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A comparison of some bunching models for rural traffic flow
A COMPARISON OF SOME BUNCHING FOR RURAL TRAFFIC FLOW
MODELS
M. A. P. TAYLOR Country Roads Board of Victoria. Australia A. J. MILLER C.S.I.R.O. Division of Mathematical Statistics, Sydney, Australia K. w. OGDEN Department of Civil Engineering. Monash University, Victoria. Australia (Receioed 13 November 1972) Abstract-On two-lane two-way roads overtaking opportunities are often limited by traffic and highway factors so that vehicles may be forced to queue behind slower vehicles before overtaking can take place. To explain this clustering or bunching of traffic several bunching models have been proposed. One such model which has been used is the Borel-Tanner distribution of bunch sizes. The paper examines this model. and others. and compares them using both experimental and simulated bunch size data. From the comparison it is suggested that a more general model than the Borel-Tanner model is the two-parameter Miller distribution of bunch sizes which can be fitted to a wide range of traffic conditions.
INTRODU0ION
In this paper the three models mentioned above are compared in terms of their fit to experimental and simulated queue length data (Taylor, 1971). The. terms queue, bunch and platoon have all been taken as synonymous. They represent a group of vehicles travelling in the same direction with a short headway (termed the tracking headway) between each vehicle. A single vehicle travelling without other vehicles close behind or in front is considered to be a queue of one vehicle. The vehicles in a bunch are then travelling in a car-following situation, in which their speeds, spacings and accelerations are dependent on the vehicle immediately in front of them The leading vehicle in a bunch is assumed to be travelling free of the influence of other vehicles (including those behind him in the bunch).
The phenomenon of the clustering or bunching of traffic on rural two-lane roads has long been noted and has been the subject of considerable empirical and theoretical study. Bunching models may provide a more acceptable concept of traffic flow on a two-lane highway than do any of the single vehicle headway models, such as those of Buckley (1962) and Greenberg
(1966). A single vehicle headway model usually requires an assumption of independence between the headways and speeds of consecutive vehicles. Where vehicles cannot overtake freely it is doubtful if this assumption is ever met. Several bunching models have been postulated and a typical model consists of a distribution of bunch sizes and an assumed distribution of headways between bunches. Three bunch size distributions have been used. These are: (a) the geometric distribution (Miller. 1961); (b) the Borel-Tanner distribution (Tanner. 1953. 1963 and Miller, 1963, 1965, 1968); and (c) the Miller distribution (Miller, 1961). The Borel-Tanner distribution is the most widely-accepted of the three bunch size distributions. The usual form for the inter-bunch headway distribution is a negative exponential. This was explicitly stated for Miller’s “random bunches” model (1963) and implied in the other models.
CONDITIONS
CAUSING
BUNCHING
OF TRAFFIC
occurrence of bunching is largely due to speed differentials between vehicles and restrictions on overtaking. Vehicles overhauling slower units may not be able to pass them immediately. Overtaking may be restricted both by traffic and highway factors. A driver will need to find a suitable gap in the opposing traffic and will require some clear sight distance ahead before attempting to overtake. Three studies of the sizes of acceptable gaps for overtaking on two-lane two-way roads in Australia have The
1
2
M. A. P. TAYLOR,A. J. MILLERand K. W.
been reported. In two studies log-normal distributions of acceptable critical gaps were fitted to experimental data using maximum likelihood estimators. Complications were found in both cases and a simple log-normal distributfon was insufficient to describe the observed behaviour of drivers. Miller and Pretty (1968) studied acceptable gaps for overtaking in the face of opposing traffic, with no restrictions on visibility or sight distance. They found a small proportion of drivers who would not overtake under any circumstances. Later Troutbeck, Szwed and Miller (1972) investigated acceptable gaps when sight distance was the sole restriction on overtaking. A small but significant proportion of drivers who would accept extremely small sight distances was found. In a third study, Ashton, Buckley and Miller (1968) determined the average number of vehicles which would overtake in a given size of headway in the opposing traffic. There may be other factors which influence the driver’s decision to overtake. Many drivers appear content to track behind vehicles whose travel speeds are only marginally below the desired speed of the tracking driver. Perhaps the number of vehicles in the queue also influences the size of gap in the opposing traffic stream accepted by overtaking vehicles. On level roads, the sight distance to the next curve is the limiting geometric factor. On grades there are also vertical sight distances to restrict overtaking. The limits of vehicle performance on grades will also affect overtaking opportunities. Cars may not be greatly affected by gradient but trucks may find considerable difficulties when attempting to overtake on grades. Moskowitz and Newman (1965) recognized this in
OGDEN
their study of traffic flow on rural freeways where the truck-passing-truck manoeuvre was critical in capacity considerations. DATA SOURCES
Data collected by various workers in Australia and Europe on the bunching of traflic on two-way roads has been used in this study. Eight sets of English data, six sets of Swedish data and eight sets of data for Australian conditions were used In addition, use was made of queue length data from a simulated no-passing zone, generated in a recent simulation study of traffic flows on grades (Taylor, 1971). The experimental data are presented in Tables 1 and 2. One of the major difficulties with the use of experimental bunch size data is the difference in queue discrimination criteria adopted. Buckley (1962), Underwood (1963). Miller (1963). and Daou (1966) have discussed queue discrimination methods. More recently Pahl and Sands (197 1) have suggested that the vehicle interaction phase can be described in terms of a time headway. Another problem is that of whether overtaking vehicles should be included in a bunch. The variability of tracking headways also causes difficulties. Queue discrimination is easier with simulated data. Both the position of each vehicle in its bunch, and whether or not the vehicle is delayed are known immediately. BUNCHING MODELS
The mathematical forms of the three bunch size distributions listed previously are: (a) the geometric distribution (1 - so.r’-i;
P, =
E(r) = _&
Table I. English/Swedish queue length data Frequency of a bunch of size r*
English
Swedish
5
6
7
I
2 4
5
1
2
3
4
2 3 4
184 467 392 536
38 117 87 130
19 32 29 41
12 9 20
6 5 7
5 6 7 8
229 451 333 414
55 107 91 136
18 30 25 38
16 13 6 15
6 2 10
1 2 3 4 5 6
119 99 121 112 107 175
29 29 32 23 16 25
19 6 9 13 5 6
6 2 10 3 7 1
5 4 1 0 -I 2
Data set 1
* Last entry is for bunches & r
1
-
7 3 6 8
8
7 3
4
5
2 7
5
12
(1)
Bunching models for rural traffic
3
Table 2. Australian queue length data Frequency of a bunch of size r* Data set 1 28 3 4 5 6 7 8
1
2
3
4
5
6
7
x21
d.o.f.:
P
296 127 66 50 67 46 51 62
44 53 5 8 6 11 12 12
16 29 0 3 2 2 9 3
6 21 1 0 0 2 3
2 5
2 4
1 6
4 6
1
0
2.80 18.1 0.16 0.36 0.43 0.43 3.70 0.09
1 1 1 0
0 2
1
1
1.330 2.057 1.111
2
1.290
: 3 2
1.184 I.403 1.632 I.388
* Last entry is for bunches Z r. t The x2 scores are for the Bore&Tanner distribution fitted to the data. 1 Degrees of freedom. p Set 2 is the onlv set for which the Borel-Tanner distribution provides a poor fit. The twoparameter Miller distribution provides a good fit for this set (see text and Appendix). For the 2-parameter Miller distribution fitted to this data l2 = 4.05 with 4 d.o.f.
(b) the Borel-Tanner P, =
(rat?_yr!
(b) for the one-parameter
distribution
1*f?-# ; E(r) = A;
and
(2)
I
l)!(r-l)!. l)!
