Multi-Stage Pedestrian Crossings and Two-Stage Bicycle Turns: Delay Estimation and Signal Timing Techniques for Limiting Pedestrian and Bicycle Delay

Signalized intersections sometimes involve multistage pedestrian 
crossings, in which pedestrians cross to one or more islands and then wait there 
for a signal to continue. When signals are timed without attention to 
pedestrian progression, pedestrian delay at multistage crossings can be very 
long. This paper addresses two issues. First, pedestrian delay at multistage 
crossings is rarely evaluated because there are no tools in the industry for 
that purpose except microsimulation. We present a numerical method for 
determining crossing delay with any number of stages and with the possibility 
of multiple WALK intervals per cycle. The same method can be applied to single 
stage crossings, to diagonal two-stage crossings where pedestrians may have 
path choice, and bicycle two-stage turns. This method has been implemented in a 
freely available online tool. Second, we describe several signal timing techniques 
for improving pedestrian and bicyclist progression, and thus reducing their 
delay, through multistage crossings. They include reservice for selected 
crossing phases, left turn overlaps, having pedestrian phases overlap each other, and bidirectional 
bicycle crossings which create path options for two-stage turns. Examples show 
the potential for large reductions in pedestrian delay, often with little or no 
increase in vehicular delay. In one example, the addition of a short pedestrian 
overlap phase reduced average pedestrian delay at a 3-stage crossing by 82 s 
while average vehicular delay increased by only 0.5 s.

strians cross to an island and are expected to wait there for a signal there to resume crossing. Two-stage crossings are common; three-and four-stage crossings are also used.
While multistage crossings can improve an intersection's efficiency, they can also lead to large pedestrian delays if pedestrian progression through the different stages is poor. While the Manual on Uniform Traffic Control Devices [1] specifies that all of the safety provisions that apply to signalized crosswalks must be applied to each stage of a multistage crossing individually, it is silent on whether or how those stages are to be coordinated. Unfortunately, in current practice, there are no easily applied tools for evaluating pedestrian delay at multistage crossings. Using standard formulas for pedestrian delay at each stage and adding them is incorrect, because those formulas assume random arrivals, and pedestrian arrivals at every stage after the first is not at all random. Because of the lack of analysis methods, intersection designs with multistage crossings are routinely advanced and implemented without any evaluation of pedestrian progression or delay. No wonder then that pedestrian delay is often very long.
For bicyclists, two-stage left turns [2] are a form of multistage crossing. A cyclist turning left may do so in a "vehicular manner", i.e. using the same left turn lane as cars, in a single stage; however, more and more bike lanes-particularly protected bike lanes-are designed with the idea that cyclists will turn left in two stages, as a pedestrian would. Again, standard methods of delay evaluation do not apply, and methods are needed to ensure that cyclist delay is properly measured. Part 1 of this paper describes a method for evaluating pedestrian and bicyclist delay at multistage crossings. This method has been implemented in a freely available software tool. Part 2 describes signal timing techniques for achieving good pedestrian progression through multistage crossings and thus limiting pedestrian delay. Examples show that by using these techniques, intersections with multistage crossings can offer pedestrians a good level of service while retaining their efficiency advantages for vehicular traffic.

