6.5Determining Which Routes to Include Inside BRT Infrastructure

An almost indispensable skill for any creative person is the ability to pose the right questions. Creative people identify promising, exciting, and, most important, accessible routes to progress—and eventually formulate the questions correctly.Lisa Randall, theoretical physicist, 1962–

At this point one should have some basic idea of how to calculate the best vehicle size and how to calculate the necessary fleet for a variety of alternative services one might want to run inside the BRT infrastructure. One now has enough information to start doing the basic service planning for a proposed BRT corridor.

The first question that needs to be answered is what of the existing bus services currently using the corridor should one include, at least in some form, as part of the new BRT services. If properly designed, new BRT infrastructure should increase the speed of all the vehicles that use the BRT infrastructure. Ideally, then, all the customers currently using the BRT corridor should be served by the BRT system’s new operations. The simplest way to ensure this is being done is to simply include as new BRT services all of the services currently using the corridor. Most of these will be direct services that use the BRT corridor for only a part of their route.

However, there are a variety of circumstances where the benefits to the customers of specific bus services are outweighed by the disadvantages that allow this bus route to use the BRT system would impose on the remainder of the BRT customers. There are three reasons to exclude some of the existing services:

  1. Avoidance of station saturation;
  2. Lack of administrative authority over an operator with incompatible operations;
  3. Minimal overlap with the corridor.

If a preexisting bus route is excluded from the BRT infrastructure, there are a few things that can happen to it. First, it can be allowed to continue to operate in the mixed traffic lanes along the BRT corridor. Most of the examples in this first section assume that if the route is not allowed to use the BRT infrastructure, it will operate in the mixed traffic lanes. The route could also simply be cancelled, and its demand transferred onto some similar service.

However, there are other alternatives. It could be allowed to operate inside the BRT corridor but not stop at BRT stations. This option is worth considering in situations when there are no important stations on a particular route along a BRT trunk corridor. Finally, the route could be converted into a feeder route. This option will be considered in the next section on trunk-and-feeder routes.

The best BRT systems minimize boarding and alighting delay by requiring special vehicles that have a clean interface with the station, as described below. Further, the image and quality of service of the BRT system matters.

On any given corridor, there may be a wide variety of bus and minibus operations owned by different companies or government agencies and regulated and administered by different government entities. For instance, a major arterial might have school buses, charter buses, intercity buses, regional buses, private express buses, and local buses.

The best BRT systems limit access to special BRT infrastructure to prescribed operators that operate with the specific permission of a BRT authority and provide bus services following detailed technical specifications required by the BRT authority. These are sometimes called “closed systems.” In general, the highest-quality examples of BRT, such as Bogotá, Lima, Guangzhou, Brisbane, and Curitiba, take advantage of the possibility of restricting access to the new BRT infrastructure to leverage improvements in bus services. In Bogotá and Lima, companies compete for the right to provide public transport services in the BRT system through a process of competitive tendering. These systems only permit vehicles with highly defined specifications operating under a specific contract to a single public authority to operate on the corridor. Because BRT systems try to maintain a higher quality of service than regular bus services, BRT operating contracts are generally far tougher than regular bus operating contracts, and the number of vehicles able to use the BRT infrastructure is normally limited to levels that will avoid saturation of the busway.

By contrast, systems that have implemented a simple busway system open to some bus operations not under the full administrative control of a BRT authority are known as “open systems.” Very few BRT systems have completely open access: Regional or intercity buses, charter buses, and school buses are rarely allowed access to BRT infrastructure. As a rule of thumb, if the BRT authority cannot regulate the vehicle specification and the operation of the bus operator, it is not a good idea to allow that service to operate inside special BRT infrastructure.

Many cities with simple busways that do not qualify as BRT using The BRT Standard, or qualify in some cases as “basic” BRT, utilize an open-system structure, where any bus regardless of type is allowed to use the bus lane. One of the major problems with open busways is that the lack of regulation tends to lead to lower service quality. For instance, in the Delhi busway, which is open to a wide variety of bus operations reporting to different regulatory bodies, some of them with extremely weak maintenance oversight, frequent bus breakdowns tend to plague the services inside the busway. Another problem typical of open busways is that they are more likely to become congested, as the number of buses is harder to control at levels that will avoid saturation.

Fig. 6.28 Open busway congestion in Delhi.

In general, a closed structure is more conducive to efficient traffic operations. Since the number of operators and the number of vehicles are controlled, a closed system can be designed around the optimum conditions for customer movement.

Furthermore, limiting access to new BRT infrastructure is often effectively used by governments to leverage industry modernization and higher quality of service, as discussed in Volume IV: Business Plan, especially Chapter 13: Business Structure. By placing certain minimum requirements on potential bus operators as a condition to bid for a BRT operating contract, the BRT administrative authority can use this leverage in a variety of ways, from encouraging ownership structures that include adversely impacted former operators, to requiring companies to comply with best modern business practices, and so forth.

Fig. 6.29 Bogotá originally operated an open busway system. Individually owned, unreformed informal “collectivo” bus services with merely a route license, but no formal operating contract plied the corridor. The bus operators were largely unaccountable. Vehicles were polluting, poorly maintained, and of a variety of types that did not interface cleanly with BRT infrastructure. Frequencies were too high to avoid congestion.
Fig. 6.30 By contrast, the development of the closed TransMilenio system along the same corridor, where schedules are fixed and where operators meeting certain minimum corporate standards have to comply with tight technical specifications and maintain quality-of-service standards, has dramatically improved travel times and customer comfort

The vehicle types allowed will also greatly affect several performance indicators, including boarding and alighting times and station congestion levels. A single small bus with a very small door can badly congest an exclusive BRT lane, and for this reason, such buses are incompatible with high-speed, high-capacity BRT systems. Specifying maximum vehicle age and maintenance practices can also affect performance. Breakdowns contribute to corridor congestion. Thus, weak regulatory control over the vehicle fleet is incompatible with consistent high-speed, high-capacity, and high-quality service. Tight regulation of emissions, operating speeds, and noise is also important to protecting the environmental quality of the corridor.

Prior to developing its TransMilenio system, Bogotá actually operated a median busway on its Avenida Caracas corridor. The Avenida Caracas busway operated as an open system, permitting all existing operators to utilize the infrastructure. The result was excessive busway congestion and average commercial speeds of approximately 10 kph (Figures 6.29 and 6.30). The busway was partially effective in improving conditions for mixed traffic, but did little to improve travel conditions for public transport customers. It should be noted, though, that having a closed system is a necessary but insufficient condition for ensuring good system performance.

Emergency vehicles, such as ambulances, are generally permitted access on most BRT systems (Figures 6.31 and 6.32), whether they are open or closed systems. This public service provides an additional motivation for approving a BRT project, especially since many rail options are not compatible with emergency vehicles. In many cities, mixed traffic congestion significantly inhibits emergency access and delivery. By facilitating rapid emergency services for the injured and critically ill, the BRT system is in effect helping save lives.

Fig. 6.31 In cities such as Quito, the exclusive busways also permit emergency vehicles.
Fig. 6.32 Allowing emergency vehicles to avoid traffic delays should provide quicker responses to those in need.

Some cities also permit “official” vehicles to utilize the busway. This usage may include presidential and ministerial motorcades, as well as travel for low-ranking public officials (Figure 6.33). The justification for such usage can be questionable. Certainly, for the highest-ranked officials, such as a president or prime minister, the exclusive busway does allow for potentially safer movements. The usage by lower-ranking officials is harder to justify and can ultimately have a highly detrimental impact on system speeds and capacity. In Quito, sometimes the appropriation of busway space even extends to public utility vehicles, such as garbage trucks (Figure 6.34). The presence of such vehicles can do much to hinder proper BRT operation.

Fig. 6.33 When usage of the exclusive busway by lower-ranking government officials becomes commonplace, there can be negative impacts on the efficiency of the system.
Fig. 6.34 In Quito, even garbage trucks take advantage of the exclusive busway.

On most BRT corridors, some existing public transport routes overlap the corridor for a short segment. If the segment is very short, it may not be worth incorporating the service into the BRT system. Usually, incorporating a bus route into the BRT system requires buying special new vehicles for that route, and if the benefits of including the route in the BRT system are low because it only overlaps with the BRT for a short segment, these benefits may not outweigh the cost of buying a new BRT vehicle.

