Guide to Pedestrian Analysis (2022)

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A-1   Estimation of Pedestrian Delay: Uncontrolled Crossings The method presented here is a revised version of the one given in the sixth edition of the Highway Capacity Manual (HCM 6th ed.), for estimating pedestrian delay at uncontrolled crossings (1). This method predicts pedestrian delay at two-way stop-controlled intersections and midblock crossings where pedestrians cross up to four through lanes on the major street. The revisions address discontinuities found in the predicted pedestrian delay when examined over a range of traffic volumes. A method for predicting pedestrian delay at uncontrolled crossings was first introduced in Chapter 18 of the 2000 edition of the Highway Capacity Manual (HCM 2000) (2). The method- ology estimates the average number of pedestrians that will cross when the vehicle headway exceeds the minimum (i.e., critical) headway. This predicted pedestrian group size is then used to estimate the minimum headway the group needs to cross the street. The distribution of vehicle headways is assumed to follow the negative exponential distribution. An equation is provided for predicting pedestrian delay (i.e., the delay incurred while waiting for a headway to exceed the group minimum headway, at which time pedestrians are able to enter the cross- walk and begin crossing). The key analytic elements of this methodology are described in TRB Special Report 165 (3). For the Highway Capacity Manual 2010 (4), developed by NCHRP Project 03-92, the method- ology in Chapter 18 of the HCM 2000 was updated to consider motorists that yield the right-of- way to pedestrians crossing the street (J. Parks, NCHRP Project 03-92 correspondence, 2009). The HCM 2000 methodology is based on the conservative assumption that no drivers will yield. However, many innovative pedestrian crossing treatments have been found to successfully induce most drivers to yield to pedestrians. To incorporate this behavior, the methodology was updated to include equations to estimate the delay associated with two delay-producing scenarios. The first scenario represents the delay incurred before the first yielding driver. The second scenario represents the delay incurred if no driver yields. The methodology uses a weighted average of the two scenarios to estimate the average delay. The updated methodology was also included in the HCM 6th ed. in 2016. NCHRP Project 17-87 evaluated the HCM 6th ed. method by examining the sensitivity of the predicted delay to traffic volume. The findings indicate a discontinuity in the predicted pedestrian delay when it is evaluated over a range of traffic volumes. More importantly, the methodology over-predicted delay when the proportion of yielding motorists was high. Detailed information about the evaluation is provided in the project’s final report (5). Revisions to the method to address these issues have been incorporated into the method described below and into the computational engine implementing the method, described in Appendix B. A P P E N D I X A Calculation Details for Analysis Method

A-2 Guide to Pedestrian Analysis The method’s computational steps are as follows: • Step 1. Identify two-stage crossings. • Step 2. Determine critical headway. • Step 3. Estimate probability of a delayed crossing. • Step 4. Calculate average delay to wait for an adequate gap. • Step 5. Estimate average pedestrian delay for the crossing stage. • Step 6: Calculate average pedestrian delay. Step 1: Identify Two-Stage Crossings Step 1 does not include any calculations. Instead, the analyst decides whether pedestrians cross the entire street in a single stage or use the median as a refuge to complete the crossing in two stages. When pedestrians cross in two stages, pedestrian delay is estimated separately for each stage of the crossing and the two delay values are summed to produce the total delay incurred in crossing the street. Step 2: Determine Critical Headway This step computes the critical headway for a group of pedestrians waiting to cross the street. If the pedestrian volume is high, the procedure assumes pedestrians will cross in groups during each crossing opportunity. First, the critical headway for a single pedestrian is calculated with the following equation: = + Equation A-1t L S tc p s where tc = critical headway for a single pedestrian (s); Sp = average pedestrian walking speed (ft/s); default = 3.5 ft/s; L = crosswalk length (ft); and ts = pedestrian start-up time and end clearance time (s); default = 3.0 s. The average number of pedestrians waiting to cross is computed using the following equation: ( ) = + + ( ) − − Equation A-2N v e ve v v e c p v t vt p v v t p c c p c where Nc = total number of pedestrians in the crossing platoon, vp = pedestrian flow rate (p/s), v = conflicting vehicular flow rate (veh/s) (combined flows for one-stage crossings, separate flows for two-stage crossings), and tc = single pedestrian critical headway (s). Pedestrians waiting to cross are assumed to form rows, with the first row in position to cross and subsequent rows lined behind the first row. The number of rows is computed with the following equation: =     max 8.0 ,1.0 Equation A-3N N W p c c

Calculation Details for Analysis Method A-3   where Np = spatial distribution of pedestrians (number of rows of pedestrians), Nc = total number of pedestrians in the crossing platoon, Wc = crosswalk width (ft), and 8.0 = default clear effective width used by a single pedestrian to avoid interference when passing other pedestrians (ft). Finally, the group critical headway is computed with the following equation: ( )= + −2 1 Equation A-4,t t Nc G c p where tc,G is the group critical headway (s) and all other variables are as previously defined. Step 3: Estimate Probability of a Delayed Crossing The probability that a given lane cannot be crossed is equal to the probability that the vehicle headway in the subject lane does not exceed the group critical headway. This probability is computed using the following equation: = − − 1 Equation A-5 , P eb t v N c G L where Pb = probability of a blocked lane, NL = number of through lanes crossed, tc,G = group critical headway (s), and v = conflicting vehicular flow rate (veh/s) (combined flows for one-stage crossings, separate flows for two-stage crossings). A crossing can occur when each of the lanes crossed has a vehicle headway in excess of the group critical headway. A delayed crossing occurs when the headway in one or more of the lanes crossed is less than the group critical headway. The probability of a delayed crossing is computed with the following equation: ( )= − −1 1 Equation A-6P Pd b NL where Pd is the probability of a delayed crossing and all other variables are as previously defined. Step 4: Calculate Average Delay to Wait for an Adequate Gap The average delay per pedestrian to wait for an adequate headway (i.e., a headway longer than the minimum critical headway) is computed with the following equation: ( )= − −1 1 Equation A-7,,d v e vtg vt c Gc G where dg = average pedestrian delay (s), tc,G = group critical headway (s), and v = conflicting vehicular flow rate (veh/s) (combined flows for one-stage crossings, separate flows for two-stage crossings; the lower limit of this variable is set to 0.0001).

