase difference for one of these runs between the two buses, measured from each with respect to the other (so they always sum up to 360°) as a function of time. They start off antipodally staggered and very quickly one bus “goes chasing after” the other “slower one”. Here, the no-boarding policy is implemented if Δθ > θ₀ = 225° which prevents them from bunching, and they settle on a steady state where their phase difference fluctuates around the median of 204.5° or mean of 209.5°. From the simulations, the average waiting time is 0.294±0.163 units, certainly much shorter than the bunched case of 0.515 units. It turns out that according to Eq (15), it is the median phase difference of θ_eff = 204.5° corresponding to an average waiting time of units, which closely matches the simulation value. In contrast, the mean phase difference plugged into Eq (15) gives 0.308 units, which is a slight overestimate.

These all generalise to various implemented angles θ₀ as shown in Fig 4. Shown here is a collection where each simulation data point is an independent run with different (but fixed) θ₀. In each plot of Fig 4(a), 4(c) and 4(d), the (red) dash line is the analytical result from Eq (15), whilst the (blue) dots are simulations points. Fig 4(a) shows a “naïve” plot of the simulation results for the average waiting time versus the implemented angle θ₀. There is some conspicuous shift between the analytical line and the simulation points. This is because the effective phase difference θ_eff is smaller than the implemented one θ₀, with the exception of the “horizontal tail” of points for θ₀ ∼ 360°. That horizontal tail appears because for those extremely large θ₀, the no-boarding policy is unable to prevent bunching and once bunched the buses stay bunched, i.e. their effective phase difference is 360°. But what is a suitable representative value of θ_eff < θ₀ in general? Fig 4(b) plots the relationship between θ₀ and the mean as well as median angles, as measured from the steady state part of the simulations. The mean seems to be biased, in the sense that the gradient of the “mean line of points” is not parallel to y = x, i.e. as the implemented angle θ₀ gets smaller, the degree of shift on the mean angle is smaller. On the other hand, the median angle shows a more consistent shift across various implemented angles θ₀. Consequently in Fig 4(c), when the simulation data points are adjusted based on the mean phase difference, there is still a noticeable deviation from the analytical line. Nevertheless, this difference is not too glaring when adjusted based on the median phase difference in Fig 4(d).

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Fig 4. Graphs of versus: (a) θ₀, (c) the mean, as well as (d) median phase differences, respectively.

The effective phase difference θ_eff during steady state being the median phase difference between the two buses more closely matches the predicted value from Eq (15), compared to the mean phase difference. Note that below the lower bound θ₀ = θ_min, blows up by two orders of magnitude (not plotted): at θ₀ = 191°, at θ₀ = 190°, at θ₀ = 189°, …. Similar graphs can be obtained for any value of k. (b) The relationship between θ₀ and the mean as well as median phase differences as measured from the simulations.

https://doi.org/10.1371/journal.pone.0230377.g004

Furthermore, we run separate simulations for this bus system where the no-boarding policy is implemented at various (but fixed) angles θ₀ ∈ {180°, 181°, …, 360°}, for various (but fixed) values of k ∈ {1/100, 1/99, …, 1/5}. With these extensive simulation data, we find that the analytical theory’s predictions are in excellent agreement with the simulation results. In particular, it holds for any (fixed) k which comprises both lull and busy periods. In fact, the analytical curve for θ_min(k) in Eq (15) fits nicely as displayed in Fig 5, where simulations show that if θ₀ < θ_min, then the waiting times for the passengers blow up to multiple (tens of) revolutions. As explained, this is because the buses are disallowing boarding too frequently whenever their phase difference deviates too much from the staggered configuration and Δθ > θ₀ executes no-boarding, to the point where the bus system is not meeting the demand for service. The number of people at the bus stop would grow unbounded as time goes to infinity, when θ₀ < θ_min. If θ₀ > θ_min, then this number remains bounded although it may not always drop to zero since the no-boarding policy would leave some people there.