In Table 6.1, existing routes are shown to the degree to which the route overlaps the planned BRT infrastructure. If the overlap is less than 20 percent of the total length of the route, it may not be worth incorporating it into the BRT service plan.

In many cases, BRT service planners start by mapping the existing route and then exclude the routes that overlap the planned BRT corridor for less than 20 percent of their route. This is not a hard-and-fast rule; it is just what is normally done. In situations of low station saturation and high-frequency routes, the percentage may be as low as 10–15 percent, but in capacity-constrained settings, a range of 20–30 percent is more typical.

If the part of the route that overlaps with the BRT corridor is downtown, on a highly congested road, or it includes some very high-demand stations, there may be significant benefits to incorporating the route into the BRT system, but in most cases 20 percent overlap is a reasonable rule of thumb for excluding a route. In case of doubt, the operational cost savings and time savings that would accrue from using the BRT infrastructure could be compared to the additional costs associated with buying special vehicles, if the BRT requires new vehicles. If the overlap is below 20 percent, but there is no particular need for the route to stop on the BRT corridor, one might consider allowing the route to use the BRT corridor for this short segment, while not stopping at any of the BRT station stops.

Once the routes that have only a tangential relationship to the BRT trunk service have been excluded from the proposed BRT system services, ideally the BRT infrastructure should be designed to accommodate the demand from all of the remaining bus routes. However, this may be impossible. If the capacity of what can be designed is less than the remaining demand, it may be necessary to make further decisions about which routes to include in the BRT services, and which to leave in mixed traffic, or reroute off the corridor. In this sense, service planning is iterative with infrastructure planning. The next chapter provides the formulas needed to calculate the capacity and speed of a BRT corridor depending on The BRT Standard elements used. The normal situation where one needs to exclude some bus routes in the manner described in this section is when a single-lane BRT with no sub-stops or passing lanes has been designed.

Having many routes is not a problem per se; it only becomes a problem if frequencies are high enough that stations begin to become saturated, and the speed of the BRT system begins to slow. Since new BRT infrastructure will allow vehicles to increase their speed, ideally as many customers as possible should be able to use it. Usually, a corridor will be chosen for BRT that already has a lot of bus or minibus services along it, and normally these preexisting services are a reasonable match to public transport demand patterns. As such, the first principle of service planning is to incorporate as many preexisting bus routes as possible into the services that will use the new BRT infrastructure. This will maximize the number of beneficiaries and minimize the disruption of service.

In specific cases where current bus services already take up more than one mixed traffic lane, the BRT system may begin to become congested if too many routes are brought into the dedicated infrastructure, at which point the speeds will slow. Eventually, as saturation worsens, speeds inside the BRT system can drop to levels below the original mixed traffic speeds. Then, unless the design of the busway can be modified to accommodate all of the bus routes, the service planner will need to exclude some of the bus routes from the system.

An example of this problem occurred with the Seoul BRT. The system was initially designed with insufficient capacity to handle the vehicle demand in the corridor. As a result, in the first several months of operation, the corridor became saturated, and vehicle speeds dropped below the pre-BRT speeds. After diverting some of the routes back to the mixed traffic lanes so the busway was not saturated, the overall system yielded significant user benefits. The methodology sketched out below should be used in similar circumstances when deciding which routes to put back into mixed traffic.

The process of determining how many routes to include in the services for the planned BRT system has two basic steps. First, calculate which future BRT station or stations are most likely to become saturated. This will be the station projected to have the highest frequency and boarding and alighting volumes, because high frequency and high boarding and alighting volumes are the most likely causes of station saturation.

In nearly all cases, BRT system saturation happens at stations where the capacity is too low to handle the frequency and boarding and alighting volumes, so vehicles begin to bunch. The formulas for calculating station saturation are covered in Chapter 7: Capacity and Speed.

Second, a calculation should be performed to determine, given the characteristics of each route (demand, vehicle type, frequencies, and demand to that particular station, etc.), how many routes and which routes should be included as part of the new BRT services and which should be excluded. To do this, the routes should be ordered based on the seconds consumed at that specific station multiplied by the number of customers that are passing by that bottleneck station. In other words:

The first criteria used to select routes for inclusion in the BRT system is frequency and proportion of overlap with the BRT corridor. If the route only uses the BRT corridor for a short distance, or carries few customers, the benefits may not justify the cost of purchasing a BRT-infrastructure-compatible bus.

For systems where there is a danger that the busway will become saturated and speeds will slow, a second method should be used: routes should be ranked based on the number of customers that a route can move through the bottleneck station for each second of dwell time they consume at the bottleneck station. Routes moving more customers per second of dwell time should continue to be included until the total delay caused to the busway of adding the last route is greater than the time savings benefits of adding the last route.

The existing bus route that uses the busway most efficiently will be the route that carries the most customers with the fewest customers getting off at the bottleneck station. For instance, if the bottleneck station in a BRT system is Times Square, and there is an express route, “T1,” that carries large numbers of customers through Times Square but does not stop at Times Square, it should certainly be included in the busway, as it does not contribute at all to the bottleneck at Times Square station. By contrast, a route that carries relatively few customers, yet all of them get off at Times Square, should be the first route to be excluded.

In this section we provide formulas to determine how many and which vehicle routes to include in the BRT system. Given that every effort has already been made to design the system with a capacity that will avoid saturation, but, for one reason or another, design compromises had to be made, and the design has already been fixed in a way that does not accommodate all public transport customers who want to use the corridor. It is assumed that if some existing routes are not excluded, the bus speeds in the corridor will slow down. This is fairly common in the developing world, but not so common in the United States or Europe. Each scenario is progressively more complex—and more realistic; however, they do not cover every possibility. Service planners will need to understand the fundamentals laid out in this chapter and make modifications to fit a given situation.

In these scenarios, there are two possible ways for running buses:

  1. BRT corridor: The BRT corridor is generally described in text as “BRT infrastructure” or “busway” and is indicated in formulas with the subscript “inside.”
  2. Mixed-traffic lane: The mixed-traffic lane is assumed to run parallel to the BRT corridor. The mixed-traffic lane is where bus routes that are excluded from the BRT infrastructure will run. The mixed-traffic lane is referred to in text as “outside the BRT corridor” and is indicated in formulas with the subscript “outside.” It is assumed that speeds in the mixed-traffic lane do not vary based on the number of buses in it, since buses generally constitute a relatively small share of total traffic, and it is generally possible for buses to pass each other in the mixed traffic lanes. If bus volumes in the mixed traffic lanes are likely to affect mixed traffic speeds, a more detailed analysis of the mixed traffic lanes should be made.

In the following scenarios, one should aim to minimize aggregate travel time for all customers (\(ATT_\text{total}\)), both inside and outside the BRT corridor. One assumes that there is a total number of bus routes, both inside and outside the BRT corridor operating at a combined total frequency, indicated by \(Fi_\text{total}\). Bus routes operate as units, so that one cannot remove a few buses of the route “f” from the busway; the entire route must be removed.

In all scenarios below, the part of the corridor that will receive BRT infrastructure is assumed to be 5 kilometers long and the speed in the mixed traffic lanes along this corridor is assumed to be 10 kph. The speed within the busway will vary by the scenario, while the speed in the mixed traffic lane will be assumed to stay at 10 kph.

DepartureArrivalTravel TimePassengers
6:00 a.m.8:00 a.m.2:0025
6:15 a.m.8:15 a.m.2:0025
6:30 a.m.8:30 a.m.2:0025
6:45 a.m.8:45 a.m.2:0025
7:00 a.m.9:00 a.m.2:0025
7:15 a.m.9:15 a.m.2:0025
7:30 a.m.9:30 a.m.2:0025
7:45 a.m.9:45 a.m.2:0025

Further, the peak hour was already identified as between 6:15 and 7:14, the peak hour ridership is a hundred, and the peak hour travel time is two hours. It is assumed that all bus routes travelling through the corridor before the busway was introduced took a half hour to pass through the corridor. It means that at any given moment, with the hourly frequency of four, two buses of route A would be visible operating on the BRT corridor; the moment the bus ahead leaves the segment (after half an hour), another bus would get in at the beginning (being thirty minutes behind).

Before the BRT system is implemented, all bus routes are operating “outside” the BRT. Therefore, during one hour we would have seen Aggregate Travel Time OUTSIDE the BRT, operating along a planned BRT corridor, of two hours for this route. That is simply four bus trips per hour, each taking half an hour.