A-4 Guide to Pedestrian Analysis The average delay for any pedestrian who is unable to cross immediately upon reaching the intersection (e.g., any pedestrian experiencing nonzero delay) is computed with the following equation: = Equation A-8d d P gd g d where dgd is the average delay for pedestrians who incur nonzero delay and all other variables are as previously defined. Step 5: Estimate Average Pedestrian Delay for the Crossing Stage When a pedestrian arrives at a crossing and finds the vehicle headway is shorter than needed to cross, that pedestrian is delayed until either a headway greater than the critical headway is avail- able or motor vehicles yield and allow the pedestrian to cross. Equation A-7 estimates pedestrian delay when motorists on the major approaches do not yield to pedestrians. When motorist yield rates are significantly higher than zero, pedestrians will experience considerably less delay than that estimated by Equation A-7. In the United States, under most circumstances in both marked and unmarked crosswalks, motorists are legally required to yield to pedestrians. However, the actual yielding behavior of motorists varies considerably. Table 3-4 summarizes motorist yielding rates given in the literature. Consider a pedestrian waiting for a crossing opportunity at an uncontrolled crossing. Vehicles in each conflicting through lane arrive at an average of h seconds apart. In other words, a potential yielding event occurs every h seconds. For any given yielding event, each through lane is in one of two states: • Clear—no vehicles are arriving within the critical headway window or • Blocked—a vehicle is arriving within the critical headway window. The pedestrian may cross only if vehicles in each blocked lane choose to yield. If vehicles do not yield, the pedestrian must wait an additional h seconds for the next yielding event. On average, this process will be repeated until the wait exceeds the expected delay required for an adequate headway in traffic (dgd), at which point the average pedestrian will receive an adequate headway in traffic and will be able to cross the street without having to depend on yielding motorists. Average pedestrian delay can be calculated with Equation A-9, where the first term in the equation represents expected delay from crossings occurring when motorists yield and the second term represents expected delay from crossings when pedestrians wait for an adequate headway. The following equation computes the appropriate headway h (6): ∑∑ ( ) ( ) ( )= − + −      == 0.5 Equation A-9 00 d h i P Y P P Y dp i d i i n gd i n with ( ) [ ] [ ] = − + − − − 1 1 exp 1 exp Equation A-10, , , h v t v vt vt c G c G c G [ ] = −      int 1 exp Equation A-11 , n vtc G

Calculation Details for Analysis Method A-5   where dp = average pedestrian delay (s), i = potential yielding event (i = 0 to n), h = average headway of all headways less than the group critical gap (s), dgd = average delay for pedestrians who incur nonzero delay, tc,G = group critical headway (s), v = conflicting vehicular flow rate (veh/s) (combined flows for one-stage crossings, separate flows for two-stage crossings), P(Yi) = probability that motorists yield to pedestrian on potential yielding event i (this value is zero when there are no crossing events), Pd = probability of a delayed crossing, and n = average number of potential yielding events before an adequate headway is available. One-Lane Crossing For a one-lane crossing, the probability that motorists yield to the waiting pedestrians is calculated with the following equation: ( )( ) = − −1 Equation A-121P Y P M Mi d y y i where My is the motorist yield rate (decimal)—a maximum value of 0.999 or less—and i is the potential yielding event (i = 0 to n). When i = 1, P(Yi) is equal to the probability of a delayed crossing Pd multiplied by the motorist yield rate My: ( ) = Equation A-131P Y P Md y Two-Lane Crossing For a two-lane pedestrian crossing, P(Yi) requires (a) motorists in both lanes to yield simul- taneously if both lanes are blocked or (b) a single motorist to yield if only one lane is blocked. Because these cases are mutually exclusive, where i = 1, P(Yi) is given by Equation A-14: ( ) ( )= − +2 1 Equation A-141 2 2P Y P P M P Mb b y b y where Pb is the probability of a blocked lane. For i > 1, Equation A-15 gives the probability that motorists yield to waiting pedestrians: ∑ ( ) ( ) ( )( ) [ ]= −       − +   = − 2 1 Equation A-15 0 1 2 2 P Y P P Y P P M P M P i d j j i b b y b y d where P(Yj) is the probability that motorists yield to pedestrians on potential yielding event j. Three-Lane Crossing A three-lane crossing follows the same principles as a two-lane crossing. The probability of all blocking vehicles yielding on the first potential yielding event is given by Equation A-16: ( ) ( ) ( )= + − + −3 1 3 1 Equation A-161 3 3 2 2 2P Y P M P P M P P Mb y b b y b b y For i > 1, Equation A-17 gives the probability that motorists yield to the waiting pedestrians: ∑ ( )( ) ( ) ( )= −       + − + −   = − 3 1 3 1 Equation A-17 0 1 3 3 2 2 2 P Y P P Y P M P P M P P M P i d j j i b y b b y b b y d

A-6 Guide to Pedestrian Analysis Four-Lane Crossing A four-lane crossing follows the same principles as above. The probability of all blocking vehicles yielding on the first potential yielding event is given by Equation A-18: ( ) ( ) ( ) ( )= + − + − + −4 1 6 1 4 1 Equation A-181 4 4 3 3 2 2 2 3P Y P M P P M P P M P P Mb y b b y b b y b b y For i > 1, the probability that motorists yield to the waiting pedestrians is given by Equa- tion A-19: ∑ ( )( ) ( ) ( ) ( ) = −       × + − + − + −     = − 4 1 6 1 4 1 Equation A-19 0 1 4 4 3 3 2 2 2 3 P Y P P Y P M P P M P P M P P M P i d j j i b y b b y b b y b b y d Step 6: Calculate Average Pedestrian Delay In the case of a one-stage crossing, the average pedestrian delay for the crossing is the average pedestrian delay dp calculated in Step 5. In the case of a two-stage crossing, the average pedestrian delay for the crossing is the sum of the average pedestrian delays calculated for each crossing stage. Estimation of Pedestrian Delay: Signalized Crossings This section describes a methodology for computing pedestrian delay when an intersection leg at a signalized intersection is being crossed. The methodology recognizes that pedestrian delay can be influenced by the phase sequence, signal operation, and pedestrian travel paths at the subject crossing location. The methodology addresses the following cases: • Pedestrians cross one leg of the intersection during one signal phase (i.e., a one-stage crossing), • Pedestrians cross one leg of the intersection during two signal phases (i.e., a two-stage cross- ing), and • Pedestrians cross two legs of the intersection during two signal phases (i.e., a diagonal crossing). A computational engine implementing this method is described in Appendix B. One-Stage Crossing Procedure This section summarizes the equation offered in Chapter 19 of the HCM 6th ed. (1) for comput- ing the delay of pedestrians that cross a specified intersection leg during one phase (referred to hereafter as a “one-stage crossing”). This equation is based on the following three assumptions: (a) pedestrian arrivals to the crossing location (i.e., street corner) are random, (b) the signal operation is such that pedestrians can cross the entire width of the intersection leg (corner-to- corner) during one signal phase, and (c) random arrivals over a large number of signal cycles can be modeled deterministically by using a uniform arrival rate. The HCM 6th ed. provides the following equation for computing this delay: ( ) = − 2 Equation A-20Walk, 2 d C g C p i where dp = pedestrian delay (s/p), C = cycle length (s), and gWalk,i = effective walk time for Phase i serving the subject pedestrian movement (s).