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Fig 5. Graph of θ_min versus k.

https://doi.org/10.1371/journal.pone.0230377.g005

3.3 N = 2 buses serving many bus stops

With more than M = 1 bus stops, each additional bus stop systematically contributes towards an additional stoppage due to τ where the bus allows alighting/boarding. Obviously such intermediate stoppages at each such bus stop in between would bump up the waiting time. The overall trend corresponds to the analytical result for N = 2 buses serving M = 1 bus stop [see Fig 6(b), below]: the average waiting time decreases linearly with a decrease in θ₀ where no-boarding is implemented if Δθ > θ₀. In particular, the general gradient of the simulation points matches the analytical line but is shifted upwards due to additional stoppages at other bus stops. On top of that, in Fig 6(b) with M = 12 staggered bus stops being 30° apart from each other, we observe discrete jumps at 210°, 240°, 270°, 300°, 330°, where these additional bus stops are located between the two buses.

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Fig 6.

(a) Simulation results for N = 2, 3, 4, 5, 6, 7, 8 buses, respectively, serving M = 1 bus stop. The demand level is fixed at k = 1/16. (b) Simulation results for N = 2, 3, 4, 5, 6, 7, 8 buses, respectively, serving M = 12 bus stops. The demand level is fixed at k = 1/100. The analytical curves are for those serving M = 1 bus stop. The simulation points have average waiting times which are higher than their corresponding analytical curves, since these buses serve M = 12 bus stops. Furthermore, there are discrete jumps on the simulation points whenever an extra bus stop is present. (c) Graphs of the piecewise continuous curves for N = 2, 3, 4, 5 as given by Eq (18) with subtracted away, together with the corresponding smooth curves Eq (18) that pass through the boundary points of each line segment, for the respective N. These smooth curves are hyperbolas, and approach an “L”-shaped asymptotic curve as N → ∞. (d) Legend for (a) and (b).

https://doi.org/10.1371/journal.pone.0230377.g006

3.4 N buses serving M bus stops—Analytical theory and simulations

We can directly generalise the analytical theory from N = 2 buses to any N ≥ 2 buses serving M = 1 bus stop. Here are the results:

1. The stoppage duration for each bus is (16)
   The geometric series expansion is valid, since k < 1 and N ≥ 2.
2. The lower bound to θ₀ is (17)
3. The average waiting time for passengers at the bus stop before a bus arrives is as follows. For N buses, there are N − 1 piecewise continuous line segments. The i-th line segment, where i = 1, 2, …, N − 1, is given by (18) where 1/(i + 1) ≤ x ≤ 1/i and x ≔ θ_eff/2π.

Eq (16) is derived by having a total of s(T + τ) people per revolution being carried by the N buses, so each bus carries s(T + τ)/N and they do so over an average time interval of τ/2 (since the other τ/2 is for people to alight). This gives s(T + τ)/N = lτ/2, which leads to Eq (16). For Eqs (16) and (17) though, we directly calculated all these expressions individually and case-by-case by hand up to N = 6 in the manner similar to the N = 2 buses serving M = 1 bus stop case in Section 3A, and then write down the generalisation to any arbitrary N = 2, 3, … buses serving M = 1 bus stop. In particular for Eq (18) the calculation steps are exactly those for N = 2 buses, which are applied systematically and tediously to more and more buses, one by one, calculating the luckiest and unluckiest people to board each bus, taking note of the proportion of load carried by each bus or each group of bunched buses.

Let us illuminate the content of Eq (18). For N buses, there are N − 1 piecewise continuous line segments labeled by i = N − 1, N − 2, …, 1, corresponding to increasing x from x_min to 1. The gradient of the line segment labelled by i = N − 1 is the steepest, and these line segments monotonically become less steep as we decrease i until i = 1 (but all have positive slope). The line i = N − 1 is the situation where all N buses are unbunched. For i = N − 2, there is a pair of bunched buses, with the remaining all unbunched. For i = N − 3, there is a triad of three buses bunched into one unit, with the remaining all unbunched. This goes on until i = 1, where all N buses bunch into one single unit.

