| POISSON (Random) EVENTS | 
   | 
   | 
   | 
   | 
  3 | 
   | 
 
 
  | Poisson
  arrivals produce headways that are negative exponential distributed. | 
   | 
 
 
   | 
   | 
   
  
   | 
   | 
   | 
 
 
   | 
  u= | 
  0.167 | 
   | 
   | 
 
 
   | 
  Headway | 
  P(H=t) | 
   | 
   | 
 
 
   | 
  0 | 
  0.167 | 
   | 
   | 
 
 
   | 
  1 | 
  0.141 | 
   | 
   | 
 
 
   | 
  2 | 
  0.119 | 
   | 
   | 
 
 
   | 
  3 | 
  0.101 | 
   | 
   | 
 
 
   | 
  4 | 
  0.086 | 
   | 
   | 
 
 
   | 
  5 | 
  0.072 | 
   | 
   | 
 
 
   | 
  6 | 
  0.061 | 
   | 
   | 
 
 
   | 
  7 | 
  0.052 | 
   | 
   | 
 
 
   | 
  8 | 
  0.044 | 
   | 
   | 
 
 
  | IV. | 
  P(H = t) = ue-ut | 
   | 
   | 
 
 
   | 
   | 
   | 
   | 
 
 
   | 
  u = lambda | 
   | 
   | 
  Notice that short
  headways are more likely than long headways | 
   | 
 
 
   | 
   | 
   | 
   | 
 
 
  | What
  is the probability of observing a headway that is 2.5
  seconds long if the mean arrival rate is 600 vph? | 
 
 
   | 
  u = (600 veh/hour)(1 hour/3600 sec) = 0.167 veh/hr | 
   | 
   | 
 
 
   | 
  P(H = 2.5)=(0.167 veh/sec)e-(0.167 veh/sec)(2.5 sec) =
  0.110 | 
   | 
 
 
   | 
   | 
 
 
   | 
   | 
   | 
 
 
  | V. | 
  P(H < t) =
  1-e-ut | 
   | 
 
 
  | What
  is the probability of observing a headway that is less than
  2.5 seconds if u = 600 vph? | 
   | 
 
 
   | 
  P(H < 2.5)= 1 - e-(0.167
  veh/sec)(2.5 sec) = 0.341 | 
   | 
 
 
   | 
   | 
 
 
   | 
   | 
   | 
 
 
  | VI. | 
  P(H >= t)
  = e-ut | 
   | 
 
 
  | What
  is the probability of observing a headway that is greater than or equal to 2.5 sec if u = 600 vph? | 
   | 
 
 
   | 
  P(H >= 2.5)= e-(0.167
  veh/sec)(2.5 sec) = 0.659 | 
   | 
 
 
   | 
   | 
 
 
 
   | 
   | 
   | 
   | 
   | 
   | 
   | 
   | 
   | 
   | 
   | 
   | 
   | 
   |