Hsiu-Jy Ho, Wei-Ming Lin, University of Texas at San Antonio, weiming.lin@utsa.edu

Abstract

When traffic load approaches the capacity of a freeway, not only does the expected delay for motorists exponentially increases but also the overall throughput suffers dramatically. Engineers have thus started employing stoplights (ramp-meter) on access ramps to freeways in highly congested areas to control the traffic injection into the freeway in order to reduce the overall delay for the motorists. The goal of this paper is to develop a comprehensive Autonomous Real-time Traffic Injection Control system (ARTIC) capable of minimizing the overall delay for motorists according to the then traffic input load and freeway congestion situation without replying on any history data as all other known approaches do. We first propose a general methodology that provides an autonomous self-adjusting process to optimize performance for many real-time resource-sharing applications, the Autonomous Performance Optimizing Control Methodology (APOCoM). This methodology which ARTIC is based on can be used to automatically optimize performance of most resource-sharing system in real time through periodical self-adjustment of some system settings. Simulation results show that the system employing the ARTIC thus designed easily outperforms the system without using the control approach. In one of the simulation for daily twin-peak rush hours, about two fifth of congestion time (70 out of the original 167 minutes) is eliminated, and about one third of travel time (9 out of the 27 minutes) is saved per peak cycle.

Introduction

Freeway systems have become a necessity in our daily-life commute as well as for leisure travel since the past three decades. Freeway systems provide a very efficient means to relieve cross-traffic blocking which is otherwise inevitable in typical city road systems. However, due to the ever-increasing population in metropolitan areas and the use of cars, congestion remains a problem in most freeways systems, especially in certain sections at certain periods of time. One particular problem that exists in almost all freeways that are prone to heavy congestion is that, even when traffic load (incoming rate of cars to the system) has approached (or even exceeded) the freeway’s capacity, cars are still allowed to enter the congested freeway, which further aggravates the congestion problem. Eventually, cars can only at best crawl at a very low bumper-to-bumper speed on freeways while more and more cars cram into the jam. There have been many mechanisms proposed in the past to alleviate these problems. These mainly focus on using some types of performance measurement systems to assist freeway users to foresee problems ahead or to help traffic system managers to make real-time adjustment decisions. Such decisions could be short-term or long-term, with the former including mostly traffic guidance signaling and on-ramp traffic light control, and the latter one involving lane-use control (e.g. high-occupancy vehicle lanes) and others.

Since over a couple of decades ago, engineers have started employing stoplights on access ramps to freeways in highly congested areas to control traffic injection into the freeway in order to reduce the overall delay for the motorists. This is based on a notion that, if a car is delayed by a certain period of time before it is allowed to get onto the freeway, it may eventually save more time overall by traveling on a less congested freeway. That is, the goal is to ensure that the “investment” in lengthening the waiting time on the on-ramp for the cars would lead to a reduction in the cars’ traversing delay on the freeway large enough to offset the “investment”. The question is how long and in what situation this “delay” should be set at in order to achieve the best possible performance in real time. There have been many research activities that targeted on improving traffic congestion by controlling traffic injected to the freeways. Ramp metering is the most popular mechanism to control and upgrade freeway traffic, typically employed in freeway network systems activated via installation of on-ramp traffic lights. A comprehensive overview of most popular ramp metering techniques is given in [15]. Various positive effects are achievable if ramp metering is appropriately applied. Regulating strategies for local ramp usually depends on information from measurement of vicinity section. The “fixed-time” ramp metering strategy is first proposed by Wattleworth [19] and later by [16] with a strategy based on constant historical demands on on-ramp cars without the use of any real-time measurement. Due to the lack of real-time information, this type of approaches is prone to suffer from either freeway congestion or under-utilization. “Reactive” ramp metering strategies have also gained attention to attempt to attune the traffic condition of freeway system close to some pre-specified values, based on data accrued in real-time. In [10], befitting ramp metering values are determined with the assistance of flow information gathered in the very vicinity of the ramp. A strategy based on the relationship between throughput (flow) and freeway density is used to determine the metering setting so as to control flow (throughput) of freeway system to reach a set capacity of freeway using an open loop control approach. A closed-loop ramp metering strategy ALINEA, first proposed in [13] and later further modified in [3, 14, 17], is proposed to maximize the mainline throughput by maintaining a desired occupancy on the downstream mainline freeway. It may prevent congestion by stabilizing the traffic flow at a high throughput compared to the open loop approaches. METALINE [7, 12], a strategy proposed to improve local ramp regulating by obtaining semi-global information from several on-ramps in the vicinity. In general, METALINE performs better than ALINEA in non-recurrent congestion situations due to more comprehensive measurement information. Several nonlinear optimal ramp metering strategies are proposed in [6, 9, 15] with the use of nonlinear traffic flow dynamics. Furthermore, extensions were proposed in [8, 17], a combination of optimal coordination model-based ramp metering control with ALINEA. Integrated optimal control is proposed in [5] with a control strategy consisted of the use of ramp metering, inter-freeway control and route guidance. Another control technique is proposed in [1] with ramp metering considering re-route condition. A very successful of implementation of metering control was also reported in [18] using fuzzy logic approach.