Fig 6(a) shows the simulation results for N = 2, 3, 4, 5, 6, 7, 8, respectively, serving M = 1 bus stop in a loop, with the corresponding analytical curves Eq (18). Note that the gap between the analytical curve and the simulation points is due to this plot being with respect to the implemented angle θ₀ instead of the effective angle θ_eff. With N buses, there are N − 1 independent local phase differences between adjacent buses, since these N local phase differences sum up to 2π. Eq (18) with N > 2 assumes that all N − 1 independent effective angles between any local pair of adjacent buses are equal, whereas in the actual system, they are not and sometimes even differ substantially. Making a single representative θ_eff out of those different local phase differences would introduce bias, sometimes giving grossly wild results especially with larger N. Plotting against θ₀ would cleanly circumnavigate this and not artificially tamper with the data points, though we would have that shift between the analytical curve and the simulation points since θ_eff < θ₀.

As mentioned, each of these line segments in Eq (18) represents different situations corresponding to different number of bunched buses. For example with N = 3, when the implemented angle for no-boarding θ₀ is such that θ_min < θ_eff < 180°, the three buses are not bunched. However, if θ₀ is larger such that 180° < θ_eff < 360°, then two of these buses would bunch such that the system becomes a two-bus system where one of them comprises two bunched buses with twice the loading rate. It turns out that this is actually less efficient than a system with two unbunched buses where they have the same loading rate. Intuitively, this is because in the former the bunched pair is so-called “faster” due to its double loading rate and chases after the lone “slower” bus which has to implement no-boarding. The lone bus only picks up 1/3 of the demand, whilst the bunched pair picks up 2/3 of the demand (1/3 each). Similar ramifications are true for N = 4, 5, 6, 7, 8 buses, as shown in Fig 6(a). The bunched pair becomes “faster” and grows into larger bunched group of buses, if the implemented angle θ₀ is large. A bus system with larger bunched group of buses performs less efficiently compared to smaller bunched group of buses.

With many bus stops, each bus stop adds additional waiting time, just like the case of N = 2 buses discussed in the preceding subsection. Fig 6(b) shows simulation results for N = 2, 3, 4, 5, 6, 7, 8 buses, respectively, serving M = 12 bus stops. In these simulations, we set k = 0.010. Nevertheless, similar results hold for any (fixed) value of k. As the number of buses increases, since we keep the demand level to be the same lull k = 0.010, having more buses would spread out the load and so each bus would spend shorter at the bus stops [Eq (16)]. Therefore, the jump at each bus stop diminishes with larger N.

We do not proceed to repeat the laborious calculations for the general case of N buses serving M bus stops, because the analytical results from N buses serving 1 bus stop are sufficient to describe and understand the behaviour as θ₀ is changed. As the simulation results indicate in Fig 6(b), the precise corrections from what would be painstakingly going through each piece of individual case-by-case calculations only amounts to adjusting for the small shifts due to additional bus stops without affecting the already known behaviour of the system.

There is one important remark regarding decreasing the implemented angle θ₀ towards θ_min. For each N, as , after the linear decrease, the average waiting time begins to increase before eventually blowing up by several orders once θ₀ < θ_min [see Fig 6(a) and 6(b)]. These increases in the average waiting time when are more prominent with more buses, suggesting that with more buses the system is more susceptible to perturbations which nudge the buses to implement no-boarding too soon as its Δθ(t) gets larger than θ₀. In particular, adding more buses does not always seem to lead to the expected improved average waiting time near θ_min. For instance in Fig 6(b), the simulated best average waiting time for N = 8 buses is higher than 0.2 units, way above its theoretical minimum of less than 0.1 units. In contrast, the N = 3 and N = 4 systems do achieve simulated best average waiting times below 0.2 units, which are better than the N = 8 system. Therefore, whilst it is desirable to implement θ₀ as small as possible, one needs to beware that setting it too close to θ_min would risk having the system implement no-boarding too frequently due to fluctuations in Δθ(t).

