The consul distribution as a bunching model in traffic flow

Transpn.Res.-B, Vol.25B,No. 5, pp. 365-372, 1991 Printedin Great Britain.

THE CONSUL

0191-2615/91 $3.00+.00 © 1991PergamonPressplc

DISTRIBUTION AS A BUNCHING IN TRAFFIC FLOW

MODEL

M. N. ISLAM Department of Economics & Statistics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 and

P. C. CONSUL Department o f Mathematics and Statistics, University o f Calgary, Calgary, Alberta, Canada T2N IN4

(Received 28 September 1990) A b s t r a c t - T h e Consul distribution introduced by Consul and Shenton (1975), and modified by Islam and Consul (1990) is derived as a bunching model in traffic flow through the branching process and also through the birth-and-death process. Some applications of the model to vehicle bunch size data are also considered.

I. INTRODUCTION

The Consul distribution introduced by Consul and Shenton (1975), and modified by Islam and Consul (1990) is defined by the following probability function P(X = x)=

x-1(xm-Xl)OX-~(1 - O). . . . +',

x = 1,2 . . . . .

(1.1)

and zero otherwise, where (i) m e N +, 0 < 0 < 1, such that 1 _< m < 0 -I, or (ii) m < 0, 0 < 0 such that mO _< 1. It reduces to the geometric distribution when m = 1. The Borel (Borel-Tanner) distribution with parameter ot is a limiting case of the Consul distribution when m --, oo ( m e N +, 0 < 0 < 1), or m --, - oo (m < 0, 0 < 0), 0 - , 0, such that mO = c~. Consul and Shenton (1972) introduced a class of Lagrangian probability distributions using the Lagrange expansion of a probability generating function (pgf)f(t) under the transformation t = ug(t), when g(t) is another pgf. By considering f(t) = t, and g(t) = (1 - 0 + Ot)", where (i) m e N +, 0 < 0 < 1, or (ii) m < 0, 0 < 0, the Consul distribution is defined with support { 1,2 . . . . . }. Islam and Consul (1990) proved that the parameters m and 0 must satisfy (i) 1 _< m -< 0 -l, when m is a positive integer, or (ii) mO <_ 1, when both m and 0 are negative. They studied some properties o f the model, estimation of the parameters, and also some applications to automobile insurance claims data. In this paper we derive the model (1.1) through the Galton-Watson branching process and as well as through a generalized birth-and-death process. Finally, we discuss some applications to experimental data on bunch sizes in traffic flow. Consider a traffic flow on a two-lane (one stream in each direction) road which is uninterrupted by traffic signals, intersections etc. A vehicle either can travel freely at the desired speed (without being affected by other vehicles) or is forced to travel behind a slow moving vehicle. Overtaking is possible but may be partially restricted, either by the presence o f vehicles in the opposing lane, or by bends, hills, road conditions etc. The slow moving vehicle and those following it are described as a bunch (platoon or queue). A single vehicle which is neither following nor is followed immediately by other vehicles, is called a bunch o f one vehicle (Taylor and Young, 1988). A slow moving vehicle with two others following is a bunch of three vehicles and so on. Bunching may provide a measure o f quality traffic flow. In heavily bunched traffic, the drivers experience more stress and the tendency to leave the bunch at any time T~cs~ 2s:s-R

365

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M. N. ISLAM and P. C. CONSUL

increases. The degree or extent of bunching is usually measured by the percentage o f vehicles following, which is given by (1 - 1//~)100, where/~ is the mean bunch size. A mean bunch size o f two vehicles means half of the vehicles are following in bunches. Bunching models m a y be used to obtain a good representation of traffic flow. Let the r a n d o m variable X denote the bunch size with support { 1,2 . . . . }. Up to now, the bunching models available in the literature m a y be listed as follows: (a) The geometric distribution P(X=x)

= (1 - 0 ) 0 x-~,

0<0

< 1.

(1.2)

0 < c~ < 1.

(1.3)

(b) The Borel (Borel-Tanner) distribution

P ( X = x) ---

(xoO ~-~ e x p ( - ax)

x!

,

(c) The Miller (1961) distribution

P ( X = x) =

(m + 1)(s + x 1)l(m + s + 1)l , s!(m + s + x + 1)!

m > 0, s > - 1 .

(1.4)

Both the geometric and Borel-Tanner distribution can be derived f r o m queueing theory, while the Miller distribution can be derived f r o m a beta mixture of the geometric distribution. We now derive the Consul distribution as a bunching model through the G a l t o n Watson branching process and also through the generalized birth-and-death process.