(m + r +
’
E(r)=
‘Y$
(3)
(nt is not necessarily an integer).
COMPARISONS OF THE BUNCH DISTRIBUTIONS
SIZE
A comparison between the three common bunch size distributions can be made by considering the behaviour of each distribution over the range of P (mean bunch size). Bunches are referred to as “small” if of size r = 1 or 2. (A vehicle travelling by itself is considered to be a bunch of size r = 1.) “Intermediate” bunches are of size 3-6, while large bunches are of size r > 6 (in theory r is unlimited). The geometric distribution predicts smaller frequencies of both small and large bunches than either of the other distributions. The Miller distribution has the longest tail of the three for a given mean and so predicts smaller frequencies of bunches of intermediate size. Most of the available experimental data is for f < 1.6, and this has greatly influenced the acceptance of particular queue length distributions, For instance, if the proportion of single vehicles bunches (p,,l) is examined, then (a) for the Borel-Tanner distribution p, =e-‘he-’
as
P+X,;
Pl
Z-----d:
1
m+2
Miller distribution as
f--r co;
and
(5)
(c) for the geometric distribution
(c) the Miller distribution (one-parameter) p =(m+l)(m+
m+
(4)
Pl = (1 -a)+0
as
P-+co.
(6)
The three distributions thus show a wide divergence of behaviour as mean bunch size is increased. Figure 1 shows this divergence and also shows p1 plotted against P using both the simulated data and the experimental data from England Sweden and Australia. It can be seen that the Borel-Tanner and one-parameter Miller distributions tend to overpredict the proportion of single vehicle bunches for larger f (f > 2). The observed data seems to lie between the curves for the geometric and Borel-Tanner distributions of bunch sizes. In order to explore in more detail the behaviour of the three bunch size distributions, synthetic bunch size data was generated by the simulation of a length of road on which overtaking was impossible The road consisted of segments of both level road and grades. The length of each segment was variable and was specified in the simulation run. A level road was defined as one on which all unimpeded vehicles could maintain their desired travel speeds. Trafik arrivals were sampled from various headway distributions. The distributions used were: (i) negative exponential distribution; (ii) displaced negative exponential distribution; and (iii) Borel-Tanner bunches with a constant headway(r) between bunched vehicles and inter-bunch headways consisting of r plus a negative exponential headway.
M. A. P. TAYLOR.A. J. MILLERand K. W. OCUEN
4
h 3’1
I
I
2
I
3
1
L
I
I
4 5 6 Mean bunch size,
7-
IO
7
of
Fig. I. Probability singlevehicle bunches (p,) against mean bunch size (P): (I) Predicted curve for oneparameter Miller dist; (2) Predicted curve for Borel-Tanner dist; (3) Predicted curve for geometric dist.
Vehicle free (desired) speeds were sampled from a gamma distribution with a coefficient of variation of @lg. For the case of bunched arrivals a bunch speed was generated in the following manner. For a bunch of n vehicles, n free speeds were generated and the slowest of the n speeds became the arrival speed of the bunch. All vehicles in a bunch were assumed to travel at the same speed but each vehicle also had its own desired speed. The simulation models have been fully described by Taylor (1971). Eight sets of synthetic bunch size data were generated for use in the present study. These represented noovertaking zones varying in length from O-20 to 3.22 km, and average bunch sizes of from 1.245 to 6.728 vehicles. The simulated data thus covered a wide range of traffic situations (Table 3). The Borel-Tanner, geometric and one-parameter Miller distributions were each fitted to the synthetic bunch size data. The parameters of the distributions
were determined using maximum likelihood techniques; these are described in the Appendix. The distributions thus obtained were checked for goodness-of-fit to the eight sets of synthetic data using a chi-squared test. Table 3 shows the chi-squared scores which were obtained. These scores suggest that the geometric distribution provides a reasonable fit for bunches of vehicles leaving a no-passing zone of length X when X z- 1.6 km. On the other hand the BorelTanner distribution is acceptable for X < 04 km. These results appear to be true for one-way (single lane) traffic flows in the range of 7.5-800 vehicles/hr (Taylor, 1971). Mean desired speeds for the synthetic traffic were in the range 68-80 km/hr with a coefficient of variation of 0.18. For J > 3 the fits are not very good but this is tempered because of the small sample sizes and the occasional occurrence of very long queues. The Borel-Tanner distribution has been used to represent bunches of vehicles on a two-lane .road with
Table 3. Chi-squared scores for some simulated data Chi-squared values for* Data set number
Road length X (km)
Mean bunch size (I)
1 2 3 4 S 6 7 8
3.22 1.61 1.61 0.80 080 @40 0.40 0.20
1.622 6.7’8 1.317 1.245 2.648 2.744 1.617 I.539
Totals
Miller Borel-Tanner 7.5 (3) 95.6 (9) 2.4 (2) 1.9 (I) 16.2 (8) 23.8 (9) 6.5 (5) 2.315) 156.2 (42)
* Number of degrees of freedom shown in parentheses.
Geometric 2.6 (3) 11.7 (13) 1.8 (2) 0.2 (2) 29.2 (7) 52.6 (8) 15.5 (4) 55.2 (4) 171.7 (43)
l-parameter 20.4 (5) 158.3 (7) 9.4 (4) 5.9 (3) 45.6 (8) 42.9 (9) 19.0(6) 7.4 (6) 338.9 (48)
2-parameter 3.6 (3) -1.9(II) 0.8 (3) 0.2 (2) 11.1 (7) 7.3 (8) 5.1 (-I) 4.1 (1) JO.6 141b
Bunching models for rural traffic
5
limited overtaking opportunities (Miller, 1963). It and 111+ s + 1 should be expected that short lengths of no-passing E(r) = . (7) zone can be related to a two-lane road on whieh 111 limited overtaking can occur. For example, the limited By fitting two-.parameters (e.g. sample p1 and P) it is (equilibrium) overtaking road may be pictured as conpossible to match the theoretical distribution to sistingofalternate sections on which overtaking cannot observed data in a more flexible manner than for the occur followed by a short length on which wholesale one-parameter distributions described above, The twoovertaking is possible. The lengths of no-passing zone parameter distribution thus provides a much closer fit would be both static (e.g. sections of road between to the data as is shown in Table 3 for the simulated sharp curves, bends, crests and dips) and dynamic (the data described in the previous section. Moreover, the section of road between bunches travelling in opposing distribution is shown in the Appendix to fit the simudirections and visible to each other). The relevant geolated data over the range of P for all X. metric design parameter in the static case is the overIn the original derivation (Miller, 1961) the parataking sight distance. A vehicle travelling at the design meter was found to be practically zero for the data speed on a road where overtaking is generally possible tested and so the parameter was excluded; this prowill not be able to overtake safely if the sight distance duced the one-parameter distribution described above. falls below the overtaking sight distance. Values of However results for the simulated data and the Engovertaking sight distance are of the same order as the lish/Swedish experimental data suggested that s may length of road in which the Borel-Tanner distribution be significant. particularly for large mean bunch sizes. provides a good fit to queue length data. For example, Table4 provides an interesting comparison between the National Association of Australian State Road the Borel-Tanner and two-parameter Miller distribuAuthorities (1970) has set overtaking sight distances tions based on this data. The values of the parameters varying from 115 m at 32 km/hr to 754 m at 96 km/hr. used were maximum likelihood estimators and the Thus it is not surprising that the Borel-Tanner distritable lists chi-squared scores for each distribution. The bution fits queue length data from short sections of noformulation of the maximum likelihood parameters is passing zone. given in the Appendix. The one-parameter Miller distribution was also fitThe differences between the fits obtained for the data ted to the data. From Table 3 it can be seen that the from the two sources may be due to the different queue distribution provides a poor fit to the data. This result is not unexpected as the one-parameter Miller distriTable 4. Comparisons of two queue length distributions fitted to some European data bution tends to over-estimate the proportions of small bunches (I = 1 or 2). For example, from equation (5) 2-parameter the minimum value of p1 was shown to be 0.5 for the Bore&Tanner Miller distribution. With this restriction the one-parameter Miller distribution cannot adequately describe do.f. Data set d.o.f.* x2 %? observed bunch size data sets with large mean bunch English 1 7.0 2 6.4 2 sizes (P > 15).
