AbstractResearch by Dowling, Singh and Cheng (1,2) focused on refining the speed estimates of network assignment models by using the Akçelik speed-flow model (3). This research demonstrated that the Akçelik model produces significantly improved traffic assignment run times and provides more accurate speed estimates. The Akçelik speed-flow model is as follows:t = to + {0.25T[(x-1) + {(x-1)2 + (8Jax/QT)}0.5]} where:
The Akçelik model is more steeper than the Metropolitan Transportation Commission Bay Area
(MTC) speed-flow function between v/c ratios of 1.0 and 1.5, beyond which the MTC function is steeper.
For freeways (for free-flow speed of 60 mph) Akçelik has a speed of approximately 45 mph at a
v/c ratio of 1.0 compared to MTC which has a speed of 50 mph. Beyond v/c ratios of 1.0, the travel time
for Akçelik increases linearly, whereas for MTC the travel time increases non-linearly. Since a
non-linear growth in travel time is contrary to queuing theory, the Akçelik model appears to be
theoretically more appealing.
The Webster formulation (4) is highly regarded and used to represent delay, but it requires
that the degree of saturation be much less than 1.0. Beyond 1.0 the delay estimate approaches infinity.
As such, this model cannot be used for equilibrium assignments as demand often exceeds capacity in
forecasting future year demand. Deterministic queuing models or oversaturation models offer an
alternate way to model oversaturated conditions. Given that these two models can deal with
undersaturated and oversaturated conditions, what is needed is a model that can bridge the gap between
these two models for intermediate conditions with v/c ratios around unity. The Akçelik
relationship is a good example of the combined model which follows the Webster's formulation for v/c
well below 1.0, the oversaturation model (deterministic queuing model) for v/c well above 1.0, and has
a model for v/c ratios around unity.
Earlier Singh (5) demonstrated that the MTC model [Congested Speed = (Free-Flow Speed)/(1+0.20[volume/capacity]10)] which was similar to the 1994 Highway Capacity Manual was an improvement to the BPR curve (6). The Akçelik relationship appears to be a further improvement to the MTC model. This paper investigates the Akçelik model from a forecasting standpoint and compares various forecast years (2000, 2020) to analyze how the steepness of the Akçelik curve impacts speeds for future years. |
This paper investigates the Akçelik link congestion function, primarily with the idea of comparing it in
more detail with the MTC curve (i.e., link congestion function or speed flow function). Previous research
(1,2) has shown that the results of the highway assignment for the year 1990 using the Akçelik
link congestion function compare well with the results of the highway assignment using the MTC link congestion
function (5).
The Akçelik link congestion function has the added advantage of better simulating link travel times for
oversaturated conditions (7). Though the 1994 Highway Capacity Manual (HCM)(8) offers techniques for
simulating values of volume-to-capacity ratios (v/c) greater than 1.0, they may be used with some caution for
values of v/c upto 1.2 for signalized intersections.
A speed-flow function predicts facility speed as a function of traffic flow. They are based on empirical
research (9).
The 1994 HCM presents a speed-flow function (see Figure 1) which is derived empirically. The drawback of using
this function is it's inability to predict speeds for volume-to-capacity ratios in excess of 1.0. This limits it's
use in planning models where demand can exceed capacity resulting in volume-to-capacity ratios in excess of 1.0.
The 1985 HCM speed-flow relationship is also parabolic in shape with higher sensitivity of speeds to increasing
flows in the low flow region compared to 1994 HCM curve. The 1985 HCM has the same basic shape as the 1965 HCM,
though the sensitivity of speeds to increasing flows was lesser for low flows.
Traditionally the BPR function (6)(see Figure 1) has been used for planning models. This curve was based
on the 1965 HCM which was parabolic in shape, and speed was fairly sensitive to increasing flows. The BPR curve is
as follows:
Congested Speed = (Free-Flow Speed)/(1+0.15[volume/capacity]4)
The problems with the BPR curve is that it overestimates speeds for volume-to-capacity ratios in excess of 1.0
and underestimates speeds for volume-to-capacity ratios less than 1.0.