Incidentally, note that in the expression for the average waiting time in Eq (18), the k-dependence solely arises from . If we define the quantity , then this quantity is purely a function of x ∈ [x_min, 1], for each N. We can immediately write down some properties of V(x, N) [see also Fig 6(c)]:

1. The set of boundary points for each of the N − 1 line segment constituents for each curve of constant N is (19)
   The leftmost boundary point is when i = N, i.e. (20) whilst the rightmost boundary point is when i = 1, i.e. (21)
   The rightmost boundary point for any curve of constant N is the same point. This is where all buses bunch into a single unit, and the average waiting time is half units of T (plus ).
2. The curve (22) passes through all those points in Eq (19). This equation shows that the overall decrease in average waiting time as x decreases (or as θ₀ approaches the staggered configuration) is in fact, hyperbolic.
3. With Eq (22), it is clear that as N → ∞, the curves shift upwards and leftwards, approaching an “L”-shaped asymptotic curve defined by the horizontal line y = 1/2 (from x = 0 to x = 1) and the vertical line x = 0 (from y = 0 to y = 1/2). This is a manifestation of a larger group of bunched buses being less efficient than a smaller bunched group due to greater asymmetry in the system. Higher N also allows for further reduction in waiting time, since there are more buses—with diminishing returns, however.
4. The set of leftmost boundary points for each curve of constant N, viz. (23) as well as their common rightmost boundary point (1, 1/2), all lie on the straight line y = x/2.
5. Eq (18) readily extends to include the degenerate situation of N = 1 bus. Here, θ_eff = 2π as measured from itself to itself since there is just one bus serving the loop, and the average waiting time is always half units plus . So Eq (18) is just the point for N = 1, with i = 0, x = 1 (or θ_eff = 2π), . In this case, k has to be less than 1/2 in order for that lone bus to meet the demand for service.

5.1 Deviation from the analytical curve in the busy phase

It is interesting to note that unlike the previous setup with buses having identical natural frequency, buses with frequency detuning can potentially unbunch. Consequently for the no-boarding policy by looking at the bus immediately ahead, here the buses do not get stuck in the (N − 1)-1 bus system since buses can unbunch. This is manifested by the N = 3 and N = 6 systems in the busy phase [middle and bottom plots on the right column of Fig 8(a)], where the simulation points do not lie on the expected curve for an (N − 1)-1 system but we see that they manage to unbunch themselves to more efficient configurations. Furthermore for the N = 6 system, having six buses with different frequencies turn out to be better than forcefully implementing the no-boarding policy over a wide range of implemented angle θ₀. On the other hand, if the no-boarding policy is based on looking at the bus immediately behind [Fig 8(b)], then the no-boarding policy generally improves the waiting time in the busy phase (right column), since it never allows any bunching in the first place.

6.1 Passenger arrival rates for the 12 bus stops in the NTU loop campus shuttle bus service

Ref. [1] presented data on the NTU loop campus shuttle bus service for the Blue route, measured over the full working week from 16th to 20th of April, 2018 (live data are found here: https://baseride.com/maps/public/ntu/). More specifically, there are two phases: 1) lull, from 4 pm to 5 pm served by 3 buses; and 2) busy, from 9 am to 10 am served by 6 or 7 buses. In that paper, loop averages are obtained where the values for k are averaged over all 12 bus stops. These loop averages produced data points which have relatively little noise, as the inhomogeneity amongst the bus stops are averaged away. The lull period for that week was measured to have a loop average of k = 0.024±0.004 whilst the busy period at peak demand was measured to have a loop average of k = 0.065±0.017. This peak demand was due to a selection of ten time series with the largest demand to obtain a representative value for the highest demand, where each time series is a tracking of one bus during that time interval. As discussed in the previous section, a phase transition between lull and busy was theoretically predicted to occur at , given buses with natural frequencies f_i ∈ [0.93, 1.39] mHz. This value denotes the critical value of k where the buses do not experience sustained bunching if but clusters of phased-locked buses emerge when .

