- Email: [email protected]
Comments on flow characteristics on acceleration lanes
owl-26L?7/87 u.aI+ .a0 0 1987 Perpmon Journals Ltd.
Transpn. Res:A Vol. 21A. No. 1, PP. 39-46, 1987 Printed in Great Britain.
COMMENTS
ON FLOW CHARACTERISTICS ACCELERATION LANES
ON
ABISHAI POLUS and MOSHE LIVNEH Department of Civil Engineering and Transportation Research Institute, Tech&n-Israel Institute of Technology, Haifa 32OW,Israel (Received 18 January 1985; in revisedform 10 April 1986)
Abstract-This paper deals with driver behavior while travelling on and merging from acceleration lanes. Two possible groups of drivers were identified: drivers who always perform the merging maneuver during the second part of the acceleration lane, regardless of whether an appropriate gap or lag was available to them previously. Three components of the aggregated delay for the merging process were suggested and evaluated. A method of estimating the random delay and travel time on acceleration lane was proposed and evaluated against an aggregated empirical data obtained on three freeway acceleration lanes. An evaluation of the contribution of the ramp volume to the traffic delay was also performed and dkcussed and a graph which may be of practical use for road and traffic engineers in assessing the expected influence of various ramp and freeway volume combinations is presented. INTRODUCTION AND BACKGROUND
line at which the size of rejected lags was determined. The study analyzed only lags in order to overcome a potential bias that may be caused by the overrepresentation of cautious drivers in the population of all gaps considered. The first investigator to point out this possible bias probably was Raff (1950) in a study of volume warrants for urban stop signs; it was later quantified by Ashworth (196f3) in England and discussed further by Miller (1971) in Australia in an evaluation of estimators of gapacceptance parameters. In gap-acceptance and merging studies, it is normally assumed that a driver on a ramp measures each lag, c(or gap), in the right lane of the through road and either merges-i.e. accepts the lag if c > r-or continues the trip along the acceleration 1ane-i.e. rejects the lag if I < t, where t is assumed to be the driver’s critical lag for that decision. Raff (1950) defined the critical lag-a fixed parameter for the entire population of drivers-as the size of a lag having the property that the number of accepted lags shorter than the critical lag is equal to the number of rejected lags longer than the critical lag. The principal use of this parameter is to simplify the computation of delay values by assuming that all intervals shorter than the critical value are rejected, and all longer intervals accepted. A more complete description of the delays to vehicles would be obtained if the gap distribution were incorporated in the analysis, rather than the constant critical gap or lag. Blunden, Clissold and Fisher (1%2) evaluated the delay in crossing and turning maneuvers under the assumption that the critical lag distribution was of the Erlang type and suggested that the mean of the acceptance-gap distribution was an appropriate measure of the critical gap. Drew (1967) also fitted in Erlang (gamma) distribution to data of observed gaps; he attributed its conceptional
The
study of driver behavior while travelling on and merging from acceleration lanes holds great importance for both theoretical and practical investigations: among others, traffic flow and simulation studies; geometric design, such as determination of the required lane length; and driver performance in relation to elements of freeway surveillance and control or road safety. Merging is defined simply as the absorption of the stream of ramp vehicles from the acceleration lane to the freeway shoulder lane during and following a gap-acceptance process. This process, however, is very complex. The factors involved consist of both internal factors, such as driver attitude and vehicle characteristics, and external environmental parameters, which may include speed and flow on the main stream, speed of entering vehicles and percentage of commercial vehicles. The purpose of the study reported on in this paper was to investigate and evaluate the characteristics of flow on and merging from acceleration lanes in order to shed more light on driver behavior and the gap acceptance process. Such an evaluation, it was felt, may lead to design recommendations for the length of acceleration lanes, based on delay characteristics to merging vehicles. Details of the study itself, which was conducted by the Transportation Research Institute in Israel, were contained in a recent report by Livneh, Polus and Factor (1984). The first stage of the study, an analysis of vehicle-flow characteristics on four acceleration lanes, led to several conclusions, among them that acceleration rates were rather low and that the speed differences between merging and mainstream vehicles ranged between 10 and 15 km/h on the average. Lag-acceptance distribution was found to vary considerably with the location of a reference 39
40
ABISHAI
POLUSand MOSHELI~NEH
appeal as being due to its nonnegative restriction and inherent element of randomness, exemplified by its formulation from the Poisson distribution. Drew later calculated merging delay in terms of highway flow, critical gap and the Erlang constant and also compared observed average merging delay to theoretical merging delay. Investigation by Ashworth (1969) of the capacity of priority-type intersections with a nonuniform distribution of critical acceptance gaps showed a considerable reduction in the capacity of the minor road as a result of replacing the constant gap-acceptance step function with a distribution of critical gaps. Specific values evaluated for normal distribution demonstrated a capacity loss of about 30% at high majorroad volumes. The effect at lower road volumes, however, was found to be less significant. The present study did not concentrate on finding the exact distribution of critical lags. Rather, it made use of the critical value as defined by Raff (1950), and attempted to evaluate merging patterns for two main population groups of drivers and to estimate some measure of correction for the obtained values because of this nonuniform driver population.