THE TWO PARAMETER
2 3 4 5 6 7 8
4.8 0.7 11.0 4.6 3.2 6.9 4.4
3 2 3 4 4 3 4
4.6 2.8 12.0 8.3 1.0 6.4 1.8
4 3 3 4 4 3 3
1 2 3 4 5 6
7.5 2.3 6.4 1.8 4.8 1.2
5 2 4 2 2 2
2.2 1.2 4.9 24 5.2 0.5
3 3 2 2 2
Totals English Swedish
42.6 24.2
25 17
43.3 16.5
27 14
Overall
66.8
42
59.8
41
MILLER
DISTRIBUTION On the basis of the above empirical evidence, it can be concluded that the three common bunch size distributions have some validity. However, none of the distributions provides a satisfactory fit to the data over the full range of average bunch sizes, f. A more general distribution which can provide a good fit for the whole range off is obtained by reverting to the original form of the Miller distribution (Miller, 1961) which is a twoparameter distribution:
p = (m + l)(s + r - l)!(m + s + l)! I
s!(nr + s + r + l)!
Swedish
+ Degrees of freedom.
M. A. P. TAYLOR, A. J. MILLERand K. W. OGDEN
6
discrimination criteria used For the English data, bunches were assessed subjectively. The assessment of the Swedish data was based on a maximum tracking headway of 8 set and maximum speed differentials between successive vehicles of + 10 km/hr and -5 km/hr. The speed differentials effectively separate vehicles engaged in overtaking into a separate bunch from the vehicles being overtaken. Some further data was gathered as part of a study of bunching and delays on grades (Taylor, 1971). This data is presented in Table 2. The data is for bunches both at the crest of the grade for selected points along the grade. One sample (Set 2) showed significant bunching with a mean bunch size of slightly greater than 2-O.This was the only sample which was not well represented by the Borel-Tanner distribution. The two-parameter Miller distribution fitted this data well. For the other data there was little difference between the two distributions. Bunches were discriminated according to two sets of criteria. For data gathered at the top of grades bunches were discriminated on the basis of headways of 8 set or more. For data from lower reaches of the grades discrimination was based on headways of 5 set or more and a speed differential of +8 km/hr. The different criteria were adopted because: (a) many slow moving vehicles were bunched at the top of the grades but generated long tracking headways because of their speed; and (b) vehicles overtaking bunches at the foot of the grade needed to be considered separately from those bunches. The even numbered data sets (Table 2) are from the top of grades. Sets 7 and 8 are from data collected at two locations on a grade with climbing lanes.
DETERMINATIOKOF m AND s PARAMETERS For the two-parameter Miller distribution lowing relations hold: J?l+ 1 5=tn+s+2 where 5 is the expected proportion bunches and m+s+l cc= 111
the fol(8)
of single vehicle (9)
where p is the expected mean bunch size. Now both 4 and p are constrained to certain ranges of permissible values These ranges are : O<<
(10)
and fi>
1.
(11)
Equations (10) and (11) then detine permissible regions for m and s. These regions are: (i) (m>O,s>--1) and (ii) (m < -1, s G 1). Note that it is possible to have negative values of both m and s, and that zero values of s can also occur, as long as the pair of values fall inside one of the permissible regions. To determine values of the parameters given p and 5, equations (8) and (9) can be rearranged to solve for mands.
(12) s=+-1)-l.
(13)
So far the only constraint upon the m and s parameters is that they fall in the permissible regions above. However, by reference to Fig. 1 some further inferences can be drawn about likely pairs (m, s) for particular values of p and t, for single lane traffic on a twolane two-way road The graph suggests that the value of < is dependent to some extent on the value of b and that i’ lies between the values producted by the geometric distribution (for which 5 = l/p) and the BorelTanner distribution {for which 5 = exp[ -(p - 1)/p]}. That is: 1 lc:Cexp
(p - 1) I --’ n I
Mean bunch sizes can be estimated from methods described by Miller (1963, 1965) and Taylor (1971) for various traffic and highway factors. The range of the proportion of single vehicle bunches is then defined and estimates of m and s parameters can be made. Taylor (1971) has compared maximum likelihood estimates (m, s) and estimates made from sample (?, pl) for both experimental and synthetic data. The introduction of a two-parameter distribution of bunch sizes for traffic on a two-lane road on which overtaking is either restricted or prohibited introduces yet another parameter to be known or estimated in order to describe the traffic flow. This parameter involves the proportion of bunches of one vehicle in the traffic. However, the weight of experimental evidence suggests that this proportion (0 is not a single-valued function of mean bunch size b) and is dependent upon other traffic and highway factors. On this basis estimates of t cannot be made from $ and thus 5 has to be determined separately. The two-parameter Miller distribution has great flexibility and can be calibrated
Bunching models for rural tragic
to reproduce a wide range of bunch distribution for road traffic flow. which the one-pa&meter distributions cannot do. HEADWAYS
BETWEEN
BUNCHES
The usual assumption for the distribution of headways between bunches is that they follow the form of a negative exponential distribution. This has been used by Miller (1961. 1963) for bunching under limited (equilibrium) overtaking conditions. Cowan (1971) has found the distribution of inter-bunch headways on exit from a no-passing zone. His distribution is not an exponential. However it can be approximated by an exponential distribution, and the approximation is closer as traffic volume is increased. Thus even under this extreme condition the negative exponential is a reasonable distribution to represent inter-bunch headways Experimental headway distributions were measured during collection of the data in Table 2. Figure 2 shows the sample frequencies of headways, together with an exponential distribution based on the sample average headway. A good agreement between the observed fre-
quencies and the theoretical
distribution
was obtained.
7
of platoon sizes under some conditions. A more general distribution is the two-parameter Miller distribution. The major difficulty with the application of bunching models in road traffic flow is that knowledge of several parameters (e.g. traffic volume and mean bunch size) is required for use of the models. However methods of determining parameters. such as mean bunch size, in terms of traffic, highway and alignment factors are available. The following authors have suggested methods for estimation of mean bunch size. (i) Tanner (1961); (ii) Miller (1963, 1965); (iii) Underwood (1963); and (iv) Taylor (1971). With these methods bunching models may be used to provide greater insight into traffic flow on rural highways. In particular the two-parameter Miller distribution appears capable of representing bunching in trtic under a wide range of highway and traffic conditions. Acknowledgement-This paper is presented with the permission of the Chairman of the Country Roads Board of Victoria. Views expressed are those of tire authors, and not necessarily those of the organization they represent.
Inter_- bunch hwdwoy.
f set
Fig. 2. Inter-bunch headways for the data of Set 2, Table 2. The figure shows an exponential distribution based on the sample mean headway for comparison with the observed headways. CONCLUSIONS
REFERENCES
Bunching models provide a good representation of the flow of traffic on two-lane rural roads. In this paper some of the common
models for the bunching
of traffic
have been examined in the light of some experimental and synthetic data covering a wide range of flow conditions A model combining negative exponential interbunch headways with a bunch size distribution provides a good representation
Tanner distribution
of rural traffic. The Borel-
gives a reasonable representation
Ashton H. R., Buckley D. J. and Miller A. J. (1968) Some aspects of capacity and queueing in the vicinity of slow vehicles on a rural two lane road. Proc. A.R.R.B. 4. 595-
612. Buckley D. J. (1962) Road traffic headway distributions Proc. A.R.R.B.
1,153-183.