If this route is included inside the BRT infrastructure where average speed (without congestion that is the aim) is 25 kph, it would take only 12.5 minutes to cross the corridor. In this case, the Aggregate Travel Time INSIDE the BRT is 12.5 times 4 or 50 minutes. This is a net improvement over the no-build scenario of one hour and ten bus minutes. Another way to think of this is that after only 12.5 minutes the first bus will exit the BRT, but the next bus would not appear on the segment for another 2.5 minutes, so if we look at the corridor once a minute over an hour, only 50 times in 60 times will we see a bus riding in the segment. In this situation the Aggregate Travel Time INSIDE the BRT added 50 minutes for this route (or 5/6 of hour or 0.833 hour).

For each of the examples, the total dwell time of each bus (Td) at a station inside the busway, assuming boarding and alighting through all doors will be a function of the fixed dwell time (T0), also known as dead time or the time the bus takes to pull up to the station, open and close its doors, and pull away, and the average boarding time per customer for buses with the given configuration (Tb) multiplied by the number of boarding customers and the average alighting time per customer for buses with the same configuration (Ta) times the number of alighting customers, or:

Eq. 6.22

\[ T_d=T_0+T_b+T_a \]

Or

\[ Td =T_0 + t_b * P_b + t_a * P_a \]

Where:

  • \(T_d\): Total dwell time;
  • \(T_0\): Fixed dwell time (or “dead time”);
  • \(T_b\): Total boarding time per vehicle ((given by \(tb * Pb\));
  • \(t_b\): Boarding time per customer;
  • \(P_b\): Number of boarding customers;
  • \(T_a\): Total alighting time per vehicle (given by \(ta * Pa\));
  • \(t_a\): Alighting time per customer;
  • \(P_a\): Number of alighting customers.

As discussed in Chapter 7: Capacity and Speed, boarding and alighting times per customers are a function of bus configuration and bus station interface (number and width of doors, at-level boarding or boarding via several steps, internal or external fare collection, position of turnstiles, etc.) and bus occupancy at the station. A linear proxy, i.e., using average customers per second of surveyed boarding and alighting times under conditions similar to those being designed is a more accurate way of estimating dwell time per station than using a flat average dwell time per station, so long as the busway is not beginning to become saturated (saturation below 0.4). Avoiding saturation is the design goal of the examples. Outside the BRT infrastructure, it is assumed that buses’ speeds are the current commercial speed, which already include dwell times at station stops.

In this scenario, all vehicles, routes, and stations inside the BRT infrastructure have the same operational characteristics. In other words, they have roughly the same number of customers getting on and off at each stop; have the same number of doors; have the same frequency; use the same vehicle type; have floors level with the vehicle platform; and hence have the same dwell time per customer. As a result, one should not care which routes are included or excluded, one should only care about the number of routes one includes. In this simplified scenario, since all of the routes have the same dwell time per customer, and the same number of customers benefitting from the busway, there is no need to rank the bus routes. Given this uniform demand, all stations will become saturated equally. Vehicles operating inside the BRT infrastructure will become congested if there are too many vehicles using the bus lane. These assumptions allow for busway congestion to be isolated as the only factor that would cause a variance in total travel time when all else is constant.

In this scenario, given that all bus routes have the same demand and vehicle size, it is assumed that all routes have the same frequency. In later examples the frequency will vary by route. All routes are also assumed to have the same dwell time, since they have the same demand and vehicle size. It is assumed that the total dwell time Td is the same for all vehicle routes, because both the fixed and variable dwell times are the same for all vehicles:

\(L_\text{corridor}\): Corridor length = 5 km;

\(V_\text{outside}\): Velocity outside the busway = 10 km/h;

\(V_\text{inside}\):Velocity inside BRT at free-flow speed = 25 km/h;

\(N_\text{stations}\) = 10;

\(F_\text{total}\): Total frequency of all services = 200 vehicles/h.

\(T_d\): Dwell time per vehicle at each station = 18 seconds = 0.005 h

\(T_b\): Time per customer boarding = 3 seconds;

\(T_a\): Time per customer alighting = 2 seconds;

\(T_0\): Fixed dwell time per vehicle = 12 seconds.

To determine the optimal number of vehicle routes to include in a BRT corridor, as many buses as possible should be brought into the BRT corridor until the point where time-savings benefit for the last bus added to the corridor is less than the congestion delay it causes to the remaining vehicles in the busway.

By measuring the total vehicle travel time over the course of one hour for all vehicles on a single corridor (\(ATT_\text{total}\)) both inside (\(ATT_\text{inside}\)) and outside (\(ATT_\text{outside}\)) the BRT infrastructure, it is clear that it is often beneficial to leave some routes out of the BRT corridor. The ATT both inside and outside the BRT infrastructure will be the number of buses per hour (\( F_\text{inside} \) and Foutside) multiplied by the travel time per bus (\(TT_\text{inside}\) and \(TT_\text{outside}\) respectively). In all cases one will assume that the frequency outside the BRT corridor does not affect the travel time outside the corridor. Therefore, all of the total travel times (ATT) are expressed as a function of the frequency inside the busway (\(F_\text{inside}\)).

Eq. 6.23

\[ATT_\text{total}=ATT_\text{outside}+ATT_\text{inside}\]

Or:

\[ATT_\text{total}=TT_\text{inside}*F_\text{inside}+TT_\text{outside}*F_\text{outside} \]

Where:

  • \(ATT_\text{total}\):Total vehicle travel time over the course of one hour for all vehicles on a single corridor;
  • \(ATT_\text{outside}\): Total vehicle travel time over the course of one hour for all vehicles on a single corridor outside of the BRT infrastructure (\(ATT_\text{outside}=TT_\text{outside}*F_\text{outside}\));
  • \(ATT_\text{inside}\): Total vehicle travel time over the course of one hour for all vehicles on a single corridor inside of the BRT infrastructure(\(ATT_\text{inside}=TT_\text{outside}*F_\text{outside}\));
  • \(TT_\text{inside}\): Travel time per bus inside the busway;
  • \(F_\text{inside}\): Number of buses per hour inside the busway;
  • \(TT_\text{outside}:\) Travel time per bus outside the busway;
  • \(F_\text{outside}:\) Number of buses per hour outside the busway.

Because \(TT_\text{outside}\) is fixed, one can easily calculate it as:

Eq. 6.24a

\[TT_\text{outside}={L_\text{corridor} \over V_\text{outside}}\]

Where:

  • \(TT_\text{outside}\): Travel time per bus outside the busway;
  • \(L_\text{corridor}\): Length of the corridor;
  • \(V_\text{outside}\): Velocity outside the busway.

So, using the values defined above for the corridor:

\[TT_\text{outside}={5\text{km} \over 10 \text{km/hr} } * 1 \text{hr} ={ 5 \text{hr} \over 10 } = 0.5 \text{hours} \]

\(TT_\text{outside}\) will remain 0.5 hours (30 minutes) per bus no matter how many buses are operating outside the corridor.

Travel time inside the BRT infrastructure (\(TT_\text{inside}\)), however, varies in this simplified example as a simple function of frequency. That is, with each new vehicle added to the busway, a slight congestion is introduced and the total travel time inside increases.

Travel time for a bus within a busway is the sum of the time spent:

  1. In motion (free running time);
  2. At intersections;
  3. At stations;
  4. In congestion.

In most conditions, the stations become saturated long before the traffic signal or the busway itself is saturated, so generally one should assume that the bottleneck is the station and not the intersections or the busway. Even at quite low frequencies, it often occurs that one vehicle is unable to approach the station, because another vehicle is already occupying that location, and these delays can rapidly become very significant.

Design techniques for reducing dwell time at stations are discussed extensively in Chapter 7: Capacity and Speed. In this chapter, these design issues are not discussed; instead, one should assume that the best possible design has been used, so the boarding and alighting time at stations per passenger (\(T_b + T_a\)) is already fixed by these design characteristics. In this scenario, the only factor that varies is (“In congestion”) and it varies based only on vehicle queuing at stations (\(T_q\)). Scenario II will consider variations in dwell times, but for now one should assume a fixed dwell time for all vehicles on the corridor.