Calculation Details for Analysis Method A-7   Equation A-20 does not account for the possibility that some arrivals may come from the intersecting crosswalk as part of a diagonal crossing. Figure A-1a indicates the number assigned to each crosswalk. Figure A-1b indicates the number assigned to traffic movement at each intersection. The numbers shown in Figure A-1b are established to coincide with the signal phase that serves the corresponding traffic movement. Notably, the pedestrian movement and the adjacent through vehicle movement share the same number because they are served during the same phase. For example, Vehicle Movement 2 is a through movement on the left side of the intersection. This movement is served by Signal Phase 2. Pedestrian movement 2P crosses in Crosswalk 2. This pedestrian movement is also served by Signal Phase 2. On the basis of the preceding explanation of traffic movement numbers and signal phases, the delay to pedestrian movement 2P is based on the cycle length C and effective walk time for Phase 2 gWalk,2. Equation A-20 computes this delay. The delay value describes the delay to pedes- trians crossing in either direction in Crosswalk 2 (i.e., from Corner C to Corner B, and from Corner B to Corner C). Two-Stage Crossing Procedure This section describes the procedure for computing pedestrian delay when a specified inter- section leg is being crossed in two phases. This procedure is used to estimate delay to a given direction of travel in a specified crosswalk. The procedure is separately applied to evaluate the other direction of travel in the specified crosswalk or to evaluate other crosswalk locations. The procedure is based on that developed by Wang and Tian (7). This procedure is based on the following two assumptions: (a) pedestrian arrivals to the first crossing location (i.e., street corner) are random and (b) the signal operation is such that pedes- trians will need two phases to complete the crossing (waiting on the median before crossing the second half of the street). With regard to the first assumption, this procedure does not account for the delay associated with diagonal crossings (i.e., all pedestrians are assumed to arrive randomly at the first corner). Major Street Minor Street Vehicle Movements Pedestrian Movements 5 2 4P 3 8 2P 1 6 8P 7 4 6P 6 8 2 4 Corner A Corner BCorner C Corner D (a) Crosswalk numbering scheme (b) Traffic movement numbering scheme Figure A-1. Intersection traffic movement and crosswalk numbering scheme.

A-8 Guide to Pedestrian Analysis The procedure described in this section is based on the vehicle movement numbering scheme shown in Figure A-2a. These vehicle movement numbers correspond to the signal phase that serves the movement (i.e., Vehicle Movement 2 is served by Signal Phase 2), which follows the traditional eight-phase dual-ring structure. The procedure is also based on the crosswalk number- ing scheme shown in Figure A-2a and the pedestrian movement scheme shown in Figure A-2b. With a two-stage crossing, the signal operation accommodates pedestrians crossing an inter- section leg by providing pedestrian service during two signal phases. During the first phase, pedestrians cross from the first corner to the median. During the second phase, they cross from the median to the next corner. The first phase to occur is denoted by the letter X and the second phase to occur is denoted by the letter Y. The crossing direction of interest and the phase sequence are considered to determine which phase is Phase X and which is Phase Y. The two phase numbers of interest are identified in Figure A-2a by the two-digit crosswalk number associated with the crosswalk of interest. For example, if the crosswalk between Corner C and Corner B is of interest, Phases 1 and 2 are used to define Phase X and Phase Y. The crossing direction and phase sequence are considered in the following manner: • If the crossing direction is clockwise (i.e., from Corner B to Corner C) and Phase 1 leads Phase 2 in the phase sequence, then Phase X is Phase 1 and Phase Y is Phase 2. • If the crossing direction is clockwise (i.e., from Corner B to Corner C) and Phase 1 lags Phase 2 in the phase sequence, then Phase X is Phase 2 and Phase Y is Phase 2. • If the crossing direction is counterclockwise (i.e., from Corner C to Corner B), then Phase X is Phase 2 and Phase Y is Phase 1, regardless of whether Phase 1 leads or lags Phase 2. The required data for the two-stage crossing procedure and the diagonal crossing procedure (described in the next section) are as follows: • Cycle length (s), • Phase sequence (list of phases in order of occurrence), • Phase duration (sum of the duration of the green, yellow change, and red clearance intervals) for all phases (s), D4 B12 C12 B2 D56 A6 D6 A56 D34 C2 C34 C4 A78 A8 B8 B78 Corner A 8 122 4 656 Corner BCorner C Corner D 34 78 (a) Crosswalk numbering scheme (b) Pedestrian movement numbering scheme Figure A-2. Pedestrian movement and crosswalk numbering scheme for two-stage crossing.

