The following notation is used in the model: ????, ?? = Network of freeways ?? = Set of nodes in network ?? ?? = Set of links ???? ??n network ?? ???? = Set of links ???? ??n netwotk
Traffic Incidents as a Cause of Non-Recurring Congestion
Traffic congestion is a persistent topic in transportation research because it affects daily life, and incidents—non-recurring events that reduce roadway capacity—are a major contributor to delay They account for more than half of urban non-recurring congestion and nearly all delay in rural areas In the literature, an incident is defined as any non-recurring event that affects roadway capacity, with examples including disabled vehicles, stranded motorists, debris in the roadway, spilled loads, vehicle crashes, work zones, obstructions to traffic, dead animals, and other hazards.
Traffic incidents account for roughly 25% of congestion on U.S roadways, causing lane blockages during peak hours and adding about four minutes to travel time due to delays, according to the National Traffic Incident Management Coalition About 20% of crashes are caused by a preceding crash, and the time needed to clear the initial incident increases the risk of a secondary crash The U.S Department of Transportation reports that roughly 14–18% of crashes result from other incidents, highlighting how incident chains contribute to overall roadway risk.
To restore network performance quickly, organizations should implement systematic incident response procedures that guide the detection, response, and clearance of incidents State agencies and transportation professionals have achieved substantial delay reductions by deploying incident management programs that streamline recovery from traffic incidents Undoubtedly, rapid clearance of incidents is the key factor in restoring network performance, highlighting the value of proactive incident management for minimizing travel delays and maintaining roadway efficiency.
Traffic Incident Management
Effective incident management procedures can significantly mitigate the adverse impacts of incidents by guiding policies and strategies that shorten incident clearance duration Traffic Incident Management (TIM) aims to recognize, report, and remove incidents quickly to restore normal traffic flow and reduce congestion For TIM to succeed, programs must be well designed and tailored to the local region, dynamically managed, structurally planned, inter-jurisdictional, multidisciplinary, and thoroughly documented In practice, TIM comprises seven steps: detection, verification, response, site management, traffic management, clearance, and recovery.
Incident delay is the sum of Detection Time, Verification Time, Response Time, Clearance Time, and Recovery Time The Response Time—the interval from incident detection to the incident management team's arrival on site to clear the freeway—depends heavily on the incident type Detection Time and Response Time together constitute a large share of the total delay, but these durations can be significantly reduced with a well-planned incident management strategy.
Traffic management uses different approaches to quickly respond to unexpected incidents such as variable message signs, ramp metering, temporary shoulder use or other strategies [12]
New York State offers a successful example of Traffic Incident Management (TIM) through a program fostered by NYSDOT that coordinates human, institutional, mechanical, and technological resources in a systematic, planned way to shorten incident duration and minimize impacts The Strategic Highway Safety Plan for New York outlines TIM goals to protect motorists, crash victims, and responders; to ensure appropriate response, investigation, and safe clearance of incidents; to improve interagency collaboration for planned events; and to restore traffic flow as quickly as possible while managing conditions until normal operations return The Federal Highway Administration notes that, in a typical year, deaths among responders on New York highways average 12 fire and rescue personnel, 5 police officers, 60 towing and recovery operators, and more than 100 transportation professionals from DOTs, public works, and safety service patrol programs.
Strategies for implementing or continuing traffic incident management (TIM) programs include ensuring the correctness and use of TIM data, reducing incident clearance times through improved coordination between responders and motorist-assistance programs, and strengthening coordination among responders through enhanced training and communication They also involve supporting and expanding the Highway Emergency Local Patrol (HELP) program with increased operations and the creation of a HELP truck operator academy and curriculum, establishing regional TIM committees where they do not yet exist, educating emergency responders and the public on laws and best practices, promoting high-visibility apparel for emergency responders, highway workers, and tow operators, increasing the number of TIM training classes and identifying target audiences, establishing statewide end-of-queue notification protocols in coordination with ITS/TSMO operations, promoting awareness of the Move Over law, improving public knowledge of steer it/clear it best practices, and continuing to investigate and implement best practices for informing travelers before and through temporary traffic control zones.
Emergency Traffic Patrol
Examples of Emergency Patrol Programs
many metropolitan regions such as Los Angeles, Chicago, and Dallas-Fort Worth, implement patrol programs Examples of patrol programs are the following:
• H.E.L.P (Highway Emergency Local Patrol; New York)
• CHART (Coordinated Highways Action Response Team; Maryland)
• HERO (Highway Emergency Response Operators; Georgia)
In North Carolina, Freeway Service Patrols (FSPs), also known as Incident Management Assistance Patrols (IMAP), provide a key service by reducing the congestion impact of highway incidents while protecting the safety of motorists and involved emergency responders The FSP mission requires operators, dispatchers, and system managers to continuously adapt to dynamic highway conditions within the constraints of their agencies By operating within these constraints, agencies can maximize FSP effectiveness through deployment strategies that weigh the costs and benefits of FSP implementation at each location, improving overall incident management and traffic flow.