This loop average value of k was obtained by fitting the equation τ = kΔt to the data points (Δt, τ), where the components are averages of the time headways Δt_ij and stoppages τ_ij over all 12 bus stops, respectively. Note that in the NTU buses, passengers alight and board simultaneously via different doors, and so the stoppage is primarily due to the number of people accumulated during Δt_ij at the bus stop, who are then boarding during τ_ij. The error in k is the square root of the mean squared deviation of each data point’s corresponding k from the best fit line’s k. The relatively small errors in the loop average values of k are consistent with the fact that they have reasonable fit with coefficient of determination r² values of 0.59 and 0.77 for the lull and busy periods, respectively. As mentioned earlier, the loop average irons out the inhomogeneity amongst the various bus stops’ k_j.

If we do this linear fit for each individual bus stop however, the measured values of Δt_ij and τ_ij display a much greater error and deviation from the best fit line. The results are summarised in Table 1. We find that the individual bus stop’s error in k_j are generally much larger than their loop average values for both the lull and busy periods, respectively. In fact, the variance is more pronounced during the busy period: Some bus stops like H14/15, CH, H8 have noticably small k_j, with H4 conspicuously having a negative k_j. However, their Δk_j is very large with their r² values being close to 0. This may be due to surges of people (students) collectively heading out to the bus stops at preferred times, say for example, to meet the 9.30 am lectures. The number of people going to the bus stops is thus not uniform over time, but experiences times when many people appear at the bus stops, as well as times when fewer people are there. The great stochasticity on individual bus stops may also be due to people crossing the road to take the bus service in the opposite direction when that bus arrives, since this is a loop service and people travelling antipodally (or near antipodally) may take a bus in either direction. The purpose of this crude linear fit is to set up a simulation environment (next subsection) to illustrate the adaptive algorithm on dynamically determining the best θ₀ in a stochastic and non-stationary environment (two subsections later).

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Table 1.

(a) The values of k_j and their errors, for each of the 12 bus stops in the NTU loop campus shuttle bus service, during the lull period (4 pm to 5 pm). These results are obtained from data measured over the entire working week of 16th to 20th of April, 2018. (b) The corresponding values during the busy period (9 am to 10 am).

https://doi.org/10.1371/journal.pone.0230377.t001

The τ_ij-intercepts for each bus stop are primarily around the order of + 10 seconds to + 30 seconds, which is due to the fact that buses have to wait for clear traffic before rejoining the road (in fact, there are zebra crossings right in front of some of the bus stops, so some people would be crossing the road right after they alighted—impeding the bus from departing) and that the positional data are only updated once in approximately every 10 seconds (which implies that any delay would be recorded in multiples of + 10 seconds). This is consistent with the loop average’s τ-intercept values being about + 20 seconds, as discussed in Ref. [1].

6.2 A simulated environment to model the NTU system

For our simulated environment to model the NTU system, we assume that the buses are programmable such that they all move with the same natural period of T = 720 seconds (excluding stoppages). We generate the rates of people arriving at each of the 12 bus stops according the their k_j values from Table 1 for simulations on the lull and busy periods, respectively. To create some variations in k_j with time, at every fixed time interval, we sample k_j,sample for each bus stop according to a normal distribution with mean k_j and standard deviation Δk_j. Note however that this may allow for negative values of k_j,sample to be sampled, which is invalid and unphysical. To overcome this, we truncate the normal distribution to take only values from 0 to 2k_j in order to maintain the distribution to center around k_j. (For H4 during the busy period with negative k_j, we take |k_j| as the mean). These sampled k_j,sample will be taken as the values for s_j ≔ average rate of people arrival at the respective bus stops, with l ≔ loading/unloading rate set at 1 person per second. Once k_j,sample have been sampled, the number of people arriving at bus stop j is then determined by a Poisson distribution with λ_Poisson = k_j,sample. As mentioned, the values for k_j,sample are resampled every fixed time interval to reflect the varying demand level with time, and this fixed interval is a hyperparameter for this model, i.e. one is free to set any value as desired for the model. We let this interval be the period of each bus T = 720 seconds, i.e. k_j,sample for each bus stop are resampled from their respective truncated normal distributions every 720 seconds or 12 minutes, which seems realistic.

