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This paper examines many-to-many carpooling services with advance reservations, and constraints on waits and detours. An analytic model yields approximate formulas for the percent of requests matched, the expected vehicle-distance driven, and the passenger-distance traveled in some idealized scenarios. Simulations of these scenarios validate the formulas. In the most favorable cases carpooling reduces the vehicle-kilometers driven by all users by a few percent. The paper also shows how the formulas can be used by service providers to optimize offerings, and by city governments to design regulatory policies that will perform as intended. A simple example illustrates the ideas.

This paper explores whether upper bound guarantees to detour distances can be introduced in ride sharing services. By ride sharing we mean taxi ride aggregation services such as Uber-Pool. The paper develops an analytical model that for a given demand relates the guarantee levels to (i) the percent of rides that can be matched, (ii) the expected vehicle distance traveled; (iii) the expected passenger distance traveled; (iv) the fleet size required, and (v) the average passenger trip time including waiting and riding. The formulas developed reveal that for the full range of feasible fleet sizes, ridesharing with detour distance guarantees outperforms both ordinary ride-sharing and ordinary taxi. This suggests that there is a business opportunity that is not currently being exploited.

Author(s): Itani, Ibrahim, MS; Cassidy, Michael J., PhD; Daganzo, Carlos F., PhD | Abstract: An agent-based, multichannel simulation of a downtown area reveals the impacts of both redistributing traffic demand with time-dependent congestion pricing, and supplying extra capacity by banning left turns. The downtown street network was idealized, and loosely resembles central Los Angeles. On the demand-side, prices were set based on time-ofday and distance traveled. On the supply side, left-turn maneuvers were prohibited at all intersections on the network. Although both traffic management measures reduced travel costs when used alone, the left-turn ban was much less effective than pricing. When combined with pricing under congested conditions, however, the left-turn ban’s effectiveness increased considerably—it more than doubled in some cases. Furthermore, the two measures combined reduced travel costs in synergistic fashion. In some cases, this synergistic effect was responsible for 30% of the cost reduction. This strong synergy suggests that turning bans should be considered as an added option when contemplating congestion pricing.

The paper presents a general analytic framework to model transit systems that provide door-to-door service. The model includes as special cases non-shared taxi and demand responsive transportation ( DRT ). In the latter we include both, paratransit services such as dial-a-ride ( DAR ), and the form of ridesharing ( shared taxi ) currently being used by crowd-sourced taxi companies like Lyft and Uber. The framework yields somewhat optimistic results because, among other things, it is deterministic and does not track vehicles across space. By virtue of its simplicity, however, the framework yields approximate closed form formulas for many cases of interest.

As every user knows buses tend to bunch. To alleviate this problem, transit agencies introduce slack into their schedules and then hold buses back to schedule at pre-established control points along their routes. Unfortunately, this practice retards buses and only works with low frequency systems; i.e., when the headways are long. For higher frequency systems, which effectively operate without a schedule, headway-based control strategies show promise but unfortunately, they also retard buses. To alleviate bus retardation in all its forms, transit signal priority (TSP) is commonly used. Curiously however, the potential of TSP to enhance bus control practices has not been explored in detail. This paper evaluates a form of TSP designed for this purpose. In this version of TSP, which we call conditional signal priority (CSP), buses send priority requests only when the requests improve reliability. The evaluation includes both scheduled (low frequency) systems, and unscheduled (high frequency) systems operated by headways. A mathematical model based on Brownian motion is proposed for the former. This model allows us to optimize the CSP strategy and to develop formulas for the expectation and variance of the steady state deviations from the schedule-with CSP, with conventional TSP and with neither. Simulations confirm the accuracy of these findings. It is found that CSP improves reliability considerably and that it is better for traffic than TSP because it reduces the number of priority requests. For high frequency systems operated with a fixed number of buses CSP is shown to not just improve reliability and reduce the number of priority requests of TSP (by about 50%) but also to reduce the average headway by speeding up the buses. This paper has been accepted for a podium presentation at the 23rd International Symposium on Transportation and Traffic Theory (ISTTT23) July 24-26, 2019 in Lausanne, CH.