MJ%RGINCFROM AN ACCELERATION LANE The process of merging from an acceleration lane was treated by Blumenfeld and Weiss (1971), who developed a theoretical model to describe vehicle flow and gap acceptance. In this model, vehicles on the acceleration lane are assumed to travel at a constant speed u, (Note Fig. 1.) The merging driver is assumed to continue moving along the acceleration lane until a suitable gap is found or to continue to the end of the lane, where the driver’s speed goes instantaneously to 0. The merge, it is assumed, occurs at some time t. Blumenfeld and Weiss defined the delay, D, as the relative time difference between a vehicle on the main stream and a merging vehicle. Thus
D = (1 - $1 = pt,
where all variables have been defined above, and l3=1-;. They further devised a formula that evaluated the total delay to a vehicle in the acceleration lane as the sum of two random variables: the time spent travelling on the lane and the delay while stopped. Numerical solutions to this formula enabled them to relate the probability of reaching the end of the acceleration lane without merging to the length of the lane. Figure 2, a graph based on the model developed by Blumenfeld and Weiss (1971), presents the probability of not merging as a function of travel time on the acceleration lane for a critical gap of 4 seconds and constant Al3 products. The product h$ was established to be constant in this graph, since as A (the volume on the freeway shoulder lane) increases, the speed V on the main road decreases, u being constant. It is assumed therefore, that a constant hf3 product may represent a large, although not full, spectrum of speed-volume relations for a given road. Other products may be expected to exist if different flow conditions occur (e.g., wet pavement, accident situations) or on another highways. Two points stand out at once in Fig, 2: the sharp decrease of the probability not to merge with an increase in travel time and the relatively low probabilities at time 0. This inadequate boundary condition is underscored if one assumes reasonable volumes and speeds on acceleration lanes and freeways in, say, suburban conditions. For example, assume the conditions at the Netanya site as reported in the Transportation Research Institute report (1984), whereby u and Vequal approximately 64 and 85 km/h, respectively, and h equals 600 vph in the freeway right lane. The result is that A * p equals, approximately, 150, which suggests (note Fig. 2) a probability of some 84% of merging at time 0, which mean at the merging-end point (note Fig. 1). The distribution of merging locations presented by Polus, Livneh and Factor (1984) shows, however, that in the context of real-life experiment this is not so. The
_
k+ -
Vohmuon
p
-
Romp
VOIUW
V
-
Spnd
on Fmway
v
-
S~rd
Fmwoy
Right
Right
on Accrlrmtion
MXELENATION
LANE
Lone
Lone LOM
Fig. 1. Schematic diagram of freeway lanes and acceleration lane.
41
Flow characteristics on acceleration lanes I
I
I
I
X - Volume en Freeway Right Lone
v - Speed on Acceleration V-Speed T - Critical
Lane
on Freeway Right Lane Gap
wu .g ._ a
x
e
a
0
2 TL- Travel
4
6
lime
on Acceleration
6
IO
Lane,(rec.)
Fig. 2. Probability of not merging as a function of travel time on acceleration lane.
difference, of course, stems from several assumptions of the Blumenfeld and Weiss model, such as that speed is constant on acceleration lanes and that drivers’ behavior is completely rational; that is, they accept the first available appropriate gap. The model does not consider the volume coming onto the freeway from the ramp, and therefore the analysis is suitable for low ramp volumes only. This paper will attempt to generalize the assumption of rational drivers by suggesting a method for adjusting the travel time on the acceleration lane. The influence of ramp volume on computed delay is also discussed. The assumption of constant speed along the acceleration lane is adopted for the analyses since it was found that this is a reasonable assumption; speed changes along well-designed lanes were found to be rather small (Polus, Livneh and Factor, 1984) and ranged between 1.2 and 10.5%. The process of merging from an acceleration lane is much more complex, however. One can treat this process in either an aggregated or a disaggregated way. In the first approach, some statistical summaries (mean, variance, etc.) of the two possible pop ulations of drivers should be specified: those who accept the first available appropriate gap or lag, and those who remain on the acceleration lane for reasons of convenience or acceleration and merge during the second half of the lane. This approach is discussed at length in a previous paper by Polus, Livneh and Factor (1985). In the disaggregate approach, the behavioral attributes of individual drivers should be examined and some probability functions to best represent overall behavior should be developed. COMMENTS ON DELAY CALCULATION ACCELERATION
FROM
LANES
Delays to vehicles during a gap-acceptance have been discussed by several researchers.
process Among
others were Blunden, Cl&old and Fisher (1962) and Major and Buckley (1962) in Australia; Drew (1%7), who studied gap-acceptance characteristics of freeway ramps in Texas; and Ashworth (1969)) in Britain, who analyzed the capacity of priority-type intersections with a tmmmiform distribution of critical acceptance gaps. In general, one can assume that the total delay to each driver on an acceleration lane is composed of three parameters: the geometric delay (g), the traffic delay (d) and some random delay (a). The geometric delay may be observed, if significant, and is the result of some necessity to travel at a lower speed on the acceleration lane beyond the merging-end point even if no traffic is present on the freeway shoulder lane. The traffic delay was first discussed by Adams (1936) for pedestrians waiting to cross a road. It was proved to be applicable to vehicles having a fixed critical gap, T, which accepts gaps in a major stream with a flow of A vehicles per unit time and was given as d, = $P
- XT - I),
where d, is the traffic delay under the above assumptions and all other parameters are as defined above. Note that gaps smaller than the critical gap are assumed to be rejected by all drivers, and longer gaps accepted. The random delay on acceleration lanes is more difficult to assess. It is caused by a tendency of some drivers to merge at a point along the second half of the acceleration lane (note Fig. 1) even though an appropriate gap or lag was previously available. This tendency was observed during the authors’ study of flow characteristics, and it was estimated that the percentage of such drivers may reach about JO%, depending on location, type and flow at the con-
ABISHAI POLUS
42
4.00 3 1: _
0
and MOSHELWNEH
p- Romp Volume A - Volumr on Freeway Right Lone T- Critical
Gap
200 400 600 000 1000 A - Volumr on Freewoy Right Lone , (vph)
Fig. 3. Calculated traffic delay for various critical gaps and freeway right-lane volumes.