So in this scenario the first three aspects of travel time in the busway are fixed and constitute the base speed (\(_\text{Vinside-no-congestion}\)) and travel time for any vehicle in the busway before the busway begins to become congested. Further, if only one vehicle is in the busway, there is no possibility of congestion, and that vehicle will experience this base travel time as:

Eq. 6.24b

\[TT_\text{inside-no-congestion}={L_\text{corridor} \over _\text{Vinside-no-congestion}}\]

Where:

  • \(TT_\text{inside-no-congestion}\): Travel time per bus inside the busway without delays at stations;
  • \(L_\text{corridor}\): Length of the corridor;
  • \(V_\text{inside-no-congestion}\): Velocity inside the busway, when there is no congestion delay at stations.

If one wants to know the speed inside the busway without any vehicle congestion, one should calculate the initial value of \(TT_\text{inside}\), with only one vehicle in the corridor:

\[TT_\text{inside-no-congestion}={5\text{km} \over 25 \text{km/hr} } = { 5 \text{hr} \over 25 } = 0.2 \text{hours} \]

Usually, when performing this type of analysis, if one is planning to build a Silver or Gold Standard BRT, one would normally assume that the average speed of the BRT corridor before saturation would be around 20 kph in a dense urban area or downtown and around 25 on a major arterial, based on empirical observation of BRTs in different conditions. As soon as any additional vehicles are added to the busway, some possibility of queuing emerges. One should measure queuing delay on a per-station basis; however, in this scenario, it can be assumed that the queuing delay (\(T_q\)) will be the same at every station. One must thus expand Equation 6.17b to all vehicles as:

Eq. 6.24c

\[ TT_\text{inside}={ L_\text{corridor} \over V_\text{inside-no-congestion}}+T_q*N_\text{stations}\]

Where:

  • \(TT_\text{inside}\): Travel time per bus inside the busway;
  • \(L_\text{corridor}\): Length of the corridor;
  • \(V_\text{inside-no-congestion}\): Velocity inside the busway without delay at stations;
  • \(T_q\):Delay for a single vehicle at a single station due to queuing;
  • \(N_\text{stations}\): Number of stations.

Because queuing in a busway (\(T_q\)) is generally a direct result of station saturation, one should begin with the formula for station saturation. Detailed methodologies for calculating station saturation under different scenarios are provided in Chapter 7: Capacity and Speed, but for now one should use the basic consideration for station saturation under these extremely simplified conditions.

In this example, saturation at each station is expressed by the equation below.

Eq. 6.25

\[ x= T_d * F_\text{inside} \]

Where:

  • \(x\): Saturation at each station for a specific frequency inside the busway;
  • \(T_d\): Dwell time per bus;
  • \(F_\text{inside}\): Number of buses per hour inside the busway.

If one includes 80 vehicles in the BRT corridor, and one knows the dwell time is 18 seconds (0.005 hours) per bus, station saturation, \(x\), will be 40 percent.

\[ x = 0.005 * 80 = 0.4 \]

In this case, “\(x\)” is the average percentage of time that the station is occupied by vehicles loading and unloading, but it can be thought of as the probability that a vehicle approaching a station will find the station occupied. In our example, if a person (or a BRT vehicle) about whom we know nothing else arrives at the station, he/she/it has a 40 percent chance of finding a station with BRT vehicles using it and a 60 percent chance that no buses are there at that exact moment.

Let us now focus our attention on this 40 percent of time that the station is occupied, the probability of a BRT vehicle arriving at the station when another vehicle is already there is still 40 percent of the time the station is occupied, if nothing else is known. So the chance of a vehicle being a second vehicle queueing at the station is 40 percent of 40 percent (or 16 percent). This means that 16 percent of the time a bus will have to wait until the docking bay is cleared by at least one bus. Thus, 40 percent of these 16 percent, or 6.4 percent of the wait, will be for two or more buses to clear the station and so on.

Station saturation, as described above, begins to result in delay when vehicles are forced to queue up to the station waiting to dock. The probability that a vehicle will find the station occupied is approximately given by x, and the chance that the next vehicle will also face a queue is x2, and the vehicle after that would face a probability of x3. The average queue in such a situation would be given by:

Eq. 6.26

\[\text{Average Queue Size}={x^2 \over(1-x)} \]

Where:

  • \(Average Queue Size\): Number of vehicles on average in queue;
  • \(x\): Probability a vehicle will find the station occupied.

If arrivals and departures are random, the average waiting time in queue per vehicle would be given by:

Eq. 6.27:

\[ \text{Average Waiting time} = { x^2 \over (1-x) } * {1 \over F_\text{inside}} \]

Where

  • \(\text{Average Queue Size}\): Number of vehicles on average in queue;
  • \(x\): Probability a vehicle will find the station occupied;
  • \(F_\text{inside}\): Number of buses per hour inside the busway.

This is a derivation of what is known as Little’s Law: If the average waiting time in a queue is two hours and customers arrive at a rate of three per hour (Frequency) then, on average, there are six customers in the queue, or:

Eq. 6.28:

\[ \text{Average Queue Size}=\text{Average Waiting Time}*\text{Frequency of Arrivals}\]

Where:

  • \( \text{Average Queue Size} \): Number of vehicles on average in queue;
  • \( \text{Average Waiting Time} \): Average waiting time in a queue;
  • \( \text{Frequency of Arrivals} \): Number of customers arriving per a given amount of time.

In our example, the average queue is known as 0.2667 vehicles (= 4/15 vehicle) and there are an average of 1.333 vehicles arriving per minute (80 vehicles per hour = 4/3 per minute), so in average they must be waiting 1/5 of a minute (= 12 seconds).

Considering this distribution of arrivals and departures, theoretical queuing time per bus at each station would be given by:

Eq. 6.29

\[ T_q={0.5(Irr_\text{arrival}+ Irr_\text{departure})* x^2 \over 1-x } {1 \over F_\text{inside}} \]

Where:

  • \( T_q \): One vehicle queueing time at each station;
  • \( x \): Saturation of the station;
  • \( F_\text{inside} \): Number of buses per hour inside the busway;
  • \( Irr_\text{arrival} \): Irregularity of arrivals (\( Irr_\text{arrival}={\text{Variance}_\text{arrivals'intervals} \over \text{Mean}_\text{arrivals'intervals}^2}\));
  • \( Irr_\text{departure} \): Irregularity of departures (\( Irr_\text{departure}={\text{Variance}_\text{departures'intervals} \over \text{Mean}_\text{arrivals'intervals}^2}\)).

The mean of arrival and departure intervals is the headway and the variance would be similar to boarding and alighting variance if there were no traffic lights and starting schedules were followed to the letter. Equation 6.20 is the particular case where irregularities for arrival and departure are random (\(Irr_\text{arrivals} = Irr_\text{departures} = 1)\).

Empirical observation shows that using the coefficient of 0.7 mimics the series of probabilities in high-frequency (above 80 vehicles per hour) busways in urban conditions. In fact, empirical observation of busways with full BRT characteristics do tend to saturate at around 80 vehicles per hour, and for a busway 0.4 is considered the beginning point of station saturation, so service planners will avoid designing services with frequencies where x > 0.4. In any case, queuing time per vehicle can be expressed (as a portion of an hour by):

Eq. 6.30a

\[ T_q={0.7 * x^2 \over 1-x } {1 \over F_\text{inside}} \]

Where:

  • \( T_q \): One vehicle queueing time at each station;
  • \( x \): Saturation of the station;
  • \( F_\text{inside} \): Number of buses per hour inside the busway;

Since empirical observation shows that this phenomenon is just as well captured by the much simpler formula above, it is not necessary to use the theoretical formula. It may also be interesting to note that this equation can also be written as a function of dwell time and saturation, or only as a function of dwell time and frequency inside the busway, as the three are related by Equation 6.25:

Eq. 6.25

\[ x=T_d*F_\text{inside}\Leftrightarrow Td= {x \over F_\text{inside}} \]

Where:

  • \( T_d \): Dwell time per bus;
  • \( x \): Saturation of the station;
  • \( F_\text{inside} \): Number of buses per hour inside the busway;

Eq. 6.30b

\[ T_q = {0.7 * x^2 \over 1-x } {1 \over F_\text{inside}} = {0.7 * x \over 1-x } {x \over F_\text{inside}} = {0.7 * x \over 1-x } * T_d \]

\[ T_q = {0.7 * (T_d*F_\text{inside})^2 \over 1-(T_d*F_\text{inside})} {1 \over F_\text{inside}} = {0.7 * T_d ^2 * F_\text{inside} \over 1- (T_d *F_\text{inside} )} \]