Although these systems have provided a significant improvement in the efficiency of freeway usage, unexpected and spontaneous traffic bursts could still easily lead to heavy congestion and undesirably long traffic delay. The main drawbacks in these designs are in that they mostly rely on some model(s) derived from profiling long-term traffic data and thus are unable to respond to unexpected traffic patterns. A preset “optimal” flow value is assumed to be available a priori for the metering control to adjust according to. For example, most of these systems are based on an optimal flow value established through traffic observation. Figure 1 shows the “Fundamental Diagram”, a well-known freeway traffic phenomenon, displaying a relationship between flow and density, that has been employed by many designs to control metering so as to reach the optimal flow value known in advance. A metering mechanism thus designed suitable for a particular section on a particular freeway may not be applicable to other locations, and may easily be outdated when modifications are made to the freeway systems.

In today’s ramp metering systems, such a “Traffic Injection Delay” (TID) is usually preset according to some traffic profiling results or managed in real time through some global manual control. Neither of the two approaches can effectively handle spontaneous changes in traffic, temporal-and/or spatial-wise. Note that general profiling approaches using history information as some known techniques for specific applications have used is purposefully excluded in the proposed method since we believe that the database thus produced does not truly reflect the real-time nature (in terms of perturbation in load) of the target systems nor is it capable of handling any potential changes of the system characteristics. Expert systems, neural networks or other intelligent control techniques that rely on training with known patterns similarly cannot address these intrinsic problems. Fuzzy logic approaches relying on pre-setting numerous default parameters are still tailored to a specific freeway with a specific traffic pattern.

In this paper, we will first propose in Section 2 an automatic control process, the Autonomous Performance Optimizing Control Methodology (APOCoM), which allows automatic adjustment on certain system parameter setting based on observed performance changes without relying on any pre-determined optimal value known beforehand. This process, unlike the close-loop control process, adjusts the setting based on an unbiased approach in order to disengage the dependence of performance on the varying input. This is needed since the performance observed may be influenced by the then input rate. In Section 3 we then apply the proposed APOCoM to develop a comprehensive Autonomous Real-time Traffic Injection Control (ARTIC) system that is capable of minimizing the overall delay for motorists according to the then traffic input load and freeway congestion situation. Simulation results are presented in the following section.

APOCoM – Autonomous Performance Optimizing Control Methodology

Our goal in this paper is to develop a traffic control mechanism that can self-adapt to the constant changing input rate to optimize performance in real time. In this section, we briefly describe a general methodology that is to be based on for such a development.

Figure 2 gives a general block diagram description of a closed-loop control system, or so called feedback-control system. In closed-loop control, also known as feedback control, the controlled input I_C depends on the comparison between the output P and a goal G known beforehand. A perfect real-life example is the cruise control system in a modern-day vehicle, where a set goal (a set speed in this case) is known beforehand, and the control system is to adjust the input I_C (gasoline and/or brake in this case) according to the feedback information (the current speed sensed).

Our target problem instead involves a real-time performance optimization process, where there is no optimal goal known beforehand to reach at any given time. All known control techniques, such as traditional tracking, input command shaping, receding horizon control, etc. are not capable of handling this problem. Fuzzy logic control also relies on a known relationship between the adjustment direction and a goal to reach. Employing a stop light on an access ramp to freeway to control traffic injection is one perfect real-life example. The ultimate question is: how long should this “delay” be set at any given moment in order to achieve the best possible performance in real time? In this case, there is not an “optimal” delay values known beforehand, nor should it be a fixed value under constantly changing traffic (load) situation. Other examples exist in many resource sharing applications where two or more algorithms (techniques), each known to be more suitable for certain input/load situation, need to be combined to exploit the benefit of each under different input scenario. A combination “setting” among these algorithms needs to be adjusted in real time so as to accommodate different input/load then presented. Similarly, this setting parameter does not have a fixed optimal value as a reference to adjust according to.