Estimating Multistage Crossing Delay
Existing Methods for Estimating Crossing Delay At single stage crossings, average pedestrian delay can be evaluated rather easily. Except where pedestrian volumes are so great that pedestrians queue up several people deep, pedestrians can be assumed to cross without interference from other pedestrians, making pedestrian counts unnecessary for estimating delay. The signal cycle can be divided into two intervals: effective pedestrian green, in which pedestrians are allowed to start crossing, and the rest of the cycle, called effective pedestrian red. Effective pedestrian green consists of the WALK interval plus the first few seconds following it. Legally, pedestrians are supposed to begin crossing only during the WALK interval, but observation indicates that most pedestrians are still willing to begin crossing a few seconds after it ends.
The Highway Capacity Manual (HCM) [ where C = cycle length and r ped = length of effective pedestrian red [3]. It is based on the assumption that pedestrians are equally likely to arrive at any moment in the cycle (uniform arrivals). The first ratio is the fraction of pedestrians arriving on red, and the second is the average delay to those that do arrive on red. Those who arrive during pedestrian green, of course, have no delay.
With multistage crossings, it would be wrong to evaluate pedestrian delay by simply applying Equation (1) to each crossing stage and summing, because at every stage except the first, pedestrians do not arrive uniformly; instead, they arrive in platoons as released by the previous stage's signal [5]. Wang and Tian [6] and Ma et al. [7]  This method is readily expandable to any number of stages, any number of walk intervals per cycle, and path choice. It is computationally fast. It lends itself well to graphical representation, making it easy to understand, easy to check for errors, and easy to explain to the public.
Tracing a pedestrian trajectory for a given initial arrival time involves the following steps. Assume the crossing has n stages. Initial arrival time, t in , is given. Let t arr and t dep denote arrival time and departure time, respectively, at the current stage.
1) Initialize. Crossing Stage = 1. t arr = t in . Accumulated Walk = 0. 2) Departure time. If the signal state for Crossing Stage at time t arr is effective green, t dep = t arr . Otherwise, t dep = next moment at which the signal state becomes effective green.
3) If Crossing Stage < n, advance to the next step. Otherwise, set t out = t dep ; calculate Time in System = t out − t in ; calculate delay = Time in System-Accumulated Walk. STOP. 4) Walking time. t walk = distance to the next stage/Walk Speed; augment Accumulated Walk by t walk . Distance to the next stage is the crossing length for Crossing Stage plus the distance across the island to the stopline for the next crossing stage. A standard walking speed such as 4.5 ft/s, intended to represent an average walking speed, can be applied. (It is also possible to use a distribution of walking speeds to account for a mix of faster and slower pedestrians. We tested this approach and found that it yielded no meaningful difference in results compared to using average walking speed). 5) Advance to the next stage. t arr = t dep + t walk . Augment Crossing Stage by 1.Go to step 2.
We have programmed this method using MatLab as the computational engine. It is freely available for download as the Northeastern University Ped & Bike Crossing Delay Calculator [8]. (Using the program does not require a Mat-Lab license). It has one module for sequential crossings and another for diagonal crossings that may involve path choice. Users enter walk distances (i.e. crosswalk lengths and island widths) and signal timing data (start time and duration of each walk interval). User-adjustable parameters are walking speed (default = 4.5 ft/s) and extra time after the WALK interval that is still considered effective green (default = 4 s). The output is average delay at each stage of the crossing and overall, and a progression diagram with trajectories illustrating pedestrian progression. Pedestrian volume data is not needed, because of the assumption that pedestrians will not be substantially delayed by each other.  ABCD accumulate while waiting a long time at A until there's a WALK signal, and then walk in a platoon to island B; there, they wait almost 30 s, and then travel en masse to island C, where they wait roughly 60 s for a WALK to finish their crossing. Overall, average delay is 124 s in direction ABCD, and 123 s in direction DCBA-more than double the 60 s limit used in the Highway Capacity Manual to define level of service "F", the worst possible level of service.
It is highly unlikely that this timing plan, implemented in 2016, would have been approved if these average delays had been calculated and reported to the public and to approving agencies. Later in this paper, we show that with a modification to the timing plan, pedestrian delay can be reduced drastically with no discernable impact on vehicular delay.
To confirm the proposed method, the two-stage crossing described by Wang and Tian [6], crossing Boulder Highway at Flamingo Road in Las Vegas, was evaluated. They found the delay to be 55.8 s in each direction; our method finds the average delay to be 55.8 s eastbound and 55.9 s westbound. The 0.1 s discrepancy is approximation error due to using finite time steps. Confirmation was Evaluating Delay at Intersections with Actuated Control Our method, like those cited in the literature, treats cycle length and pedestrian interval timing as given, while many intersections use actuated control logic in which the length of relevant intervals is not fixed. For isolated actuated signals, expected values of cycle length and phase lengths can be estimated following the methods described in the HCM or in Furth, Cesme, and Muller [9].
For coordinated-actuated signals, cycle length is fixed, but analysts may have to estimate the start time and duration of WALK intervals. During peak periods, for which evaluation is often most critical, actual operations often match nominal timings closely as phases tend to run to their maximum split. Where flexible control logic makes the signal operation highly variable, microsimulation may be the only reliable way to estimate performance.
Estimating Delay for Diagonal Crossings with Path Choice Pedestrians crossing to a diagonally opposite corner of an intersection normally make their crossing in two stages, passing through another corner en route. Where such a movement is important for transportation planning-for example, for a major crossing from a railroad station-designers will be interested in estimating and limiting the delay for this 2-stage movement.
Diagonal crossings usually involve path choice. If the corners of an intersection are labeled clockwise A-B-C-D, then a pedestrian crossing from A to C will typically have the choice of path A-B-C (clockwise) or A-D-C (counterclockwise). Where there is no choice-say, because one of the crossings is closed-a diagonal crossing can be treated as a sequential multistage crossing. But where there is path choice, additional logic is needed.
We assume that pedestrians identify the feasible paths and take whichever path offers a WALK signal first. With this assumption, there is a unique chosen path for every arrival moment, so that trajectories for an arrival in each time step can readily be constructed and delay calculated, as before. In the unlikely event that both paths offer WALK at the same time at the first stage, demand is split equally between the paths.
An example calculation is given in Figure 2 for a hypothetical intersection with 8-phase, dual ring control, leading lefts, concurrent crossings, and an 80 s cycle. Other relevant timing parameters are shown in the figure. As the figure shows, with path choice, average delay from corner A to corner C is 33 s. Another calculation (not shown) finds that if pedestrians were restricted to a single path, average delay would instead be 53 s for path A-B-C and 52 s for path A-D-C.  rides, are unidirectional, in which case a cyclist making a two-stage left turn has only one legal path. In such a case, cyclist delay for the left turn movement can be calculated by treating it as a sequential two-stage crossing. However, if bicycle crossings are marked as bidirectional-a tactic recommended in the Dutch bikeway manual [10]-cyclists can take advantage of path choice, just like pedestrians making a diagonal crossing, thereby reducing their delay.
Using the same example shown in Figure 2, but with cyclist speed set to 12 ft/s and cyclists allowed to start through the entire vehicular green, their average delay from corner A to corner C will be 42 s if crossings are bidirectional, versus 64 s if crossings are unidirectional.
It is interesting to compare these results with delay for a vehicular left-turn, made in a single stage from the left turn lane. If overflow delay can be ignored, the HCM's uniform delay formula may be used to estimate delay [3]. Assuming that the left turn phase has 10 s of effective green per cycle and a volume-capacity ratio of 0.8, average delay for a single-stage crossing will be 38 s. In this example, then, providing bidirectional crossings reduces two-stage turning delay to the point that it is comparable to delay associated with a vehicular-style, single-stage left turn.