Where:

  • \( T_q \): One vehicle queueing time at each station;
  • \( x \): Saturation of the station;
  • \( F_\text{inside} \): Number of buses per hour inside the busway;

Going back to Equation 6.24c

Travel time inside the corridor can be expressed as function of the frequency inside the corridor:

Eq. 6.24c

\( TT_\text{inside}={L_\text{corridor} \over V_\text{inside-no-congestion}}+T_q*N_\text{stations}\) \( TT_\text{inside}={L_\text{corridor} \over V_\text{inside-no-congestion}}+{0.7 * T_d ^2 * F_\text{inside} \over 1- (T_d *F_\text{inside} )}*N_\text{stations}\)

Where:

  • \( TT_\text{inside} \): Travel time per bus inside the busway;
  • \( L_\text{corridor} \): Length of the corridor;
  • \( V_\text{inside-no-congestion} \): Velocity inside the busway if there is no queueing at stations;
  • \( T_d \): Dwell time per bus at each station (equal for all buses in all stations in this scenario);
  • \( F_\text{inside} \): Number of buses per hour inside the busway;
  • \( N_\text{stations} \): Number of stations along the busway.

Because it is assumed in this case that all stations will saturate equally (because the boarding and alighting customer volumes and frequencies are assumed to be constant), one can simply add the free-flow speed (\(L_\text{corridor} \over V_\text{inside-no-congestion}\)) to the queue delay at each station and multiply it by the number of stations (\(N_\text{stations}\)). Later, it will be necessary to calculate the queue delay at each station and sum it across stations.

On the BRT corridor, if our sample frequency of 80 vehicles inside the corridor is tried, the travel time per vehicle inside the corridor is:

\[ TT_\text{inside}(F_\text{inside}=80)=0.2 + ({ {0.7*0.4^2 \over (1-0.4)} \over 80}*10) = 0.2233 \text{hours}\]

By fixing the other values in Equation 6.17 as proposed for our scenario (\(L_\text{corridor} = 5 \text{km}, V_\text{inside-no-congestion}=25 \text{km/hour}, T_d = 18 \text{seconds (or 0.005 hour)} \) and \( N_\text{station}=10\)), one can calculate the travel time per bus inside the corridor (\( TT_\text{inside} \) shown in the graphic on Figure 6.35), as a function of the bus frequency inside the corridor, up to the maximum \(F_\text{total} = 200 \). By considering that \( F_\text{outside} = 200 – F_\text{inside}\) and using equations 6.16 and 6.17a, one may also calculate total aggregate time as a function of frequency inside as shown in Table 6.18.

On the figure, the orange line, \(TT_\text{outside}\), remains constant at 0.5 hours, as described earlier, regardless of the frequency of buses outside the corridor. The blue line, however, increases as a function of bus frequency, from a minimum of 0.2 hours to a maximum reaching toward infinity (i.e., the vehicles are not moving at all) if all 200 vehicles are included in the BRT corridor.

Fig. 6.35 Scenario I travel time inside the BRT corridor (blue line) and outside the BRT corridor (orange line) as a function of bus frequency brought inside the corridor.

Figure 6.35 shows that the travel time within the bus lane reaches the travel time outside of the bus lane at approximately \(F_\text{inside} = 179\). Thus, if 179 vehicles were included inside the corridor, there would be no benefit at all. Above 179 vehicles inside the busway, the travel time would be slower than the mixed traffic.

One should keep in mind that the travel times will affect all vehicles and that it is not actually the per bus travel times we are interested in but rather the aggregate travel times inside and outside the corridor (\(ATT_\text{inside}\) and \(ATT_\text{outside}\)). More precisely, we are most interested in the aggregate travel times for customers inside and outside the corridor. However, in this simplified case, where customer volumes are the same from one bus to another, we leave customer volumes out of the equation with no consequence.

Aggregate travel times give the full picture of the overall benefit to all customers inside and outside the corridor each time a vehicle is added to the corridor. By optimizing the aggregate travel times inside and outside of the corridor so that the total aggregate travel time is minimized, we find the best \( F_\text{inside} \) for the corridor.

Continuing the example, one can now calculate the aggregate travel times both inside and outside the corridor, assuming that one will include 179 vehicles in the BRT corridor. The average travel time per vehicle when there are 179 vehicles in the corridor is 0.5 hours. So using Equation 6.23, one should multiply the per vehicle travel time inside the BRT corridor at a frequency of 179 vehicles by 179, to get the aggregate travel time for 178 vehicles inside the corridor. That is:

\[ ATT_\text{inside}(F_\text{inside}=179)=0.5 \text{hours}*179=89.5 \text{hours} \]

If one includes 179 vehicles inside the BRT corridor, then 21 are left out. The travel time per bus outside of the BRT corridor (\(TT_\text{outside}\)) is fixed at 0.5 hours (again, equal to \(TT_\text{inside} \) only in this case). So one should multiply the per bus travel time outside the BRT corridor by 21 to get the aggregate travel time for 21 buses outside the corridor. That is:

\[ ATT_\text{outside}(F_\text{inside}=179)=0.5 \text{hours}*21=10.5 \text{hours} \]

Thus,

\[ ATT_\text{total}(F_\text{inside}=179)=89.5+10.5=100\text{hours}\]

Exploring visually the properties of Equation 6.16, Figure 6.36 shows the aggregate travel times both inside and outside the BRT corridor when 179 vehicles are included inside the corridor (\( F_\text{inside} = 178\)). Aggregate travel times are indicated by the blue (\(TT_\text{inside}\)) and orange (\(TT_\text{outside}\)) shaded areas. Note that the shaded areas are rectangular in shape. This is because every bus in either category (\( F_\text{inside} \) or \(F_\text{outside}\)) experiences a travel time equal to every other bus in its category.

Fig. 6.36 Scenario I aggregated travel times, obtained by multiplying by the fixed travel time for all buses (0.5) by the number of buses inside the busway (179, represented by the blue shading) and outside the busway (21, represented by the orange shading), one should get ATT_\text{inside}(F_\text{inside}=179) = 89.5 \text{hours} and ATT_\text{outside}(F_\text{inside}=179) = 10.5 \text{hours} , respectively.

One can already tell that \(F_\text{inside} = 179 \) is not the optimal frequency for this corridor, since one knows that total area (an \( ATT_\text{total} \) of 100 hours) is not the lowest aggregate travel time that is achievable in this example and can be diminished. Figure 6.37 shows an example of a reduced area for \( F_\text{inside}= 150\) totaling 70.75 hours, i.e., \( ATT_\text{total} (F_\text{inside}=150) = 70.75 \text{hours} \).

Fig. 6.37 Scenario I aggregated travel times, obtained by multiplying by the fixed travel time for all buses inside the busway (0.305 hour) by the number of buses inside the busway (150, represented by the blue shading) added with the travel time outside the busway (0.50) multiplied by number of buses outside (50, represented by the orange shading), one should get AT_\text{Tinside} (F_\text{inside}=150) = 45.75 \text{hours} and ATT_\text{outside} (F_\text{outside}=179) = 25 \text{hours} , respectively.

We determine the lowest aggregate travel time achievable by evaluating the time savings and time losses of shifting an additional bus route (or the bus frequencies of that route) onto the corridor. Each time a vehicle is added to the corridor, some time savings are realized to the customers on the vehicle that has been shifted, due to the higher speed of the bus lane. At the same time, delay is created for all the customers on vehicles already in the bus lane, which now suffer some new (marginal) congestion due to a larger number of vehicles travelling there.

In order to determine the savings, one should subtract the travel time for one vehicle inside the bus lane from the travel time for one bus outside the bus lane. When calculating the travel time per vehicle inside the bus lane, one does so based on congestion incurred by the new frequency (i.e., \( F_\text{inside} + 1 \)), since the new travel time caused by that vehicle is ultimately the travel time that vehicle will experience. One therefore defines the benefit of adding one vehicle to the corridor as:

Eq. 6.31

\[ ATT_\text{savings}(1)=TT_\text{outside}–TT_\text{inside-after-shift} \]

\[ ATT_\text{savings}(1)=TT_\text{outside}–TT_\text{inside}(F_\text{inside}=F_\text{inside-before}+1)\]

Where:

  • \( ATT_\text{savings}(1) \): Benefit of adding one vehicle to the busway;
  • \( TT_\text{outside} \): Travel time outside the busway;
  • \( TT_\text{inside-after-shift} = TT_\text{inside}(F_\text{inside} = F_\text{inside-before} + 1)\): Travel time per bus inside the busway based on congestion incurred by the new frequency of adding one vehicle inside the busway.