These problems possess two critical characteristics very different from those suitable for typical control processes. For one, not having a reference value for the control system to adjust according to renders all the known control techniques very limited in applying to these problems. Secondly, the input (or, more precisely, the “load” to the system) is constantly changing, thus any possible reference value dynamically produced may become out-dated very quickly. These input load characteristics, for example, those of generation rate, task sizes, etc., are known to be simply random processes in most real-life circumstances. Perfect examples include: incoming tasks in a computer system, incoming traffic for a local area network, incoming calls to a telephone system, and incoming traffic to a highway system, etc.

Similar to the simple feedback control process, our proposed algorithm would incur an automatic adjustment to some system setting(s) periodically based on performance observed in a set “window” of time. Since there is no optimal “goal” known beforehand to adjust according to, we propose a very simple automatic adjustment algorithm as a base algorithm that aims at approaching the optimal setting gradually and autonomously. Figure 3 displays the algorithm proposed for this purpose. In the current window, performance P ^t is observed and compared with the performance from last window, P ^t-1. If it leads to an improvement, the adjustment operator op ^t remains the same as the previous operator op ^t-1, otherwise, the operator is “reversed”. A simple example for such an operator is “increase” versus “decrease” as the two reversed operators. Other more complex operators may be needed. For the example of freeway system, the operator corresponds to “increase delay by x seconds” versus “decrease delay by y seconds”. The exact control setting applied to the input at the end of this window, d ^t, is then derived from adjusting the previous value d ^t-1 by a margin Δd with the new operator. For a simple example, d ^t = d ^t-1 +Δd if op ^t is “increase”.

There are four main components in this proposed control process: (1) Adjustment Parameter(s), (2) Performance Indicator(s), (3) Performance Sampling Window Period and (4) Performance Measurement as described in the following.

Adjustment Parameter(s)

Different adjustment parameters may be used for different applications. There may be more than one viable parameter for the same application. For example, mixing weights for a combination of techniques for all generic resource sharing systems, delay time setting for all traffic control systems (such as freeway, router, etc.), and many others are potential choices. Some parameters may be easier to adjust than others, while their adjustment may not be as effective in improving target performance. Proper selection of such parameter(s) has a critical impact on the success of resulted design.

Through testing, the amount of adjustment on the parameter (Δd as shown in Figure 3) very much depends on how it is quantified and how sensitive the adjustment needs to be. For example, in the entrance traffic injection control for a freeway system, if the adjustment amount is too large, the control systems tend to over-adjust in more situations, which may become disastrous when the load is near saturation; on the other hand, if this amount is set too small, the system becomes slow to respond to real-time changes. A dynamic amount of adjustment may be needed under different situations to avoid these problems. Adjustment damping may also be needed to avoid any flip-flop effect.

Performance Indicator

There may be various “performance” data that can be measured to indicate the capability of the technique (i.e. with the current setting value) being observed; however, some may be misleading or even fallacious. One has to be careful in finding the one that is revealing enough while requiring reasonable amount of time to obtain. For example, in the freeway system aforementioned, using the speed of cars traveling on the freeway as the system performance would easily lead to an adjustment that completely blocks the cars from entering the freeway in order to achieve the higher “performance”. On the other hand, using car density in the freeway as the performance again creates a biased goal for the adjustment to reach. Properly selecting the correct performance data to adjust according to is a must for our system to succeed. In some cases, more than one performance indicators may be needed to provide a more significant measurement reference.

Performance Sampling Window Period

Finding a proper window size for performance sampling between two adjacent adjustments remains a tough issue to deal with. When a small window size is used, performance data sampled tends to be affected more by the randomness factor; on the other hand, a large window size deters the system from adapting effectively to the change of the load.

Performance Measurement

The algorithm proposed so far seems straightforward; however, many pitfalls remain unsolved. When measuring performance in a window, one has to be careful about whether or not the performance measured is not wrongly influenced by the then input rate (load) and thus the comparison between two windows is biased. Note that, since the input rate (load) is essentially random, it is unreasonable to assume that a similar load exists in all windows. Consider a simple real-life example (as shown in Figure 4) where two banks having different customer-serving capacities, µ_A and µ_B, respectively, and µ_B < µ_A. If the corresponding customer incoming (load) rates λ_A and λ_B satisfy the condition of λ_A < λ_B < µ_B < µ_A under the assumption that there has not been any load accumulation, then performance reading demonstrated by the throughput P_A and P_B would lead to P_A < P_B which implies a wrong indication of the capacity superiority between the two banks.