Cowan R. J. (1971) A road with no overtaking. Aust. J. qf Sm. 13,94-l 16. D aou A. (1966) On flow within platoons. J. Amt. Rd Res.
2,4-13.
8
M. A. P.
TAYLOR. A.
J.
MILLER
Greenberg H. (1966) The log-normal distribution of headways. J. Aust.Rd Res. 2, 14-18. Miller A. J. (1961) A queueing model for road traffic flow. J. Roy. Stats. Sac., Series B U,64-75.
Miller A. J. (1963) An analysis of bunching in two-lane rural
A.R.R.B. 6.286301.
Underwood R. T. (1963) Traffic flow and bunching. J. Aust. Rd Rrs 1.8-25.
N.A.A.S.R.A. (19701Policyfor Roads. 4th Ed.. Sydney. LIST r r P, J e m,s X EO 5 p 5
the Geometric
Design of‘Rurul
Let
L = In (IJ
then
(‘4.2) r=
1
(a) For the Borel-Tanner -3 -1 (rre Y e-’ PI = r.I
distribution
1
J-1 I., = $J f, (r - i)ln(a) - c(r - In 7 I= 1
1
+ M’.
If L is maximized by taking the partial derivative with respect to n and equating to zero, then
&=&’ J
where P is the mean bunch size of the observed sample. Thus for the Borel-Tanner distribution the sample mean is the maximum likelihood estimator of the population mean. A similar result exists for the geometric distribution. However the Miller distribution is less well behaved. (b) For the two parameter Miller distribution: p = (m + l)(m + s + l)!(s + r - l)!; I s!(m + s + r + l)!
OF SYMBOLS
size of platoon (a single vehicle is r = 1) sample mean bunch size probability of a bunch size r bunching parameter in Borel-Tanner or geo,metric distributions base of Napierian logarithms bunching parameters in Miller distributions length of a no-passing zone (km) denotes the expected value of the variable in the parentheses expected proportion of bunches of 1 vehicle expected mean bunch size headway between bunched (tracking) vehicles.
OGDEN
where M is some arbitrary constant and the @,) are the set of expected probabilities of bunches of size r. The equation is simply a product of the probabilities of occurrence of the bunches.
traffic. Ops. Res. 11,235-247. Miller A. J. (1967) Queueing in rural traffic. 3rd International Symp. on the Theory of Traffic Flow, New York (1965) Published as Vehicular Traffic Science, pp. 122-137, Elsevier, New York. Miller A. J. (1968). Studies of speeds and delays on rural ;oads in Australia, 9th International Study Week in Traffic and Safety Engng., Munich. Miller A. J. and Pretty R. L. (1968) Overtaking on two-lane rural roads. Proc. A.R.R.B. 4 (I), 582-591. Moskowitz K. and Newman L. (1965) Effects of grades on service volumes. H.R.B. Record 99, 224-243. Pahl J. and Sands T. (1971) Vehicle interaction criteria from time series measurements. Trans. Sci. $403-417. Tanner J. C. (1953) A problem of interference between two queues. Biometrika 40, 58-69. Tanner J. C. (1961) Delays on a two-lane road. J. R. Stats. Sot. Series B X%,38-63. Taylor M. A. P. (1971) A simulation study of traffic flow on grades. Thesis for M.Eng.Sc., Monash Univ., Clayton, Vie., Australia (unpublished). Troutbeck R. J., &wed N. and Miller A. J. (1972) Overtaking sight-distances on a two-lane rural road. Pror
and K. W.
P=
m+s+l m
; <=
m-f
and
1
m+s+2’
The log likelihood function is now L = i L ln(m + 1) - i ln(m + s + k + 1) ,=I I k=,
k=1
+ i
In(s + k - 1) - In(s)
1.
It is necessary to equate both dL,‘am and dL/b zero : 2L
z
= 0
to
(A.4)
and Esfimatiny
hunch size
tliscrihutiorl purametrrs
Maximum likelihood methods have been used to find estimators of the parameters in the bunch size distribution. For a given set v,) of the observed frequencies of quelles of length r. (r = I. 2.. x ). The general likelihood function is given by
(A.3 An iterative technique is required to solve equations (A.3) and (A.4) simultaneously. The maximum Iikelihood estimators r&and S are not related simply to the sample mean bunch size. The initial estimates of m and s can be taken from the sample P and pl.
Bunching models for rural traffic Table 5 Predicted Bore-Tanner frequency
Predicted 2 par. Miller frequency
Bunch size
Observed frequency
1
127 53 29 21 5 4
148 45 21 II 7
127 59 29 15 8
1
7
I
5
7
6
2 3 4 5 6 7
>8
The estimation of the parameter nt in the one-parameter Miller distribution is similar to the above. The European data for which chi-squared scores were given earlier is presented in Table 1. Some Australian data is given in Table 2. There is little difference between the Bore]-Tanner and the two-parameter Miller distributions for the Australian data except for data set 2. The two-parameter Miller distribution fits this set well, while the Borel-Tanner distribution provides a poor fit. This is a significant result and agrees closely with the result found using the simulated data.
The queue length frequences in Table 5 were predieted for data set 2 of Table 2. Table 3 summarizes chi-squared scores for distribution fitted to this data. A significant result is that the geometric and Bore]-Tanner distribution fit the overall data poorly. They are however reasonable for limited ranges of the data. The geometric distribution is reasonable for data from long sections of no-passing zone and for large F. The Bore]-Tanner distribution is satisfactory for small X ( < 0.4 km) and F. On the other hand the two-parameter Miller distribution covers the ranges of X and F satisfactorily.
ResumP-Sur les routes bidirectionelles a deux voies, les occasions de dtpassement sont souvent limit&es par le trafic et les caracteristiques de la voirie en sorte que les vehicules peuvent itre obliges d’attendre derriere les vehicules plus lents avant que le dtpassement devienne possible. Pour expliquer ce regroupement ou ce tassement de la circulation plusieurs modkles de regroupement ont ete proposes. Un modele de ce type, qui a Cte utilis6, est la distribution de Borel-Tanner des tailles de groupes de vehicules. Cet article analyse ce modP1e et les autres, et les compare a l’aide de donntes concernant les tailles des groupes de vihicules a la fois experimentales et simultes. De cette comparaison on tire la suggestion qu’un modtle plus global que le modele Borel-Tanner est constitui par la distribution des tailles de groupes de vehicules. a deux parametres de Miller; ce modtle peut &tre ajuste pour un eventail assez large de conditions de circulation. Zusammenfassung-Auf zweisputigen Strassen mit Gegenverkehr werden iiberholmiiglichkeiten haufig durch verkehrliche und strassenbauliche Gegebenheiten beschnitten, so dass Fahrzeuge gezwungen sein konnen, hinter langsameren Fahrzeugen zu verbleiben, ehe der iiberholvorgang beginnt. Zur Beschreibund der daraus resultierenden Kolonnenbildung sind verschiedene Modelle entwickelt worden. Eines dieser Modelle, und zwar das von Bore1 und Tanner. befasst sich mit der Verteilung der Kolonnenllnge. In der vorliegenden Abhandlung wird es untersucht und mit anderen AnsStzen verglichen, wobei sowohl auf experimentell gewonnene als such auf simulierte Werte fur die Kolonnenllnge zuriickgegriffen wird. Aus dem Vergleich ergibt sich, dass die von Miller entwickelte zweiparametrische Verteilung der Kolonnenlange allgemeiner verwendbar ist als das Model1 von Bore1 und Tanner und einer grossen Spannbreite von Verkehrsbedingungen angepasst werden kann.