This paper shows that severe congestion on streets controlled by traffic signals can be reduced by dynamically adapting the signal offsets to the prevailing density with a simple rule that keeps the signals’ green-red ratios invariant. Invariant ratios reduce a control policy’s impact on the crossing streets, so a policy can be optimized and evaluated by focusing on the street itself without the confounding factors present in networks. Designed for heavy traffic with spillovers, the proposed policies are adaptive and need little data - they only require average traffic density readings and no demand forecasts. A battery of numerical experiments simulating the dynamics of rush hour traffic on a congested, homogeneous circular street reveals that the proposed form of adaptation reduces the duration of the rush and overall congestion compared with pre-timed control strategies. Eighteen different adaptation policies were considered. All inspect the street densities periodically and simulataneously, and retime the signals immediately thereafter. The period is a fixed multiple of the cycle. The street is evenly divided into sections that contain a set number of consecutive blocks and signals. The offset is the same for all blocks in a section. Three inspection intervals and six section sizes were tested. The latter ranged from a single block/signal to the whole street. It was found that adaptation worked best when sections were large and adaptation frequent. The effects were considerable across all scenarios. For a short street with a short rush and high input flows the probabilistic incidence of gridlock was reduced from 10% to 0%, and the average duration of a trip from 216s to 181s. For a long street with a long rush and high input flows the gridlock probability was reduced from 23% to 0% and the average trip duration from 2037s to 1143s.

This paper considers a signalized street of uniform width and blocks of various lengths. Its signals are pretimed in an arbitrary pattern, and traffic on it behaves as per the kinematic-wave/variational theory with a triangular fundamental diagram. It is shown that the long run average flow on the street when the number of cars on the street (i.e. the street’s density) is held constant is given by the solution of a linear program (LP) with a finite number of variables and constraints. This defines a point on the street’s macroscopic fundamental diagram. For the homogeneous special case where the block lengths and signal timings are identical, all the LP constraints but one are redundant and the result has a closed form. In this case, the LP recipe matches and simplifies the so-called “method of cuts”. This establishes that the method of cuts is exact for homogeneous problems. However, in the more realistic inhomogeneous case the difference between the two methods can be arbitrarily large. The paper uses the LP method to obtain the macroscopic fundamental diagrams arising under four different traffic coordination schemes for streets with four different block length configurations. It is found that the best scheme depends on the prevailing density. Curiously, the popular scheme in which all the traffic green phases are started synchronously wins only in rare circumstances. Its performance is particularly underwhelming when the street’s blocks are long. The paper also presents density-aware numerical methods to optimize the signal offsets for 1-way and 2-way streets. For 1-way streets operated with a common cycle the method reduces to a simple graphical construction . In this case the resulting flow matches the flow that would arise if all of the street’s intersections except one with the shortest green phase had been eliminated.

Incorporating network configurations in bridge management problems is computationally difficult. Because of the interdependencies among bridges in a network, they have to be analyzed together. Simulation-based numerical optimization techniques adopted in past research are limited to networks of moderate sizes. In this paper, a simple framework is developed to determine optimal maintenance plans for large networks with many bridges. The objective is to minimize disruption, specifically, the extra travel distance caused by potential bridge failures over a planning horizon and under a budget constraint. It is conjectured and then verified that the expected increase in vehicle-miles traveled due to failures can be approximated by the sum of expected increases due to individual failures. This allows the network-level problem to be decomposed into single-bridge problems and tackled efficiently. The computational effort increases linearly with the number of bridges.

A growing literature exploits macroscopic theories of traffic to model congestion pricing policies in downtown zones. This study introduces trip length heterogeneity into this analysis and proposes a usage-based, time-varying congestion toll that alleviates congestion while prioritizing shorter trips. Unlike conventional trip-based tolls the scheme is intended to align the fees paid by drivers with the actual congestion damage they do, and to increase the toll’s benefits as a result. The scheme is intended to maximize the number of people that finish their trips close to their desired times. The usage-based toll is compared to a traditional, trip-based toll which neglects trip length. It is found that, like trip-based tolls, properly designed usage-based tolls alleviate congestion. But they reduce schedule delay more than trip-based tolls and do so with much smaller user fees. As a result usage-based tolls leave most of those who pay with a large welfare gain. This may increase the tolls’ political acceptability.