Whilst the 12 bus stops are not perfectly staggered around the loop in the NTU campus, they are quite reasonably spaced out [1]. We place them equally spaced out in our simulated environment. Also, we ignore traffic conditions and let the buses move with constant speed when they are not at a bus stop. These should not be considered as unrealistic simplifications, because of the fact that the simulation environment repetitiously draws out k_j,sample from the truncated normal distribution every T = 720 seconds. They are then further subjected to Poisson distributions to determine how many people actually arrive at each bus stop, thus already generating pretty high stochasticity in the simulated environment.

If buses move with identical natural speeds, they may not be able to unbunch once they are bunched at a bus stop. To overcome this so that the buses may continue to explore other values of θ₀ instead of remaining bunched, we dictate in this simulated environment that if buses bunch at a bus stop, then they may randomly decide to leave. This mechanism thus allows buses with identical natural speeds to unbunch and carry on as if the system gets reset with bunched buses getting repositioned.

6.3 A simple general adaptive algorithm for dynamically determining θ₀

Our proposed simple adaptive algorithm for selecting the angle to implement no-boarding θ₀ motivated by the classical multi-armed bandit in reinforcement learning [45]. In this setup, each bus is an agent, and they only experience one state repeatedly. Each time, they can select one out of a set of actions, measure the reward, and subsequently arrive at the same state to select one action, ad infinitum. The set of actions comprises discrete integer values of θ₀ ∈ (360°/N, 360°), where N buses are serving the loop. This range of allowed θ₀ would ensure that θ₀ ≤ 360°/N never gets picked, which would otherwise always implement no-boarding, as we have found in the analytical theory. Once θ₀ is picked, this is the value where no-boarding would be implemented if Δθ > θ₀ (until the next time the Q-value gets updated and a new θ_0new gets selected, described below). At the start of the run, a value of θ₀ is randomly chosen from its allowed integer values. Each value of θ₀ has a Q-value associated with it, also randomly initialised to some sufficiently high value. These Q-values represent the average waiting time of people at the bus stop for a bus to arrive, associated with that θ₀. Hence, we aim to minimise the average waiting time here, instead of typically maximising the reward in reinforcement learning [45]. One may choose to let each bus (agent) have its own set of Q-values, or all buses share one single Q-table. Here, we adopt the simple setup with one shared Q-table.

When a bus arrives at a bus stop, it allows passengers who wish to alight there to do so. Once this has completed, this bus measures the phase difference Δθ from the bus immediately ahead of it. If Δθ ≤ θ₀, then it proceeds to allow boarding until there is nobody left to board and leave, otherwise Δθ > θ₀ triggers the implementation of no boarding and the bus leaves. For people who are boarded, their waiting times are recorded so that the average waiting time is subsequently calculated. For the case where nobody is boarded and the bus just leaves, the waiting times thus far for people denied boarding at the bus stop are recorded instead. This ensures that the bus receives feedback on performance, even in cases where no-boarding is always implemented and nobody gets boarded. In reality, the information on how long a person has waited at a bus stop for a bus to arrive may be collected by a mobile app, where a person registers the intention to board a bus after just arriving at a bus stop. For the NTU campus shuttle bus system, perhaps a nifty way to do so instead, would be to install WiFi routers at bus stops since essentially students, staff, and anybody who regularly present themselves in NTU would access the university WiFi service. By installing WiFi routers at bus stops, not only does the university provide internet service for people spending time at bus stops, but the routers also count how many people are present at a bus stop waiting for a bus to arrive, automatically and in real time. Of course, this is just an approximate data collection mechanism, since some people may have multiple devices or there may be guests who do not log in to the NTU WiFi automatically. Nevertheless, such an approximation should suffice to obtain a representative average waiting time which requires no effort from the passengers since their devices connect to the WiFi routers automatically.