New York State's coordinated approach demonstrates that high levels of interagency cooperation can boost the effectiveness of field verification conducted by on-site responders In the Hudson Valley, Highway Emergency Local Patrol (HELP) vehicles are equipped with a live video streaming system that transmits footage to the Traffic Management Center (TMC) at the New York State Department of Transportation and the State Police These dash cameras provide real-time incident information to dispatchers, enabling faster and more appropriate deployment of personnel and equipment The use of streaming video has proven especially valuable for remote transportation and law enforcement teams by helping them assess incident characteristics and determine subsequent response needs.
Problem Statement
Three core challenges define the FSP problem in patrol planning: beat configuration, fleet size constraint, and truck allocation Beat configuration partitions the freeway network into patrol beats, with every link assigned to at least one beat The fleet size constraint determines the optimal number of trucks needed to fully cover the network while considering the cost of adding more trucks Truck allocation then assigns trucks to beats to minimize incident delay, leveraging patrols’ awareness of incidents through mean detection–response times This study proposes a mixed-integer programming model that simultaneously addresses these three issues and incorporates several additional aspects of the patrol program.
Report Structure
This paper is organized as follows: Section 2 surveys existing studies on freeway service patrols and clarifies their contributions to the field; Section 3 presents the proposed mathematical model and explains its components; Section 4 applies the model and the accompanying heuristics to a subset of Maryland’s freeway network managed by the Coordinated Highways Action Response Team (CHART); Section 5 analyzes extensions to the model, and Section 6 summarizes the results, presents the main conclusions, and outlines directions for future research.
The current state of research on freeway patrolling can be organized into two main strands Evaluation studies analyze the costs and benefits of both existing and proposed patrol programs, helping decision-makers weigh efficiency, safety, and cost-effectiveness Network design studies, by contrast, develop mathematical frameworks to guide the design and allocation of patrol resources, optimizing coverage, response times, and routing Together, these two categories provide a comprehensive view of how freeway patrolling can be evaluated and optimally designed, bridging practical impact with theoretical models.
Evaluation Studies
Research across multiple regions shows that Freeway Service Patrol (FSP) programs deliver clear economic and operational benefits Economic analyses indicate benefit-cost ratios ranging from 2.1 to 36.2 nationwide, underscoring substantial financial gains from FSP deployment Patrol trucks often detect incidents themselves, significantly reducing detection time, as illustrated by the San Francisco–Oakland FSP, which located about 92% of incidents In the Puget Sound region, Nee and Hallenbeck found that lane-blocking incidents take 7.5 minutes to respond to without FSP, but only 3.5 minutes with FSP in service, with patrol programs reducing incident response times by 19% to 77% The overall reduction in incident duration and motorist delay is the primary driver of the benefits attributed to FSP programs.
To estimate delay, researchers compare the effects of incidents with and without an FSP program [31] Non-recurrent congestion delay has been estimated using several approaches, including analytical methods based on deterministic queuing diagrams [32], shock-wave theory [33], heuristic methods [34], and simulation methods [35].
Skabardonis and Mauch developed a model to estimate the benefit-to-cost ratio of providing FSP service using empirical data and added a second model to predict the cost-effectiveness of proposed FSP beats that currently have no FSP service Evaluation studies show that patrol programs are cost-effective based on selected MOEs before and after implementation, with benefits driven by beat geometry, traffic characteristics, and the frequency and type of assisted incidents Moore and colleagues report that secondary incidents on Los Angeles freeways where FSP is implemented occur less frequently than what the literature commonly suggests Moreover, reducing response time is linked to shorter incident durations, with Khattak et al noting that a one-minute decrease in response time can yield a 0.6- to 1-minute reduction in incident clearance time Taken together, a substantial body of performance-evaluation studies confirms the effectiveness of incident-management programs in mitigating congestion caused by incidents.
Researchers estimate static or dynamic thresholds in space and time to define secondary incidents as a measure of FSP benefits Static thresholds apply a fixed spatial-temporal boundary to classify secondary incidents, such as within one mile upstream of a primary incident, while Moore et al defined secondary incidents as those occurring within two hours and within two miles of a primary incident using California Highway Patrol data In 2007, Sun and Chilukuri established a dynamic threshold method by tracking the back-of-queue location throughout the entire incident duration, showing that dynamic methods can change the count of secondary incidents by up to 30% In another study, Chou and Miller-Hooks proposed a simulation-based secondary incident filtering method (SBSIF) using the CORSIM microscopic simulation model.
A regression model was implemented for corner-point identification alongside the SBSIF method In a Virginia study by Zhang and Khattak [47], cascading incident event duration was analyzed, addressing not only single-pair events (one primary and one secondary incident) but also large-scale cases with a single primary incident and multiple secondary ones, classified as contained or extended using a deterministic queuing method—an extended event occurs when the last secondary incident is cleared, otherwise it is contained Later, Zhang and Khattak [47] advanced an incident management integration tool to estimate dynamic incident duration, secondary incident occurrence, and incident delays Similarly, Chung [48] presented a process to recognize secondary crashes caused by diverse types of primary crashes in the impact area and developed a method to distinguish non-recurring congestion from recurring congestion.