Buses are known to be an unstable system, and as a result they struggle to stay on schedule and/or to maintain regular headways. In current practice, transit agencies add significant slack to the schedule and hold buses until their scheduled departure time at certain stations along the route. These practices can smooth out small disruptions but are not robust to large disruptions. Various headway-based strategies have been proposed but these also rely on holding buses and therefore reduce their average pace. Transit signal priority (TSP) is commonly used to reduce the signal delay that buses experience but the potential for signals to serve as an additional control agent to enhance bus reliability has not been systematically evaluated. We are especially interested in evaluating conditional signal priority (CSP), in which buses send priority requests only when a condition is met. A mathematical model based on Brownian motion is proposed for bus systems with a schedule where pairing is not a concern and is used to develop formulas for the expectation and variance of steady state deviation. Simulation is used to confirm these findings and to evaluate the case of bus systems operated by headway where pairing is a concern. The analytical and simulation results show that CSP can be used not just to accelerate buses but to improve their reliability. CSP is as good as TSP in some scenarios and better in others. In all scenarios it is better for other traffic because it sends fewer priority requests.

Conventional wisdom states that transit riders are averse to transfers and that consequently bus networks should be designed to limit their number. Probably as a result of this belief, many real bus systems try to connect as many origins and destinations as possible without transfers, so they are usually composed of long, circuitous routes with redundant overlapping sections – and the resulting bus map is hard to understand. If coverage is extensive, many routes are needed. Economics then prevents an agency from populating all routes with sufficient buses to provide attractively frequent service. This low frequency and the complicated circuitous map discourage transfers, perpetuating the belief that people are averse to transferring. Not surprisingly, the percentage of bus trips that includes a transfer has been reported to be: 1.5% for Boston, 3% for New York, 13% for London, and 16% for Melbourne. The Nova Xarxa in Barcelona was designed with a different paradigm. It was designed and deployed to cover the whole city on the belief that if a bus map is easy to understand, and has direct lines with frequent service and ubiquitous transfer points then the bus system would become more appealing, people would transfer more freely and become users of the network rather than its single lines. With this design paradigm, a city can be covered with fewer lines, which can be depicted on a simple map. The lines can in turn be economically populated with sufficient buses to deliver the high frequency required to encourage transfers. Could this work? To answer this question and see whether there is truth in the beliefs underlying the new paradigm, this paper examines data from the first three deployment phases of the Nova Xarxa (from 2012 to 2015). It is found that the Nova Xarxa is already attracting more demand than the network it replaced. This attests to its appeal. Furthermore, this demand has increased disproportionately with the number of lines opened for service in each phase, revealing that some people are using the Nova Xarxa as a network. The paper further shows that this growth is underpinned by transfers -- at the end of 2015, the percentage of trips that involved a transfer was approximately 26%, and it reached a maximum of 57% for line V7. These numbers should increase considerably (to 44% and 66%, respectively) once the Nova Xarxa is completed in 2018 and passengers have even more opportunities for transferring. The numbers disprove the conventional wisdom. They strongly suggest that transit providers can attract more demand by providing transfer-friendly networks that can be used as such and not as an inefficient aggregation of individual lines.

According to Euler–Lagrange duality principle of kinematic wave (KW) theory any well-posed initial value traffic flow problem can be solved with the same methods either on the time–space (Euler) plane or the time vs vehicle number (Lagrange) plane. To achieve this symmetry the model parameters and the boundary data need to be expressed in a form appropriate for each plane. It turns out, however, that when boundary data that are bounded in one plane are transformed for the other, singular points with infinite density sometimes arise. Duality theory indicates that solutions to these problems must exist and be unique. Therefore, these solutions should be characterized. The paper shows that the only added feature of these solutions is a new type of shock that can contain mass and we call a supershock. Nothing else is different. The evolution laws of these shocks are described. Solution methods based on these laws for problems with singularities are also presented. The methods apply to problems with monotone speed–density relations and not necessarily concave fundamental diagrams. In accordance with duality theory they can be used with both, the Euler and Lagrange versions of a problem.