2. BUNCHING OF CARS MAY BE FORMED THROUGH A BRANCHING PROCESS The bunching of cars always starts f r o m a single car when one is moving slowly either due to heavy weight or some fault or otherwise and the other cars are unable to overtake it due to the unsafe road conditions (bends, nonvisibility or incoming vehicles etc.). The first car starting the bunch can be said to belong to the 0th generation so that X0 = 1. Let this one car generate X~, in a certain period of time, vehicles to join the bunch. We assume that the pgf of X~ is g(t) = (1 - 0 + 0t) m, when (i) m e N +, 0 < 0 < 1, or (ii) m < 0, 0 < 0 (restrictions on the parameters will be set later). Then each one of the vehicles, unless everyone of them is able to overtake the car )to, will generate the second generation X2 cars to join the bunch and so on there will be X3,X4 . . . . . . We assume that each X.. n = 1,2 . . . . . has the same pgf g(t). Let g.(s) = E[sX]. Then the iterates of the pgf g(s) will be defined by

go(S) = s,

g~(s) = g(s)

and for n = 2,3 . . . . .

g.+l(s) = Z P(X.+, = k)s* k=O

= ~ k=O

s~ ~

p(x.+,

= ~ I .t:. = j? p ( x .

j=O

oo

= ~ P(x. = j? ( g ( s ) y = g.(g(s)). j=O

= ~?

The Consul distribution as a bunching model

367

Also,

g,(s) = g , ( g ( s ) ) = e(g(s)) = g ( g l ( s ) ) . Similarly, by induction

gn+l(S) ~--- g(gn(S)).

(2.1)

We assume that the process o f branching reaches a steady state (i.e. the growth stops) after the nth generation. Let Y, = X0 + .,I"1 + • • • + Xn (where X0 = 1), and let G,(s) be the pgf o f Z, = X~ + • • • +X~. Then Gz(s) = g~(s) = g(s)

and the pgf o f Y, is E[s Yx] = s E [ s z'] = sg(s) = RI(s) say.

Since each vehicle in Xj will start a new generation o f vehicles for the bunch, in the same period, the pgf o f Z2 becomes G2(s) = g(sG~(s)) = g(R~(s))

and similarly, for n = 2,3 .....

G.(s) = g(R~_,(s)).

(2.2)

Thus the pgf o f Yn = 1 + Zn is R.(s) = sG~(s) = sg(R._,(s)).

Taking the limit as n increases without limit G(s) = lira R~(s) = sg(lim R._,(s)) = sgG(s). n~oo

(2.3)

n~oo

Substituting G(s) = t in (2.3), we obtain (2.4)

t = sg(t).

A Lagrange expansion o f t as a function s may be obtained (Islam and Consul, 1990), by putting f ( 0 = t, to give the probabilities o f the total number o f cars in a bunch generated by a single car as

P(X=x)

x1 ( x T 1 ) 0 ~ - I ( 1 - O ) '~-'+1,

x=

1,2 . . . .

where the restrictions on the parameters are (i) m e N +,

0 < 0 < 1,

1 < m s 0 -~,

or (ii) m < 0,

0 < 0,

mO < 1.

,

(2.5)

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M . N . ISLAMand P. C. CONSUL 3. BUNCHING AS A BIRTH-AND-DEATH PROCESS

The addition o f a car or vehicle to the bunch can be considered as a birth and as soon as a car f r o m the bunch succeeds in overtaking the car which initiated the bunch, it m a y be regarded as a death. Also, it m a y be noted that the larger the size o f a bunch o f cars, the greater the possibility (mean rate) of an increase in the bunch. Similarly, the larger the size of a bunch, the greater the desire a m o n g the drivers to get out of the bunch and they will make greater effort to get out which will increase the mean death rate. At time t (t _> 0) let N ( t ) be the total numbers of cars in a bunch. The birth-anddeath process probabilistically describes how N(t) changes with the change in t. Though the cars join a bunch and leave a bunch randomly, their mean occurrence rates depend only on the current state o f the system and on some parameters dependent upon the area and conditions of the road. The assumptions can be stated as follows: 1. Given N ( t ) = n, the probability distribution of the remaining time until the next birth o f a car in the bunch is exponential with parameter )~ (n = 1,2 . . . . . ) given by X~ = (ran + m - n + 1)~ 0, ( m n - n + 2)(~-1)

(3.1)

where m and 0 are the two parameters depending on area, road conditions, auto conditions etc. and c(~) = I'(c + n ) / F ( c ) . Here 3,~ increases with n but rather slowly. 2. Given that N ( t ) = n, the probability distribution of the remaining time until the next death f r o m the bunch is exponential with parameter/~n (n = 2,3 . . . . . ) given by ~t~ = n(1 - O)l-re.