cemed site. The reasons mentioned for this tendency were twofold: the need to accelerate in comfort and the search for some added convenience resulting from 8 late merge 8t 8 faster speed. A method Of estim8ting the needed 8djustment in such 8 parameter 8s critical 18g or gap was suggested and diScussed. It is of interest to note that an Australian study by Miller and Pretty (1%8) on overtaking on twolane roads assumed that some drivers behaved in 8 manner completely different from others. They found that 8 small proportion of drivers would not overtake, however large the gap in the oncoming traffic. This finding is perhaps analogous to the findings here that some drivers will not merge at the reference line (established for 8 determination of the rejected lag size), regardless of the gap or lag size available t0 them. The mean traflic delay, dp, may also be evaluated for vehicles arriving at random at the acceleration lane, with 8 mean rate of arrival, p, by adopting aspects of queueing theory discussed by Kendall (1950) and later by Major and,Buckley (1%2) and ‘. others. Thus,
where d, is the mean tmffic delay and the mean and variance of time spent at the head of the queue 8re d, and d, respectively. This equation is a standard result in queueing theory, for Poisson arrivals and general service times, assuming that service times are independent. In the c8se of merging, the service times are the times taken for drivers at the head of the queue to accept gaps in the freeway traffic lanes. For the independence condition, one must 8ssume
here that only one driver can merge in one g8p, which is probably 8 reasonable assumption for medium to heavy flow conditions on the freeway. This equation also considers the flow from the ramp when assessing vehicle traffic delay. Note that when p + 0, d, + d, is obtained. A plot of the calculated traffic delay for a fixed ramp volume and various critical gaps (or lags) T, and freeway right-lane volumes A, is presented in Fig. 3; one can note the increase in delay with an increase in T and A. A plot of the calculated traffic delay for a fixed critic81 g8p of four seconds and various main flow A or ramp p volumes is presented in Fig. 4. It can be seen that for low volumes on the right lane of the main road, the delay is almost not affected by the
6.00
I
I
I
1
I
I
1
/
I
/
1
p - Romp Volume Vokuoron Frrewoy
A -
T-
0
I
I
I
I
I
I
Gap
Critical
I
I
200 400 600 600 p-Romp Volume , (vph)
Fig. 4. Calculated delay for various ramp volumes and freeway right-lane volumes.
43
Flow characteristics on acceleration lanes
ramp volume, whereas delay is greatly influenced by it for higher main road volumes. For example, when the freeway right-lane volume is equal to 800 vph, the delay to merging vehicles almost doubles when ramp volume increases from 100 to 600 vph. One must keep in mind, therefore, that when estimating merging delay, it is important to consider the volume on the feeding ramp. This point is, of course, also true for calculations of merging probabilities, such as presented in Fig. 1. In order to estimate further the influence of ramp volume on the calculated delay, the following expression of y, the relationship between the two equations of delay, with and without consideration of the ramp volume, was expressed as
’
d C-r d’*
where y may be considered as a measure of rampvolume influenced and dq and d, were defined above. Figure 5 presents the relationship between freeway right-lane volume, A, and ramp volume, p, for these equal values of y: 1.01, 1.02, 1.05 and 1.10. The curves may serve as a guide for geometric designers or traffic planners who need to determine the values of both A and p for which a predetermined y value is established. Thus, for values of A between 100 and 1000 vph, ramp volumes may range approximately from 115 to 40 vph, respectively, if the difference between dq and d, is not to exceed 5%. An increase in y will allow, of course, a respective increase in combinations of ramp and freeway right-lane volumes. One may now note that the geometric delay, g, which represents the added delay due to the restricting geometry, sometimes coincides with the traffic delay d,. This occurs since driving at a low speed on
p-Ramp fig. 5.
the acceleration lane may-come about, due to both adverse geometry (e.g., steep slope or sharp curve) and oncoming freeway traffic. Hence, if one assumes a constant speed on the acceleration lane, it follows that
where all the parameters have been defined above. For the four sites studied, the ramp volumes were rather small, in the order of less than 120 vph or 0.033 vps. It can be assumed, therefore, that dq + d,, as noted above. However, since the freeway rightlane volume relative to the driver moving in the acceleration lane is A@, the d, equation is adjusted accordingly and the merging delay, d, is given as
APT - 11,
,=&e=
where T is the fixed critical gap or lag, depending on the particular case. Hence the travel time on the acceleration lane from the merging-end point to the moment of merge is given by
dsm
tl = -,
P
where all the parameters have been defined above. The calculated travel time and merging delay according to these formulas is presented in Table 1 along with the volume and speed data from four acceleration lanes where vehicle-flow characteristics were observed. The speed u was the average speed along the acceleration lane assumed to be linearly distributed between the initial speed (at the mergingend point, note Fig. 1) and the final speed at the
Volume
, (vph)
Relationship between freeway right-lane volume and ramp volume for equal values of y.
63.6 63.1 63.7 56.9
Netanya Yavneh Geha Haifa
0.1 0.0833 0.0625 0.075
(lcmv/h)
Speed on acceleration lane
Site
Volume on freeway right lane
83.3 74.8 73.1 67.8
(ltmv/h)
Speed on freeway right lane
0.236 0.156 0.128 0.161
B 3.95 3.65 3.60 3.60
Critical Ia8 T W) 4.83 . 1O-2 1.37 * lo-’ 0.67 . 1O-2 1.27. 1O-2
Estimated time at head of queue d, &c)
Table 1. Calculated travel time and merging delay for four acceleration lanes
20.47 8.79 5.23 7.94
* 1O-2 . W2 . lo-* . 10-Z
Calculated merging delay d (sZ)
0.867 0.563 0.408 0.493
Calculated travel times f &c)
45
Flow characteristics on acceleration lanes instance of the actual merge. One should note, however, that the changes between the initial and final speeds along the acceleration lanes studied were rather small (up to 10.5% at the Netania site, 1.2% at Yavneh, 3.5% at Geha and - 6.3% at Haifa site). The speed V was calculated as the average speed on the freeway shoulder lane at a point opposite the merging-end point. EVALUATIONOF OBSERVEDDELAY From the discussion above, it follows that in order to compare observed merging delay and calculated traffic delay, one has to perform two adjustments: one is to estimate the random delay and the second, if applicable, is to adjust for ramp volume. Random delay is caused by various tendencies of the random driver (e.g. lack of awareness, hesitance, habits) to merge along the second half of the acceleration lane even if no traffic is on-coming on the freeway shoulder lane. Since, as mentioned above, the ramp volumes at the four sites in this study were rather small, the second adjustment was ignored for the evaluation of observed delay. The random delays, however, were observed to be significant, and therefore an estimate of their magnitude was sought. Observation of flow characteristics on the four acceleration lanes studied showed that some merging maneuvers occurred when no traffic was present for an extended distance upstream on the highway outside lane. It was decided to use these maneuvers to estimate the proportion of random delay by estimating the contribution of “late merging” vehicles to the average total delay. For this estimation, a parameter a was defined as