Modifications to the BPR curve were investigated (10) as follows:
Congested Speed = (Free-Flow Speed)/(1+[volume/capacity]10)
This function had the same speed at volume-to-capacity ratio of 1.0 as the 1985 HCM (30 mph less than the
free-flow speed). The results of this study were validated with results of operational models for test freeway and
arterial sections.
The 1994 HCM speed-flow relationship had a more gradual slope with constant speed for higher level of flows (see
Figure 1). For volume-to-capacity ratio of 1.0, the congested speed is only 5 mph less than free-flow speed. To
account for the 1994 HCM speed-flow relationship, the BPR curve was updated (5) as follows and called the
"MTC" curve (see Figure 1):
Congested Speed = (Free-Flow Speed)/(1+0.20[volume/capacity]10)
The coefficient was changed to 0.20 instead of 0.15, and the exponent was changed to 10 instead of 4.0. Also
capacity values at level-of-service "E" (operations at capacity according to 1994 HCM) were used instead of
practical capacity (level-of-service "C" according to 1965 HCM). This function followed the 1995 HCM speed-flow
relationship very closely and gave good results for speed and volume validation when applied to the full MTC model
system. To more closely reflect local conditions, the speed drop at v/c ratio of 1.0 was 10 miles instead of 5
miles, e.g., for a free-flow speed of 65 mph, the congested speed at a v/c ratio of 1.0 is 55 mph.
Further work (11) recommended a separate curve for arterials as follows:
Congested Speed = (Free-Flow Speed)/(1+0.05[volume/capacity]10)
The speed-flow relationship for the arterial component was changed to reflect a coefficient of 0.05 instead of
0.20. This was called the "Updated BPR" curve.
The Akçelik curve was compared to the MTC curve based on the root-mean square error (RMS). The RMS error
was based on a comparison of observed speeds (using floating car runs) with speeds predicted by the models for 119
selected freeway segments over the San Francisco Bay area. The 119 selected freeway segments varied in length from
1 mile to 9 miles and provided good coverage by including 550 miles (41 percent) of the total 1,340 center-line
freeway miles in the highway network. The RMS error for the MTC curve was 10.1, compared to BPR which was 10.8,
Updated BPR was 10.4, and Akçelik was 9.83. As such, the Akçelik curve appears to be encouraging.
A comparison of the travel times in Figure 2 shows that the BPR curve is fairly insensitive to increasing flows
beyond v/c ratios of 1.0. The travel time for the Akçelik curve increases linearly beyond v/c ratios of 1.0,
whereas the MTC curve travel time increases non-linearly beyond v/c ratios of 1.0. For v/c ratios below 1.5 the
Akçelik curve predicts higher travel times than the MTC curve. However, as presented in Figure 3, beyond v/c
ratios of 1.5 the travel time for the MTC model increases non-linearly which is contrary to queuing theory compared
to the Akçelik curve which increases linearly. As such, the MTC curve overpredicts travel time for links
with v/c ratios in excess of 1.55.
Delay at an intersection is categorized as uniform and random delay. If demand is less than capacity and
vehicles arrive uniformly at a constant rate, this type of delay is referred to as uniform delay. When the arrival
pattern is not uniform but random, which is more likely to happen at isolated intersections, and the flow is close
to capacity, this type of delay is referred to as random delay.
The Webster delay model (4) is one of the earliest models and has the following formulation:
Delay = uniform delay + random delay
= C[1 - (g/C)]2 + (v/c)2
2[1-(v/s)] 2v[1-(v/c)]
where:
C = cycle length (sec)
g/C = ratio of effective green to cycle length
v/s = ratio of demand flow rate v to (adjusted) saturation flow rate s
v/c = (v/s)/(g/C)
The first term in the model represents uniform delay, and the second term represents random delay (Poisson
arrivals i.e., exponential interarrival times). The second term is also referred to as the "overflow delay" as the
randomness leads to vehicles overflowing from one cycle to the next, even though the v/c is less than 1.0. A
correction factor of 0.9 is applied to the overall delay and is based on simulation results.
