We consider Bayesian modelling of the inter-arrival times of vehicles on a road. We give particular attention to model checking and propose diagnostic plots to assess whether the model captures features observed in the data. Models for such processes are important in predicting the behaviour of road junctions. Two-component mixtures have often been used, with a variety of distributions for the components and non-Bayesian methods for fitting. The components are often regarded as representing ‘free flowing’ and ‘congested’ vehicles. We propose a gamma-exponential mixture and compare this with some other models. To allow for the serial dependence of headways, we consider the use of a hidden Markov chain for the mixture component allocation.
Adams WF
(1936)
Road traffic considered as a random series
. Journal of the Institution of Civil Engineers,
121–32
.
2.
Ashton W
(1971)
Distribution for gaps in road traffic
. Journal of the Institute of Mathematics and Its Applications, 7,
37–46
.
3.
Baras JS
,
Dorsey AJ
and
Levine WS
(1979)
Estimation of traffic platoon structure from headway statistics
. IEEE Transactions on Automatic Control, AC-24(4),
553–59
.
4.
Box GEP
(1980)
Sampling and Bayes’ inference in scientific modelling and robustness
(with discussion). Journal of the Royal Statistical Society, A, 143,
383–430
.
5.
Calabria R
and
Pulcini G
(1994)
Bayes credibility intervals for the left truncated exponential distribution
. Microelectronics and Reliability, 34,
1897–907
.
6.
Cowburn GJ
(2003) Bayesian mixture modelling with application to road traffic flow. Ph.D. thesis,
University of Sunderland
.
7.
Cox DR
and
Miller HD
(1965) The Theory of Stochastic Processes.
London: Chapman & Hall
.
8.
Dempster AP
,
Laird NM
and
Rubin DB
(1977)
Maximum likelihood from incomplete data via the EM algorithm
(with discussion). Journal of the Royal Statistical Society B, 39,
1–38
.
9.
Gilks WR
,
Richardson S
and
Spiegelhalter DJ
(1996) Markov chain Monte Carlo in practice.
London: Chapman & Hall
.
10.
Griffiths JD
and
Hunt JG
(1991)
Vehicle headways in urban areas
. Traffic Engineering and Control, 32,
458–62
.
11.
Katti BK
and
Pathak RH
(1986)
A study on headway distribution models for the urban road sections under mixed traffic conditions
. Highway Research Bulletin (New Delhi), 26,
1–39
.
12.
Madi TM
and
Leonard T
(1996)
Bayesian estimation for shifted exponential distributions
. Journal of Statistical Planning and Inference, 55,
345–51
.
13.
Miller AJ
(1961)
A queueing model for road traffic flow
. Journal of the Royal Statistical Society B, 23,
64–90
.
14.
O'Hagan A
(2003) HSSS model criticism (with discussion). In
Green, P.J.
,
Hjort, N.L.
,
Richardson, S.T.
eds. Highly structured stochastic systems.
Oxford: Oxford University Press
, 423–53.
15.
Ohno K
and
Mine H
(1972)
Traffic light queues with dependent arrivals as a generalisation to queueing theory
. Journal of Applied Probability, 19,
630–41
.
16.
Robert CP
,
Rydén T
and
Titterington DM
(1999)
Convergence controls for MCMC algorithms, with applications to hidden Markov chains
. Journal of Statistical Computation and Simulation, 64,
327–55
.
17.
Rubin DB
(1981)
Estimation in parallel randomized experiments
. Journal of Educational Statistics, 6,
377–401
.
18.
Rubin DB
(1984)
Bayesianly justifiable and relevant frequency calculations for the applied statistician
. Annals of Statistics, 12,
1151–72
.
Schuhl A
(1955) The probability theory applied to the distribution of vehicles on two-lane highways. In Poisson and traffic.
Washington DC: The Eno Foundation
, 59–75.
21.
Scott SL
(2002)
Bayesian methods for hidden Markov models: recursive computing in the 21st century
. Journal of the American Statistical Association, 97,
337–51
.
22.
Spiegelhalter DJ
,
Thomas A
,
Best NG
and
Gilks WR
(1995) BUGS: Bayesian inference using Gibbs sampling, version 0.50.
MRC Biostatistics Unit, Cambridge
.
23.
Tamura Y
and
Chisaki T
(1983)
Time headway distribution model based on the composition of free and constrained flowing vehicles
. Transactions of the Japanese Society of Civil Engineers, 15,
448–51
.
24.
Titterington DM
,
Smith AFM
and
Makov UE
(1985) Statistical analysis of finite mixture distributions.
Chichester: Wiley
.
25.
Trader RL
(1985)
Bayesian inference for truncated exponential distributions
. Communications in Statistics: Theory and Methods, 14(3),
585–92
.
26.
Troutbeck RJ
(1986)
Average delay at an unsignalised intersection with two major streams each having a dichotomised headway distribution
. Transportation Science, 20(4),
272–86
.
27.
Troutbeck RJ
and
Kako S
(1999)
Limited priority merge at unsignalised intersections
. Transportation Research (Part A: Policy and Research), 33A (3/4),
291–304
.