where a = correction factor for an estimation of random delay, & = number of drivers who do not merge despite no traffic on freeway right lane, & = number of drivers who accept the first appropriate gap (rational drivers). Note that 5, is estimated from merging maneuvers which occur when there is no traffic on the freeway shoulder lane, a situation which may frequently occur on suburban freeways during off-peak hours. The estimated value of the average random delay, t,, is no+ estimated as t. = t, - a,
where r, is the observed (empirical) travel time the merging-end until the merge into the right of the highixray. The observed average travel may now be adjusted in order to be compared the calculated travel time, t, = t,(l - a),
where t, is the adjusted empirical travel time, and t, and a were defined above. One should also note that fc = t, + t,. In other words, the observed travel time is composed of the adjusted travel time and the random delay. It is important to note here that it is not claimed that both t and f, represent the true values of the random delay and travel time, respectively; rather, it is suggested that these parameters be estimated according to the procedure outlined in this paper. This approach is believed to be adequate for practical issues involving evaluation of delay or travel time on acceleration lanes. In order to compare the average values of the adjusted observed travel time and the calculated travel time, as well as to estimate the values of the random delay, Table 2 was constructed with data from three acceleration lanes where vehicle delays were observed; no delay observations could be conducted at the Geha site due to the unique method of data collection (aerial video-photography from an helicopter). By observing the last columns of Table 1 and Table 2, one can notice that there is a reasonable agreement between the calculated travel time and the adjusted observed travel time at Yavneh and Haifa sites. This suggests that the proposal method of adjustment and estimation of the random delay may represent a valid approach. It is acknowledged, however, that a much larger sample is needed to further strengthen the findings regarding the proposed method. A second noticeable finding is the relatively large disagreement between calculated and adjusted observed travel time at the Netanya site. On this lane, however, the greatest violations of a basic assumg tion of the model (which suggests that the speed u remains constant) were found. Perhaps an overestimation of u was adopted which led to an underestimation of the calculated travel time.
Table 2. Observed delav. estimated random delav and adiusted travel time for three acceleration lanes
Site
Observed delay G (set)
Netanya Yavneh Haifa
4.33 2.14 1.68
from lane time with
Correction factor u
Estimated random delay 1. (set)
Adjusted Observed travel time L (=I
0.644 0.697 0.650
2.79 1.49 1.09
1.54 0.65 0.59
46
ABISHAI
Porus and MOSHELWNEH
A third prominent finding (note Table 2) is that random delays are considerably higher than the adjusted observed delay, caused by a lack of appropriate gaps in the traffic on the freeway right lane. This finding is believed to be due to the relatively
low volumes at the three sites, which resulted in an extended, more comfortable stay on the acceleration lane before the actual merging maneuver.
evaluation of the various components of delay, and particularly random delay; and (b) an examination of the behavioral attributes of individual drivers and the identification of parameters leading to various merging maneuvers. Acknowledgement-me authors wish to express their deep gratitude to the Israel Public Works Department, Ministry of Building & Housing, which commissioned and sponsored this research.
CONCLUSIONS AND RECOMMENDATiONS The merging from an acceleration lane is a rather complex process. Two possible groups of drivers can be identified: drivers who accept the first possible gap as soon as it becomes available, and who may be termed rational drivers; and drivers who always perform the merging maneuver during the second part of the acceleration lane, regardless of whether an appropriate gap or lag was available to them previously. Three components of the aggregated delay for the merging process are suggested: geometric delay, which is caused, at times, by the necessity to travel at a lower speed along the acceleration lane even if no traffic is present; traffic delay, caused by the lack of appropriate gaps or lags owing to traffic on the right lane of the freeway; and random delay, which is caused by the desire to accelerate in comfort and to perform a late merge. A method of estimating the third type, random delay, was proposed and evaluated against the aggregated data of empirical results obtained on three sites. An evaluation of the contribution of the ramp volume to the traffic delay was also performed and discussed, and a graph is presented which may be of practical use for road and traffic designers in assessing the expected influence of various ramp and freeway volume combinations. It is recognized that the comments and proposals made in this paper do little more than illuminate the fringes of the complex phenomenon of gap acceptance from acceleration lanes; however, it is believed that they may serve as a basis for further research. Proposed studies in this connection are (a) a further
REFERENCES
Adams W. F. (1936) Road traffic considered as a random time series. J. Inst. Civ. Engrs 4,121. Ashworth R. (1968) A note on the selection of gap acceptance criteria for traffic simulation studies. Tranrpn Res. 2, 171-175. Ashworth R. (1969) The capacity of priority-type intersections with a non-uniform distribution of critical acceptance gaps. Tivnspn Res. 3,273-278. Blumenfeld D. E. and Weiss G. H. (1971) Merging from an acceleration Jane. ltruupn Sci. S(2). 161-168. Bhmden W. R., Chssoid C. M. and Fisher R. B. (1962) Distribution of acceptance gaps for crossing and turning maneuvers. Rot. Austral. Road Res. Board 101. . I. 188205. Drew D. R. (1967) Gap acceptance characteristics for rampfreeway surveillance and control. Hiahwuv _ - Ru. Ret: l57, l&-135. Kendall D. G. (1931) Some problems in the theory of queues. J. Roy. Statist. Sot. Ser. B 13, 151. Livneh M., Pohrs A. and Factor J. (1984) Analysis of vehicle flow characteristics on acceleration lanes. Research Report 8640, lfantportatioa Research Institute, Technion-Israel Institute of Technology, Haifa. Major N. G. and Buckky D. J. (1962) Entry to a traffic stream. Proc. Austrai. Road Res. Board l(l), 206428. Miller A. J. and Pretty R. L. (1%8) Overtaking on twolane rural roads. Proc. Austral. Road Res. Board 4(l). 58-594. Miller A. J. (1971) Nine estimators of gap-acceptance parameters. In Traffic Flow and Transoortation, Proc. 5th Int. Symp. on th;’ Theory of Traffic ‘Flow and Transportation, pp. 215-235. Polus A., Livneh M. and Factor J. (1985) Vehicle flow characteristics on acceleration lanes. Transoortation Engineering Journal of ASCE, Ul(6). 5954%. FUff M. S. (1950) A Volume Warrant for Urban Stop Signs. Eno Foundation, Saugatuck, CN.