| Even though the Webster formulation or steady-state model (Figure 4) is most quoted and highly regarded, it requires that the degree of saturation be much less than 1.0. As such, this model cannot be used for equilibrium assignments as demand often exceeds capacity during the equilibration process. For Webster's formulation, the delay estimate approaches infinity at capacity. This is because this formulation is for the "steady state" of a stochastic system. As Hurdle (12) points out "if vehicles continued to arrive at a rate v nearly equal to capacity c, the giant queues really would form, but in reality the peak period ends and v decreases long before a steady state is reached". |
|
Deterministic queuing models or oversaturation models (Figure 4) offer an alternative way to model oversaturated
conditions (14,15). These models are "time-dependent" as the overflow delay is directly proportional to the
duration of the flow period (12). Given that we have formulations to model undersaturated and oversaturated
conditions, the major issue then is what model to use for reconciling intermediate conditions with v/c ratios
around unity.
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As discussed in reference 13, it would be desirable to have a model which follows the Webster formulation for v/c well below 1.0 and the oversaturation model for v/c well above 1.0. The dashed curve in Figure 4 shows a combined model which could take on the best features of both the models. The Akçelik curve is a good example of the combined model (Figure 5). The TRANSYT program delay equation is another example of the combined model (Figure 5). Both curves approach the oversaturation model asymptotically (13). Hurdle (12) points out that "No claim is made that the formulas are correct, but rather that they yield answers which do not violate elementary logic in the troublesome region of v/c near unity where neither the steady-state nor the oversaturation models can be expected to yield reasonable results." |
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The Akçelik function is as follows (3):
t = to + {0.25T[(x-1) + ((x-1)2 + (8Jax/QT))0.5]}
where:
t = average travel time per unit distance (hours/mile)
to = free-flow travel time per unit distance (hours/mile)
T = flow period, i.e., the time interval in hours, during which an average arrival (demand) flow rate, v, persists
Q = Capacity (veh/hour)
x = the degree of saturation i.e., v/Q
Ja = the delay parameter
Akçelik provided the following representative parameter values for Ja:
Table 1: Representative Parameters for Akçelik 's Congestion Function
|
Facility Type
|
Capacity (vphpl)
|
Free-Flow Speed (mph)
|
Ja
|
tc/to
|
| Freeway |
2000
|
75
|
0.1
|
1.587 |
| Arterial (uninterrupted) |
1800
|
62
|
0.2
|
1.754 |
| Arterial (interrupted) |
1200
|
50
|
0.4
|
2.041 |
| Secondary (interrupted) |
900
|
37
|
0.8
|
2.272 |
| Secondary (high friction) |
600
|
25
|
1.6
|
2.439 |
Source (6)
Note: tc is travel time at capacity
The Akçelik speed-flow curve for freeways is presented in Figure 1. The Akçelik curve has a speed
of approximately 45 mph at a v/c ratio of 1.0 compared to the MTC curve which has a speed of 50 mph at a v/c ratio
of 1.0. Also between v/c ratios of 1.0 and 1.5, the Akçelik curve has a higher sensitivity of speeds to
increasing flows compared to the MTC curve. A comparison of the travel time functions for freeways (Figure 2 and 3)
shows how travel time increases linearly for the Akçelik curve beyond v/c ratios of 1.0.
A comparison of the speed-flow curves and travel time curves for arterials are presented in Figures 6 and 7. The
Updated BPR curve for arterials is also presented. Compared to the MTC curve, the Updated BPR curve has a lesser
sensitivity of speeds to increasing flows for the high flow region. The Akçelik curve on the other hand, has
a higher sensitivity of speeds to increasing flows compared to the MTC curve for the high flow regions. The travel
time comparisons show the non-linear growth of travel times for the BPR curves compared to the linear growth in
travel time for the Akçelik curve.
The Akçelik equation is a time-dependent modified form of the Davidson's function (3) to model
flows near and above capacity. Davidson's function is as follows:
t = to [1 + (JDx/1-x)]
where:
t = average travel time per unit distance
to= minimum (zero-flow) travel time per unit distance
JD= a delay parameter (or 1-JD = a quality of service parameter)
x = q/Q = degree of saturation
q = demand (arrival) flow rate (in veh/h)
Q = capacity (in veh/h)
As pointed out by Akçelik (3), "Davidson derived this function from concepts of queuing theory but
a direct derivation has not been clearly demonstrated".
