A model for setting services on auxiliary bus lines under congestion

Abstract

In this paper, a mathematical programming model and a heuristically derived solution is described to assist with the efficient planning of services for a set of auxiliary bus lines (a bus-bridging system) during disruptions of metro and rapid transit lines. The model can be considered static and takes into account the average flows of passengers over a given period of time (i.e., the peak morning traffic hour). Auxiliary bus services must accommodate very high demand levels, and the model presented is able to take into account the operation of a bus-bridging system under congested conditions. A general analysis of the congestion in public transportation lines is presented, and the results are applied to the design of a bus-bridging system. A nonlinear integer mathematical programming model and a suitable approximation of this model are then formulated. This approximated model can be solved by a heuristic procedure that has been shown to be computationally viable. The output of the model is as follows: (a) the number of bus units to assign to each of the candidate lines of the bus-bridging system; (b) the routes to be followed by users passengers of each of the origin–destination pairs; (c) the operational conditions of the components of the bus-bridging system, including the passenger load of each of the line segments, the degree of saturation of the bus stops relative to their bus input flows, the bus service times at bus stops and the passenger waiting times at bus stops. The model is able to take into account bounds with regard to the maximum number of passengers waiting at bus stops and the space available at bus stops for the queueing of bus units. This paper demonstrates the applicability of the model with two realistic test cases: a railway corridor in Madrid and a metro line in Barcelona.

Appendix: A simple comparison of bulk service type queues and M/M/s queues for bus stops

In this appendix, discrepancies are shown between the application of M/M/s queueing models and a bulk-service model evaluated by simulation that accurately reproduces the congestion effects on a bus stop served by a single bus line.

A steady-state queueing model M/M/1 for the average waiting time φ(ρ) of passengers at the bus stop queue as a function of the loading factor ρ of the bus stop following TCRP Web Document 6 (1999), would be as follows:

$$ \varphi_{M/M/1}(\rho) = \frac{ E[h]}{ 2}\biggl(1+\frac{ \operatorname{Var}[h]}{ E^2[h]}\biggr) + \frac{ \rho}{ \mu (1-\rho)}$$

(60)

The first term in (60) is the average waiting time per passenger at the bus stop resulting from bus headway characteristics of dispersion reflected in the coefficient of variation $\operatorname{Var}^{1/2}[h]/E[h]$. If only this term applied, then implicitly, all passengers present at the bus stop could ride on the arriving bus, and no passenger would wait for a second bus (or third bus, …). The second term is the contribution to the average queueing time per passenger of those passengers that cannot be served by the first-arriving bus. This second term depends on the load factor ρ and also on the rate of service μ. Under an exponentially distributed headway, then the first term would reduce to E[h], as $\operatorname{Var}^{1/2}[h]/E[h]=1$ and that μ=c/E[h], with c as the average passenger capacity of buses arriving at the bus stop. Assume now that c is not random and that c=30 pax and that E[h]=90 seconds so that the total alighting capacity of the line servicing the bus stop is 1200 pax/h.

To illustrate this comparison, a simulation model was developed in order to evaluate the average waiting time per passenger during a finite period of service H. The arrival of passengers at the stop was assumed to be Poissonian, and strict FIFO discipline of passengers was assumed to be observed. It was assumed that buses were empty on arrival and that the dwell time model at the stop was deterministic without holding. The probability distribution for headways could be chosen among several options. Figure 7 shows the simulation results for headways exponentially distributed and finite H=180 minutes, (120 bus arrivals). These results appear as a scatterplot in which each point is a simulation shot. A point’s x-coordinate is the resulting loading factor of the bus stop, and its y-coordinate is the waiting time in minutes. Function φ _M/M/1(ρ) appears overlapped on the scatterplot, showing clearly that serious discrepancies of both models appear for ρ>0.6, i.e., under high congestion, and that the M/M/1 model strongly underestimates passenger waiting times. In addition, the scatter of points presents a heavy dispersion as long as the load factor approaches 1. This dispersion is the result of the moderate but realistic horizon length H chosen.

Fig. 7

A comparison between the average waiting times per passenger at the bus stop for bus arrivals with exponentially distributed headway for a period of 3 hours and the average waiting times for the M/M/1 queueing model. The capacity of arriving buses was assumed to be constant

Another set of simulation runs was made with strictly deterministic headways of arriving buses. In this case, the comparison with an M/M/1 model was made by applying formula (60) with $\operatorname{Var}^{1/2}[h]/E[h]=0$ but with same rate of service μ as before. (Notice that if φ _M/M/1(⋅) had been applied in this case under the form φ _M/M/1(ρ)=1/μ+ρ/μ(1−ρ)=1/μ(1−ρ), then c/μ>E[h]/2= the uncongested waiting time at the bus stop.) The simulation results appear in Fig. 8 below showing that model M/M/1 overestimates the waiting times in this case. For a bus stop where buses arrive with random capacity, average waiting times per passengers would have been greater than those shown by the previous simulations. To evaluate the effect of randomness in passenger capacity of arriving buses, a steady-state bulk-service queue model M/M ^[Y]/1 with random capacity was solved analytically for various probability distributions for the capacity. The effects of randomness in capacity of arriving buses on average waiting time per passenger appeared low or moderate, as shown in Fig. 9. This figure shows at right, the graph for function ξ(ρ), i.e., the normalized delay per passenger and per service for three distributions of the capacity of arriving buses. All them are truncated normals on the interval [0, 100] with an average of 50 passengers of capacity. The coefficients of variation for these distributions are 0.0, 0.2 and 0.5. For the M/M ^[Y]/1 queueing model, Table 8, below, shows coefficients $\tilde{\beta}$ and $\tilde{\gamma}$ of the approximation (54) developed in Sect. 6.1 for several coefficients of variation of the capacity distributions for the arriving buses.

Fig. 8

A comparison between average waiting times per passenger at the bus stop for bus arrivals with strict constant headway for a period of 3 hours and the average waiting times for the M/M/1 queueing model

Fig. 9

The values of function ξ(ρ) in (right) for a M/M ^[Y]/1 bulk-service queue with random server capacity Y. Y is assumed to follow a truncated normal distribution on the interval [0,100] and with an average value of 50 pax for several coefficients of variation ranging from 0 to 0.5 (left)

Table 8 The coefficients $\tilde{\beta}$ and $\tilde{\gamma}$ in (54) for a bulk service queue M/M ^[c]/1. The residual bus capacity c is distributed following a truncated normal on [0,100] with an average value of 50 pax. Values are reported for several coefficients of variation for c, C _c , in the range 0.0 to 0.5