Calculation Details for Analysis Method A-9   • Duration of “Walk” interval for Phase X and Phase Z (s), • Distance crossed during Phase X [i.e., distance from first corner to far side of median (two- stage crossing) or second corner (diagonal crossing)] (ft), • Duration of yellow change interval for Phase X and Phase Y (two-stage crossing) or Phase Z (diagonal crossing) (needed only if rest-in-walk is enabled or no pedestrian signal head is provided) (s), • Duration of red clearance interval for Phase X and Phase Y or Phase Z (needed only if rest- in-walk is enabled or no pedestrian signal head provided) (s), and • Pedestrian clear duration for Phase X and Phase Y or Phase Z (only needed if phase is actuated and rest-in-walk is enabled) (s). If the signal control is fully actuated, an average value is used for the cycle length and the green interval durations. If the signal control is semiactuated, then an average value is used for the green interval duration of the actuated phases. Step 1. Determine the Effective Walk Time During this step, the analyst determines the effective walk time for Phase X and for Phase Y. The following guidance is provided to estimate the effective walk time for a given phase. This guidance is derived from the HCM 6th ed. (1). If the subject phase is either (a) actuated with a pedestrian signal head and rest-in-walk is not enabled or (b) pretimed with a pedestrian signal head, then the following equation is used to compute the effective walk time: = +Walk 4.0 Equation A-21Walk,g i i where gWalk,i is the effective walk time for Phase i serving the subject pedestrian movement (s) and Walki is the duration of the “Walk” interval for Phase i (s). If the phase providing service to the pedestrians is actuated with a pedestrian signal head and rest-in-walk enabled, then the following equation is used to compute the effective walk time: = − − − + 4.0 Equation A-22Walk, , ,g D Y R PCi p i i c i i where Dp,i = duration of Phase i (s), Yi = duration of yellow change interval for Phase i (s), Rc,i = duration of red clearance interval for Phase i (s), and PCi = duration of pedestrian clear time for Phase i (s). For all other situations (e.g., there is no pedestrian signal head), Equation A-23 computes the effective walk time: = − − Equation A-23Walk, , ,g D Y Ri p i i c i For crosswalk sections associated with two phases (i.e., the section has a two-digit number), the time to cross the section is provided to pedestrians during one or both phases. If they are served during both phases, then an overlap is used. When Equation A-22 or Equation A-23 is used for a crosswalk section served by two phases (i.e., when overlap is used), the duration of Phase i, Dp,i, used in either equation must equal the sum of the duration of both phases that are parent to the overlap. The yellow change, red clearance, and pedestrian clear values are equal to those for the parent phase that occurs last in the overlap pair. For example, when Equation A-22 is used to compute the effective walk time for Crosswalk Section 12, the variable Dp,i in this equa- tion must equal the sum of the durations for Phases 1 and 2 (i.e., Dp,12 = Dp,1 + Dp,2).

A-10 Guide to Pedestrian Analysis Step 2. Determine Crossing Time During First Phase The time required to cross from the first corner to the median is determined in this step and is computed with the following equation: = Equation A-24t L S X X p where tX = time for pedestrians to cross during Phase X (s), LX = distance from the first corner to the far side of the median (measured along the path of the pedestrian crossing) (ft), and Sp = average pedestrian crossing speed (ft/s). The HCM 6th ed. (1) recommends the use of 4.0 ft/s for the pedestrian crossing speed, Sp, when less than 20% of the pedestrians are elderly (i.e., 65 years of age or older). If the percentage of elderly pedestrians exceeds 20%, then a walking speed of 3.3 ft/s should be used. Step 3. Determine the Start of the “Walk” Intervals This step determines the relative time in the cycle when the subject “Walk” interval starts. Specifically, the start time for “Walk” intervals is associated with Phase X and Phase Y. To establish the relative start time for a given “Walk” interval, TWalk, one phase in the sequence will be established as Time 0 (i.e., the start of the cycle). The start time of all subsequent phases will be established by using the cumulative duration of preceding phases. With the relative phase start times established in this manner, the relative start time for a phase’s “Walk” interval can be established by summing the preceding phase durations. In general, a “Walk” interval’s relative start time is equal to the relative start time of its parent phase. However, if an LPI is used, then the “Walk” interval’s relative start time equals the relative start time of the phase minus the duration of the leading interval. Similarly, if a lagging pedestrian interval is used, then the “Walk” interval’s relative start time equals the relative start time of the phase plus the duration of the lagging interval. The guidance provided for this step can be illustrated by considering an analysis of the pedestrian crossing from Corner B to Corner C in Figure A-2a, where the intersection has the phase sequence shown in Figure A-3. As shown in the numbering scheme in Figure A-3, this Protected Movement Permitted Movement Pedestrian Movement Φ1 Φ2 Φ3 Φ4 Φ5 Φ6 Φ7 Φ8 Barrier Ring 1 Ring 2 Barrier Time 1 6P 65 2 2P +12 3 7 4P 4 8 8P 0 CDp1 Dp1+Dp2 Dp1+Dp2+Dp3 12 Figure A-3. Example phase sequence for two-stage crossing shown using dual-ring structure.

Calculation Details for Analysis Method A-11   crossing is served by Phases 1 and 2. On the basis of Figure A-3, Phase 1 occurs first for the subject crossing direction (X = 1) and Phase 2 occurs second (Y = 2). The start time of Phase X is 0. The start time of Phase Y is equal to the duration of Phase 1, Dp1. Although not needed for this illustration, the start time for Phase 3 is shown in Figure A-3 to equal the sum of the dura- tion of Phase 1 and Phase 2. Both “Walk” intervals start with their parent phase for this illustration, so the relative start time of the “Walk” interval for Phase X, TWalk,X, is 0 and that for Phase Y, TWalk,Y, is equal to Dp1. Other values will likely be obtained for other phase sequences. The illustration can be continued by considering an analysis of the pedestrian crossing from Corner C to Corner B in Figure A-2a, where the intersection has the phase sequence shown in Figure A-3. On the basis of the numbering scheme shown in Figure A-2a, this crossing is served by Phases 1 and 2. On the basis of Figure A-3, Phase 2 occurs first for the subject crossing direction (X = 2) and Phase 1 occurs second (Y = 1). The start time of Phase X is equal to Dp1. The start time of Phase Y is equal to 0. Step 4. Compute Delay for First-Stage Crossing This step uses Equation A-25 to compute the delay for the first-stage crossing incurred by pedestrians waiting at the first corner: 2 Equation A-25,1 Walk, 2( ) = − d C g C p X where dp,1 = pedestrian delay at corner for Stage 1 (s/p), C = cycle length (s), and gWalk,X = effective walk time for Phase X serving the subject pedestrian movement (s). Step 5. Compute Delay for Second-Stage Crossing Given Arrival Is During “Don’t Walk” Interval During this step, the second-stage crossing delay is computed for one portion of the pedes- trian stream. This delay is incurred by pedestrians waiting on the median who arrived at the first corner during a “Don’t Walk” indication (flashing or solid). The other portion of the second-stage crossing delay is computed in the next step. A. Compute the Time Between “Walk” intervals. The time between the “Walk” intervals for Phases X and Y is computed with the following equation: ( )= −modulo Equation A-26Walk, Walk, ,t T T CYX Y X where tYX = time between start of “Walk” intervals (s), TWalk,X = relative start time of the “Walk” interval for Phase X (s), TWalk,Y = relative start time of the “Walk” interval for Phase Y (s), and C = cycle length (s). The modulo function in Equation A-26 ensures that the value for tYX is a nonnegative number less than the cycle length. When used, the equation in parenthesis is computed and the resulting value is compared with the range 0 to C. If this value is outside the range, the value is changed by adding (or subtracting) one cycle length and then reassessing range satisfaction. The value is changed by adding or subtracting additional cycle length increments until it is within the range 0 to C.