Signal Timing Techniques for Limiting Delay at Multi-Stage Crossings
Just as traffic signals along an arterial can be timed to offer good progression for vehicles, they can also be timed to offer good progression to pedestrians passing through multiple crossing stages. This section describes several signal timing techniques that can be used to improve pedestrian progression. Technique 1: Offset Crossing Phases with Short Cycles For crossing a wide street with a median, a single-stage crossing demands a long crossing phase, reducing traffic capacity. To limit that traffic capacity im-Journal of Transportation Technologies pact, the crossing can be configured as a pair of half-crossings, timed with only enough pedestrian clearance time to reach the end of the half-crossing, forcing pedestrians to wait in the median for the next WALK phase. If the WALK phases for the two half-crossings are simultaneous, pedestrians will have to wait almost a full cycle to cross, and if the cycle is long, the average delay will be quite large.
The first way to lower this delay is to reduce the cycle length. Because a half-crossing requires a shorter phase than a full crossing, it should be possible to provide the same vehicular capacity with a shorter cycle. A second way, relevant mainly at mid-block crossings, is to offset the two crossing phases by half a cycle. Pedestrians will have to wait in the median after the first half-crossing, but only for a limited time because the second crossing stage begins only half a cycle after the first stage. If the cycle is short, the wait in the middle can be very short.
The crossing of Huntington Avenue at Opera Place in Boston offers an illustrative example (see Figure 3   At a 4-leg intersection with channelized right turns, pedestrians have to make 3-stage crossings-a short crossing to a delta island, then a main crossing to another delta island, and then another short crossing. If the short crossings across the right turn lanes are signalized, then providing reservice is critical to having good progression, since pedestrians crossing to any given delta island may be continuing along either the north-south or east-west axis, and with only a single WALK interval per cycle, it is impossible to provide good progression for crossings along both axes.