Network Design
Patrol programs have been studied broadly, but most research concentrates on evaluating overall performance and cost-benefit after implementation, with only a few efforts offering a formal mathematical framework to design the patrol network efficiently The deployment and placement of response patrol trucks are central to program efficiency, yet the literature lacks rigorous analytical methods to optimize this aspect Nevertheless, some ambiguous approaches have been proposed that claim to enhance patrol performance This section reviews general incident-response models and highlights more specific patrol-program models that appear in the current literature.
Some studies use historical data to identify patrol route locations For example, Khattak and Rouphail [28] relied on historical crash metrics—crashes per 100 million vehicle miles, crashes per mile per year, and lane-specific average annual daily traffic (AADT)—to pinpoint candidate routes They then assign each road segment a score based on these predicted parameters and rank the potential routes by computing the average score across segments.
Simulation studies continue to illuminate effective freeway service patrol (FSP) strategies Pal and Sinha developed a simulation model to evaluate and enhance FSP performance, focusing on total vehicle-hours in the system and conducting sensitivity analyses that show how fleet size, beat design, dispatch policies, patrol areas, and hours of operation can be adjusted to improve outcomes Their work highlights key parameters—fleet size, beat design, dispatch policies, patrol areas, and hours of operation—that warrant careful examination in case studies In a follow-up, Pal and Sinha proposed a mixed-integer programming model to determine optimal locations for incident response units with the goal of minimizing operating costs More recently, Ma et al used the Paramics microsimulation tool to quantify how different FSP service strategies influence incident duration and to guide dispatch decisions They compared two policies: (1) FSP vehicles following predetermined routes responding to incidents in the current travel direction, and (2) FSPs turning around at the next opportunity to serve incidents identified in the opposite direction The comparison under varying patrol headways showed that giving FSPs the ability to turn around to assist in the opposite direction offers greater benefits as headways increase (fewer patrol vehicles) and smaller benefits as headways decrease (more patrol vehicles) Taken together, these microsimulation-based findings illustrate how modeling can assess FSP patrol strategies and vehicle allocation options.
Different versions of the optimal freeway patrol service design problem have been formulated using mathematical programming techniques Sherali et al present two mixed-integer models to optimally assign multiple response units to multiple incidents, accounting for operation and opportunity costs Kim et al develop an integer-programming model to minimize total incident-incurred delay by optimizing the deployment locations of incident response units Daskin proposes a mixed-integer model to determine the dispatching policy and routing for incident response units These studies focus on optimal locations and dispatch policies but do not consider patrolling of incident-response units Early efforts, such as the Tennessee HELP program and the Maryland CHART program, use traffic and incident indices to identify important locations that should be covered in their networks.
Zografos et al propose a districting model to minimize incident-induced delay by optimally locating emergency response units, transforming freeway corridors into homogeneous-demand sections and assuming that each section’s demand is concentrated at its centroid Zhu et al evaluate incident-response performance under three allocation strategies: placing units near high-frequency incident locations, distributing units evenly across the network, or situating units at traffic operation centers to dispatch when an incident occurs In a separate study, Zhu et al develop a methodology to assess patrolling and dispatching strategies for emergency-response-unit allocation using field data from the I-495/I-95 Capital Beltway, concluding that the best strategy depends on factors such as incident frequencies, traffic characteristics, and available detection methods.
Petty developed a model that blends traffic theory with marginal benefit analysis to identify tow-truck locations that maximize the expected reduction in congestion Yin proposed a minimax bi-level programming model to determine fleet allocation that minimizes the maximum system travel time that may arise from incidents Together, these studies illustrate two distinct truck-allocation strategies driven by different objectives Our research extends this work by presenting a methodology to optimally allocate tow trucks that minimizes incident duration while also accounting for operating costs.
Khattak et al [62] present a method to determine, evaluate, and compare the most beneficial candidate facilities for expanding the FSP network by integrating incident indexes (including incident type distribution and delay estimation) with spatial analysis and average hourly freeway traffic volumes They assume that high-priority locations are already covered The approach does not aim to design beats or allocate trucks; it only ranks candidate locations where expansion would be most beneficial if pursued.
Yin (63) formulated a mixed integer nonlinear programming model to allocate patrol trucks across beats, aiming to minimize the expected loss under high-consequence incident scenarios within the FSP system, thereby advancing patrol optimization Daneshgar et al (64) proposed a hybrid model that combines deterministic and probabilistic approaches to estimate average response time and optimize patrol program performance by minimizing total response time and identifying the best beat configuration among existing beat structures in Tarrant County, Texas Building on this, Daneshgar and Haghani (65) developed a joint mixed-integer model to determine beat configuration and fleet size with a single depot, optimizing total response time, though they do not present a heuristic algorithm suitable for large networks Generally, these studies address beat configuration and fleet sizing to reduce response times and potential losses in patrol operations.