The day-long system optimum (SO) commute for an urban area served by auto and transit is modeled as an auto bottleneck with a capacitated transit bypass. A public agency manages the system’s capacities optimally. Commuters are identical except for the times at which they wish to complete their morning trips and start their evening trips, which are given by an arbitrary joint distribution. They value unpunctuality – their lateness or earliness relative to their wish times – with a common penalty function. They must use the same mode for both trips. Commuters are assigned personalized mode and travel start times that collectively minimize society’s generalized cost for the whole day. This includes unpunctuality penalties, queuing delays, travel times and out-of-pocket costs for users, as well as travel supply costs and externalities for society. It is shown that in a SO solution there can be no queuing and that the set of SO solutions forms a convex set. Furthermore, if the schedule penalty that users suffer due to unpunctuality is separable into morning and evening components, then the set of commuters traveling by the same mode arrive at work and depart from work in the order of their wishes. These orders are in general different in the morning and the evening. It is also shown that there always is a SO solution in which users are at all times, and on both modes, either punctual or flowing at capacity. These problem properties are used to identify search methods, both, for SO solutions and for time-dependent tolls and transit fares that preserve the solutions as Nash equilibriums. In every case studied, these prices exist. They must peak concurrently for the two modes in both periods. In special cases involving only one mode, only one period or concentrated demand the solution to the complete problem decomposes by period conditional on the number of transit users, and this facilitates the solution. In these cases the day-long SO cost is the sum of the SO costs for the two peaks considered separately. However, this is not true in general – the solution obtained by combining the two single-period solutions can be infeasible. When this happens, the optimum day-long cost will exceed the sum of the single-period costs. The discrepancy is about 40% of the total schedule penalty for an example representing a large city. Thus, to develop realistic policies the day-long problem must be addressed head on. An approximate method that yields closed form formulas for the case with uniformly distributed wishes is presented.

This paper extends Vickrey's (1969) commute problem for commuters wishing to pass a bottleneck for both cars and transit that share finite road capacity. In addition to this more general framework considering two modes, the paper focuses on the evening rush, when commuters travel from work to home. Commuters choose which mode to use and when to travel in order to minimize the generalized cost of their own trips, including queueing delay and penalties for deviation from a preferred schedule of arrival and departure to and from work. The user equilibrium for the isolated morning and evening commutes are shown to be asymmetric because the schedule penalty in the morning is the differ- ence between the departure and wished curves, and the schedule penalty in the evening is the difference between the arrival and wished curves. It is shown that the system optimum in the morning and evening peaks are symmetric because queueing delay is eliminated and the optimal arrival curves are the same as the departure curves. The paper then considers both the morning and evening peaks together for a single mode bottleneck (all cars) with identical travelers that share the same wished times. For a schedule penalty function of the morning departure and evening arrival times that is positive definite and has certain properties, a user equilibrium is shown to exist in which commuters travel in the same order in both peaks. The result is used to illustrate the user equilibrium for two cases: (i) commuters have decoupled schedule preferences in the morning and evening, and (ii) commuters must work a fixed shift length but have flexibility when to start. Finally, a special case is considered with cars and transit: commuters have the same wished order in the morning and evening peaks. Commuters must use the same mode in both directions, and the complete user equilibrium solution reveals the number of commuters using cars and transit and the period in the middle of each rush when transit is used.

This paper extends Vickrey's (1969) commute problem for commuters wishing to pass a bottleneck for both cars and transit that share finite road capacity. In addition to this more general framework considering two modes, the paper focuses on the evening rush, when commuters travel from work to home. Commuters choose which mode to use and when to travel in order to minimize the generalized cost of their own trips, including queueing delay and penalties for deviation from a preferred schedule of arrival and departure to and from work. The user equilibrium for the isolated morning and evening commutes are shown to be asymmetric because the schedule penalty in the morning is the difference between the departure and wished curves, and the schedule penalty in the evening is the difference between the arrival and wished curves. It is shown that the system optimum in the morning and evening peaks are symmetric because queueing delay is eliminated and the optimal arrival curves are the same as the departure curves. The paper then considers both the morning and evening peaks together for a single mode bottleneck (all cars) with identical travelers that share the same wished times. For a schedule penalty function of the morning departure and evening arrival times that is positive definite and has certain properties, a user equilibrium is shown to exist in which commuters travel in the same order in both peaks. The result is used to illustrate the user equilibrium for two cases: (i) commuters have decoupled schedule preferences in the morning and evening, and (ii) commuters must work a fixed shift length but have flexibility when to start. Finally, a special case is considered with cars and transit: commuters have the same wished order in the morning and evening peaks. Commuters must use the same mode in both directions, and the complete user equilibrium solution reveals the number of commuters using cars and transit and the period in the middle of each rush when transit is used.