\(ATT_\text{savings}(1)\) indicates that one vehicle is added to the corridor. Were one to be complete, one would multiply this benefit by the number of customers on the vehicle shifted into the bus lane. Again, since customer volumes are the same from one vehicle to another, this can be left out for now.

If expanded to shifting multiple vehicles (\(N_\text{shift}\)) to the corridor, one must calculate \(TT_\text{inside}\) based on the old frequency plus the number of new vehicles (i.e., \( F_\text{inside} + N_\text{shift}\)). Additionally, the benefit is realized not just by the customers on one vehicle but by the customers on all the vehicles shifted to the corridor. So the entire savings calculation must be multiplied by the number of vehicles added (and by the number of customers shifted in later scenarios).

Eq. 6.32

\[ATT_\text{savings}(N_\text{shift}) =TT_\text{outside}–TT_\text{inside-after-shift}*N_\text{shift}\]

\[ATT_\text{savings}(N_\text{shift}) =TT_\text{outside}–TT_\text{inside}( F_\text{inside} = F_\text{inside-before}+N_\text{shift})*N_\text{shift} \]

Where:

  • \( ATT_\text{saving}(N_\text{shift}) \): Aggregated travel time savings of adding Nshift vehicles to the busway;
  • \( TT_\text{outside} \):Travel time per bus outside the busway (it does not change with frequency);
  • \( TT_\text{inside-after-shift}( F_\text{inside} = F_\text{inside-before}+N_\text{shift}) \): Travel time per bus inside the busway based on the old frequency plus the number of new vehicles;
  • \( N_\text{shift}:\) Number of shifting vehicles inside the busway.

The time losses of shifting one vehicle onto the busway are felt by all the customers on the other vehicles already travelling within the BRT infrastructure, since every new vehicle adds some congestion to the busway. This is calculated by subtracting the travel time before the shift from the travel time after the shift, as this is the marginal cost to each previously existing vehicle of shifting new vehicles. Next, one should multiply this by the total number of previously existing vehicles to get the full cost of adding new vehicles. The cost is defined as:

Eq. 6.33

\[ ATT_\text{losses}(N_\text{shift})=(TT_\text{inside-after-shift}–TT_\text{inside-before-shift}) * F_\text{inside-before-shift}\]

\[ ATT_\text{losses}(N_\text{shift})=(TT_\text{inside} (F_\text{inside}=F_\text{inside-after-shift})–TT_\text{inside}(F_\text{inside} = F_\text{inside-before-shift}))* F_\text{inside-before-shift}\]

\[ ATT_\text{losses}(N_\text{shift})=(TT_\text{inside} (F_\text{inside}=F_\text{inside-before-shift} + N_\text{shift})–TT_\text{inside}(F_\text{inside} = F_\text{inside-before-shift}))* F_\text{inside-before-shift}\]

Where:

  • \( ATT_\text{losses} (N_\text{shift})\): Aggregated time losses by moving \(N_\text{shift}\) vehicles onto the busway;
  • \( TT_\text{inside-in-situation} \): Travel time per bus inside the busway as a function of the frequency inside the busway in given situation;
  • \( TT_\text{inside} (F_\text{inside} = K)\): Travel time per bus inside the busway based on a frequency of \(K\) vehicles inside the busway;
  • \( F_\text{inside-before-shift} \): Number of buses per hour inside the busway before moving vehicles;
  • \( F_\text{inside-after-shift} \): Number of buses per hour inside the busway after moving vehicles.

Continuing the example, suppose the decision was initially made to include 80 vehicles in the corridor, and now one wants to know whether it would be better to include 90. So 10 vehicles are being shifted onto the corridor.

The aggregated time savings can be calculated using Equation 6.32:

\[ATT_\text{savings}(10)=TT_\text{outside}–TT_\text{inside}(90)*10\]

\[TT_\text{outside}=0.5\]

\[TT_\text{inside}(90)={L_\text{corridor} \over V_\text{inside-no-congestion}}+({0.7 * T_d ^2 * F_\text{inside} \over 1- (T_d *F_\text{inside} )}*N_\text{stations})=0.2286\]

So,\(ATT_\text{savings}(10)=(0.5-0.2286)*10=2.714 \text{hours} \)

The aggregated travel time losses can be calculated using Equation 6.25:

\[ATT_\text{losses}(10)=TT_\text{inside}(90)-TT_\text{inside}(80)*80 \]

above, \(TTinside(90)=0.2286 \)

And from the first moment of this example (saturation):

\( T_\text{Tinside}(80)=0.2233\).

So,\(ATT_\text{losses10}=0.2286-0.2233*80=0.424 \text{hours}\).

On the corridor, the benefits of increasing the number of vehicles inside the busway from 80 to 90 outweigh the costs by a total of \( 2.714 - 0.424 = 2.29 \text{hours}\). The 10 buses should therefore be shifted in.

In Figure 6.38, the blue box shows the \(ATT_\text{inside}\) for 80 vehicles and the light orange box shows the \( ATT_\text{outside} \) for the remaining 120 buses outside the corridor. If another 10 vehicles are added to the corridor, the blue box expands to 90, and the orange box shrinks to 110, with the purple area being transferred from one to the other. The benefit of including the 10 new vehicles in the corridor is shown by the green rectangle where the aggregate travel time savings is realized. This benefit is realized for the customers on the 10 new vehicles that can now use the bus lane.

Meanwhile, the “cost” of including 10 new vehicles in the corridor is shown in the yellow sliver between 0.2233 and 0.2286, where the 80 vehicles that previously travelled the corridor now need to accommodate an additional 10 vehicles, slowing down the travel time for all the customers on those buses. The cost is therefore on the customers on the 80 vehicles now suffering the new congestion.

Fig. 6.38 The losses to the buses inside the busway, ATT_\text{loss} as represented by the light yellow rectangle, of shifting 10 buses into the corridor, amount to 0.424 hours. The aggregated time savings of those 10 buses, ATT_\text{savings} as represented by the green rectangle, amounts to 2.714 hours. Since the time savings are greater for those 10 buses than the losses to other buses already inside the corridor, it is still beneficial to shift in more buses.

To minimize aggregated travel time, the goal is to maximize the white area under the orange line for any given frequency inside the BRT in Figure 6.38, which in the shifting example includes the green area and loses the yellow area. So long as the area of the green rectangle (the time gains by customers on buses outside the BRT, or \( ATT_\text{savings} \)) is greater than the yellow rectangle (the time lost to the customers on buses inside the BRT due to congestion, or \( ATT_\text{loss} \)), more buses should be shifted in. For lower vehicle frequencies, the benefit of adding more vehicles to the BRT lane is even larger and the losses smaller. The benefit is larger than the cost because the new vehicles are going faster inside the BRT, and they are not yet adding much congestion to the bus lane (i.e., the curve is flatter at lower frequencies). Conversely, at higher frequencies, the benefit drops, and the costs increase. An optimal frequency is reached when the benefits equal the marginal costs of the last vehicle added, or where the area of the white rectangle is maximized.