The most appealing part of Akçelik 's link congestion function is it's ability to deal with varying flow
periods (the time-dependent equation) and model periods of oversaturation (i.e., v/c >1.0). Speed-flow curves
and travel time curves for 1-hour and 2-hour flow periods are presented in Figures 8 and 9 respectively. The
Akçelik speed-flow curve for 2-hour flow period has a higher sensitivity of speeds to increasing flows
beyond v/c ratios of 1.0 compared to 1-hour flow period curve. The travel time comparisons show that the delay
increases two-fold for the 2-hour flow period compared to the 1-hour flow period for v/c ratios in excess of 1.0.
To model a two-hour flow period, it should be noted that the input trip table should be for one-hour, the facility
capacity should be hourly and "T" in the Akçelik equation should be coded as 2.0.
The Year 2000 MTC highway network covering the San Francisco Bay area has 32,114 links and 15,730 nodes. The
Year 2020 highway network has 32,476 links. There are a total of 1120 zones of which 21 are external zones. The
highway network has eight facility types and six area types as presented in Table 2.
The facility types are: freeway-to-freeway connectors, freeways, expressways, collectors, freeway ramps, dummy
connectors, major arterials, and metered ramps. The area types are: core, central business district, urban business
district, urban, suburban, and rural.
The "area type" is based on the density of development within each zone and is used to determine the free-flow
speed and capacity of links of a certain facility type. The area density is defined as follows:
Area Density = (Population + 2.5*Employment)/Developed Acres
where:
Developed Acres = Acres used for residential or commercial/industrial purposes.
The free-flow speeds and capacities are based on the Highway Capacity Manual and take into account the capacity
decreases due to heavy vehicles and weaving.
Two speed-flow curves were analyzed for the MTC highway assignment are:
1) MTC curve (5) which uses a coefficient of 0.2 and an exponent of 10. This curve is very
similar to the 1994 Highway Capacity Manual curve for freeways.
2) Akçelik (3) which uses the Akçelik curve and values of Ja as follows:
freeways=0.1, freeway-to-freeway connectors=0.1, freeway ramps=0.167, expressways=0.2, arterials=0.4, metered
ramps=0.4, and collectors=1.2. These are based on Akçelik 's representative parameter values (7).
The software used for this analysis is MINUTP (1993 version). Coding speed-flow curves which are of the
functional form of BPR curves are relatively simple to code for this version. However, to code the Akçelik
function, capacity restraint factoring curves had to be coded. Since each facility type and area type has a
distinct curve, forty-eight capacity restraint factoring curves had to be coded. This involved coding the curve
volume/capacity values and corresponding factor to multiply the link base impedance by to obtain the congested
impedance.
This section discusses the results of the various curves investigated for this analysis. Comparison between the
different curves is conducted based on computing times, convergence achieved, systemwide average speeds, speeds and
volumes on selected facilities, vehicle-miles traveled, vehicle-hours traveled, speeds by facility types, and
vehicle-miles by facility type.
The highway assignment for 1990 was conducted for a.m. peak hour. For years 2000 and 2020 the highway assignment
was conducted for a 2-hour a.m. peak period. For the peak period highway assignment, the facility capacities were
doubled and the trip table used was for a 2-hour time period.
A significant area of concern is the computing time it takes using the Akçelik curve compared to the MTC
curve. As presented in Table 3, the computing times are fairly similar. Also the Theta factor which reflects the
degree of convergence appears to be similar or slightly better for the Akçelik curve compared to the MTC
curve.
The average systemwide speed for the Akçelik curve is lower as compared to the MTC curve for all
scenarios. As shown in Figure 10, the average systemwide speed for the highway network as a result of the highway
assignment is consistently lower by 3 to 6 miles/hour for the Akçelik curve compared to the MTC curve.
The results of the Akçelik curve are very similar to the results of the MTC curve for both years 2000 and
2020. Volume comparisons for both curves are presented in Table 4 for all San Francisco Bay Area bridges and
county-to-county screenlines. These results are very encouraging and show that the use of the Akçelik curve
does not cause large differences in the assigned volumes compared to the MTC curve. The use of the Akçelik
curve may warrant some changes to the trip table for the base year so that the resulting assigned volumes are
closer to the observed.