Contribution
Among the few studies to design the network for patrol programs, nearly all of them attempt to either design the beats or allocate trucks into the pre-designed beats and perform these two steps separately while these are truly interrelated Therefore, our research aims to present a model to merge these problems and determine the beat configuration, fleet size, and truck allocation together According to the literature, only one study by Lou [69] attempted a similar strategy The current study aims to present an improved and comprehensive model, and as a result, here, we explain what is completed in Lou’s work and explain significant contributions that are made by the current study Lou presented a non-linear model to determine beat configuration and fleet allocation with the objective of minimizing the overall average incident response time However, in developing this non-linear model, many simplistic assumptions are made such as assuming the number of beats is given, or a total number of trucks (fleet size) is assumed They proposed a non-linear model [69] which aims to minimize only the response time as part of the total delay and does not consider truck’s expenses Our research aims to present a comprehensive mixed-integer programming model to design the network for freeway service patrol programs This model aims to concurrently determine the optimal beat configuration along with the optimal fleet size and trucks allocation to minimize incident-incurred delay while the operational cost is taken into account, as well
The proposed model and heuristic approaches, as well as the examples, experiments, and results presented in sections 3-6 and 8, are part of the doctoral dissertation [70].
Consider a directed graph, G(N,A), representing a network of freeways where N and L represent sets of nodes and links, respectively We assume t ij is the travel time, and f ij is the number of incidents during the planning horizon, for each link ij There are two major decision variables in the model that need to be determined The first variable is 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 which determines whether link ij is covered by beat b and the second decision variable is 𝑉𝑉 𝑏𝑏 which determines the number of trucks that must be assigned to each beat b As a result, the fleet size can be determined, too The following notation is used in the model:
𝑁𝑁 =Set of nodes in network 𝐺𝐺
= Set of links 𝑖𝑖𝑖𝑖 𝑖𝑖n netwotk 𝐺𝐺 𝑝𝑝lus dummy links from the hypothetical origin node to each node
𝐵𝐵= Maximum possible number of patrol beats
𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 =� 1 if link 𝑖𝑖𝑖𝑖 ∊ 𝐿𝐿 𝑖𝑖s covered by beat 𝑏𝑏
𝑃𝑃𝑠𝑠 =Probability of patrol trucks being busy on another incident at the time of an incident occurrence
𝑓𝑓 𝑖𝑖𝑖𝑖 = 𝑇𝑇otal number of incidents on link 𝑖𝑖𝑖𝑖
𝑓𝑓 𝑖𝑖𝑖𝑖 𝑝𝑝 = Number of incidents on link 𝑖𝑖𝑖𝑖, detected by patrol trucks
𝑓𝑓 𝑖𝑖𝑖𝑖 𝑑𝑑 = 𝑁𝑁umber of incidents on link 𝑖𝑖𝑖𝑖,𝑛𝑛ot detected by patrol trucks
𝑉𝑉𝑏𝑏 =Number of patrol trcuks assigned to beat 𝑏𝑏
𝛼𝛼= 𝐶𝐶oefficient to monetize the benefit of incident duration reduction
𝛽𝛽= Coefficient to monetize the nonservice time spent by trucks to travel between beat and depot
𝑅𝑅 𝑖𝑖𝑖𝑖 𝑏𝑏 =Average response time in case of an incident on link 𝑖𝑖𝑖𝑖 in beat 𝑏𝑏
𝑆𝑆 𝑖𝑖𝑖𝑖 𝑏𝑏 = Average service time for an incindent on link 𝑖𝑖𝑖𝑖 in beat 𝑏𝑏
𝑆𝑆 𝑖𝑖𝑖𝑖 = Average service time for an incident on link 𝑖𝑖𝑖𝑖 𝑎𝑎ssuming only one truck provides the assist
𝐶𝐶 𝑖𝑖𝑖𝑖 𝑏𝑏 = Variables defined to resolve non-linearity of the model: 𝑆𝑆 𝑖𝑖𝑖𝑖 𝑏𝑏 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏
ℎ𝑟𝑟 =Patrol trucks operating hours per day
𝑈𝑈𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑚𝑚𝑖𝑖 𝑏𝑏 =Binary varibles defined to resolve non-linearity of the model: 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 𝑉𝑉 𝑚𝑚𝑖𝑖 𝑏𝑏
𝑊𝑊 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 𝑏𝑏 = Binary variables defined to resolve non-linearity of the model∶ 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏
𝑂𝑂 𝑖𝑖𝑖𝑖𝑚𝑚𝑖𝑖 𝑏𝑏 = Binary varibles defined to resolve non-linearity of the model∶ 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 𝑉𝑉𝑚𝑚𝑖𝑖 𝑏𝑏
𝑟𝑟 𝑖𝑖𝑖𝑖 𝑑𝑑 = 𝑆𝑆hortest distance from depot 𝑑𝑑 to link 𝑖𝑖𝑖𝑖
𝑆𝑆𝐷𝐷 𝑑𝑑 𝑏𝑏 = Shortest distance from depot 𝑑𝑑 to beat 𝑏𝑏 (𝑀𝑀𝑖𝑖𝑛𝑛 𝑟𝑟 𝑖𝑖𝑖𝑖 𝑑𝑑 |𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 = 1)
𝑑𝑑 𝑖𝑖 𝑏𝑏 =� 1 if node 𝑖𝑖 is covered by beat 𝑏𝑏
𝑉𝑉 𝑚𝑚𝑖𝑖 𝑏𝑏 ,𝑍𝑍 𝑖𝑖 𝑏𝑏 =Binary variables defined to determine 𝑉𝑉 𝑏𝑏