A-12 Guide to Pedestrian Analysis B. Compute the Delay Given Arrival Is During “Don’t Walk” Interval. The delay incurred by pedestrians waiting on the median who arrived at the first corner during a “Don’t Walk” indi- cation is computed with the following equation: if 0 if Equation A-272,DW1 Walk, Walk, = < − ≥ −     d t t C g t C g Y Y with ( )= −modulo , Equation A-28t t t CYX X where d2,DW1 = delay on median for Stage 2, given arrival is during a “Don’t Walk” indication at corner (s/p); t = waiting time on median when pedestrians reach median during a “Don’t Walk” indi- cation (s); gWalk,Y = effective walk time for Phase Y serving the subject pedestrian movement (s); tX = time for pedestrians to cross during Phase X (s); and all other variables are as previously defined. Step 6. Compute Delay for Second-Stage Crossing Given Arrival Is During “Walk” Interval During this step, the second-stage crossing delay is computed for the second portion of the pedestrian stream. This delay is incurred by pedestrians waiting on the median who arrived at the first corner during the “Walk” indication. There are two sets of equations that can be used to compute the second-stage crossing delay. The correct set of equations is determined by comparing the value of t [waiting time on median when pedestrians reach the median during a “Don’t Walk” indication (s)]; with the effective walk time for Phase X, as described in the following paragraphs. When t < gWalk,X, compute the second-stage crossing delay with the following equation: 0.5 If 0.5 If 0.5 If Equation A-292, 1 2 Walk, Walk, Walk, Walk, 2 Walk, Walk, Walk, Walk, 2 Walk, Walk, ( ) ( ) ( ) ( ) ( ) ( ) = + + − + < ≤ + ≤ − + >            d a t a C g g t g g t g g t g C C g g t g C W X X Y X X X Y Y X Y with = − − Equation 30Walk, Walk,a g g tX Y where d2,W1 is the delay on the median for Stage 2, given the arrival is during the “Walk” indica- tion at the corner (s/p), a is the undefined intermediate variable, and all other variables are as previously defined.

Calculation Details for Analysis Method A-13   When t ≥ gWalk,X, compute the second-stage crossing delay with the following equation: ( ) ( ) ( ) ( ) ( ) ( ) = − + < + − ≤ + ≤ + + > +        0.5 If 0.5 If 0 If Equation A-312, 1 Walk, Walk, 2 Walk, Walk, Walk, Walk, Walk, Walk, d t g t g C b b t g g C t g C g t g C g W X Y X X Y X Y X with = − − + Equation A-32Walk, Walk,b g g t CX Y where d2,W1 is the delay on the median for Stage 2, given arrival is during the “Walk” indication at the corner (s/p); b is the undefined intermediate variable; and all other variables are as previously defined. Step 7. Compute Delay for Second-Stage Crossing The pedestrian delay for a two-stage crossing is computed with the following equation: [ ]( )= + + −1 Equation A-33,1 2,DW1 DW1 2, 1 DW1 2d d d P d Pp p W with ( ) = − Equation A-34DW1 Walk,P C g C X where dp = pedestrian delay (s/p); dp,1 = pedestrian delay at corner for Stage 1 (s/p); d2,DW1 = delay on median for Stage 2, given arrival is during “Don’t Walk” indication at the corner (s/p); d2,W1 = delay on median for Stage 2, given arrival is during “Walk” indication at corner (s/p); and PDW1 = proportion of arrivals during “Don’t Walk” indication at corner (s/p). Diagonal Crossing Procedure This section describes a procedure for estimating the delay to pedestrians that cross two inter- section legs during two phases of one cycle to complete a diagonal crossing. Delays crossing (a) the first crosswalk and (b) both crosswalks as a system are computed. The delay when crossing the second crosswalk is computed by subtracting the first crosswalk delay from the system delay. The procedure is based on that developed by Zhao and Liu (8). A diagonal crossing at the typical four-leg intersection has two possible travel paths, depending on whether the major street leg is crossed first or second. These two paths are referred to herein as the “clockwise path” and the “counterclockwise path.” The procedure described herein is used to estimate the delay to a given path of travel when crossing from one corner to the diagonally opposite corner by using two crosswalks. The procedure is applied separately to evaluate the other travel path between the two diagonal corners or to evaluate diagonal crossings for other corner combinations.