Thus, judging superiority between two sampling windows would face the same challenge. In order to come up with a fair performance comparison between adjacent windows, a notion of “qualified sampling point” first proposed. A point (time instance) is only considered “qualified” and then the performance at which point can be measured for comparison when the load at that point exceeds a certain predetermined threshold value.

When comparing the two techniques (or, more precisely, two setting values) in adjacent windows with each having a capacity of µ_A and µ_B, respectively, as shown in the example in Figure 5, the best threshold value would be the smaller of µ_A and µ_B. When the input load falls below the smaller of µ_A and µ_B, it is virtually impossible to tell which “technique” is the better one since both would produce the same throughput assuming no residual effect. Theoretically, only when the input load rises above the smaller of µ_A and µ_B is the performance measurement more likely to reveal the superiority of one over the other. These points are shown as the “best qualified time frame” in the figure. In general, the higher such a “qualifying threshold value” is set, the more revealing is the performance measured on the actual capacity/capability of the techniques being observed. Listed below are several important factors that need to be addressed when trying to decide how a point is considered qualified.

·          Input Load Indicator Selection: The input load indicator selected has to be independent of the technique; that is, whether a point is considered qualified can not be in any way dependent on the technique being observed; otherwise, the qualification criteria may be biased. For example in the freeway ramp-metering system, if one chooses to use the queue length as the load indicator, a worse setting may lead to a longer queue length and thus is mistakenly regarded as having a higher input load than one under a better setting.

·          Qualification Threshold Selection: The threshold to be used for qualification purpose has to be selected properly. Performance measurement using a low threshold leads to a comparison result that is less likely informative or reliable. On the other hand, with a high threshold, one may have a problem finding sufficient qualified points to have a reliable performance reading in a window, and thus may in turn require a window too large to allow for a timely responsive adjustment.

Proper load-independent performance measurement to ensure unbiased reference is one of the most important goals to reach. Several different approaches have been tested as shown in the following.

·          Inter-Window Load-Independent Measurement: Let f(i) denote the “input load” measured at time instance i and l_th be the threshold for the load to be considered “qualified”. Let f(i) be the performance observed at time instance i. Thus, the performance measured for a technique during a window w^t is calculated according to the following formula:

where the denominator is the total number of qualified points in the window. P ^t is essentially the average performance among all qualified points in the window w ^t. Performance comparison is then carried out between P ^t and P ^t-1 as shown in the control algorithm (Figure 3) for adjusting parameters.

·         Intra-Window Load-Independent Measurement: One simpler way to achieve a degree of load-independent performance measurement is to have window w ^t divided into two equal-sized sub-windows, w ^ta and w ^tb, with performance observed in each denoted as P ^ta and P ^tb, respectively. Performance comparison for adjustment is done between P ^ta and P ^tb, instead of P ^t and P ^t-1, to determine the direction for adjustment. This approach is simpler than the inter-window one but reliability will be compromised.

·         Multi-Window Neighborhood-Averaging Load-Independent Measurement: In many applications, where performance measured locally may be tempered by input load from imprecise temporal and/or spatial locality. For example, in a freeway system, performance measured closed to an on-ramp is actually a reflection of not only the local on-ramp input load but also from other on-ramps in different time windows. In order to relieve effects from such an imprecision, we propose a multi-window neighborhood-averaging load-independent measurement approach in which P ^t is derived as a weighted combination of different performance values observed in a series of windows and/or from neighboring locations.

·         Variation of System Utilization/Throughput: Another approach that can be used to lead to a reliable performance measurement with minimal load disturbance is to use the variation of system utilization and/or throughput. This is based on a concept that a better technique (a setting with a higher capacity) can temporally “follow” the load more closely. As shown in Figure 6, technique A has a higher capacity than B has, thus the latter one needs to catch up with the load in more time instances. This leads to a general scenario that technique A delivers an observed performance (system utilization) with a higher variation. That is, measuring the standard deviation of the system performance may be a very reliable yardstick to determine the superiority of a setting over the other, and has proved to be less sensitive to load dependency.