Through the use of a profit-maximizing continuum approximation model, this paper systematically analyzes the development and structure of informal transit systems as a function of the network, user, and modal characteristics. This study examines the evening commute problem along a linear corridor where passengers originate uniformly from a central business district and have destinations uniformly distributed along the corridor. Informal transit drivers who are profit-maximizing will be compared against the traditional case of coordinated, government service that aims to maximize the total welfare. Policies, such as fare regulation and vehicle licensing schemes, will be presented to help rationalize informal transit service using a government-operated service as the baseline.

The morning commute problem for a single bottleneck is extended to model mode choice in an urban area with time-dependent demand. This extension recognizes that street space is shared by cars and public transit. It is assumed that transit is operated independently of traffic conditions, and that when it is operated it consumes a fixed amount of space. As a first step, a single fixed-capacity bottleneck that can serve both cars and transit is studied. Commuters choose which mode to use and when to travel in order to minimize the generalized cost of their own trip. The transit agency chooses the headway and when to operate. Transit operations reduce the bottleneck’s capacity for cars by a fixed amount. The following results are shown for this type of bottleneck: 1. If the transit agency charges a fixed fare and operates at a given headway, and only when there is demand, then there is a unique user equilibrium. 2. If the transit agency chooses its headway and time of operation for the common good, then there is a unique system optimum. 3. Time-dependent prices exist to achieve system optimum. Finally, it is also shown that results 2 and 3 apply to urban networks.

This paper considers the optimization of public infrastructure systems, recognizing that these systems serve multiple user classes. Under the guidance of a policy-making body, the analyst chooses both the system design, including its layout and control, and the prices to be charged for the service. The goal of the optimization is to maximize society's welfare recognizing that the system's performance will in general depend on the system's demand, and vice versa. The optimization problem is first formulated in its full complexity. Under defined circumstances, the problem decomposes into three sub-problems that can be solved sequentially. The resulting design is independent of how net user benefits are measured. If the policy-making body does not specify cost or demand targets, and instead assesses benefits by means of consumer surplus then the optimum design is still the solution of a conventional design problem with fixed demand. In this case, however, the demand has to be obtained iteratively using a marginal cost pricing rule.

A model with few variables is said to be parsimonious. If it is also analytically tractable, physically realistic, and conceptually insightful, it is said to be effective. Effective parsimonious models have long been used in fields such as economics and applied physics to describe the aggregate behavior of systems as opposed to the behavior of their individual parts. In transportation, these models are particularly well suited to address big picture questions because they provide insights that might be lost when focusing on details. This paper presents an abbreviated history of effective parsimonious models in the transportation field, classified by sub-area: regional and urban economics, traffic flow, queuing theory, network dynamics, town planning, public transportation, logistics, and infrastructure management. The paper also discusses the benefits of these models—fewer data requirements, reduced computational complexity, improved system representation, insightfulness—and ways of constructing them. Two examples, one from logistics and one from urban transportation, are used to illustrate these points. Finally, the paper discusses ways of expanding the application of effective parsimonious models in the transportation field.

Recently cities have been converting traditional one-way downtown street networks to two-way operation partly because one-way networks are seen as confusing and as less conducive to economic activity and a livable environment and they require vehicles to travel longer distances on average. However, one of the main disadvantages of such conversions is thought to be a reduction in the network's ability to serve vehicles. Intersections in two-way networks can serve fewer vehicles per unit time than their one-way counterparts. Several studies have assessed the differences between these two types of networks, but most studies are site specific and do not consider the best possible two-way networks. This paper presents an analytical model that uses macroscopic analysis techniques to compare various one-way and two-way networks using their trip-serving capacities. This metric is a key indicator of network performance. Two-way networks can serve more trips per unit time than one-way networks when average trip lengths are short. This study also found that two-way networks in which left-turn movements were banned at intersections could always serve trips at a higher rate than one-way networks could, even long trips. Thus, the trip-serving capacity of a one-way network can actually be increased when it is converted to two-way operation if left turns are banned. In this way, livability and efficiency objectives can be achieved simultaneously. This framework can be used by planners and engineers to determine how much a network's capacity changes after a conversion, and also to unveil superior conversion options.

In theory midblock presignals can be used to increase the capacity of signalized intersections [The authors define “presignal” as “a set of signal heads that are installed in the middle of a block upstream of an intersection.”—Ed.]. The capacity is increased because presignals can reorganize how traffic is stored between a presignal and an intersection downstream. However, different vehicle classes have different acceleration characteristics, and the effectiveness of presignals hinges on the assumption of linear superposition; that is, the total time to discharge a mixture of distinct vehicle classes equals the sum of the times to discharge each vehicle class separately. This assumption has not been tested in the field. In this study, results from a natural experiment are used to validate the assumption for the case of cars and buses. The effectiveness of presignals to increase intersection capacity is also demonstrated.