MODELS
M. A. P. TAYLOR Country Roads Board of Victoria. Australia A. J. MILLER C.S.I.R.O. Division of Mathematical Statistics, Sydney, Australia K. w. OGDEN Department of Civil Engineering. Monash University, Victoria. Australia (Receioed 13 November 1972) Abstract-On two-lane two-way roads overtaking opportunities are often limited by traffic and highway factors so that vehicles may be forced to queue behind slower vehicles before overtaking can take place. To explain this clustering or bunching of traffic several bunching models have been proposed. One such model which has been used is the Borel-Tanner distribution of bunch sizes. The paper examines this model. and others. and compares them using both experimental and simulated bunch size data. From the comparison it is suggested that a more general model than the Borel-Tanner model is the two-parameter Miller distribution of bunch sizes which can be fitted to a wide range of traffic conditions.
INTRODU0ION
In this paper the three models mentioned above are compared in terms of their fit to experimental and simulated queue length data (Taylor, 1971). The. terms queue, bunch and platoon have all been taken as synonymous. They represent a group of vehicles travelling in the same direction with a short headway (termed the tracking headway) between each vehicle. A single vehicle travelling without other vehicles close behind or in front is considered to be a queue of one vehicle. The vehicles in a bunch are then travelling in a car-following situation, in which their speeds, spacings and accelerations are dependent on the vehicle immediately in front of them The leading vehicle in a bunch is assumed to be travelling free of the influence of other vehicles (including those behind him in the bunch).
The phenomenon of the clustering or bunching of traffic on rural two-lane roads has long been noted and has been the subject of considerable empirical and theoretical study. Bunching models may provide a more acceptable concept of traffic flow on a two-lane highway than do any of the single vehicle headway models, such as those of Buckley (1962) and Greenberg
(1966). A single vehicle headway model usually requires an assumption of independence between the headways and speeds of consecutive vehicles. Where vehicles cannot overtake freely it is doubtful if this assumption is ever met. Several bunching models have been postulated and a typical model consists of a distribution of bunch sizes and an assumed distribution of headways between bunches. Three bunch size distributions have been used. These are: (a) the geometric distribution (Miller. 1961); (b) the Borel-Tanner distribution (Tanner. 1953. 1963 and Miller, 1963, 1965, 1968); and (c) the Miller distribution (Miller, 1961). The Borel-Tanner distribution is the most widely-accepted of the three bunch size distributions. The usual form for the inter-bunch headway distribution is a negative exponential. This was explicitly stated for Miller’s “random bunches” model (1963) and implied in the other models.
CONDITIONS
CAUSING
BUNCHING
OF TRAFFIC
occurrence of bunching is largely due to speed differentials between vehicles and restrictions on overtaking. Vehicles overhauling slower units may not be able to pass them immediately. Overtaking may be restricted both by traffic and highway factors. A driver will need to find a suitable gap in the opposing traffic and will require some clear sight distance ahead before attempting to overtake. Three studies of the sizes of acceptable gaps for overtaking on two-lane two-way roads in Australia have The
1
2
M. A. P. TAYLOR,A. J. MILLERand K. W.
been reported. In two studies log-normal distributions of acceptable critical gaps were fitted to experimental data using maximum likelihood estimators. Complications were found in both cases and a simple log-normal distributfon was insufficient to describe the observed behaviour of drivers. Miller and Pretty (1968) studied acceptable gaps for overtaking in the face of opposing traffic, with no restrictions on visibility or sight distance. They found a small proportion of drivers who would not overtake under any circumstances. Later Troutbeck, Szwed and Miller (1972) investigated acceptable gaps when sight distance was the sole restriction on overtaking. A small but significant proportion of drivers who would accept extremely small sight distances was found. In a third study, Ashton, Buckley and Miller (1968) determined the average number of vehicles which would overtake in a given size of headway in the opposing traffic. There may be other factors which influence the driver’s decision to overtake. Many drivers appear content to track behind vehicles whose travel speeds are only marginally below the desired speed of the tracking driver. Perhaps the number of vehicles in the queue also influences the size of gap in the opposing traffic stream accepted by overtaking vehicles. On level roads, the sight distance to the next curve is the limiting geometric factor. On grades there are also vertical sight distances to restrict overtaking. The limits of vehicle performance on grades will also affect overtaking opportunities. Cars may not be greatly affected by gradient but trucks may find considerable difficulties when attempting to overtake on grades. Moskowitz and Newman (1965) recognized this in
OGDEN
their study of traffic flow on rural freeways where the truck-passing-truck manoeuvre was critical in capacity considerations. DATA SOURCES
Data collected by various workers in Australia and Europe on the bunching of traflic on two-way roads has been used in this study. Eight sets of English data, six sets of Swedish data and eight sets of data for Australian conditions were used In addition, use was made of queue length data from a simulated no-passing zone, generated in a recent simulation study of traffic flows on grades (Taylor, 1971). The experimental data are presented in Tables 1 and 2. One of the major difficulties with the use of experimental bunch size data is the difference in queue discrimination criteria adopted. Buckley (1962), Underwood (1963). Miller (1963). and Daou (1966) have discussed queue discrimination methods. More recently Pahl and Sands (197 1) have suggested that the vehicle interaction phase can be described in terms of a time headway. Another problem is that of whether overtaking vehicles should be included in a bunch. The variability of tracking headways also causes difficulties. Queue discrimination is easier with simulated data. Both the position of each vehicle in its bunch, and whether or not the vehicle is delayed are known immediately. BUNCHING MODELS
The mathematical forms of the three bunch size distributions listed previously are: (a) the geometric distribution (1 - so.r’-i;
P, =
E(r) = _&
Table I. English/Swedish queue length data Frequency of a bunch of size r*
English
Swedish
5
6
7
I
2 4
5
1
2
3
4
2 3 4
184 467 392 536
38 117 87 130
19 32 29 41
12 9 20
6 5 7
5 6 7 8
229 451 333 414
55 107 91 136
18 30 25 38
16 13 6 15
6 2 10
1 2 3 4 5 6
119 99 121 112 107 175
29 29 32 23 16 25
19 6 9 13 5 6
6 2 10 3 7 1
5 4 1 0 -I 2
Data set 1
* Last entry is for bunches & r
1
-
7 3 6 8
8
7 3
4
5
2 7
5
12
(1)
Bunching models for rural traffic
3
Table 2. Australian queue length data Frequency of a bunch of size r* Data set 1 28 3 4 5 6 7 8
1
2
3
4
5
6
7
x21
d.o.f.:
P
296 127 66 50 67 46 51 62
44 53 5 8 6 11 12 12
16 29 0 3 2 2 9 3
6 21 1 0 0 2 3
2 5
2 4
1 6
4 6
1
0
2.80 18.1 0.16 0.36 0.43 0.43 3.70 0.09
1 1 1 0
0 2
1
1
1.330 2.057 1.111
2
1.290
: 3 2
1.184 I.403 1.632 I.388
* Last entry is for bunches Z r. t The x2 scores are for the Bore&Tanner distribution fitted to the data. 1 Degrees of freedom. p Set 2 is the onlv set for which the Borel-Tanner distribution provides a poor fit. The twoparameter Miller distribution provides a good fit for this set (see text and Appendix). For the 2-parameter Miller distribution fitted to this data l2 = 4.05 with 4 d.o.f.
(b) the Borel-Tanner P, =
(rat?_yr!
(b) for the one-parameter
distribution
1*f?-# ; E(r) = A;
and
(2)
I
l)!(r-l)!. l)!
(m + r +
’
E(r)=
‘Y$
(3)
(nt is not necessarily an integer).