Global vs. Local Optimization

For a system that requires multiple locations of setting control, e.g. multiple entrances of a freeway, multiple routers in a network, etc., when the proposed APOCoM is applied to control local settings based on strictly local performance measurement, the system risks running into a “feast-and-famine” phenomenon where performance of two local points continue to adjust in opposite directions due to the performance information continuously misguided by each other. Such a process easily leads to a degenerated global performance. In order to somewhat remedy this problem, one can establish a semi-local scope of performance measurement to include spatially dispersed performance measurements for each local control. Weighing may also be needed to optimally combine these performance readings.

           ARTIC – Autonomous Real-time Traffic Injection Control System

One direct application of the proposed APOCoM is in designing an Autonomous Real-time Traffic Injection Control System (ARTIC) for freeway traffic injection control. It has been well established [11] that freeway traffic demonstrates a specific relationship between the average traffic flow (throughput) and the average speed, as shown in Figure 7. This indicates that, as in the upper part of the curve when the traffic load is low where all vehicles are in relatively high speed, flow is not actually maximized. On the other hand, as in the lower part of the curve when the traffic load is high where vehicle speeds are in general low, throughput is also degraded. Only when the vehicle speed reaches the “critical point” will the flow be optimized.

In general, this can be accomplished by “adjusting” density on the freeway so as to control the average speed through two possible “Performance Improvement Sequences” (PIS) processes, “PIS-1” and “PIS-2”. Many research works have focused on using this critical point to adjust traffic injection to optimize performance assuming that the value of the point is known in advance. The problem is that the critical point usually obtained through some statistical analysis of past traffic data collected is very much environment dependent, e.g. the various design aspects of the freeway, human driving characteristics, performance and design of vehicles, etc. Relying on a critical point thus obtained is only applicable to the freeway region where data were collected and only valid for a short time frame before it becomes out-dated.

Applying the proposed APOCoM methodology to building the ARTIC system would require several sensors for performance reading and load assessment. The basic architecture of the system for each on-ramp (as shown in Figure 8) is composed of four components: a performance sensor (PS) on the freeway to gather both the flow and speed information of cars passing a certain sampling point on the freeway, a load sensor (LS) on the on-ramp to gather load and queuing information of cars on the on-ramp, a traffic-injection delay (TID) control signal and a local controller in charge of controlling the TID based information from the PS and LS. Note that all these required components are either already available or easily attainable with modern-day technology. Information from both local sensors and potentially neighboring sensors are to be used to measure the performance, namely the throughput and travel delay. Information gathered from sensors LS are also used to assist in decision making based on changes of queues.

Our APOCoM control algorithm for this application will control each of the TID control signal for the desired injection delay, as the control parameter d ^t shown in the algorithm in Figure 3. When TID is set to 0, no delay is imposed to any incoming car. The adjustment margin (Δd in Figure 3) is fixed at 0.2 second (one simulation time unit) in our process, with the operator OP ^t being either “increase” or “decrease”. Performance measured by each of the PS’s during a time window has two components: (1) the overall throughput (flow), and (2) the average speed of cars passing through. Qualification criteria for scheduling control are to follow what is described in the general APOCoM although our simulation does not actually call for any load qualification due to the complex local performance dependence on actual traffic “load” presented from both the local on-ramp and from internal traffic on the freeway itself. We believe that a more precise performance measurement with load qualification would lead to an even more reliable control. Another generic problem that happens specifically to this type of application is in how the global optimization can be reached through local optimization on each entrance controller. The aforementioned “feast-and-famine” process would occur if this problem is not carefully addressed. Some “semi-global” performance measurement mechanism has to be incorporated to relieve this problem. The complete control algorithm for each TID is shown in Figure 9. The general control paradigm presented in APOCoM is followed in this algorithm, with the darkly shaded boxes and lines corresponding to the original algorithm. Some additional details and modifications are needed to avoid unnecessary perturbation and undesired localization problem. Not only the speed information gathered in the local sensor PS is used, but also that of two immediate neighboring PS sensors is collected. Listed below summarizes the necessary modifications incorporated into the original APOCoM algorithm:

·          Change of performance observed in the two windows, P ^t-1 and P ^t, is not considered “significant” unless the change is more than a set relative threshold, δ as specified in the algorithm. Otherwise, the control setting remains unchanged in the next adjustment.

