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Publication Number: FHWA-HRT-98-107
Date: February 1998

Capacity Analysis of Pedestrian and Bicycle Facilities

Recommended Procedures for the "Pedestrians" Chapter of the Highway Capacity Manual

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6. APPENDIX: ADDITIONAL FORMULAS FOR COMPUTING RECOMMENDED MEASURES OF EFFECTIVENESS

Interrupted Pedestrian Facilities

Signalized Crossings

Delay: Griffiths et al.'s Method. Griffiths et al. (1985) derived the following expression for pedestrian delays with two-way vehicle volumes below 1,500 vehicles/h:

equation 1

For vehicle volumes at or above 1,500/h (with pedestrian noncompliance less likely at these high vehicle volumes), they found that the following formula best fit their simulation results:

  equation 2
where:
  dp = mean overall delay to pedestrians, s;
  V = vehicle mean two-way arrival rate, veh/h;
  µ = pedestrian mean arrival rate, ped/s;
  dT = µFDWd{a + b + e + f + d/2} + µR(e + f) {a + b + (e + f) /2} + µR(a + b)2/2
  + (a + b)e-{µFDWd + µR(e+f)}
  y2 = a + b+ c+ k+ (1/_µR)e-{µFDWd + µR(e+f)}
  µFDW = pedestrian flow rate during flashing DON'T WALK, ped/s;
  µR = pedestrian flow rate during steady DON'T WALK, ped/s;
  a = vehicular yellow time, s;
  b = all red period, s;
  c = WALK time, s;
  d = flashing DON'T WALK time during vehicle red indication, s;
  e = flashing DON'T WALK time during vehicle "yield to peds" indication, s;
  f = vehicle green time, s;
  k = pedestrian effective red time = d + e + f

Under vehicle actuation, they found the following best matched simulation results:

  equation 3
where all variables and parameters are as before, except:
  equation 4
  k = d + e + fmin; and
  fmin = minimum vehicular green, s.

Space: Virkler et al.'s Method. Virkler, Elayadath, and Geethakrishnan (1995) offer the following expression for required effective green time (Greq):

  Greq = t0 + t1 + t2
where:
  equation 5
and where:
  R = effective red time, s;
  equation 6
  qA = flow rate of peds approaching the queue, ped/min;
  qC = flow rate of peds leaving the queue, ped/min;
  kB = density of arriving pedestrians, ped/m2;
  kA = density of platooned pedestrians, ped/m2;
  Lq = maximum depth of the standing queue (waiting to cross), m;
  L = walking distance, m;
  uC = pedestrian walking speed before meeting opposing platoon, m/s; and
  uD = pedestrian walking speed after meeting opposing platoon, m/s.

Regarding space-based methods at signalized intersections, Virkler assumes 1.2 m/s platoon flow speeds in the following calculation, which reflects this:

  WALK interval = 3.2 s + (0.19 s/ped) * N1
where:
  N1 = number of people in the primary movement who arrive before the WALK indication and exit the curb during the WALK indication.

For larger effective crosswalk widths, he offers the following modification:

  WALK interval = 3.2 s + (0.57 s/ped/m) * (platoon size/W)
where:
  W = effective crosswalk width, m.

Virkler then offers the following method of determining sufficient total crossing time (WALK plus flashing DON'T WALK), which accounts for the effects of dispersion of platoons larger than 15 persons, for crosswalks with effective widths up to about 3 m:

equation 7

Recommendation. This report recommends the above methods of determining sufficient total crossing time(WALK plus flashing DON'T WALK) proposed by Virkler into the HCM.

Unsignalized Crossings

Delay: Griffiths et al.'s Method. Griffiths et al. (1985) established the following expression for pedestrian delay at a zebra crossing:

  equation 8
where:
  equation 9
  G = mean pedestrian group size, ped.

The exponential portion of the expression reflects the authors' observation that pedestrian groups experience no delay when their arrival at the curbside occurs before a preceding pedestrian group has reached about halfway across the road.

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