𝑄𝑄 𝑖𝑖𝑖𝑖 𝑏𝑏 =Variables defined to assure connectivity of beats
𝑆𝑆 𝑖𝑖𝑖𝑖𝑏𝑏 𝑖𝑖 ,𝐶𝐶 𝑖𝑖𝑖𝑖𝑏𝑏 𝑖𝑖 ,𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖 1 ,𝑎𝑎 𝑖𝑖𝑖𝑖𝑖𝑖 2 = Dummy variables defined to calculate 𝑆𝑆𝑖𝑖𝑖𝑖
ℎ 𝑖𝑖𝑖𝑖 𝑏𝑏 = 𝐵𝐵inary variable defined to assign beats to depots
Unlike most studies that assume patrol trucks are immediately available and not busy when an incident occurs, our research explicitly accounts for this possibility We define Ps as the probability that, at the moment an incident happens, the patrol trucks on the same beat could be busy with another case One way to estimate Ps is to analyze historical incident log data and determine how often the truck serving an incident was already attending another case at the time of the subject incident occurrence This data may be available if patrol trucks record log data about the incidents they serve.
Patrolling Response Time
Well-designed patrol programs can significantly reduce response times and the delays experienced by users, making response-time reduction a fundamental consideration in FSP network design In patrol operations, response time typically includes detection and verification time when incidents are detected by patrol trucks themselves Let V_b denote the number of patrol trucks assigned to beat b, and assuming patrol trucks maintain a constant headway, the average response time on each beat can be estimated using a headway-based approach, highlighting how fleet size and scheduling influence incident detection, verification, and overall responsiveness.
Within this network optimization model, X_ij^b denotes whether link ij is included in beat b, V_b represents the number of patrol trucks assigned to beat b, and t_ij is the average travel time on link ij To introduce a linear term in the formulation, the response time is reformulated as a linear function of these variables, enabling efficient optimization of beat assignments, link inclusion, and patrol routing This linear reformulation preserves the core relationships among linkage, patrol resources, and travel time, while allowing tractable computation for network-wide decision making.
Equation (2) initially calculates the average response time based on one truck on the beat
(V b =1) and reduces the response time for each additional truck assigned to the beat Given equation (2) we may calculate the following statement:
All variables are as defined before Note that each truck could be allocated only to one beat and for each beat 𝑉𝑉𝑏𝑏= ∑ ∑ 𝑉𝑉𝑚𝑚𝑖𝑖 𝑏𝑏
𝑚𝑚 Equation (3) is presented to linearize the statement 𝑋𝑋 𝑖𝑖𝑖𝑖 𝑏𝑏 𝑅𝑅 𝑖𝑖𝑖𝑖 𝑏𝑏 which will be applied in the objective function.
Non-Patrolling Detection: Response Time
Average response time calculations are based on incidents detected by patrol trucks during their regular patrol on the assigned beat However, incidents can also be detected by other sources, after which trucks are notified to respond Recognizing these different detection sources is important for accurately assessing response times and dispatch efficiency.
Patrol units do not need to follow their regular routes to detect incidents and can respond to events within their assigned beat using the shortest path Table 1 compares patrolling detection with non-patrolling detection scenarios, outlining the key differences between the two approaches Assuming incidents are addressed only by patrol trucks operating on the same beat, the analysis focuses on the average response time for non-patrolling detection.
Estimated non-patrolling response time can be modeled similarly to patrolling response time, but the average non-patrolling response time is about half of the estimated average patrolling response time This difference occurs because, in the non-patrolling scenario, the closest truck in the beat is dispatched directly to the incident, whereas in the patrolling scenario trucks are unaware of the incident in advance and must detect it along their route, as shown in Figure 2.
Table 1 - Patrolling vs Non-Patrolling Detection
Detection Path to Incidents Patrolling Detection Patrol Trucks Patrol Route
Non-Patrolling Detection Others Shortest Path
Figure 2 Patrolling vs Non-Patrolling Detection Response
Let V_b denote the number of trucks allocated to beat b If the patrolling trucks maintain a constant headway and the turnaround time is negligible, the average non-patrolling response time on each beat is determined by the truck allocation V_b and can be calculated using the following expression.