·          A condition of “low traffic load/rate” is tested by checking if current average velocity V ^t is greater a set percentage of maximum velocity, Vmax _* τ as shown in the algorithm. Note that Vmax is set to the legal maximum speed and τ is set to 85% to reflect a relatively empty freeway. This situation is indicated in the lightly shaded area as in the Figure 10 revisiting the speed-vs.-flow relationship in Figure 7. Different actions would be required to handle this situation for that either traffic injection should simply follow the traffic load/rate or the injection delay should be drastically reduced.
 

·          If the neighboring average velocity (ave(V_i-1,V _i+1) as shown in the algorithm) demonstrates the same behavior, then the “low-load” scenario is happening in the semi-global region, thus requiring a drastic reduction on the local TID by setting the new value to half of original value, i.e. d ^t = ½ d ^t-1.

·          If the neighboring average velocity does not show the same behavior, then the “low-load” scenario is happening only in the local region. That is, local performance (flow) read is likely from mostly neighboring traffic. Adjusting local TID further according to performance change would further mislead the performance trend. This instead requires the local TID to follow its own load rate by setting the new value to the average load rate coming into the ramp in the past ten windows, d ^t _* R_ave10 as shown in the algorithm.

·         Another mechanism is incorporated to handle the situation of saturation as indicated in the darkly shaded area in Figure 10: if performance (flow) is degrading and the speed is also slowing (V ^t < V ^t-1 τ), then the current trend is obviously following the opposite direction of PIS-2 in Figure 10. The TID needs to be continuously increased to avoid further congestion. This is denoted by OP ^t  ß “+” in the rounded box in the algorithm. ß

·         The adjustment margin Δd in the original APOCoM is set to one of three possible settings, 2, 4 or 6 time units, depending on how fast the performance change exceeds the threshold. Δd is set to 6 if it takes only one sampling window to deliver the change, to 4 if it takes two windows and to 2 if three or more windows. This would help adjust the setting properly depending on whether it requires fast transition or should stay in a stable mode.

Simulation Setup

In order to provide a reliable system model for simulation, a simulation environment has been constructed to simulate the traffic of a typical freeway. Main components of this model are briefly described in the following.

·          Freeway Model: one-directional one-lane of 20km with wrap-around traffic for a good approximation to a long freeway; 10 on-ramps and 10 off-ramps (one lane each) evenly separated in distance.

·         Traffic Load Model: each on-ramp has an inter-car-arrival time exponentially distributed with an average of 1/λ simulation time units (TU) and 1 TU = 0.2 seconds; the off-ramp for a car to exit is determined using a standard Gamma distribution for the number of ramps (x) between its on-ramp and off-ramp. Probability of a car exiting after x ramps is denoted as f(x) and

where γ is set to 5 to lead to a highest value around x = 4. This distribution is shown in Figure 11(A), a good approximation to typical city traffic pattern with an average distance of 4 to 5 exit ramps before a car reaches its destination ramp.

·          Speed Model: speed (meters/second) of a car follows a typical nonlinear function (see Figure 11(B)) of the distance d (in meter) between itself and the next car in front:

·         This function is obtained following a model proposed in [4] with a translation from its original speed dependence on density to our dependence on distance. This translation is needed for this simulation to ensure that per-car speed calculation strictly depends on the car in front, instead of depending on a regional density which may easily lead to transposition of cars during simulation. The original critical density value that separates the two segments of curve is now translated to a critical distance value, 15 meters as indicated in the equation.

Simulation Results

Based on the aforementioned simulation model, an extensive simulation run is carried out. A total number of 400,000 cars were tracked until they exit the freeway following the described random process to ensure stable condition. Four different simulation environments are used to compare performance among the non-controlled one, a fixed-injection-delay one, and the controlled one using our proposed ARTIC system. Both delay and throughput are shown in the results.

Simulation Setup #1: Uniform Traffic Load

In this simulation environment, all ten on-ramps are provided with a uniform traffic load with the same λ value. Figure 12 gives the comparison results. Note that the “Overall On-Ramp Arrival Rate” denotes the aggregated arrival rate among all 10 on-ramps. In (A), the ARTIC system delivers a performance relatively close to the best “fixed-injection-delay” ones, from 11 to 18 TU s while leading to a higher saturation point. All the other fixed-delay ones including the no-control one saturate much earlier. In (B), throughput value is sustained properly by the proposed ARTIC system well beyond the saturation point, while only the fixed-19-TU one is able to match it with all others either maintaining a lower throughput (e.g. fixed-20 and fixed-22) or simply slowing down to deliver only a small fraction of original throughput (e.g. w/o-control, fixed-11 and fixed-18). This clearly shows that even under the most uniform traffic load pattern, the ARTIC system still manages to deliver close to what the best possible fixed-delay one can produce and easily outperforms all the rest in terms of maintaining throughput. Note that any of the fixed-delay approaches that show better performance are applicable only to this specific simulation setup. That is, they are not capable of adapting to different environments.