A-14 Guide to Pedestrian Analysis The procedure is based on the following two assumptions: (a) pedestrian arrivals to the first crossing location (i.e., street corner) are random and (b) pedestrians will begin their crossing during the first available “Walk” interval (regardless of whether it is to cross the minor street leg or the major street leg). The second assumption reflects the pedestrian’s desire to minimize the total delay in diagonal crossing. This procedure does not address a two-stage crossing of the minor street leg or the major street leg. The procedure described in this section is based on the vehicle movement numbering scheme shown in Figure A-4a. These vehicle movement numbers correspond to the signal phase that serves the movement (i.e., Vehicle Movement 2 is served by Signal Phase 2), which follows the traditional eight-phase dual-ring structure. The procedure is also based on the crosswalk numbering scheme shown in Figure A-4a and the pedestrian movement schemes shown in Figure A-4, b through e. With a diagonal crossing, the signal operation accommodates pedestrians crossing to the diagonally opposite corner by providing pedestrian service during two signal phases. During the first phase, pedestrians cross from the first corner to the second corner. During the second phase, they cross from the second corner to the last corner. The delay incurred during a diagonal crossing is dependent on the direction the pedestrian travels around the intersection (i.e., clockwise or counterclockwise). As a result, the direc- tion of interest must be specified when this procedure is being used. The letter X denotes the first phase to occur in the subject direction of travel. The letter Y denotes the second phase to occur in the subject direction of travel. Had the pedestrian decided to cross in the other direction around the intersection, two different signal phases would provide pedestrian service. The first phase to serve travel in the other direction is denoted by the letter Z (i.e., Phase Z is the first phase to serve the pedestrian starting the diagonal crossing in a direction opposite to the direction of interest). These rules can be illustrated by considering the intersection shown in Figure A-4a. An ana- lyst desires to compute pedestrian delay when the pedestrian is traveling in a clockwise path from Corners B to D. On the basis of this information, the first phase to serve the pedestrian crossing in the subject travel direction (i.e., from Corner B to Corner C) is Phase 2, so Phase X is Phase 2 (X = 2). The second phase to serve pedestrians in the subject travel direction (i.e., from Corner C to Corner D) is Phase 4, so Phase Y is Phase 4 (Y = 4). If the pedestrian were to travel in the other direction, Phase 8 would be the first phase to provide service (i.e., from Corner B to Corner A), so Phase Z is Phase 8 (Z = 8). Step 1. Determine the Effective Walk Time During this step, the analyst determines the effective walk time for Phase X and for Phase Z. The following guidance from the HCM 6th ed. (1) estimates effective walk time for a given phase. If the subject phase is either (a) actuated with a pedestrian signal head and rest-in-walk is not enabled or (b) pretimed with a pedestrian signal head, the following equation is used to compute effective walk time: = +Walk 4.0 Equation A-35Walk,g i i where gWalk,i is the effective walk time for Phase i serving the subject pedestrian movement (s) and Walki is the duration of the “Walk” interval for Phase i (s).

Calculation Details for Analysis Method A-15   B2 B2C4 A8B2 C4 C4D6 B2C4 D6 D6A8 C4D6 A8A8B2 D6A8 B8B8A6 C2B8 C2 C2B8 D4C2 D4 D4C2 A6D4 A6 A6D4 B8A6 6 8 2 4 Corner A Corner BCorner C Corner D (b) Pedestrian movement numbering for Crosswalk 8 (a) Crosswalk numbering scheme (c) Pedestrian movement numbering for Crosswalk 2 (d) Pedestrian movement numbering for Crosswalk 4 (e) Pedestrian movement numbering for Crosswalk 6 Figure A-4. Pedestrian movement and crosswalk numbering scheme with diagonal movements.

A-16 Guide to Pedestrian Analysis If the phase providing service to the pedestrians is actuated with a pedestrian signal head and rest-in-walk is enabled, the following equation is used to compute effective walk time: = − − − + 4.0 Equation A-36Walk, , ,g D Y R PCi p i i c i i where Dp,i = duration of Phase i (s), Yi = duration of yellow change interval for Phase i (s), Rc,i = duration of red clearance interval for Phase i (s), and PCi = duration of pedestrian clear time for Phase i (s). For all other situations (e.g., there is no pedestrian signal head), Equation A-37 is used to compute effective walk time. = − − Equation A-37Walk, , ,g D Y Ri p i i c i Step 2. Determine Crossing Time During First Phase The time required to cross from the first corner to the second corner is determined in this step with the following equation: = Equation A-38t L S X X p where tX = time for pedestrians to cross during Phase X (s), LX = distance from the first corner to the second corner (measured along the path of the pedestrian crossing) (ft), and Sp = average pedestrian crossing speed (ft/s). The HCM 6th ed. recommends using 4.0 ft/s for the pedestrian crossing speed, Sp, when less than 20% of the pedestrians are elderly (i.e., 65 years of age or older). If the percentage of elderly pedestrians exceeds 20%, then a walking speed of 3.3 ft/s should be used (1). Step 3. Determine the Start of the “Walk” Intervals This step determines the relative time in the cycle that the subject “Walk” intervals start; specifically, this is the start time for the “Walk” intervals associated with Phase X, Phase Y, and Phase Z. To establish the relative start time for a given “Walk” interval TWalk, one phase in the sequence will be established as Time 0 (i.e., the start of the cycle). The start time of all subsequent phases will be established by using the cumulative duration of preceding phases. With the relative phase start times established in this manner, the relative time for the start of a phase’s “Walk” interval can be established by summing the durations of the preceding phases. In general, a “Walk” interval’s relative start time is equal to its parent phase’s relative start time. However, if an LPI is used, then the relative start time of the “Walk” interval equals the relative start time of the phase minus the duration of the leading interval. Similarly, if a lagging pedes- trian interval is used, then the relative start time of the “Walk” interval equals the relative start time of the phase plus the duration of the lagging interval. The guidance provided for this step can be illustrated by considering an analysis of the clockwise diagonal crossing from Corners B to D in Figure A-4a, where the intersection has the phase sequence shown in Figure A-5. On the basis of the numbering scheme shown in Figure A-4a, the subject travel direction is served first by Phase 2 and then Phase 4. Had a counterclockwise crossing been taken, the diagonal crossing would be served first by Phase 8.