Figure 3 depicts a four-truck patrol along a beat, where an incident triggers a response based on the incident’s location and how it is detected Trucks 1 through 4 cover the red, green, blue, and yellow sectors, respectively, with the responding unit chosen accordingly The coverage area for each truck varies depending on whether the incident is detected by the patrol fleet or by external sources that inform the trucks These response zones are dynamic and continuously change as the patrol trucks move and reposition themselves.
Figure 3 Truck Coverage for Patrolling Detection (Top) vs Non-Patrolling Detection (Down)
Service Time
Response time depends on the performance of incident management systems, including patrol programs, while clearance time is more influenced by incident severity and the quality of service delivered at the incident scene Designing a network for patrol programs solely to minimize response time, without considering incident severity, may fail to achieve optimal overall performance In a network where some segments experience more major severe incidents due to traffic characteristics and roadway geometry, while other segments have the same number of incidents but lower severity, it is evident that higher-frequency patrolling is needed on these high-risk links even though the incident distribution is similar The study does not focus on the exact mechanisms by which an effective patrol program reduces clearance time; instead, it introduces a modeling approach that incorporates clearance time by biasing coverage toward areas with a higher likelihood of severe incidents so they are patrolled more frequently.
Service time is defined as the time patrol trucks spend at the incident scene, excluding time spent by the dispatch system or other emergency units such as fire trucks, ambulances, and police vehicles to clear the incident Increasing the number of patrol trucks can reduce service time and, in turn, incident clearance time, since clearance time equals service time when only patrol trucks are involved In many cases, especially with disabled vehicles or minor incidents, the patrol system alone completely clears the scene, while other emergency vehicles assist only in severe incidents and crashes A 2012 CHART performance evaluation showed that the Coordinated Highways Action Response Team responded to more than 63,500 emergencies, with about 65% of cases providing assistance to disabled vehicles and roughly 35% involving collisions.
Assuming only one patrol truck responds to each incident while other trucks continue patrolling on their beats, the service time becomes independent of the number of trucks on a given beat In practice, however, multiple patrol trucks often respond to the same incident, and the presence of additional units can shorten the service time The extent of this reduction depends on factors such as incident severity and the type of required service, so a comprehensive study is usually needed to quantify the impact for the patrol program Nevertheless, it can be a reasonable assumption that, for certain incidents requiring rapid intervention, adding more trucks enhances response efficiency.
With a single truck, the service takes 18 minutes If two trucks operate in parallel, the time drops to 9 minutes, and with three trucks in parallel, it drops to 6 minutes If we assume that each additional truck halves the remaining service time, the pattern shows substantial time savings as more trucks are added.
Figure 4 demonstrates that adding more trucks at an incident can shorten service time The first truck initiates the response, and with two trucks the remaining service time is reduced to half of what it would be with only one truck A similar reduction occurs when a third truck—or more—arrives, with the time decrease reflecting only the aid provided by the patrol trucks and not including any additional time spent by other systems to clear the incident In Figure 4, case (a) corresponds to a single truck at the scene, case (b) to a second truck, and case (c) to a third truck joining the first to resolve the incident.
Figure 4 Additional Trucks Reduce the Service Time
Service time and clearance time in the model are shaped by operational details, including how adding more trucks can shorten the remaining service time Depending on the current operational conditions, the model can be updated to reflect these effects, enabling more accurate throughput forecasts and better fleet planning By evaluating how each additional truck accelerates subsequent service steps, the model supports scenarios with varying fleet size, dispatch patterns, and maintenance schedules, ensuring that results stay aligned with real-world conditions.
Although the idea that every additional patrol truck reduces service time can be appealing, this assumption may not be practical across different case studies; a maximum number of trucks that can influence service time can be defined For example, assuming three trucks as the maximum that can reduce service time, Table 2 presents the service time by incident type and the number of trucks on the beat In Table 2, RR is the average incident response time and Vb is the number of trucks on the beat, while Si is the average service time for incidents on link ij, assuming only one patrol truck provides the assist.
Table 2 - Service Time for each Link ij In Beat b: Additional Trucks Cause Service Time Reduction
Some studies claim that reducing response time also lowers incident clearance time Khattak et al [36] report that a one-minute reduction in response time leads to a 0.6 to 1 minute decrease in clearance time Therefore, one approach to assess the impact of patrol programs on clearance time is to treat clearance time as an input in network design and estimate the average reduction in clearance time resulting from response-time improvements, yielding savings independent of the number of patrol units Although the count of patrol units per beat might seem irrelevant in this framework, locations with more severe incidents—those that require longer clearance times—tend to be assigned more patrol trucks to push response times down and thereby shorten clearance times Consequently, high-severity locations receive an additional patrol allocation.