(3.2)

N o t e : Both kn and #~ are linear when m = 1. They are always positive even if m and 0

are negative. 3. Only one birth or death can occur during any small period of time. Due to the relationship between the exponential density function and the Poisson distribution, X~ and ~ become the mean rates of the Poisson distributions. The rate diagram for birth-and-death system o f bunching of cars is illustrated in Fig. 1. Let Pn denote the probability that the system is in state n when the process of birth and death reached a steady state. In the steady state of the system for any state (n = 1,2 . . . . . ) the mean rate (expected number of occurrences per unit time) at which the entering incidents occur must equal the mean rate at which the leaving incidents occur. This gives the principle of rate in equals rate out for each state. Thus the balance equations for each state n(n = 1,2 . . . . . ) are given by (3.3)

~2P2 = ~ l P l ,

and for n = 2,3 . . . . .

kn-i

kn

k n+l

Fig. 1. State transition rate diagram for birth-and-death system of bunching of cars.

The Consul distribution as a bunching model

369

(3.4) which gives

Pn -'~

~1 k2 ° " " kn--I

#2 # 3 " ' '

p~.

(3.5)

#~

pt can be determined using the following condition

(3.6)

~-~Jpn = 1.

n=l

Substituting the values of Xi(i = 1, 2 . . . . . n - 1 ) from (3.1) and #,(i = 2,3 . . . . . (3.2) into (3.5), and after some algebra we get ~- 1 ( m n ~ On_ 1 (1 --

Pn

n \ n - 1/

o)mn-m-n+lpl.

from

(3.7)

From (3.6) and (3.7) we obtain oo

(3.8) For It[ _< 1, the Lagrange expansion of t in terms of u, for l u[ < 1, in equation t = u(1 - 0 + Ot)m, when m e N +, 0 < 0 < 1, under the condition 1 _< m_< 0 -I, gives

n=~ n

n-1

(3.9)

Substituting t = u = 1 in (3.9) and using (3.8) we obtain Pl = ( 1 - 0 ) m,

(3.10)

which on substitution into (3.7) gives the Consul distribution (1.1). Similarly, for Itl _ 1, the Lagrange expansion of t in terms of u, l u[ ~ 1, in equation t = u(1 - 0 + Ot)m, when m < 0, O < 0, under the condition mO <_ 1, and the expression (3.8) gives Pl = (1 - 0)% which also gives the Consul distribution (1.1). 4. A P P L I C A T I O N S

Taylor, Miller and Ogden (1974) compared the geometric, Borel-Tanner and the Miller distributions (Miller, 1961) as bunching models on experimental and simulated bunch size data by considering the behaviour of each distribution over the range of the sample mean bunch size (~). They claimed that the geometric distribution did not provide a satisfactory fit as it underestimated the frequencies of both small bunch sizes (of one or two vehicles) and large bunch sizes (of seven or more vehicles) than either of the other distributions. The Borel-Tanner distribution overestimated the proportion of singlevehicle bunches when the sample mean bunch size is large (~ > 2). They concluded that the Miller distribution might be a more general representation for bunching traffic than

370

M. N. ISLAMand P. C. CONSUL

Table 1. Bunch size frequency distribution, Australian rural highways (Taylor et ai., 1974) Bunch size

Observed frequency

Frequency Borel-Tanner

Frequency Miller

Frequency Consul

1 2 3 4 5 6 >7

296 44 16 6 2 2 1

287.74 54.89 15.71 5.33 1.98 0.78 0.57

294.78 48.31 13.70 5.11 2.27 1.13 1.70

295.72 45.88 14.53 5.77 2.57 1.22 1.31

Total

367

367.00

367.00

36%00

Parameter estimates

& = .2433

X2

4.50 4 0.34

df p-value

• = 3.00 = -.02

th -- -.9234 ~ = -.2635

0.97 3 0.81

0.45 3 0.93

the Borel-Tanner distribution. Tables 1 and 2 give two typical sets (sample mean of one set is less than 2 and that of the other is greater than 2) of experimental data on bunching traffic in Australian rural highways. They are taken from Taylor et ai. (1974). We now examine the fit of the Consul distribution to these data. The parameters m and 8 of the Consul distribution have been estimated by maximum likelihood (ML) method (Islam and Consul, 1990). If an observed sample n consists of the frequencies f , (i = 1,2 . . . . . k), where f l + f 2 + • • • + f k = n , then the ML equations for 0 and m are

n-fl+nra~ln(1

k i-2

1 +~)+m~ -

m

~

if,

i=3 , = l ( m i

(4.1)