Calculation Details for Analysis Method A-17   As shown in Figure A-5, Phase 2 occurs first for the subject travel direction (X = 2) and Phase 4 occurs second for the subject direction of travel (Y = 4). Phase 8 occurs first for the other travel direction (Z = 8). The start time of Phase X is equal to the duration of Phase 1, Dp1 (since Phase 2 starts when Phase 1 ends for the sequence shown in Figure A-5). The start time of Phase Y is equal to the duration of Phases 1, 2, and 3 (Dp1 + Dp2 + Dp3) (since Phase 4 starts when Phase 3 ends for the sequence shown in Figure A-5). Similarly, the start time of Phase Z is equal to the duration of Phases 1, 2, and 7 (Dp1 + Dp2 + Dp7). Note that the dual-ring structure shown in Figure A-5 has a barrier at the end of Phases 2 and 6 that requires the duration of Phase 1 plus Phase 2 to equal the duration of Phase 5 plus Phase 6. Both “Walk” intervals start with their parent phase for this illustration, so the relative start time of the “Walk” interval is as follows: for Phase X, TWalk,X, Dp1; for Phase Y, TWalk,Y, Dp1 + Dp2 + Dp3; and for Phase Z, TWalk,Z, Dp1 + Dp2 + Dp7. Other values will likely be obtained for other phase sequences. Step 4. Compute Delay for First-Stage Crossing This step computes the delay for the first-stage crossing in the subject travel direction. This delay is incurred by pedestrians that have been waiting at the first corner since the end of the effective walk time for the other travel direction. A. Compute the End of Effective Walk Time. The end of the effective walk time for Phase X is computed with the following equation: ( )= +modulo , Equation A-39Walk, Walk,T T g CX X X where TX = relative end time of the effective walk period for Phase X (s), TWalk,X = relative start time of the “Walk” interval for Phase X (s), C = cycle length (s), and gWalk,X = effective walk time for Phase X serving the subject pedestrian movement (s). The end of the effective walk time for Phase Z is computed with the following equation: ( )= +modulo , Equation A-40Walk, Walk,T T g CZ Z Z where TZ = relative end time of the effective walk period for Phase Z (s), TWalk,Z = relative start time of the “Walk” interval for Phase Z (s), C = cycle length (s), and gWalk,Z = effective walk time for Phase Z serving the subject pedestrian movement (s). Φ1 Φ2 Φ3 Φ4 Φ5 Φ6 Φ7 Φ8 Barrier Ring 1 Ring 2 Barrier 1 6P 65 2 2P 3 7 4P 4 8 8P 0 CDp1 Dp1+Dp2 Dp1+Dp2+Dp7 Dp1+Dp2+Dp3 Figure A-5. Example phase sequence for diagonal crossing shown using dual-ring structure.

A-18 Guide to Pedestrian Analysis B. Compute the Delay for the First-Stage Crossing. The delay for the first-stage crossing is computed with the following equation: ( )= − 2 Equation A-41,1 Walk, 2 d t g t p XZ X XZ with ( )= −modulo , Equation A-42t T T CXZ X Z where dp,1 is the pedestrian delay at the corner for Stage 1 (s/p) and tXZ is the time between end of the effective walk time for Phase Z and the start of the effective walk time for Phase X (s), and all other variables are as previously defined. Step 5. Compute Delay for Entire Diagonal Crossing The delay for the entire diagonal crossing in the subject travel direction is computed in this step. This delay represents the sum of delay incurred on the first and second corners. The delay for the second-stage crossing only is computed in the next step. The diagonal crossing delay is computed with the following equation: = − Equation A-43d t tp d X with = − + ≥ ≥ − + − < − + + ≥ ≥          2 If 2 If 2 If Equation A-44 Walk, Walk, Walk, Walk, Walk, t T T T T T T T T T C T T T T T C T T T d Y X Z Y X Z Y X Z X Z Y X Z X Z Y where dp = pedestrian delay (s/p), td = time between arrival to first corner and departure from second corner (s), tX = time for pedestrians to cross during Phase X (s), TWalk,Y = relative start time of the “Walk” interval for Phase Y (s), and all other variables are as previously defined. Step 6. Compute Delay for Second-Stage Crossing During this step, the delay for the second-stage crossing in the subject travel direction is computed. This delay is incurred by pedestrians waiting at the second corner and is computed with the following equation: = − Equation A-45,2 ,1d d dp p p where dp,2 = pedestrian delay at corner for Stage 2 (s/p), dp = pedestrian delay (s/p), and dp,1 = pedestrian delay at corner for Stage 1 (s/p).

Calculation Details for Analysis Method A-19   Closing Comments The diagonal crossing procedure described in this section can be used to estimate delay in the diagonal crossing maneuver for a specified travel path (i.e., clockwise or counterclockwise) between two diagonal corners. The procedure can also be used to evaluate delay associated with a given crosswalk. As shown in Figure A-4, b through e, there are six pedestrian movements associated with each crosswalk; to be specific, each crosswalk has two directions of travel, and each direction of travel is associated with three pedestrian movements. Crosswalk 2 and pedestrians crossing from Corners C to B can be considered. As shown in Figure A-4c, the following pedestrian movements are of interest: D4C2, C2B8, and C2. Movement D4C2 represents pedestrians completing a counterclockwise diagonal crossing from Corner D to Corner B. Their delay in Crosswalk 2 can be estimated as the second-stage crossing delay of the diagonal crossing procedure (Equation A-45). Movement C2B8 represents pedestrians completing a counterclockwise diagonal crossing from Corners C to A. Their delay can be estimated as the first-stage crossing delay of the diagonal crossing procedure (Equation A-41). Finally, movement C2 represents pedestrians crossing from Corners C to B who are not destined for any other intersection corner. Their delay can be estimated using the procedure described in the HCM 6th ed. (Equation A-20). If the volume of each of these three movements is known, they can be used to compute a volume-weighted average delay for the subject travel direction of the crosswalk. The process out- lined in the preceding paragraphs can be repeated to evaluate the three pedestrian movements for the opposing travel direction of the subject crosswalk. A volume-weighted average delay for this travel direction can also be computed if the pedestrian volume is known for each of the three pedestrian movements. Finally, if both travel directions of a given crosswalk have been evaluated to produce a delay for each of the six pedestrian movements, and the volume of these six move- ments is known, then a volume-weighted average delay can be computed for the crosswalk. Estimation of Pedestrian Satisfaction: Uncontrolled Crossings The service measure for a pedestrian crossing at a mid-block or two-way stop-controlled intersection location is based on the predicted average proportion of pedestrians who would say they were dissatisfied or worse with their crossing experience. The research that developed this methodology (3) surveyed actual pedestrians who were asked to rate their experience on a scale of four potential levels of satisfaction: very satisfied, satisfied, dissatisfied, and very dissatisfied. These levels were condensed to two levels—satisfied or dissatisfied—for ease of implementation. A computational engine implementing this methodology is described in Appendix B. Equation A-46 estimates the odds that pedestrians would be satisfied with their crossing experience relative to being dissatisfied: exp 0.9951 0.0438 1.95721 0.9843 1.5496 1.9059 Equation A-46 KAADT RFFB MC MR NY( ) ( )= − + + + −O S D V I I I where O(S/D) = odds that a pedestrian would be satisfied with their crossing experience relative to being dissatisfied, VKAADT = AADT of street being crossed divided by 1,000 (thousands of vehicles), IRFFB = indicator variable for the presence of an RRFB at the crossing (1 = present, 0 = not present),