The simplest way to make this determination is to include the formulas above in a table and simply calculate \( ATT_\text{total} \) at each frequency. One can do this in increments of 5. The resulting Table 6.18 is as follows:

F inside (bus/hour)F outside (bus/hour)TT inside (hour/bus)TT outside (hour/bus)ATT inside (hour)ATT outside (hours)ATT total (hours)
02000.200000.50.00100.00100.00
101900.201840.52.0295.0097.02
201800.203890.54.0890.0094.08
301700.206180.56.1985.0091.19
401600.208750.58.3580.0088.35
501500.211670.510.5875.0085.58
601400.215000.512.9070.0082.90
701300.218850.515.3265.0080.32
801200.223330.517.8760.0077.87
901100.228640.520.5855.0075.58
1001000.235000.523.5050.0073.50
110900.242780.526.7145.0071.71
120800.252500.530.3040.0070.30
130700.265000.534.4535.0069.45
135650.272690.536.8132.5069.31
140600.281670.539.4330.0069.43
145550.292270.542.3827.5069.88
150500.305000.545.7525.0070.75
155450.320560.549.6922.5072.19
160400.340000.554.4020.0074.40
165350.365000.560.2317.5077.73
170300.398330.567.7215.0082.72
175250.445000.577.8812.5090.38
179210.498330.589.2010.5099.70
180200.515000.592.7010.00102.70
185150.631670.5116.867.50124.36
190100.865000.5164.355.00169.35
19917.165000.51425.840.501426.34

Table 6.18 shows that the \( ATT_\text{total} \) reaches its minimum somewhere around \( F_\text{inside} = 135 \). At that point, \( ATT_\text{total} = 69.3 \). It is worth noting also that at \( F_\text{inside} = 199 \), \( ATT_\text{inside} \) suddenly skyrockets. This is because 200 is where saturation, \(x\), reaches 1. At \( F_\text{inside} = 195\),\( x = 0.975\) and already \( ATT_\text{inside} \) is increasing more dramatically. But in the shift of the last five vehicles, the functioning of the busway breaks down, and at 200, it is saturated.

One can also plot this in order to see graphically the point of the minimum aggregate travel time.

Fig. 6.39 Plot of ATT_text{inside} , ATT_text{outside} , and ATT_\text{Total} for each new bus added to the corridor. The minimum aggregate travel time is reached where ATT_\text{Total} is the lowest, i.e., between 130 and 140 buses per hour.

So the approximate optimal frequency inside the busway is 135. However, one does not know whether it is precisely 135. At \( F_\text{inside} =130\), \( ATT_text{inside} = 69.5 \) and at \( F_\text{inside} =140 \), \( ATT_\text{inside} = 69.4\). It is possible that the minimum \( ATT_\text{inside} \) falls somewhere between \( F_\text{inside} =130 \) and \( F_\text{inside} =140 \). One should look more closely, and with a greater degree of precision, at \( ATT_\text{inside} \) for each of the values between \( F_\text{inside} (130) \) and \( F_\text{inside} (140)\).

F inside (bus/hour)F outside (bus/hour)TT inside (hour/bus)TT outside (hour/bus)ATT inside (hour)ATT outside (hours)ATT total (hours)
130700.265000.534.4535.0069.45
131690.266450.534.9034.5069.40
132680.267940.535.3734.0069.37
133670.269480.535.8433.5069.34
134660.271060.536.3233.0069.32
135650.272690.536.8132.5069.31
136640.274380.537.3232.0069.32
137630.276110.537.8331.5069.33
138620.277900.538.3531.0069.35
139610.279750.538.8930.5069.39
140600.281670.539.4330.0069.43

Table 6.19 confirms that the minimum aggregate travel time (69.31) is indeed at \( F_\text{inside} \) = 135.

One can also zoom in to the graph from above to see that more clearly.

Fig. 6.40 Zoom-in of Figure 6.39 above to confirm minimum ATTTotal at 130 to 140 buses per hour. Image

Therefore, as a first pass, if bus routes have similar levels of demand and fleet characteristics, services should be selected so that routes with a total frequency of about 135 vehicles per hour should be included in the BRT corridor, and routes with the remaining 55 vehicles per hour should be left to operate in mixed traffic.

In this example, customer volumes, dwell times, vehicle type, and route frequencies all vary. In this case, maximizing total benefits will require deciding which bus routes to include in the BRT system and which routes to exclude. To do this, first calculate average passenger volumes and dwell times by route so that the variation is between routes, rather than between buses. For this example we will assume that stations become saturated equally. Although this seems a too hypothetical situation, a central BRT section with high renovation factor could look somewhat like this and still keep the same loads along the section. As load is relevant to this example and we are assuming it is constant through all stations, to visualize this situation one should imagine that the number of boardings is similar to the number of alightings in every station. In this example, it is assumed that one is unable, due to lack of political will, to take sufficient road space to build stations with the necessary passing lane and sub-stops to avoid station saturation. The following is also assumed:

\(L_\text{corridor}\): Corridor length = 5 km;

\(V_\text{outside}\): Velocity outside the busway = 12 km/h;

\(V_\text{inside}\):Velocity inside BRT at free-flow speed = 25 km/h;

\(N_\text{stations}\) = 10;

In this scenario, besides varying demand levels at stations, bus types are also different. Because it is not always possible to replace the entire fleet of buses with BRT vehicles all at once, the vehicles are going to vary in size and configuration in ways that will change the dwell time per customer. Because of this varying dwell time per customer, the BRT infrastructure will saturate to different degrees depending on which routes are incorporated into the BRT system. This will be one important change from Scenario I. Now, rather than simply determining how many routes to include, it is possible to determine which routes to include. This requires using a time savings formula to be applied to the number of customers instead of to the vehicles.

The service planner would have collected two types of data about each route:

  • Frequency and occupancy surveys, which will give the system planner the number of vehicles per route per hour, and their estimated occupancy;
  • Boarding and alighting data per route, which will give the number of passengers boarding and alighting at each station stop.

Table 6.20 provides an example showing thirteen bus routes with varying dwell times and customer volumes. This example will be used to demonstrate the methodology for determining which routes to include.

Table 6.20 Example of Route Choice

In Table 6.20 the fixed dwell time (or “dead time”) per bus is varied in a very simple way: either the bus is a large bus with a time of 12 seconds, it is a smaller bus with a time of 10 seconds, or there is one very small bus with a time of four seconds. The projected boarding and alighting time per bus route also only varies based on the vehicle type. For those new BRT-type buses with at-level boarding and three or four wide doors, one should estimate that the boarding and alighting time per customer will be one second, while for typical older buses one should assume the boarding or alighting time per passenger will be three seconds. These are reasonable values. The boarding and alighting passenger volumes are collected from the boarding and alighting survey for the bottleneck station. By adding the fixed dwell time (\(T_0\)) to the boarding and alighting time per route (\(T_b + T_a\)), the total dwell time per bus (\(T_d\)) can be calculated and is shown in the table above.

Using the results of the occupancy survey, one can now define a new variable, Pax(i). This refers to the average occupancy on each bus within a single route “i,” in the peak hour passing the station most likely to face a bottleneck. With this, a “priority index” can be calculated in order to determine which routes to include first:

Eq. 6.34

\[ \text{Priority}(i)={\text{Pax}(i) \over T_d(i)} \]

Where:

  • \( \text{Priority}(i)\): Priority index of route “i”; Passengers per second of dwell time on route “i” at the bottleneck station;
  • \( \text{Pax}(i)\): Average occupancy of each bus on route “i” as the bus passes the station;
  • \( T_d(i)\): Total dwell time in seconds of each bus on route “i”.

In other words, priority should be given to those bus routes where the most passengers are able to benefit from the busway for each second used at the bottleneck station. Bus routes should be listed in descending order of this priority index, so that routes with more customers per second of dwell time consumed at the bottleneck station are included first, and so on. In Table 6.21, the routes are ranked based on the priority index listed in the column labeled “Priority index.”

Table 6.21 Ranking of Routes for Route Choice

Each row in Table 6.21 represents a new route to be included in the BRT infrastructure. Starting with the empty busway, the last column shows the benefit in customers-hours (during one hour of operation) for including the route and all routes in rows above. After route “j,” the row in blue, overall benefits start to decrease, so no more routes should be included.

Now one can proceed with the methodology used in Scenario I, noting some differences. One should begin by calculating saturation. In this case, the dwell times should still be added up per route, but since dwell times vary from route to route, one cannot simply multiply one dwell time by total frequency inside the busway. Instead, one should multiply dwell time per route by frequency per route and then perform the full summation. So, in this case, the saturation formula should be written as:

Eq. 6.35 \( x = \sum_{\text{route-inside}} (T_{d i} * F_i) / 3600 \)

Where:

  • \( x \): Saturation at station;
  • \( T_{d i} \): Total dwell time at station for route “i” in seconds;
  • \( F_i \): Frequency of route “i” in vehicles/hour.

In Table 6.21, station saturation (\(x\)) is recalculated each time a new route is added. So saturation when only Route B is part of the BRT service plan, is the number of seconds of an hour (the peak hour) that Route B consumes in total dwell time (\(=(T_0 * \text{Freq})/3600 = 10 * 20/3600 = 0.056\)), meaning that Route B uses 5.6 percent of available station time in an hour.