Transpn. Res:A Vol. 21A. No. 1, PP. 39-46, 1987 Printed in Great Britain.
COMMENTS
ON FLOW CHARACTERISTICS ACCELERATION LANES
ON
ABISHAI POLUS and MOSHE LIVNEH Department of Civil Engineering and Transportation Research Institute, Tech&n-Israel Institute of Technology, Haifa 32OW,Israel (Received 18 January 1985; in revisedform 10 April 1986)
Abstract-This paper deals with driver behavior while travelling on and merging from acceleration lanes. Two possible groups of drivers were identified: drivers who always perform the merging maneuver during the second part of the acceleration lane, regardless of whether an appropriate gap or lag was available to them previously. Three components of the aggregated delay for the merging process were suggested and evaluated. A method of estimating the random delay and travel time on acceleration lane was proposed and evaluated against an aggregated empirical data obtained on three freeway acceleration lanes. An evaluation of the contribution of the ramp volume to the traffic delay was also performed and dkcussed and a graph which may be of practical use for road and traffic engineers in assessing the expected influence of various ramp and freeway volume combinations is presented. INTRODUCTION AND BACKGROUND
line at which the size of rejected lags was determined. The study analyzed only lags in order to overcome a potential bias that may be caused by the overrepresentation of cautious drivers in the population of all gaps considered. The first investigator to point out this possible bias probably was Raff (1950) in a study of volume warrants for urban stop signs; it was later quantified by Ashworth (196f3) in England and discussed further by Miller (1971) in Australia in an evaluation of estimators of gapacceptance parameters. In gap-acceptance and merging studies, it is normally assumed that a driver on a ramp measures each lag, c(or gap), in the right lane of the through road and either merges-i.e. accepts the lag if c > r-or continues the trip along the acceleration 1ane-i.e. rejects the lag if I < t, where t is assumed to be the driver’s critical lag for that decision. Raff (1950) defined the critical lag-a fixed parameter for the entire population of drivers-as the size of a lag having the property that the number of accepted lags shorter than the critical lag is equal to the number of rejected lags longer than the critical lag. The principal use of this parameter is to simplify the computation of delay values by assuming that all intervals shorter than the critical value are rejected, and all longer intervals accepted. A more complete description of the delays to vehicles would be obtained if the gap distribution were incorporated in the analysis, rather than the constant critical gap or lag. Blunden, Clissold and Fisher (1%2) evaluated the delay in crossing and turning maneuvers under the assumption that the critical lag distribution was of the Erlang type and suggested that the mean of the acceptance-gap distribution was an appropriate measure of the critical gap. Drew (1967) also fitted in Erlang (gamma) distribution to data of observed gaps; he attributed its conceptional
The
study of driver behavior while travelling on and merging from acceleration lanes holds great importance for both theoretical and practical investigations: among others, traffic flow and simulation studies; geometric design, such as determination of the required lane length; and driver performance in relation to elements of freeway surveillance and control or road safety. Merging is defined simply as the absorption of the stream of ramp vehicles from the acceleration lane to the freeway shoulder lane during and following a gap-acceptance process. This process, however, is very complex. The factors involved consist of both internal factors, such as driver attitude and vehicle characteristics, and external environmental parameters, which may include speed and flow on the main stream, speed of entering vehicles and percentage of commercial vehicles. The purpose of the study reported on in this paper was to investigate and evaluate the characteristics of flow on and merging from acceleration lanes in order to shed more light on driver behavior and the gap acceptance process. Such an evaluation, it was felt, may lead to design recommendations for the length of acceleration lanes, based on delay characteristics to merging vehicles. Details of the study itself, which was conducted by the Transportation Research Institute in Israel, were contained in a recent report by Livneh, Polus and Factor (1984). The first stage of the study, an analysis of vehicle-flow characteristics on four acceleration lanes, led to several conclusions, among them that acceleration rates were rather low and that the speed differences between merging and mainstream vehicles ranged between 10 and 15 km/h on the average. Lag-acceptance distribution was found to vary considerably with the location of a reference 39
40
ABISHAI
POLUSand MOSHELI~NEH
appeal as being due to its nonnegative restriction and inherent element of randomness, exemplified by its formulation from the Poisson distribution. Drew later calculated merging delay in terms of highway flow, critical gap and the Erlang constant and also compared observed average merging delay to theoretical merging delay. Investigation by Ashworth (1969) of the capacity of priority-type intersections with a nonuniform distribution of critical acceptance gaps showed a considerable reduction in the capacity of the minor road as a result of replacing the constant gap-acceptance step function with a distribution of critical gaps. Specific values evaluated for normal distribution demonstrated a capacity loss of about 30% at high majorroad volumes. The effect at lower road volumes, however, was found to be less significant. The present study did not concentrate on finding the exact distribution of critical lags. Rather, it made use of the critical value as defined by Raff (1950), and attempted to evaluate merging patterns for two main population groups of drivers and to estimate some measure of correction for the obtained values because of this nonuniform driver population.