A-20 Guide to Pedestrian Analysis IMC = indicator variable for the presence of a marked crosswalk (1 = present, 0 = not present), IMR = indicator variable for the presence of a median refuge (1 = present, 0 = not present), and INY = indicator variable for the pedestrian experiencing a vehicle not yielding while using the crossing (1 = not yielding, 0 = yielding). Equation A-47 estimates the probability of a given pedestrian being satisfied with his or her crossing. The probability of a given pedestrian being dissatisfied is 1 minus the probability of being satisfied, as shown by Equation A-48: ( ) ( ) ( ) = +1 Equation A-47P S O S D O S D ( ) ( )= −1 Equation A-48P D P S where P(S) is the probability that a pedestrian would be satisfied with his or her crossing experi- ence (decimal) and P(D) is the probability that a pedestrian would be dissatisfied with his or her crossing experience (decimal), and all other terms are as defined previously. When INY = 1, Equations A-47 and A-48 produce the probabilities of being satisfied and dis- satisfied when the pedestrian is not delayed while using the crossing (i.e., either a sufficient gap exists when the pedestrian arrives to allow an immediate crossing or all blocking vehicles yield to the pedestrian). Similarly, when INY = 0, these equations produce probabilities of being satisfied and dissatisfied when the pedestrian is delayed while using the crossing. The probability of a nondelayed crossing is the sum of the probability of a sufficient gap exist- ing to allow an immediate crossing when the pedestrian arrives (i.e., 1 minus the probability of a delayed crossing), plus the proportion of the potentially delayed crossings in which all blocking vehicles yield to the pedestrian on the first potential yielding event. Equation A-49 calculates the probability of a nondelayed crossing: 1 Equation A-49nd 1( ) ( )= − +P P P P Yd d where Pnd = probability of a nondelayed crossing (decimal), Pd = probability of a potentially delayed crossing (decimal) from Equation A-6 in the uncontrolled crossing delay procedure, and P(Y1) = probability of all blocking vehicles yielding on the first potential yielding event (decimal) from Equations A-13, A-14, A-16, and A-18 for one-, two-, three-, or four-lane crossings, respectively. Over the course of the analysis period, a proportion of crossing pedestrians, Pnd, will experi- ence no delay while using the crossing; thus, the number of satisfied and dissatisfied ratings from these pedestrians will be in proportion to the respective satisfaction probabilities when no delay occurs. Similarly, the remaining proportion of crossing pedestrians, Pd, will be delayed while using the crossing, and the number of ratings in each category from these pedestrians will be in proportion to respective satisfaction probabilities when a delay occurs. The overall proportion of dissatisfied ratings is, therefore, the volume-weighted average of the probabilities of being dissatisfied under no-delay and delay conditions, as given by Equation A-50: , no delay 1 , delay Equation A-50nd nd( ) ( )( )= + −P P P D P P DD

Calculation Details for Analysis Method A-21   where PD = average proportion of dissatisfied ratings for the crossing (decimal), Pnd = probability of a nondelayed crossing (decimal), P(D, no delay) = probability of a dissatisfied rating when no delay occurs (decimal), and P(D, delay) = probability of a dissatisfied rating when a delayed crossing occurs (decimal). The value of PD can be used with Table A-1 to determine the crossing’s LOS. References 1. Transportation Research Board. 2016. Highway Capacity Manual: A Guide for Multimodal Mobility Analysis, 6th ed. Washington, DC. 2. Transportation Research Board. 2000. Highway Capacity Manual. National Research Council, Washington, DC. 3. Gerlough, D. L., and M. J. Huber. 1975. Special Report 165: Traffic Flow Theory: A Monograph. Transportation Research Board, National Research Council, Washington DC. 4. Transportation Research Board. 2010. Highway Capacity Manual 2010. Washington, DC. 5. Ryus, P., A. Musunuru, K. Lausten, J. Bonneson, S. Kothuri, C. Monsere, N. McNeil, K. Nordback, S. LaJeunesse, W. Kumfer, L. Thomas, and S. I. Guler. 2022. NCHRP Web-Only Document 312: Enhancing Pedestrian Volume Estimation and Developing HCM Pedestrian Methodologies for Safe and Sustainable Communities. Transporta- tion Research Board, Washington, DC. 6. Bonneson, J. A., and P. T. McCoy. 1993. Methodology for Evaluating Traffic Detector Designs. Transportation Research Record 1421, pp. 76–81. 7. Wang, X., and Z. Tian. 2010. Pedestrian Delay at Signalized Intersections with a Two-Stage Crossing Design. Transportation Research Record: Journal of the Transportation Research Board, No. 2173, pp. 133–138. 8. Zhao, J., and Y. Liu. 2017. Modeling Pedestrian Delays at Signalized Intersections as a Function of Crossing Directions and Moving Paths. Transportation Research Record: Journal of the Transportation Research Board, No. 2615, pp. 95–104. LOS Condition Level of Satisfaction A PD < 0.05 Nearly all pedestrians would be satisfied. B 0.05 ≤ PD < 0.15 At least 85% of pedestrians would be satisfied. C 0.15 ≤ PD < 0.25 Less than one-quarter of pedestrians would be dissatisfied. D 0.25 ≤ PD < 0.33 Less than one-third of pedestrians would be dissatisfied. E 0.33 ≤ PD < 0.50 Less than one-half of pedestrians would be dissatisfied. F PD ≥ 0.50 The majority of pedestrians would be dissatisfied. Note: LOS = level of service; PD = proportion of pedestrians giving a rating of dissatisfied or worse. Source: NCHRP Web-Only Document 312 (5). Table A-1. Level-of-service criteria for pedestrian satisfaction at uncontrolled crossings.