Speed comparisons are presented in Table 5 for 520 miles of the total freeway system (i.e., approximately 30
percent of the total freeway miles in San Francisco Bay area). The speeds for the two curves are very similar
indicating that the Akçelik curve does not cause significantly different speeds compared to the MTC curve.
The vehicle miles by facility type are shown in Table 6 and Figure 11. There is an overall increase of vehicle
miles for the Akçelik curve compared to the MTC curve. There is a decrease in vehicle miles for the freeway
system, whereas an increase in vehicle miles for the arterials. This shows that more travel is taking place on the
arterials for the Akçelik curve as compared to the MTC curve which may explain the reduction in overall
systemwide average speed for the Akçelik curve.
The vehicle hours traveled by facility type are presented in Figure 12. The vehicle hours increase for all
facility types for the Akçelik curve compared to the MTC curve.
Speeds by facility type (Table 7 and Figure 13) show that the speeds on all facility types are lower for the
Akçelik curve compared to the MTC curve.
A distribution of vehicle-miles and vehicle-hours by V/C ratio is shown in Figures 14 and 15. The VHT and the
VMT for the Akçelik curves is relatively higher for V/C ratios 0.5 to 1.1, after which the drop in VHT and
VMT is fairly steep.
Figures 16, 17 and 18 show the distribution of vehicle miles traveled (VMT) by average speed for the MTC and
Akçelik curves. The distribution for year 1990 is bimodal (two humps) whereas the distribution for years
2000 and 2020 is tri-modal (three humps). The first hump occurs at the free-flow speed for non-freeway facilities.
The second hump for year 1990 and the third hump for years 2000 and 2020 occurs at the free-flow speed for
freeways. The second hump for years 2000 and 2020 occurs at the free-flow speed for expressways.
For all the years, the Akçelik curve predicts a higher percentage of VMT operating at speeds greater than
60 miles per hour and lesser than 30 miles per hour. For the years 2000 and 2020, the Akçelik curve also
predicts a higher percentage of VMT operating at speeds in the vicinity of 50 miles per hour. This results in a net
areawide average speed reduction of 3 to 6 miles per hour for the Akçelik equation as compared to the MTC
curve.
A distribution of links by v/c ratio is presented in Figures 19 and 20 for years 2000 and 2020 respectively. The
v/c range shown in the figures is from 0.5 to 1.4. There are very few links beyond a v/c ratio of 1.4. The highest
number of links (approximately 45 percent) are within the 0-0.1 v/c range. Beyond that there is a sharp decline in
the number of links. In the 0-0.1 range the number of links are higher for the Akçelik curve. For the range
0.1-0.3 the number of links are higher for the MTC curve.
Figures 19 and 20 show that in general for both curves the number of links tend to cluster around each other. It
appears that for the Akçelik curve the number of links are greater in the v/c range 0.5-1.0 and lower beyond
v/c ratios of 1.0 compared to the MTC curve.
A comparison of volumes and speeds observed on the San Francisco Bay Area freeways shows that the Akçelik
curve performs well. A comparison of speeds for different facility types shows that the Akçelik curve
reduces speed on all facility types. The Akçelik curve is a superior curve compared to the MTC curve as each
facility type has a different value of Ja (delay parameter).
Compared to the MTC curve, there is some redistribution of vehicle-miles for the Akçelik curve. The
vehicle-miles on the freeways is reduced whereas the vehicle-miles on the arterial increases.
The computing time for the Akçelik curves is approximately the same as the MTC curves and the convergence
appears to be better (see Table 3).
The Akçelik curve is about as accurate as the MTC curve and has the advantage of predicting the linear
impact of congestion on speeds. It predicts lower speeds for congested conditions which is desirable.
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2. Dowling, R.G., R. Singh, and W.W.K. Cheng. The Accuracy and Performance of Improved Speed-Flow Curves. Road and Transport Research, Vol. 7, No.2, June 1998.
3. Akçelik , Rahmi. Travel Time Functions for Transport Planning Purposes: Davidson's Function, its Time-Dependent Form and an Alternative Travel Time Function. In Australian Road Research, 21(3), September 1991, pp 49-59.
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