MJ%RGINCFROM AN ACCELERATION LANE The process of merging from an acceleration lane was treated by Blumenfeld and Weiss (1971), who developed a theoretical model to describe vehicle flow and gap acceptance. In this model, vehicles on the acceleration lane are assumed to travel at a constant speed u, (Note Fig. 1.) The merging driver is assumed to continue moving along the acceleration lane until a suitable gap is found or to continue to the end of the lane, where the driver’s speed goes instantaneously to 0. The merge, it is assumed, occurs at some time t. Blumenfeld and Weiss defined the delay, D, as the relative time difference between a vehicle on the main stream and a merging vehicle. Thus
D = (1 - $1 = pt,
where all variables have been defined above, and l3=1-;. They further devised a formula that evaluated the total delay to a vehicle in the acceleration lane as the sum of two random variables: the time spent travelling on the lane and the delay while stopped. Numerical solutions to this formula enabled them to relate the probability of reaching the end of the acceleration lane without merging to the length of the lane. Figure 2, a graph based on the model developed by Blumenfeld and Weiss (1971), presents the probability of not merging as a function of travel time on the acceleration lane for a critical gap of 4 seconds and constant Al3 products. The product h$ was established to be constant in this graph, since as A (the volume on the freeway shoulder lane) increases, the speed V on the main road decreases, u being constant. It is assumed therefore, that a constant hf3 product may represent a large, although not full, spectrum of speed-volume relations for a given road. Other products may be expected to exist if different flow conditions occur (e.g., wet pavement, accident situations) or on another highways. Two points stand out at once in Fig, 2: the sharp decrease of the probability not to merge with an increase in travel time and the relatively low probabilities at time 0. This inadequate boundary condition is underscored if one assumes reasonable volumes and speeds on acceleration lanes and freeways in, say, suburban conditions. For example, assume the conditions at the Netanya site as reported in the Transportation Research Institute report (1984), whereby u and Vequal approximately 64 and 85 km/h, respectively, and h equals 600 vph in the freeway right lane. The result is that A * p equals, approximately, 150, which suggests (note Fig. 2) a probability of some 84% of merging at time 0, which mean at the merging-end point (note Fig. 1). The distribution of merging locations presented by Polus, Livneh and Factor (1984) shows, however, that in the context of real-life experiment this is not so. The
_
k+ -
Vohmuon
p
-
Romp
VOIUW
V
-
Spnd
on Fmway
v
-
S~rd
Fmwoy
Right
Right
on Accrlrmtion
MXELENATION
LANE
Lone
Lone LOM
Fig. 1. Schematic diagram of freeway lanes and acceleration lane.
41
Flow characteristics on acceleration lanes I
I
I
I
X - Volume en Freeway Right Lone
v - Speed on Acceleration V-Speed T - Critical
Lane
on Freeway Right Lane Gap
wu .g ._ a
x
e
a
0
2 TL- Travel
4
6
lime
on Acceleration
6
IO
Lane,(rec.)
Fig. 2. Probability of not merging as a function of travel time on acceleration lane.
difference, of course, stems from several assumptions of the Blumenfeld and Weiss model, such as that speed is constant on acceleration lanes and that drivers’ behavior is completely rational; that is, they accept the first available appropriate gap. The model does not consider the volume coming onto the freeway from the ramp, and therefore the analysis is suitable for low ramp volumes only. This paper will attempt to generalize the assumption of rational drivers by suggesting a method for adjusting the travel time on the acceleration lane. The influence of ramp volume on computed delay is also discussed. The assumption of constant speed along the acceleration lane is adopted for the analyses since it was found that this is a reasonable assumption; speed changes along well-designed lanes were found to be rather small (Polus, Livneh and Factor, 1984) and ranged between 1.2 and 10.5%. The process of merging from an acceleration lane is much more complex, however. One can treat this process in either an aggregated or a disaggregated way. In the first approach, some statistical summaries (mean, variance, etc.) of the two possible pop ulations of drivers should be specified: those who accept the first available appropriate gap or lag, and those who remain on the acceleration lane for reasons of convenience or acceleration and merge during the second half of the lane. This approach is discussed at length in a previous paper by Polus, Livneh and Factor (1985). In the disaggregate approach, the behavioral attributes of individual drivers should be examined and some probability functions to best represent overall behavior should be developed. COMMENTS ON DELAY CALCULATION ACCELERATION
FROM
LANES
Delays to vehicles during a gap-acceptance have been discussed by several researchers.
process Among
others were Blunden, Cl&old and Fisher (1962) and Major and Buckley (1962) in Australia; Drew (1%7), who studied gap-acceptance characteristics of freeway ramps in Texas; and Ashworth (1969)) in Britain, who analyzed the capacity of priority-type intersections with a tmmmiform distribution of critical acceptance gaps. In general, one can assume that the total delay to each driver on an acceleration lane is composed of three parameters: the geometric delay (g), the traffic delay (d) and some random delay (a). The geometric delay may be observed, if significant, and is the result of some necessity to travel at a lower speed on the acceleration lane beyond the merging-end point even if no traffic is present on the freeway shoulder lane. The traffic delay was first discussed by Adams (1936) for pedestrians waiting to cross a road. It was proved to be applicable to vehicles having a fixed critical gap, T, which accepts gaps in a major stream with a flow of A vehicles per unit time and was given as d, = $P
- XT - I),
where d, is the traffic delay under the above assumptions and all other parameters are as defined above. Note that gaps smaller than the critical gap are assumed to be rejected by all drivers, and longer gaps accepted. The random delay on acceleration lanes is more difficult to assess. It is caused by a tendency of some drivers to merge at a point along the second half of the acceleration lane (note Fig. 1) even though an appropriate gap or lag was previously available. This tendency was observed during the authors’ study of flow characteristics, and it was estimated that the percentage of such drivers may reach about JO%, depending on location, type and flow at the con-
ABISHAI POLUS
42
4.00 3 1: _
0
and MOSHELWNEH
p- Romp Volume A - Volumr on Freeway Right Lone T- Critical
Gap
200 400 600 000 1000 A - Volumr on Freewoy Right Lone , (vph)
Fig. 3. Calculated traffic delay for various critical gaps and freeway right-lane volumes.
