COUNTING DISTRIBUTIONS FOR ERLANG PROCESSES

WHEN THE GAPS BETWEEN CONSECUTIVE EVENTS ARE INDEPENDENTLY DISTRIBUTED WITH THEIR DENSITY FUNCTION, THE EVENTS FORM AN ERLANG PROCESS. FOR AN ERLANG PROCESS, THE SYNCHRONOUS COUNTING DISTRIBUTION IS CALLED GENERALIZED POISSON. THE ASYNCHRONOUS COUNTING DISTRIBUTION FOR AN ERLANG PROCESS IS CONSIDERED, AS PRESENTED BY MORSE AND JEWELL. THE MATHEMATICAL DEVELOPMENT IS PRESENTED OF THE ASYNCHRONOUS COUNTING DISTRIBUTION. MATHEMATICAL APPROXIMATIONS ARE ALSO PRESENTED WHICH ARISE FROM PARAMETER ESTIMATION BY THE METHOD OF MOMENTS. THE BASIC CHARACTERISTICS OF ROAD TRAFFIC WHICH INFLUENCE THE STOCHASTIC PROCESS OF VEHICLE BEHAVIOR ARE: (1) CARS GO AT VARYING SPEEDS, THE VARIATION BEING BETWEEN DIFFERENT CARS AS WELL AS FOR THE SAME CARS AT DIFFERENT TIMES, (2) THE AXIS ON WHICH EVENTS TAKE PLACE MAY BE EITHER A TIME AXIS OR A SPACE AXIS, AND (3) THE EVENTS ARE NOT PRODUCED BY ANY MECHANISM OUTSIDE THEMSELVES. THE AVOIDANCE OF MUTUAL INTERACTION IS SUFFICIENT TO DISTINGUISH BETWEEN THE PROCESS IN SPACE AND THE PROCESS IN TIME, AS CAN BE SEEN BY CONSIDERING THE CORRESPONDING GAP DISTRIBUTIONS. THE ERLANG PROCESS WITH ITS ASSOCIATED TYPE III GAP DISTRIBUTION AND ITS ASSOCIATED GENERALIZED POISSON OR MORSE COUNTING DISTRIBUTION IS APPLICABLE TO TIME MEASUREMENTS OF VEHICLES. THE ANALYSIS OF ROAD DATA HAS NEARLY ALWAYS BEEN BASED ON THE MODEL OF A RANDOM (POISSON) PROCESS, PARTLY BECAUSE OF THE SIMPLICITY OF THIS MODEL AND BECAUSE OF THE READY AVAILABILITY OF TABLES. SOME RECENT DISSATISFACTION WITH NEGATIVE EXPONENTIAL GAP DISTRIBUTIONS FOR ROAD TRAFFIC HAS LEAD TO THE SUBSTITUTION OF THE GENERALIZED POISSON COUNTING DISTRIBUTION. HOWEVER, USE OF THE MORSE-JEWELL DISTRIBUTION REPRESENTS ANOTHER ADVANCE IN THEORETICAL FORMULATION, ALTHOUGH THE PROBABILITIES INVOLVED ARE STILL MORE DIFFICULT TO COMPUTE AND THE PARAMETERS NO EASIER TO ESTIMATE. HOWEVER, IT IS HOPED THAT THE MORSE DISTRIBUTION TABLES TOGETHER WITH THIS DISCUSSION WILL MAKE THIS METHOD OF ANALYSIS AVAILABLE TO THE TRAFFIC ENGINEER.

• Supplemental Notes:
  • Vol 13, No 2, PP 91-103, 3 FIG, 2 TAB, 7 REF
• Corporate Authors:

  Annals of Inst Statistical Math /Japan

  ,    
• Authors:
  • Whittlesey, J R
  • HAIGHT, F A
• Publication Date: 0

Subject/Index Terms

• TRT Terms: Distributions (Statistics); Gap acceptance; Highway traffic; Mathematical analysis; Mathematical models; Measurement; Poisson ratio; Stochastic processes; Traffic flow; Traffic flow theory
• Old TRIS Terms: Poissons ratio; Vehicular traffic
• Subject Areas: Data and Information Technology; Highways; Operations and Traffic Management;

Filing Info

• Accession Number: 00227134
• Record Type: Publication
• Files: TRIS
• Created Date: Aug 13 2004 6:47PM