If Route F is added, the BRT service plan includes Routes B and F. The station saturation that is contributed by Route F (calculated in the same manner as above) is simply added to the saturation contributed by Route B, or in this case 8.3 percent, and so on.

As is discussed in Chapter 7: Capacity and Speed, a busway starts to saturate when \(x = 0.4\), so routes where \( x > 0.4 \) should probably be excluded, pending further analysis.

The average queue time (average for all buses using the station) is calculated using the same formula from Scenario I (Eq. 6.30a):

Eq. 6.30a

\[ T_q={0.7 * x^2 \over 1-x } {1 \over F_\text{inside}} \]

Where:

Where:

  • \( T_q \): One vehicle queueing time at each station;
  • \( x \): Saturation of the station;
  • \( F_\text{inside} \): Number of buses per hour inside the busway;

So going back to including only Route B, the queuing delay is simply:

\[T_q = {0.7* 0.056 ^ 2 \over (1-0.056)} * { 1 \over 20} = 0.00114 \text{hours}=0.4 \text{seconds} \]

For this low frequency, 20 vehicles/hour, the average delay is less than 0.5 seconds, virtually no delay; as we add more vehicles this increases visibly.

For a more refined analysis, the additional time savings and losses of shifting a route is done per customer. The final piece of information that is needed in order to determine benefits for customers is the customer load passing the station. Load is calculated by multiplying the frequency times the observed customer occupancy per bus on the link approaching the bottleneck station. In the case of including only Route B, the load is 1,000. In the case where both Routes B and F are included in the BRT service, the load is the load for both Route B and Route F, or 1,000 + 500 = 1,500. Load is the sum of all occupancies on all routes included in the BRT infrastructure.

Eq. 6.36

\[ \text{Load}_\text{inside} = \sum_\text{routes-inside} (O_{\text{route }i} * F_{\text{route }i} )\]

Where:

  • \( \text{Load}_\text{inside} \): Customer load inside the BRT infrastructure;
  • \( \text{routes-inside} \): Number of routes inside the busway;
  • \( O_{\text{route }i}\): Occupancy on route i (inside the busway);
  • \( F_{\text{route }i}\): Frequency of route i (inside the busway).

We then calculate the total aggregated time savings for including each additional route, using the same benefit formula from Scenario I, as the calculus is cumulative, i.e., started from the reference situation where there are no routes inside the busway, there is no time loss to be computed due to the addition of the new route. The time savings must be multiplied by the passenger load instead of the vehicles to determine actual benefits to passengers, by using Equations 6.24a, 6.24c, and a variant of 6.24 (6.30 below):

Eq. 6.24a

\[TT_\text{outside}={L_\text{corridor} \over V_\text{outside}}\]

Eq. 6.24c

\[ TT_\text{inside}={L_\text{corridor} \over V_\text{inside-no-congestion}}+T_q*N_\text{stations}\]

Eq. 6.37

\[ ATT_\text{savings} = (TT_\text{outside} – T_\text{Tinside}) * \text{Load}_\text{inside} \]

Where:

  • \( ATT_\text{savings} \): Aggregated Travel Time savings for including selected routes in the BRT infrastructure;
  • \( \text{Load}_\text{inside} \): Customer load inside the BRT infrastructure;
  • \( TT_\text{inside} \): Travel time per bus inside the busway;
  • \( TT_\text{outside}\): Travel time per bus outside the busway;
  • \( L_\text{corridor}\): Length of the corridor;
  • \( V_\text{outside}\): Velocity outside the busway.
  • \( V_\text{inside-no-congestion} \): Velocity inside the busway without delay at stations;
  • \( T_q \): Average delay for each vehicle at a single station due to queuing;
  • \( N_\text{stations} \): Number of stations along the busway.

We do not need, in fact, to look at the aggregate travel time outside the corridor to determine which routes must be included. Aggregated travel time savings alone can answer for the utility of the inclusion of each additional route, and the point where the benefits are highest should be selected. In this case, benefits reach a maximum in the scenario where the following routes are included in the BRT services plan: B, F, K, H, D, M, G, J, and Routes A, I, E, L, and C are left to operate in mixed traffic. The result is almost the same as simply cutting off new routes when the saturation level reaches 0.4, except one more route is added.

In this example, it is assumed that different bus routes are causing the saturation problem at different stations, and multiple stations face bottlenecks. This is the most typical of real-world situations. In this case, there may be more than one bottleneck station, and the bus routes causing the bottleneck might vary by station. This scenario attempts to include all bus routes that need to use the infrastructure based on their rankings and, like the previous scenarios, leaves out those routes for which the cost of including them is greater than the benefit.

Considering a situation of 13 routes as the previous example, there are 8,191 possible combinations of routes to include inside the busway, and one could write a program to try them all to find which one yields the maximum aggregated travel time savings. If the number of routes doubles (26 routes), the number of possibilities increases to 67 million, which can still be tested in a feasible computational time. For 30 routes there are more than 1 billion combinations and for 40 routes, there are 1 trillion combinations, which computer brute force certainly cannot test.

For determining an optimal solution in this case, one should use the simplified methodology presented here, which alone is likely to yield the ideal result or a limited and manageable number of alternatives. This is mostly because the demand profile from one route to another is usually not completely random. High levels of vehicle occupation and high dwell times are often clustered in the same areas and at the same stops across many routes.

Here is a new example to demonstrate this methodology. Let’s say the corridor has three BRT stations (X, Y, and Z), and four potential bus routes (A, B, C, and D). The average dwell time and average occupancy on each route vary between stations.

  1. For each station, make a table of bus routes including average occupancy and dwell time, and calculate the priority index, Priority (station), using Equation 6.26 from Scenario II. Using the new example, one will have three tables:

    RouteDwell timeOccupancyPriorityiY
    TdiWOiW= OiY/ TdiW
    seconds/buspax/buspax/second
    A10434.3
    B6203.3
    C10484.8
    D12554.6
    RouteDwell timeOccupancyPriorityiW
    TdiYOiY= OiY/ TdiY
    seconds/buspax/busPax/second
    A10353.5
    B9202.2
    C8303.8
    D12403.3
    RouteDwell timeOccupancyPriorityiY
    TdiZOiZ= OiY/ TdiZ
    seconds/buspax/buspax/second
    A15553.7
    B11333.0
    C7223.1
    D11302.7

  2. Determine the best priority order by averaging priority (station) for each route.

    RouteAverage Priority
    = (OiY + OiW + OiZ)/3
    pax/second
    A3.8
    B2.9
    C3.9
    D3.5

    And then reorder the routes in descending order based on the average priorities, one table per station.

    RouteAverage Priority
    = (OiY + OiW + OiZ)/3
    pax/second
    C3.9
    A3.8
    D3.5
    B2.9
  3. Maintaining the priority order obtained above, add routes to the system as in Scenario II and obtain saturation and queue time per station, travel time inside the corridor, and aggregated time savings.
  4. For each station, assume corridor length as the distance (or half the distance) to the previous and next station and real observed speed outside the corridor to determine where the total aggregate travel time savings for that section reaches its minimum. Look in greater detail at the levels of saturation at critical stations, and if they are far from the individual optimum using the average priority, consider changing the route priority from the average route priority to something that more closely optimizes the route priority around the bottleneck station.
  5. Re-create the tables by station and see if the change results in a higher total benefit (sum of benefit of all stations calculated as in Scenario II): if yes, make the change; if not, try another.

These steps essentially replicate in a crude way what linear programming would do in a systematic way, and if the planner has data, time, and a deep understanding of these formulas, she or he can put all equations and restrictions into a system and look for an optimal solution.

In conclusion, when designing BRT services for a BRT corridor with constrained capacity, the inclusion of routes into BRT services should be based on ensuring that the aggregated travel time gains that accrue to the last route included in the BRT system outweigh the losses that its inclusion imposes on the other routes already included in the BRT system. This loss results from the additional bus route to the BRT system slowing down the speeds of the vehicles already inside the system.

Further, the frequency and amount of overlap with the BRT corridor should be taken into consideration. Additionally, in order to help make decisions about which routes should be included in a new BRT service, a “priority” index could be created to measure the following: total customers on board the bus as it approaches the bottleneck stations divided by the total dwell time that that bus route uses at the bottleneck stations.