Parameters
To incorporate the benefits of incident duration reduction generated by the patrol program into the objective function, these gains must be expressed in monetary terms The first step is to quantify the traffic delay avoided by shortening incident duration, attributable to the patrol program, and to measure this avoided delay in vehicle-hours (veh-hrs) Once the delay savings are converted into an economic value, they can be integrated into optimization models as monetary gains, enabling a direct comparison of patrol strategies and more informed decision-making This approach links operational improvements to tangible cost savings and performance outcomes.
Several approaches have been proposed in the literature to estimate delay savings from traffic incidents Sun et al present a method to estimate total delay under traffic incident management (TIM) and non-TIM scenarios, enabling the calculation of delay savings, and this method requires input data on incident duration, traffic volume, and reduced capacity Khattak and Rouphail develop a method that estimates delay savings as a function of the volume-to-capacity ratio, given the area type, the number of blocked lanes, and an estimated incident duration These methods illustrate how different inputs influence delay reduction estimates and support decision-making for incident management and traffic operations.
To monetize incident duration reductions, start by multiplying the value of time by the total avoided delay in the network’s traffic volume, which yields the cost savings for each scenario in a comprehensive network evaluation A second approach uses delay-avoided figures from the literature—particularly FSP program evaluation studies—that estimate delay savings across different combinations of incident duration reductions, traffic volume, and incident types Because the avoided delay depends on these factors, different values for the parameter can emerge under different conditions, so you may apply the upper bound, lower bound, average, or another scenario-appropriate value The numerical example section provides the mathematical details on how to calculate the parameter.
Importance Factor
An importance factor I is introduced for each link based on road characteristics such as volume, capacity, road type, location, safety, and security, so that higher-priority roads are addressed more frequently Each characteristic is categorized into a small set of standard ranges, and a classification table is defined from the combination of these categories, with every class assigned an importance factor value Consequently, every road receives an importance factor value according to its class In the objective function, these importance factors are normalized so that, for each link k, the factors are scaled consistently across the network.
Objective Function - Constraints
This study introduces a mixed-integer programming model that determines the optimal beat configuration, fleet size, and allocation of patrol trucks to beats for patrol programs, with the objective of minimizing incident delay—comprising response time, service time, and the associated program costs In patrol operations, incidents are typically detected by patrol units, so the measured response time includes both detection and verification time The objective function prioritizes reducing the total response and service times, with the first term specifically capturing this combined delay while balancing the costs of implementing the patrol program.
This term minimizes the total response and service time during the planning horizon The
The original objective is non-linear and non-convex, but it can be linearized through a deliberate set of transformations The key step is to recast the response time and the service time as linear expressions, which allows the overall problem to be reformulated as a linear program By expressing these times linearly, the nonlinear terms are replaced with tractable equivalents, and the resulting linear forms for the response time and service time are provided below, enabling standard linear optimization techniques to solve the problem efficiently.
∑ ∑ X ij b fij(R b ij + S ij b + Ps S ij b
B ij∊L b=1 = ∑B ∑ij∊L X ij b fijS ij b (1 + P 2 s ) b=1 +∑B ∑ij∊L X ij b fij R b ij b=1 ∑B ∑ij∊L X ij b fijS ij b (1 + P 2 s ) b=1 + 0.5∑B ∑ij∊L∑kl∊LfijtklX kl b X ij b b=1 −
0.5∑ B b=1 ∑ ij∊L ∑ kl∊L ∑ T m=1 ∑ V e=2 f ij t kl ( e−1 1 − 1 e ) X kl b X ij b V me b (8)
Statement (8) serves to linearize statement (7) The first term in equation (8) estimates the total service time, while the second and third terms compute the total response time over the horizon For the details of the response-time calculation, refer to statement (3).
In the second step of linearizing the model, an auxiliary set of binary variables is introduced The original model is nonlinear due to cross products of binary variables, but this nonlinearity can be linearized by substituting each cross product with a newly defined binary variable and adding the corresponding linear constraints This reformulation preserves the problem’s structure while enabling the use of linear optimization techniques.
∏ j∊Q X j by a new variable X Q such that [73]:
So, the following changes are made in the model:
These dummy variables are introduced to linearize the model All variables are as defined before
Expression 14 is integrated into the objective function to capture operating costs over the planning horizon When multiple depots are available, the model assigns each beat to a depot using the proposed Constraint 15.
Statement 15 establishes the total shortest distances from each beat b to its corresponding depot d, which are monetized in the objective function by the parameter β The parameter α is introduced to translate reductions in incident duration into monetary value, thereby converting traffic delay savings into economic terms In addition, importance factors are incorporated to reflect road priorities based on influential characteristics Together, these elements define the proposed formulation, including the objective function and constraints, as follows:
In this optimization model, the objective minimizes the monetized total response and service time over the horizon plus the program costs To resolve nonlinearities, constraints 17–22 introduce a new set of binary decision variables, while constraint 23 provides the formulation for average response time and constraints 24–25 define a binary variable O to linearize and validate this average The average response times are computed under the assumption of a constant headway between patrol units and a chosen average patrolling speed, recognizing that speeds vary with traffic but using an average speed for planning; the design can accommodate multiple average patrolling speeds for different traffic conditions (e.g., peak versus non-peak hours), and the model allows patrol units to use shoulders or alternative routes to avoid congestion Constraint 26 ensures that each vehicle is assigned no more than once, constraint 27 tallies the total number of trucks in each beat, and constraints 28–29 determine the number of patrol trucks in each beat, denoted V_b Constraints 30–39 estimate the average service time on each beat, with constraint 32 computing the average service time and the remainder linearizing this calculation; the service-time formulation is general and assumes the effect of additional trucks can be unlimited Finally, constraints 40–42 assign beats to depots and determine the shortest distance between depots and their corresponding beats to address the multi-depot problem.