An example is displayed in Figure 13 showing how the ARTIC system actually saves the overall travel delay by increasing the on-ramp delay (entrance delay). λ is set to a value so as to show a scenario where some fixed-delay cases are not saturated. The entrance delay is increased to 66 seconds compared to minimal numbers in the fixed-delay cases, but it is well rewarded by a much reduced travel time on freeway, leading to a saving of up to 15% reduction in total delay.

Histogram of TID values throughout the simulation is also given in Figure 14. Distribution of TID values for three different λ values are shown. It is clear that the ARTIC system is capable of self-adjusting the TID values according to different λ values. As shown in the distribution, under the smallest λ value, the TID distribution tilts toward the smaller range more than the others with the larger λ values. As λ value increases, the distribution gradually shifts toward the larger range. Notably, other than the high occurrence point at TID= 0 where no control is needed for low traffic load, all three distribution curves peak at around 18 to 20 TUs which closely approximates the input load rate, since 2.5 cars/sec for all 10 on-ramps equals to 0.25 cars/sec per on-ramp which translates to 4 seconds per car or 20 TUs per car per on-ramp. This result clearly demonstrates that the ARTIC system is capable of maintaining a TID value resembling the load rate when needed.

Simulation Setup #2: Spatially Non-Uniform Traffic Load

In the second simulation environment, traffic load is provided for each of the ten on-ramps with a different λ value, in a range of 0.8 to 1.2 in relative value. This environment leads to a better approximation setup to a real-life freeway system where not all on-ramps incur the same traffic load. Figure 15 gives the comparison results. In (A),

the ARTIC system again delivers a performance relatively close to the best “fixed-injection-delay” ones, from 12 to 15 TUs. In this new environment with spatially variant load, fewer “fixed-TU” ones can better the ARTIC. All the other fixed-delay ones including the no-control one saturate much earlier. In (B), throughput value is again sustained nicely by the proposed ARTIC system well beyond the saturation point, while only the fixed-18-TU one is able to match it with all others either maintaining a lower throughput (e.g. fixed-20) or only a small fraction of original throughput (e.g. w/o-control, fixed-12, fixed-15 and fixed-17). This shows that, compared to the uniform load situation, the gain from using the ARTIC system is even more significant, especially in terms of maintaining throughput.

Simulation Setup #3: Temporally Non-Uniform Traffic Load

In the previous two simulation environments, although traffic load is randomly generated following an exponential distribution based on λ, load is still considered uniform from a long-term time frame to another throughout a simulation run. In this simulation environment, λ value is further deviated from time to time to have a closer approximation to the real-life traffic temporal undulation. λ value is changed every 40,000 TUs within the range of 0.8 and 1.3 in relative value with a repeating pattern of (0.8, 1.0, 1.3, 1.0, 0.9). That is, a cycle of 40,000 seconds is repeated as shown in Figure 16(A). Under this environment setup, all but three approaches saturated the system beyond recovery, i.e. the queues on the on-ramps continue to grow. Figure 17 gives the simulation result. Only the fixed-delay ones without being saturated are shown in this figure. The ARTIC system delivers the best performance (closely matched by the fixed-19-TU one) with the fixed-20 one trailing by a significant margin with about 6,000 more seconds required for the freeway to recover from the peak λ time durations, about 30% longer mired in heavy traffic than the ARTIC one. With ARTIC system in place, motorists are expected to spend an average of from 6 minutes to 20 minutes to cover an average of 10 killometers, while the fixed-20-TU one stretches the delay up to about 30 minutes during peak hours.

Simulation Setup #4: Temporally Twin-Peak Traffic Load
 

Conclusion

This paper first proposed a general methodology providing all generic resource systems a framework for designing an autonomous performance-optimizing technique. As aforementioned, this framework can be used for various applications that aim at optimizing resource sharing. The authors have also successfully applied this methodology to another application in the area of task scheduling. Based on this framework, a traffic injection control mechanism for freeway systems is then designed to minimize traffic delay in real time. From the simulation results, the ARTIC system thus designed has shown a great potential in leading to a practical system. In order to further improve its practicality, more simulations are needed on additional traffic systems with a non-uniform spatial capacity coupled with non-uniform temporal traffic load.