cemed site. The reasons mentioned for this tendency were twofold: the need to accelerate in comfort and the search for some added convenience resulting from 8 late merge 8t 8 faster speed. A method Of estim8ting the needed 8djustment in such 8 parameter 8s critical 18g or gap was suggested and diScussed. It is of interest to note that an Australian study by Miller and Pretty (1%8) on overtaking on twolane roads assumed that some drivers behaved in 8 manner completely different from others. They found that 8 small proportion of drivers would not overtake, however large the gap in the oncoming traffic. This finding is perhaps analogous to the findings here that some drivers will not merge at the reference line (established for 8 determination of the rejected lag size), regardless of the gap or lag size available t0 them. The mean traflic delay, dp, may also be evaluated for vehicles arriving at random at the acceleration lane, with 8 mean rate of arrival, p, by adopting aspects of queueing theory discussed by Kendall (1950) and later by Major and,Buckley (1%2) and ‘. others. Thus,
where d, is the mean tmffic delay and the mean and variance of time spent at the head of the queue 8re d, and d, respectively. This equation is a standard result in queueing theory, for Poisson arrivals and general service times, assuming that service times are independent. In the c8se of merging, the service times are the times taken for drivers at the head of the queue to accept gaps in the freeway traffic lanes. For the independence condition, one must 8ssume
here that only one driver can merge in one g8p, which is probably 8 reasonable assumption for medium to heavy flow conditions on the freeway. This equation also considers the flow from the ramp when assessing vehicle traffic delay. Note that when p + 0, d, + d, is obtained. A plot of the calculated traffic delay for a fixed ramp volume and various critical gaps (or lags) T, and freeway right-lane volumes A, is presented in Fig. 3; one can note the increase in delay with an increase in T and A. A plot of the calculated traffic delay for a fixed critic81 g8p of four seconds and various main flow A or ramp p volumes is presented in Fig. 4. It can be seen that for low volumes on the right lane of the main road, the delay is almost not affected by the
6.00
I
I
I
1
I
I
1
/
I
/
1
p - Romp Volume Vokuoron Frrewoy
A -
T-
0
I
I
I
I
I
I
Gap
Critical
I
I
200 400 600 600 p-Romp Volume , (vph)
Fig. 4. Calculated delay for various ramp volumes and freeway right-lane volumes.
43
Flow characteristics on acceleration lanes
ramp volume, whereas delay is greatly influenced by it for higher main road volumes. For example, when the freeway right-lane volume is equal to 800 vph, the delay to merging vehicles almost doubles when ramp volume increases from 100 to 600 vph. One must keep in mind, therefore, that when estimating merging delay, it is important to consider the volume on the feeding ramp. This point is, of course, also true for calculations of merging probabilities, such as presented in Fig. 1. In order to estimate further the influence of ramp volume on the calculated delay, the following expression of y, the relationship between the two equations of delay, with and without consideration of the ramp volume, was expressed as
’
d C-r d’*
where y may be considered as a measure of rampvolume influenced and dq and d, were defined above. Figure 5 presents the relationship between freeway right-lane volume, A, and ramp volume, p, for these equal values of y: 1.01, 1.02, 1.05 and 1.10. The curves may serve as a guide for geometric designers or traffic planners who need to determine the values of both A and p for which a predetermined y value is established. Thus, for values of A between 100 and 1000 vph, ramp volumes may range approximately from 115 to 40 vph, respectively, if the difference between dq and d, is not to exceed 5%. An increase in y will allow, of course, a respective increase in combinations of ramp and freeway right-lane volumes. One may now note that the geometric delay, g, which represents the added delay due to the restricting geometry, sometimes coincides with the traffic delay d,. This occurs since driving at a low speed on
p-Ramp fig. 5.
the acceleration lane may-come about, due to both adverse geometry (e.g., steep slope or sharp curve) and oncoming freeway traffic. Hence, if one assumes a constant speed on the acceleration lane, it follows that
where all the parameters have been defined above. For the four sites studied, the ramp volumes were rather small, in the order of less than 120 vph or 0.033 vps. It can be assumed, therefore, that dq + d,, as noted above. However, since the freeway rightlane volume relative to the driver moving in the acceleration lane is A@, the d, equation is adjusted accordingly and the merging delay, d, is given as
APT - 11,
,=&e=
where T is the fixed critical gap or lag, depending on the particular case. Hence the travel time on the acceleration lane from the merging-end point to the moment of merge is given by
dsm
tl = -,
P
where all the parameters have been defined above. The calculated travel time and merging delay according to these formulas is presented in Table 1 along with the volume and speed data from four acceleration lanes where vehicle-flow characteristics were observed. The speed u was the average speed along the acceleration lane assumed to be linearly distributed between the initial speed (at the mergingend point, note Fig. 1) and the final speed at the
Volume
, (vph)
Relationship between freeway right-lane volume and ramp volume for equal values of y.
63.6 63.1 63.7 56.9
Netanya Yavneh Geha Haifa
0.1 0.0833 0.0625 0.075
(lcmv/h)
Speed on acceleration lane
Site
Volume on freeway right lane
83.3 74.8 73.1 67.8
(ltmv/h)
Speed on freeway right lane
0.236 0.156 0.128 0.161
B 3.95 3.65 3.60 3.60
Critical Ia8 T W) 4.83 . 1O-2 1.37 * lo-’ 0.67 . 1O-2 1.27. 1O-2
Estimated time at head of queue d, &c)
Table 1. Calculated travel time and merging delay for four acceleration lanes
20.47 8.79 5.23 7.94
* 1O-2 . W2 . lo-* . 10-Z
Calculated merging delay d (sZ)
0.867 0.563 0.408 0.493
Calculated travel times f &c)
45
Flow characteristics on acceleration lanes instance of the actual merge. One should note, however, that the changes between the initial and final speeds along the acceleration lanes studied were rather small (up to 10.5% at the Netania site, 1.2% at Yavneh, 3.5% at Geha and - 6.3% at Haifa site). The speed V was calculated as the average speed on the freeway shoulder lane at a point opposite the merging-end point. EVALUATIONOF OBSERVEDDELAY From the discussion above, it follows that in order to compare observed merging delay and calculated traffic delay, one has to perform two adjustments: one is to estimate the random delay and the second, if applicable, is to adjust for ramp volume. Random delay is caused by various tendencies of the random driver (e.g. lack of awareness, hesitance, habits) to merge along the second half of the acceleration lane even if no traffic is on-coming on the freeway shoulder lane. Since, as mentioned above, the ramp volumes at the four sites in this study were rather small, the second adjustment was ignored for the evaluation of observed delay. The random delays, however, were observed to be significant, and therefore an estimate of their magnitude was sought. Observation of flow characteristics on the four acceleration lanes studied showed that some merging maneuvers occurred when no traffic was present for an extended distance upstream on the highway outside lane. It was decided to use these maneuvers to estimate the proportion of random delay by estimating the contribution of “late merging” vehicles to the average total delay. For this estimation, a parameter a was defined as
where a = correction factor for an estimation of random delay, & = number of drivers who do not merge despite no traffic on freeway right lane, & = number of drivers who accept the first appropriate gap (rational drivers). Note that 5, is estimated from merging maneuvers which occur when there is no traffic on the freeway shoulder lane, a situation which may frequently occur on suburban freeways during off-peak hours. The estimated value of the average random delay, t,, is no+ estimated as t. = t, - a,
where r, is the observed (empirical) travel time the merging-end until the merge into the right of the highixray. The observed average travel may now be adjusted in order to be compared the calculated travel time, t, = t,(l - a),
where t, is the adjusted empirical travel time, and t, and a were defined above. One should also note that fc = t, + t,. In other words, the observed travel time is composed of the adjusted travel time and the random delay. It is important to note here that it is not claimed that both t and f, represent the true values of the random delay and travel time, respectively; rather, it is suggested that these parameters be estimated according to the procedure outlined in this paper. This approach is believed to be adequate for practical issues involving evaluation of delay or travel time on acceleration lanes. In order to compare the average values of the adjusted observed travel time and the calculated travel time, as well as to estimate the values of the random delay, Table 2 was constructed with data from three acceleration lanes where vehicle delays were observed; no delay observations could be conducted at the Geha site due to the unique method of data collection (aerial video-photography from an helicopter). By observing the last columns of Table 1 and Table 2, one can notice that there is a reasonable agreement between the calculated travel time and the adjusted observed travel time at Yavneh and Haifa sites. This suggests that the proposal method of adjustment and estimation of the random delay may represent a valid approach. It is acknowledged, however, that a much larger sample is needed to further strengthen the findings regarding the proposed method. A second noticeable finding is the relatively large disagreement between calculated and adjusted observed travel time at the Netanya site. On this lane, however, the greatest violations of a basic assumg tion of the model (which suggests that the speed u remains constant) were found. Perhaps an overestimation of u was adopted which led to an underestimation of the calculated travel time.