COMPARISONS OF THE BUNCH DISTRIBUTIONS
SIZE
A comparison between the three common bunch size distributions can be made by considering the behaviour of each distribution over the range of P (mean bunch size). Bunches are referred to as “small” if of size r = 1 or 2. (A vehicle travelling by itself is considered to be a bunch of size r = 1.) “Intermediate” bunches are of size 3-6, while large bunches are of size r > 6 (in theory r is unlimited). The geometric distribution predicts smaller frequencies of both small and large bunches than either of the other distributions. The Miller distribution has the longest tail of the three for a given mean and so predicts smaller frequencies of bunches of intermediate size. Most of the available experimental data is for f < 1.6, and this has greatly influenced the acceptance of particular queue length distributions, For instance, if the proportion of single vehicles bunches (p,,l) is examined, then (a) for the Borel-Tanner distribution p, =e-‘he-’
as
P+X,;
Pl
Z-----d:
1
m+2
Miller distribution as
f--r co;
and
(5)
(c) for the geometric distribution
(c) the Miller distribution (one-parameter) p =(m+l)(m+
m+
(4)
Pl = (1 -a)+0
as
P-+co.
(6)
The three distributions thus show a wide divergence of behaviour as mean bunch size is increased. Figure 1 shows this divergence and also shows p1 plotted against P using both the simulated data and the experimental data from England Sweden and Australia. It can be seen that the Borel-Tanner and one-parameter Miller distributions tend to overpredict the proportion of single vehicle bunches for larger f (f > 2). The observed data seems to lie between the curves for the geometric and Borel-Tanner distributions of bunch sizes. In order to explore in more detail the behaviour of the three bunch size distributions, synthetic bunch size data was generated by the simulation of a length of road on which overtaking was impossible The road consisted of segments of both level road and grades. The length of each segment was variable and was specified in the simulation run. A level road was defined as one on which all unimpeded vehicles could maintain their desired travel speeds. Trafik arrivals were sampled from various headway distributions. The distributions used were: (i) negative exponential distribution; (ii) displaced negative exponential distribution; and (iii) Borel-Tanner bunches with a constant headway(r) between bunched vehicles and inter-bunch headways consisting of r plus a negative exponential headway.
M. A. P. TAYLOR.A. J. MILLERand K. W. OCUEN
4
h 3’1
I
I
2
I
3
1
L
I
I
4 5 6 Mean bunch size,
7-
IO
7
of
Fig. I. Probability singlevehicle bunches (p,) against mean bunch size (P): (I) Predicted curve for oneparameter Miller dist; (2) Predicted curve for Borel-Tanner dist; (3) Predicted curve for geometric dist.
Vehicle free (desired) speeds were sampled from a gamma distribution with a coefficient of variation of @lg. For the case of bunched arrivals a bunch speed was generated in the following manner. For a bunch of n vehicles, n free speeds were generated and the slowest of the n speeds became the arrival speed of the bunch. All vehicles in a bunch were assumed to travel at the same speed but each vehicle also had its own desired speed. The simulation models have been fully described by Taylor (1971). Eight sets of synthetic bunch size data were generated for use in the present study. These represented noovertaking zones varying in length from O-20 to 3.22 km, and average bunch sizes of from 1.245 to 6.728 vehicles. The simulated data thus covered a wide range of traffic situations (Table 3). The Borel-Tanner, geometric and one-parameter Miller distributions were each fitted to the synthetic bunch size data. The parameters of the distributions
were determined using maximum likelihood techniques; these are described in the Appendix. The distributions thus obtained were checked for goodness-of-fit to the eight sets of synthetic data using a chi-squared test. Table 3 shows the chi-squared scores which were obtained. These scores suggest that the geometric distribution provides a reasonable fit for bunches of vehicles leaving a no-passing zone of length X when X z- 1.6 km. On the other hand the BorelTanner distribution is acceptable for X < 04 km. These results appear to be true for one-way (single lane) traffic flows in the range of 7.5-800 vehicles/hr (Taylor, 1971). Mean desired speeds for the synthetic traffic were in the range 68-80 km/hr with a coefficient of variation of 0.18. For J > 3 the fits are not very good but this is tempered because of the small sample sizes and the occasional occurrence of very long queues. The Borel-Tanner distribution has been used to represent bunches of vehicles on a two-lane .road with
Table 3. Chi-squared scores for some simulated data Chi-squared values for* Data set number
Road length X (km)
Mean bunch size (I)
1 2 3 4 S 6 7 8
3.22 1.61 1.61 0.80 080 @40 0.40 0.20
1.622 6.7’8 1.317 1.245 2.648 2.744 1.617 I.539
Totals
Miller Borel-Tanner 7.5 (3) 95.6 (9) 2.4 (2) 1.9 (I) 16.2 (8) 23.8 (9) 6.5 (5) 2.315) 156.2 (42)
* Number of degrees of freedom shown in parentheses.
Geometric 2.6 (3) 11.7 (13) 1.8 (2) 0.2 (2) 29.2 (7) 52.6 (8) 15.5 (4) 55.2 (4) 171.7 (43)
l-parameter 20.4 (5) 158.3 (7) 9.4 (4) 5.9 (3) 45.6 (8) 42.9 (9) 19.0(6) 7.4 (6) 338.9 (48)
2-parameter 3.6 (3) -1.9(II) 0.8 (3) 0.2 (2) 11.1 (7) 7.3 (8) 5.1 (-I) 4.1 (1) JO.6 141b
Bunching models for rural traffic
5
limited overtaking opportunities (Miller, 1963). It and 111+ s + 1 should be expected that short lengths of no-passing E(r) = . (7) zone can be related to a two-lane road on whieh 111 limited overtaking can occur. For example, the limited By fitting two-.parameters (e.g. sample p1 and P) it is (equilibrium) overtaking road may be pictured as conpossible to match the theoretical distribution to sistingofalternate sections on which overtaking cannot observed data in a more flexible manner than for the occur followed by a short length on which wholesale one-parameter distributions described above, The twoovertaking is possible. The lengths of no-passing zone parameter distribution thus provides a much closer fit would be both static (e.g. sections of road between to the data as is shown in Table 3 for the simulated sharp curves, bends, crests and dips) and dynamic (the data described in the previous section. Moreover, the section of road between bunches travelling in opposing distribution is shown in the Appendix to fit the simudirections and visible to each other). The relevant geolated data over the range of P for all X. metric design parameter in the static case is the overIn the original derivation (Miller, 1961) the parataking sight distance. A vehicle travelling at the design meter was found to be practically zero for the data speed on a road where overtaking is generally possible tested and so the parameter was excluded; this prowill not be able to overtake safely if the sight distance duced the one-parameter distribution described above. falls below the overtaking sight distance. Values of However results for the simulated data and the Engovertaking sight distance are of the same order as the lish/Swedish experimental data suggested that s may length of road in which the Borel-Tanner distribution be significant. particularly for large mean bunch sizes. provides a good fit to queue length data. For example, Table4 provides an interesting comparison between the National Association of Australian State Road the Borel-Tanner and two-parameter Miller distribuAuthorities (1970) has set overtaking sight distances tions based on this data. The values of the parameters varying from 115 m at 32 km/hr to 754 m at 96 km/hr. used were maximum likelihood estimators and the Thus it is not surprising that the Borel-Tanner distritable lists chi-squared scores for each distribution. The bution fits queue length data from short sections of noformulation of the maximum likelihood parameters is passing zone. given in the Appendix. The one-parameter Miller distribution was also fitThe differences between the fits obtained for the data ted to the data. From Table 3 it can be seen that the from the two sources may be due to the different queue distribution provides a poor fit to the data. This result is not unexpected as the one-parameter Miller distriTable 4. Comparisons of two queue length distributions fitted to some European data bution tends to over-estimate the proportions of small bunches (I = 1 or 2). For example, from equation (5) 2-parameter the minimum value of p1 was shown to be 0.5 for the Bore&Tanner Miller distribution. With this restriction the one-parameter Miller distribution cannot adequately describe do.f. Data set d.o.f.* x2 %? observed bunch size data sets with large mean bunch English 1 7.0 2 6.4 2 sizes (P > 15).