=0

-- r)

and 0

1 m

1 = 0, m~

(4.2)

where ~ denotes the sample mean. The ML estimate ~ , if any, is obtained by the point of intersection of the graphs of Table 2. Bunch size frequency distribution, Australian rural highways (Taylor et al., 1974) Bunch size

Observed frequency

Frequency Borel-Tanner

Frequency Miller

Frequency Consul

1 2 3 4 5 6 7 >8

127 53 29 21 5 4 1 5

147.12 45.06 20.70 11.27 6.74 4.28 2.83 7.00

122.50 60.07 30.04 15.30 7.93 4.18 2.23 2.75

124.14 58.77 29.34 15.15 8.00 4.30 2.35 2.95

Total

245

245.00

245.00

245.00

& -- .51

~: = 24.51 g = 24.51

~ = 1.1230 ~ -- 0.4541

Parameter estimates X2

df

p-value

17.83 6 0.01

4.46 4 0.36

4.13 4 0.39

The Consuldistributionas a bunchingmodel

371

1

(4.3)

and k

[ - ~ /

h2(m) = exp[~mm~l - ( n - f l ) -

i-2

\7

m~~i=3,=l ~ (mi - r))l

(4.4)

and then the ML estimate ~ follows from eqn (4.2). Using these ML estimates the expected frequencies of the Consul distribution are computed and are provided in the tables. For comparison we have included in the tables the fit of the Borel-Tanner, and the Miller distributions to the above data sets by the method of maximum likelihood (different from Miller's formulation). The tables also list the x2-values along with their p-values. The Borel-Tanner distribution has been fitted using the ML estimate & = 1 - 1/-~, where ~ is the sample mean. To fit the Miller distribution we consider a random sample (XI,X2. . . . . An) of size n from the Miller distribution (1.4). Let the observed values in the sample be given by 1,2 . . . . . k, with frequencies f~f2 . . . . . fk, such that f~ + f2 + • • • + .Irk = n. Then the log likelihood function may be given by k

ln L = n l n ( m

+ I)+

[ -i

i

1

)-~,f/[~-~ ln(s + j) - ~ ln(m + s + 1 + j) - l n ( s + 0[, i=l [d=l j=l 3

which gives the following ML equations for m and s k

i

Zf[m

1

+=t

+

1

Z 1 ] /=~m+s+ 1 +j

(4.5)

and k

j:~

i

(s + y)(m + s + 1 + j)

s +

These nonlinear equations are solved for m and s by iteration method using the NEQNF subroutine of IMSL library on a IBM3081 computer programmed in Fortran-77. The initial estimates of m and s are taken from the sample mean (~) and the proportion of single-vehicle bunches. It can be seen from Table 1 that the Borel-Tanner distribution underestimated the frequency of single-vehicle bunches, and overestimated the frequency of two-vehicles bunches, while the Consul distribution improved the fit even better than the Miller distribution. The sample mean bunch size in this data set is 1.32 which is less than 2. In the data set of Table 2 the sample mean bunch size is 2.04 which is slightly greater than 2. In this data set the Borel-Tanner distribution overestimated the frequency of the single-vehicle bunches and underestimated the frequencies of bunch sizes of two, three or four vehicles, while if the Consul distribution is chosen it dramatically improves the fit. By comparing the p-values we may conclude that the Consul distribution gives the best fit to this data while the Borel-Tanner distribution gives a very poor fit, and the Miller distribution is as good as the Consul distribution. We fit the other data sets on bunching traffic in Australian rural highways given by Taylor et al. (1974) and the results are not given here. We found that the Borel-Tanner and Consul distributions fit these data closely and are much better than the Miller distribution. REFERENCES and Shenton L. R. (1972) Use of Lagrangianexpansion for generating discrete generalized probabilitydistributions.SIAMJ. Appl. Math., 23(2), 239-248.

Consul P. C.

372

M. N. ISLAM and P. C. CONSUL

Consul P. C. and Shenton L. R. (1975) On the Probabilistic Structure and Properties of Discrete Lagrangian Distributions. A Modern Course on Statistical Distributions in Scient07c Work, Vol. 1, pp. 41-57. D. Reidel, Boston, MA. Islam M. N. and Consul P. C. (1990) A probabilistic model for automobile claims Jour. Swiss Asso. Actuaries. (submitted). Miller A. J. (1961) A queueing model for road traffic flow. J. Roy. Stat. Soc., 23B, 64-75. Taylor M. A. P., Miller A. J. and Ogden K. W. (1974) A comparison of some bunching models for rural traffic. Transpn. Res. $(1), 1-9. Taylor M. A. P. and Young W. (1988) Traffic Analysis: New Technology and New Solutions. Hargreen Publishing Company, Melbourne, Australia.