Table 2. Observed delav. estimated random delav and adiusted travel time for three acceleration lanes
Site
Observed delay G (set)
Netanya Yavneh Haifa
4.33 2.14 1.68
from lane time with
Correction factor u
Estimated random delay 1. (set)
Adjusted Observed travel time L (=I
0.644 0.697 0.650
2.79 1.49 1.09
1.54 0.65 0.59
46
ABISHAI
Porus and MOSHELWNEH
A third prominent finding (note Table 2) is that random delays are considerably higher than the adjusted observed delay, caused by a lack of appropriate gaps in the traffic on the freeway right lane. This finding is believed to be due to the relatively
low volumes at the three sites, which resulted in an extended, more comfortable stay on the acceleration lane before the actual merging maneuver.
evaluation of the various components of delay, and particularly random delay; and (b) an examination of the behavioral attributes of individual drivers and the identification of parameters leading to various merging maneuvers. Acknowledgement-me authors wish to express their deep gratitude to the Israel Public Works Department, Ministry of Building & Housing, which commissioned and sponsored this research.
CONCLUSIONS AND RECOMMENDATiONS The merging from an acceleration lane is a rather complex process. Two possible groups of drivers can be identified: drivers who accept the first possible gap as soon as it becomes available, and who may be termed rational drivers; and drivers who always perform the merging maneuver during the second part of the acceleration lane, regardless of whether an appropriate gap or lag was available to them previously. Three components of the aggregated delay for the merging process are suggested: geometric delay, which is caused, at times, by the necessity to travel at a lower speed along the acceleration lane even if no traffic is present; traffic delay, caused by the lack of appropriate gaps or lags owing to traffic on the right lane of the freeway; and random delay, which is caused by the desire to accelerate in comfort and to perform a late merge. A method of estimating the third type, random delay, was proposed and evaluated against the aggregated data of empirical results obtained on three sites. An evaluation of the contribution of the ramp volume to the traffic delay was also performed and discussed, and a graph is presented which may be of practical use for road and traffic designers in assessing the expected influence of various ramp and freeway volume combinations. It is recognized that the comments and proposals made in this paper do little more than illuminate the fringes of the complex phenomenon of gap acceptance from acceleration lanes; however, it is believed that they may serve as a basis for further research. Proposed studies in this connection are (a) a further
REFERENCES
Adams W. F. (1936) Road traffic considered as a random time series. J. Inst. Civ. Engrs 4,121. Ashworth R. (1968) A note on the selection of gap acceptance criteria for traffic simulation studies. Tranrpn Res. 2, 171-175. Ashworth R. (1969) The capacity of priority-type intersections with a non-uniform distribution of critical acceptance gaps. Tivnspn Res. 3,273-278. Blumenfeld D. E. and Weiss G. H. (1971) Merging from an acceleration Jane. ltruupn Sci. S(2). 161-168. Bhmden W. R., Chssoid C. M. and Fisher R. B. (1962) Distribution of acceptance gaps for crossing and turning maneuvers. Rot. Austral. Road Res. Board 101. . I. 188205. Drew D. R. (1967) Gap acceptance characteristics for rampfreeway surveillance and control. Hiahwuv _ - Ru. Ret: l57, l&-135. Kendall D. G. (1931) Some problems in the theory of queues. J. Roy. Statist. Sot. Ser. B 13, 151. Livneh M., Pohrs A. and Factor J. (1984) Analysis of vehicle flow characteristics on acceleration lanes. Research Report 8640, lfantportatioa Research Institute, Technion-Israel Institute of Technology, Haifa. Major N. G. and Buckky D. J. (1962) Entry to a traffic stream. Proc. Austrai. Road Res. Board l(l), 206428. Miller A. J. and Pretty R. L. (1%8) Overtaking on twolane rural roads. Proc. Austral. Road Res. Board 4(l). 58-594. Miller A. J. (1971) Nine estimators of gap-acceptance parameters. In Traffic Flow and Transoortation, Proc. 5th Int. Symp. on th;’ Theory of Traffic ‘Flow and Transportation, pp. 215-235. Polus A., Livneh M. and Factor J. (1985) Vehicle flow characteristics on acceleration lanes. Transoortation Engineering Journal of ASCE, Ul(6). 5954%. FUff M. S. (1950) A Volume Warrant for Urban Stop Signs. Eno Foundation, Saugatuck, CN.