THE TWO PARAMETER
2 3 4 5 6 7 8
4.8 0.7 11.0 4.6 3.2 6.9 4.4
3 2 3 4 4 3 4
4.6 2.8 12.0 8.3 1.0 6.4 1.8
4 3 3 4 4 3 3
1 2 3 4 5 6
7.5 2.3 6.4 1.8 4.8 1.2
5 2 4 2 2 2
2.2 1.2 4.9 24 5.2 0.5
3 3 2 2 2
Totals English Swedish
42.6 24.2
25 17
43.3 16.5
27 14
Overall
66.8
42
59.8
41
MILLER
DISTRIBUTION On the basis of the above empirical evidence, it can be concluded that the three common bunch size distributions have some validity. However, none of the distributions provides a satisfactory fit to the data over the full range of average bunch sizes, f. A more general distribution which can provide a good fit for the whole range off is obtained by reverting to the original form of the Miller distribution (Miller, 1961) which is a twoparameter distribution:
p = (m + l)(s + r - l)!(m + s + l)! I
s!(nr + s + r + l)!
Swedish
+ Degrees of freedom.
M. A. P. TAYLOR, A. J. MILLERand K. W. OGDEN
6
discrimination criteria used For the English data, bunches were assessed subjectively. The assessment of the Swedish data was based on a maximum tracking headway of 8 set and maximum speed differentials between successive vehicles of + 10 km/hr and -5 km/hr. The speed differentials effectively separate vehicles engaged in overtaking into a separate bunch from the vehicles being overtaken. Some further data was gathered as part of a study of bunching and delays on grades (Taylor, 1971). This data is presented in Table 2. The data is for bunches both at the crest of the grade for selected points along the grade. One sample (Set 2) showed significant bunching with a mean bunch size of slightly greater than 2-O.This was the only sample which was not well represented by the Borel-Tanner distribution. The two-parameter Miller distribution fitted this data well. For the other data there was little difference between the two distributions. Bunches were discriminated according to two sets of criteria. For data gathered at the top of grades bunches were discriminated on the basis of headways of 8 set or more. For data from lower reaches of the grades discrimination was based on headways of 5 set or more and a speed differential of +8 km/hr. The different criteria were adopted because: (a) many slow moving vehicles were bunched at the top of the grades but generated long tracking headways because of their speed; and (b) vehicles overtaking bunches at the foot of the grade needed to be considered separately from those bunches. The even numbered data sets (Table 2) are from the top of grades. Sets 7 and 8 are from data collected at two locations on a grade with climbing lanes.
DETERMINATIOKOF m AND s PARAMETERS For the two-parameter Miller distribution lowing relations hold: J?l+ 1 5=tn+s+2 where 5 is the expected proportion bunches and m+s+l cc= 111
the fol(8)
of single vehicle (9)
where p is the expected mean bunch size. Now both 4 and p are constrained to certain ranges of permissible values These ranges are : O<<
(10)
and fi>
1.
(11)
Equations (10) and (11) then detine permissible regions for m and s. These regions are: (i) (m>O,s>--1) and (ii) (m < -1, s G 1). Note that it is possible to have negative values of both m and s, and that zero values of s can also occur, as long as the pair of values fall inside one of the permissible regions. To determine values of the parameters given p and 5, equations (8) and (9) can be rearranged to solve for mands.
(12) s=+-1)-l.
(13)
So far the only constraint upon the m and s parameters is that they fall in the permissible regions above. However, by reference to Fig. 1 some further inferences can be drawn about likely pairs (m, s) for particular values of p and t, for single lane traffic on a twolane two-way road The graph suggests that the value of < is dependent to some extent on the value of b and that i’ lies between the values producted by the geometric distribution (for which 5 = l/p) and the BorelTanner distribution {for which 5 = exp[ -(p - 1)/p]}. That is: 1 lc:Cexp
(p - 1) I --’ n I
Mean bunch sizes can be estimated from methods described by Miller (1963, 1965) and Taylor (1971) for various traffic and highway factors. The range of the proportion of single vehicle bunches is then defined and estimates of m and s parameters can be made. Taylor (1971) has compared maximum likelihood estimates (m, s) and estimates made from sample (?, pl) for both experimental and synthetic data. The introduction of a two-parameter distribution of bunch sizes for traffic on a two-lane road on which overtaking is either restricted or prohibited introduces yet another parameter to be known or estimated in order to describe the traffic flow. This parameter involves the proportion of bunches of one vehicle in the traffic. However, the weight of experimental evidence suggests that this proportion (0 is not a single-valued function of mean bunch size b) and is dependent upon other traffic and highway factors. On this basis estimates of t cannot be made from $ and thus 5 has to be determined separately. The two-parameter Miller distribution has great flexibility and can be calibrated
Bunching models for rural tragic
to reproduce a wide range of bunch distribution for road traffic flow. which the one-pa&meter distributions cannot do. HEADWAYS
BETWEEN
BUNCHES
The usual assumption for the distribution of headways between bunches is that they follow the form of a negative exponential distribution. This has been used by Miller (1961. 1963) for bunching under limited (equilibrium) overtaking conditions. Cowan (1971) has found the distribution of inter-bunch headways on exit from a no-passing zone. His distribution is not an exponential. However it can be approximated by an exponential distribution, and the approximation is closer as traffic volume is increased. Thus even under this extreme condition the negative exponential is a reasonable distribution to represent inter-bunch headways Experimental headway distributions were measured during collection of the data in Table 2. Figure 2 shows the sample frequencies of headways, together with an exponential distribution based on the sample average headway. A good agreement between the observed fre-
quencies and the theoretical
distribution
was obtained.
7
of platoon sizes under some conditions. A more general distribution is the two-parameter Miller distribution. The major difficulty with the application of bunching models in road traffic flow is that knowledge of several parameters (e.g. traffic volume and mean bunch size) is required for use of the models. However methods of determining parameters. such as mean bunch size, in terms of traffic, highway and alignment factors are available. The following authors have suggested methods for estimation of mean bunch size. (i) Tanner (1961); (ii) Miller (1963, 1965); (iii) Underwood (1963); and (iv) Taylor (1971). With these methods bunching models may be used to provide greater insight into traffic flow on rural highways. In particular the two-parameter Miller distribution appears capable of representing bunching in trtic under a wide range of highway and traffic conditions. Acknowledgement-This paper is presented with the permission of the Chairman of the Country Roads Board of Victoria. Views expressed are those of tire authors, and not necessarily those of the organization they represent.
Inter_- bunch hwdwoy.
f set
Fig. 2. Inter-bunch headways for the data of Set 2, Table 2. The figure shows an exponential distribution based on the sample mean headway for comparison with the observed headways. CONCLUSIONS
REFERENCES
Bunching models provide a good representation of the flow of traffic on two-lane rural roads. In this paper some of the common
models for the bunching
of traffic
have been examined in the light of some experimental and synthetic data covering a wide range of flow conditions A model combining negative exponential interbunch headways with a bunch size distribution provides a good representation
Tanner distribution
of rural traffic. The Borel-
gives a reasonable representation
Ashton H. R., Buckley D. J. and Miller A. J. (1968) Some aspects of capacity and queueing in the vicinity of slow vehicles on a rural two lane road. Proc. A.R.R.B. 4. 595-
612. Buckley D. J. (1962) Road traffic headway distributions Proc. A.R.R.B.
1,153-183.