As is well known, bus systems are naturally unstable. Without control, buses on a single line tend to bunch, reducing their punctuality in meeting a schedule. Although conventional schedule-based strategies that hold buses at control points can alleviate this problem these methods require too much slack, which slows buses. This delays on-board passengers and increases operating costs. It is shown that dynamic holding strategies based on headways alone cannot help buses adhere to a schedule. Therefore, a family of dynamic holding strategies that use bus arrival deviations from a virtual schedule at the control points is proposed. The virtual schedule is introduced whether the system is run with a published schedule or not. It is shown that with this approach, buses can both closely adhere to a published schedule and maintain regular headways without too much slack. A one-parameter version of the method can be optimized in closed form. This simple method is shown to be near-optimal. To put it in practice, the only data needed in real time are the arrival times of the current bus and the preceding bus at the control point relative to the virtual schedule. The simple method was found to require about 40% less slack than the conventional schedule-based method. When used only to regulate headways it outperforms headway-based methods.

A simple model of traffic flow is used to analyze the spatio-temporal distribution of flow and density on closed-loop homogeneous freeways with many ramps, which produce inflows and allow outflows. As we would expect, if the on-ramp demand is space-independent then this distribution tends toward uniformity in space if the freeway is either: (i) uncongested; or (ii) congested with queues on its on-ramps and enough inflow to cause the average freeway density to increase with time. In all other cases, however, including any recovery phase of a rush hour where the freeway’s average density declines, the distribution of flow and density quickly becomes uneven. This happens even under conditions of perfect symmetry, where the percentage of vehicles exiting at every off ramp is the same. The flow-density deviations from the average are shown to grow exponentially in time and propagate backwards in space with a fixed wave speed. A consequence of this type of instability is that, during recovery, gaps of uncongested traffic will quickly appear in the unevenly congested stream, reducing average flow. This extends the duration of recovery and invariably creates clockwise hysteresis loops on scatter-plots of average system flow vs. density during any rush hour that oversaturates the freeway. All these effects are quantified with formulas and verified with simulations. Some have been observed in real networks. In a more practical vein, it is also shown that the negative effects of instability diminish (i.e., freeway flows increase) if (a) some drivers choose to exit the freeway prematurely when it is too congested and/or (b) freeway access is regulated in a certain traffic-responsive way. These two findings could be used to improve the algorithms behind VMS displays for driver guidance (finding a), and on-ramp metering rates (finding b).

A separate turn phase is often used on the approach leg to an intersections with heavy left turns. This wastes capacity on the approach because some of its lanes cannot discharge during its green phases. The paper shows that the problem can be eliminated by reorganizing traffic on all the lanes upstream of an intersection using a mid-block pre-signal. If drivers behave deterministically, the capacity that can be achieved is the same as if there were no left turns. However, if the reorganization is too drastic, it may be counterintuitive to drivers. This can be remedied by reorganizing traffic on just some of the available lanes. It is shown that such partial reorganization still increases capacity significantly, even if drivers behave randomly and only one lane is reorganized. The paper shows how to optimize the design of a pre-signal system for a generic intersection. It also identifies both, the potential benefits of the proposed system for a broad class of intersections, and the domain of application where the benefits are most significant.

A recent study reported that the Macroscopic Fundamental Diagram of a medium size city exhibited a clockwise hysteresis loop on a day in which a major disturbance caused many drivers to use unfamiliar routes. It is shown below that, even in a perfectly symmetric network with uniform demand, clockwise loops are to be expected when there are disturbances, especially if the disturbances cause a significant fraction of the drivers to not change routes adaptively. It is also shown that when drivers are not adaptive networks are inherently more unstable as they recover from congestion than as they are loaded. In other words, during recovery congestion tends more strongly toward unevenness because very congested areas clear more slowly than less congested areas. Since it is known that uneven congestion distributions reduce network flows, it follows that lower network flows should arise during recovery, resulting in clockwise loops. Fortunately, the presence of a sufficient number of drivers that choose routes adaptively to avoid congested areas helps to even out congestion during recovery, increasing flow. Thus, clockwise loops are less likely to occur when driver adaptivity is high.