References
 

[1] T. Bellemans, B. De Schutter, and B. De Moor, “Anticipative Ramp Metering Control for Freeway Traffic Networks”, Proceedings of the 16th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2004), Leuven, Belgium, Paper 320, July 2004.
 

[2] Cambridge Systematics, Inc., Oakland, California, “Twin Cities Ramp Meter Evaluation -Final Report”, prepared for the Minnesota Department of Transportation, Feb. 2001.
 

[3] L. Chu and X. Yang, “Optimization of the ALINEA Ramp-Metering Control Using Genetic Algorithm with Micro-simulation”, TRB Annual Meeting, 2003.
 

[4] L.E. Haefner and M.-S. Li, “Traffic Flow Simulation for an Urban Freeway Corridor”, Transportation Conference Proceedings, 1998.
 

[5] A. Kotsialos, M. Papageorgiou, and A. Messmer, “Optimal Coordinated and Integrated Motorway Network Traffic Control”, In Proceedings of the 14th International Symposium of Transportation and Traffic Theory (ISTTT), Jerusalem, Israel, pp. 621-644, July 1999.
 

[6] A. Kotsialos and M. Papageorgiou, “Efficiency Versus Fairness in Network-Wide Ramp Metering”, Proc. 4th IEEE Conference on Intelligent Transportation Systems, pp. 1190-1195, 2001.
 

[7] A. Kotsialos, M. Papageorgiou, C. Diakaki, Y. Pavlis and F.Middleham, “Traffic Flow Modeling of Large-Scale Motorway Networks Using the Macroscopic Modeling Tool METALINE”, IEEE Transactions on Intelligent Transportation Systems 3(4), pp. 282292, 2002.
 

[8] A. Kotsialos, M. Papageorgiou, M. Mangeas and H. Haj-Salem, “Hierarchical Nonlinear Model-Predictive Ramp Metering Control for Freeway Networks”, The Fifth Triennal Symposium on Transportation Analysis, Le Gosier, Guadeloupe, June 2004

[9] M. Mangeas, H. Haj-Salem, A. Kotsialos and M. Papageorgiou, “Application of a Nonlinear Coordinated and Integrated Control Strategy to the Ile-De-France Motorway Network”, Transportation Research C, 2000.

[10] D.P. Masher, D.W. Ross, P.J. Won, P.L. Tuan, H.M. Zeidler and S. Peracek, “Guidelines for Design and Operating of Ramp Control System”, Stanford Research Institute Report NCHRP 3-2, SRI Project 3340, SRI, Menid Park, California, 1975.
 

[11] A.D. May, Traffic Flow Fundamentals.New Jersey: Prentice-Hill, 1990.
 

[12] A. Messmer and M. Papageorgiou, “METANET: A Macroscopic Simulation Program for Motorway Networks”, Traffic Engineering and Control, Vol. 31, pp. 466470, 1990.
 

[13] M. Papageorgiou, H. Hadj-Salem,and J.-M. Blosseville, “ALINEA: A Local Feedback Control Law for On-Ramp Metering”, Transportation Research Record, Vol. 1320, pp. 58-64, 1991.
 

[14] M. Papageorgiou, H. Hadj-Salem,and F.Middelham, “ALINEA local Ramp Metering -Summary of Field Results”, Transportation Research Record, Vol. 1603, pp. 90-98, 1998.
 

[15] M. Papageorgiou and A. Kotsialos, “Freeway Ramp Metering: An Overview”, IEEE Trans. on Intelligent Transportation Systems, Vol. 3, pp. 271-281, 2002.
 

[16] D.I. Robertson, “TRANSYT Method for Area Traffic Control”, Traffic Engineering control, Vol.10, pp. 276-281, 1996.
 

[17] E. Smaragdis and M. Papageorgiou, “A Series of New Local Ramp Metering Strategies”, 82nd TRB Annual Meeting, Washington D.C., 2003.
 

[18] C. Taylor and D. Meldrum “A Programmers Guide to the Fuzzy Logic Ramp Metering Algorithm: Software Design, Integration, Testing, and Evaluation”, Technical report, WA-RD 481.3, Universtiy of Washington, February 2000.

[19] J.A. Wattleworth, “Peak-Period Analysis and Control of a Freeway System”, Highway Research Record, Vol. 157, pp. 1-21, 1965.