After some fixed time interval U, the average waiting time of all passengers waiting at the bus stop for a bus to arrive during this time interval is calculated. This “reward” is updated to the Q-value associated with θ₀ by the rule [45]: (26)

Here, α ∈ (0, 1] is the learning rate for the Q-values. After this update, a new action is selected by a bounded ε-greedy criterion: With a probability of 1 − ε, the bus picks a new θ_0new corresponding to that whose Q-value is minimum, otherwise a new θ_0new is picked randomly from [θ₀ − lower, θ₀ + upper]. We choose lower = 30° and upper = 5° which bounds the ε-greedy exploration with a bias towards lower values of θ₀, in order to keep the exploration controllable and headed towards the prior knowledge of the optimal θ₀ to be near and above ∼360°/N, based on the analytical theory in Section 3. We set α = 0.2 and ε = 0.2. These are not decayed but kept constant, to be able to continually adapt to the non-stationary environment with varying k_j. Furthermore, we set U = 2T, so the Q-value is updated every 1440 seconds or 24 minutes. It is the average waiting time for passengers during this interval U which is sent into Eq (26).

6.4 Simulation results on the adaptive algorithm for determining θ₀ on the NTU system

Fig 9 summarises the results of our adaptive algorithm to dynamically determine θ₀, applied on the simulated environment of the NTU system, both in the lull (served by N = 3 buses) and busy periods (served by N = 7 buses), respectively. In each case, a large θ₀ is initially chosen. As the simulation runs, various θ₀ are explored according to the algorithm to minimise the average waiting time of people waiting at a bus stop for a bus to arrive. The plots (left column for lull, middle column for busy) show, as a function of time: 1) the best θ₀ to implement, together with the actual θ₀ taken (which with probability ε is not the current best action, for exploration); 2) the best Q-value (representing the best expected average waiting time, based on the present Q-table), together with the actual average waiting time. Although we start with a large θ₀, through reinforcement learning, the system is able to eventually seek the optimal θ₀, and remain near that value since it has learnt that this has the best Q-value and occasionally bumps around to the next best Q-values when stochasticity drives up the average waiting time. This algorithm is also able to adapt to non-stationary situations, since ε is kept fixed to maintain exploration.

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Fig 9.

Top row: Graphs of best and actual expected θ₀ versus time for lull (left), busy (middle), and a lull case with frequency detuning (right). Bottom: Graphs of best and actual expected average waiting time versus time for lull (left), busy (middle), and a lull case with frequency detuning (right). In each plot, the actual curve is the thin dotted curve, whereas the best expected curve is the thick curve. The red horizontal line in the bottom graphs represents the average waiting time when the no-boarding policy is not implemented.

https://doi.org/10.1371/journal.pone.0230377.g009

Furthermore, when applied to a lull situation where buses have frequency detuning, the system is able to adapt and find the optimal θ₀ to minimise waiting time. The right column in Fig 9 is the adaptive algorithm applied to “setup (1)” from the previous section, where there are N = 2 buses with different natural frequencies, and each of the M = 12 bus stops have k = 0.010. The system is given an initially small θ₀ ∼ 200°, but is able to explore and find the optimal θ₀ ∼ 340° which is slightly better than no implementation of the no-boarding policy [cf. top left plot in Fig 8(a)]. Here, we use lower = 15°, upper = 15° for the exploration, since we have no prior assumption about what the optimal θ₀ should be.

The purpose of this description is to provide a means of picking an optimal θ₀ under uncertain and stochastic real-world conditions, with some suggested hyperparameters like α = 0.2, ε = 0.2, lower = 30°, upper = 5° (or lower = upper = 15° if no prior assumption is made), U = 2T. Whilst this prescription is intended to be general enough to be applicable to generic bus systems, when more specific details of a particular bus system are known, one may certainly fine-tune the hyperparameters and even make modifications (like giving each bus its own Q-table, instead of a commonly shared one) to improve this simple algorithm for specific bus systems.