Cowan R. J. (1971) A road with no overtaking. Aust. J. qf Sm. 13,94-l 16. D aou A. (1966) On flow within platoons. J. Amt. Rd Res.
2,4-13.
8
M. A. P.
TAYLOR. A.
J.
MILLER
Greenberg H. (1966) The log-normal distribution of headways. J. Aust.Rd Res. 2, 14-18. Miller A. J. (1961) A queueing model for road traffic flow. J. Roy. Stats. Sac., Series B U,64-75.
Miller A. J. (1963) An analysis of bunching in two-lane rural
A.R.R.B. 6.286301.
Underwood R. T. (1963) Traffic flow and bunching. J. Aust. Rd Rrs 1.8-25.
N.A.A.S.R.A. (19701Policyfor Roads. 4th Ed.. Sydney. LIST r r P, J e m,s X EO 5 p 5
the Geometric
Design of‘Rurul
Let
L = In (IJ
then
(‘4.2) r=
1
(a) For the Borel-Tanner -3 -1 (rre Y e-’ PI = r.I
distribution
1
J-1 I., = $J f, (r - i)ln(a) - c(r - In 7 I= 1
1
+ M’.
If L is maximized by taking the partial derivative with respect to n and equating to zero, then
&=&’ J
where P is the mean bunch size of the observed sample. Thus for the Borel-Tanner distribution the sample mean is the maximum likelihood estimator of the population mean. A similar result exists for the geometric distribution. However the Miller distribution is less well behaved. (b) For the two parameter Miller distribution: p = (m + l)(m + s + l)!(s + r - l)!; I s!(m + s + r + l)!
OF SYMBOLS
size of platoon (a single vehicle is r = 1) sample mean bunch size probability of a bunch size r bunching parameter in Borel-Tanner or geo,metric distributions base of Napierian logarithms bunching parameters in Miller distributions length of a no-passing zone (km) denotes the expected value of the variable in the parentheses expected proportion of bunches of 1 vehicle expected mean bunch size headway between bunched (tracking) vehicles.
OGDEN
where M is some arbitrary constant and the @,) are the set of expected probabilities of bunches of size r. The equation is simply a product of the probabilities of occurrence of the bunches.
traffic. Ops. Res. 11,235-247. Miller A. J. (1967) Queueing in rural traffic. 3rd International Symp. on the Theory of Traffic Flow, New York (1965) Published as Vehicular Traffic Science, pp. 122-137, Elsevier, New York. Miller A. J. (1968). Studies of speeds and delays on rural ;oads in Australia, 9th International Study Week in Traffic and Safety Engng., Munich. Miller A. J. and Pretty R. L. (1968) Overtaking on two-lane rural roads. Proc. A.R.R.B. 4 (I), 582-591. Moskowitz K. and Newman L. (1965) Effects of grades on service volumes. H.R.B. Record 99, 224-243. Pahl J. and Sands T. (1971) Vehicle interaction criteria from time series measurements. Trans. Sci. $403-417. Tanner J. C. (1953) A problem of interference between two queues. Biometrika 40, 58-69. Tanner J. C. (1961) Delays on a two-lane road. J. R. Stats. Sot. Series B X%,38-63. Taylor M. A. P. (1971) A simulation study of traffic flow on grades. Thesis for M.Eng.Sc., Monash Univ., Clayton, Vie., Australia (unpublished). Troutbeck R. J., &wed N. and Miller A. J. (1972) Overtaking sight-distances on a two-lane rural road. Pror
and K. W.
P=
m+s+l m
; <=
m-f
and
1
m+s+2’
The log likelihood function is now L = i L ln(m + 1) - i ln(m + s + k + 1) ,=I I k=,
k=1
+ i
In(s + k - 1) - In(s)
1.
It is necessary to equate both dL,‘am and dL/b zero : 2L
z
= 0
to
(A.4)
and Esfimatiny
hunch size
tliscrihutiorl purametrrs
Maximum likelihood methods have been used to find estimators of the parameters in the bunch size distribution. For a given set v,) of the observed frequencies of quelles of length r. (r = I. 2.. x ). The general likelihood function is given by
(A.3 An iterative technique is required to solve equations (A.3) and (A.4) simultaneously. The maximum Iikelihood estimators r&and S are not related simply to the sample mean bunch size. The initial estimates of m and s can be taken from the sample P and pl.
Bunching models for rural traffic Table 5 Predicted Bore-Tanner frequency
Predicted 2 par. Miller frequency
Bunch size
Observed frequency
1
127 53 29 21 5 4
148 45 21 II 7
127 59 29 15 8
1
7
I
5
7
6
2 3 4 5 6 7
>8
The estimation of the parameter nt in the one-parameter Miller distribution is similar to the above. The European data for which chi-squared scores were given earlier is presented in Table 1. Some Australian data is given in Table 2. There is little difference between the Bore]-Tanner and the two-parameter Miller distributions for the Australian data except for data set 2. The two-parameter Miller distribution fits this set well, while the Borel-Tanner distribution provides a poor fit. This is a significant result and agrees closely with the result found using the simulated data.
The queue length frequences in Table 5 were predieted for data set 2 of Table 2. Table 3 summarizes chi-squared scores for distribution fitted to this data. A significant result is that the geometric and Bore]-Tanner distribution fit the overall data poorly. They are however reasonable for limited ranges of the data. The geometric distribution is reasonable for data from long sections of no-passing zone and for large F. The Bore]-Tanner distribution is satisfactory for small X ( < 0.4 km) and F. On the other hand the two-parameter Miller distribution covers the ranges of X and F satisfactorily.
ResumP-Sur les routes bidirectionelles a deux voies, les occasions de dtpassement sont souvent limit&es par le trafic et les caracteristiques de la voirie en sorte que les vehicules peuvent itre obliges d’attendre derriere les vehicules plus lents avant que le dtpassement devienne possible. Pour expliquer ce regroupement ou ce tassement de la circulation plusieurs modkles de regroupement ont ete proposes. Un modele de ce type, qui a Cte utilis6, est la distribution de Borel-Tanner des tailles de groupes de vehicules. Cet article analyse ce modP1e et les autres, et les compare a l’aide de donntes concernant les tailles des groupes de vihicules a la fois experimentales et simultes. De cette comparaison on tire la suggestion qu’un modtle plus global que le modele Borel-Tanner est constitui par la distribution des tailles de groupes de vehicules. a deux parametres de Miller; ce modtle peut &tre ajuste pour un eventail assez large de conditions de circulation. Zusammenfassung-Auf zweisputigen Strassen mit Gegenverkehr werden iiberholmiiglichkeiten haufig durch verkehrliche und strassenbauliche Gegebenheiten beschnitten, so dass Fahrzeuge gezwungen sein konnen, hinter langsameren Fahrzeugen zu verbleiben, ehe der iiberholvorgang beginnt. Zur Beschreibund der daraus resultierenden Kolonnenbildung sind verschiedene Modelle entwickelt worden. Eines dieser Modelle, und zwar das von Bore1 und Tanner. befasst sich mit der Verteilung der Kolonnenllnge. In der vorliegenden Abhandlung wird es untersucht und mit anderen AnsStzen verglichen, wobei sowohl auf experimentell gewonnene als such auf simulierte Werte fur die Kolonnenllnge zuriickgegriffen wird. Aus dem Vergleich ergibt sich, dass die von Miller entwickelte zweiparametrische Verteilung der Kolonnenlange allgemeiner verwendbar ist als das Model1 von Bore1 und Tanner und einer grossen Spannbreite von Verkehrsbedingungen angepasst werden kann.