Technique 3: Left Turn Overlaps
Where an intersection has a median, some crossings to and from the median can run concurrently with a left turn phase as well as a through phase, an arrangement called an overlap. That can afford both longer WALK intervals and better pedestrian progression, thus reducing delay.
Consider the two-stage crossing of Beacon Street at St. Paul Street in Brookline, MA. Beacon Street has left turn phases, while St. Paul does not. Figure 5 shows the four possible ways of sequencing the Beacon Street left turn phases. Left turn overlaps are not currently provided at this intersection or at many like it. This may be because in order to use an overlap, when there is a pedestrian call, left turn phases must not be skipped and their minimum split might have to be greater than the minimum needed for vehicles. Would constraining the left turn phase so that it could be part of a pedestrian overlap significantly affect vehicular capacity and delay? At many intersections, including this one, the answer is no. During peak periods, the left turn phase is never skipped and nearly always runs to its maximum green, which is long enough for an overlap; and outside the peak, there is excess capacity, so that extra green time taken by the left turn phase would not significantly affect the through movement.

Technique 4: Overlapping a Short All-Vehicles-Hold Phase
A final technique for improving pedestrian progression at multistage crossings is a short phase in which no vehicular movements are allowed. The idea is not to create an exclusive pedestrian phase long enough to serve pedestrians by itself, but rather to use this "hold" phase to extend pedestrian phases in multiple directions in order to create good progression.
This technique can also be used to provide a single-stage crossing for bikes.
Because bikes are faster than pedestrians, they need considerably less time for a single stage crossing. And where medians are not wide enough to safely hold a queue of bicycles, cyclists may need to have single stage crossings even where pedestrians make a two-stage crossing. This need is well understood in the Neth- This technique can be applied to the Riverway crossing that was described in Part 1 of this paper (see Figure 1). The crossing islands there are too narrow to hold queuing bikes, meaning that, in addition to having terrible service for pedestrians (average delay is 123 s), there is no safe bicycle crossing.
A new traffic signal timing plan was developed for morning peak hour volumes with the existing cycle length (100 s), improving pedestrian progression by adding a 12 s phase in which all vehicular movements are held. As shown in Figure 6, that 12 s interval begins around time 55, and contributes to elongating all three crossing phases. Figure 6 shows the progression that results, with little waiting at the median islands; that can be contrasted with the poor progression evident in the existing timing ( Figure 1). Also shown in Figure 6 is the average pedestrian delay, only 41 s-a huge drop from the 123 s that prevails today. For bikes, the existing plan has no bike crossing at all (bikes must become pedestrians to cross), while the proposed plan offers bikes a single stage crossing with average delay 42 s. At the same time, vehicular delay, evaluated using the microsimulation model VISSIM, increases by only 0.5 s, from 34.8 to 35.3 s. For more detail on the analysis of this intersection, please see the case study report [11]. Where all four left turns are of concern and there is a full set of left turn phases, there is no sequence offering good progression for all four left turns if bicycle crossings are unidirectional, as is the usual case.
However, if bicycle crossings are bidirectional, having one street's through movements lead while the other's lags allows all two-stage turns to be made with good progression, because every left turn can begin with a lagging through phase followed by a leading through phase. This is illustrated in Figure 7

Conclusions
The lack of existing methods for calculating pedestrian delay at multistage crossings means that signal timing plans with multistage crossings are sometimes developed that inadvertently lead to very long pedestrian delays because of the poor progression they offer from one stage to the next. We have demonstrated a simple numerical method with which multistage pedestrian delay can be readily calculated, even if there are multiple stages or multiple WALK intervals in a cycle, and also for diagonal crossings in which pedestrians can choose their path. This method can also be used to evaluate delay for cyclists making a two-stage left turn.
While multistage crossing delay can be very long where cycles are long and pedestrian progression is poor, numerous signal timing techniques can be used to create good pedestrian progression. In many cases, they can drastically improve service for pedestrians with little or no impact on vehicles. At one intersection with a 3-stage crossing, we show how pedestrian delay can be reduced by more than 80 s while vehicular delay remains essentially unchanged. Phasing P. G. Furth et al.