Constraint 43 ensures that exactly one beat covers each link, a rule that can be relaxed in practical implementations to allow multiple beats to cover a link or to omit coverage for links served by the dispatch system Nevertheless, in practice, assigning several beats to the same link is uncommon because it can disrupt coordination between the patrol and dispatch subsystems and degrade overall system performance.
Under the proposed patrol model, each link must be covered by exactly one patrol beat, since the system is designed for patrolling purposes only and coverage should exist unless a dispatch system covers low-incident links when incidents occur In general, patrol programs position emergency units much closer to potential incident locations, allowing them to detect and immediately respond to numerous incidents themselves, which significantly reduces detection and response times, while a dispatch system can be used for low-intensity links where continuous patrolling may not be beneficial.
Constraint 44 ensures that link ij is covered by the same beat that covers link ji, consolidating bidirectional coverage under a single patrol unit While this constraint could be relaxed to allow links on opposite directions of the same road segment to be assigned to different beats, practical network monitoring often enables patrol units to observe the opposite side while covering one side of the road To capitalize on this capability and avoid confusion between patrol units on different beats, it is more effective to assign both sides of a road to the same beat and patrol crew, ensuring consistent road-side coverage and simpler coordination.
Constraint 45 ensures that every node is covered by at least one beat Constraint 46 states that if any link covered by beat b starts or ends at node i, then node i must be included in beat b Constraints 47 through 50 guarantee that the nodes assigned to the same beat form a connected subgraph, ensuring connectivity within each beat.
To account for the number of incidents that are responded to but not detected by patrol trucks, the objective function can be revised by updating its first and second terms to explicitly include these undetected incidents, ensuring the optimization reflects both observed and unobserved responses and improves the model’s accuracy This adjustment enhances robustness and better aligns the objective with real-world patrol performance by balancing detection, response coverage, and resource allocation.
All constraints remain as before, with the only update being the service-time constraint, which must be adjusted to reflect the non-patrolling detection response time This formulation assumes that, in the event of a reported incident, the response will be carried out by trucks assigned to the same beat In the optimization model, there are two sets of decision variables: the first-stage variables X and V, which act as the primary decision variables, and the remaining variables—R, S, C, W, and other auxiliary terms—that model secondary aspects of the system.
U are second stage variables Second stage variables are calculated based on scenarios and values for the first stage variables This study presents a comprehensive model that covers important aspects of patrol programs and addresses issues as much as possible to optimize the performance of the FSP programs Part of the advantages of the current model compared to previous models in the literature is presented in Table 3
Table 3 - Advantages of the Proposed Model
Convexity of Linear Relaxation Non-Convex
Find Optimal Number of Beats Pre-specified Number of Beats
Find Optimal Fleet Size Pre-specified Number of Total Trucks Clearance Time Considered Only Response Time
Individual Cost for Each Truck Only One Cost
Trucks being Busy at the Time of Incident Not Considered
Heuristic Algorithms
For large size networks, the proposed model is combined with a number of heuristic approaches that can be used to generate near-optimal solutions Such approaches include network decomposition combined with neighborhood search, model decomposition, and beat merging Details on the heuristic procedures can be found in [70]
Overview
State of Maryland operates a patrol program which is implemented by the Coordinated
Highways Action Response Team (CHART) CHART (Figure 5) works in partnership with the
Maryland State Highway Administration (SHA), Maryland Department of Transportation (MDOT), Maryland Transportation Authority (MDTA), and the Maryland State Police (MSP)
Figure 5 Coordinated Highways Action Response Team (CHART)
CHART uses Emergency Traffic Patrols (ETP) to provide emergency motorist assistance and to relocate disabled vehicles out of travel lanes CHART Emergency Traffic Patrols deploy three different types of response vehicles to deal with the incidents.
• CHART Custom Response Vehicle – CRV
• CHART Heavy-Duty Utility Truck
As shown in Figures 6–8, the response units are equipped with tools and devices to clear incidents from the roadway, provide assistance to motorists, and warn traffic about incidents and the actions drivers should take.
Figure 6 CHART Custom Response Vehicle – CRV
Figure 7 CHART Heavy-Duty Utility Truck
CHART operates a network of five depots and seven Traffic Operation Centers (TOCs), with three TOCs permanent and four seasonal The network permanently covered by CHART, shown in Figure 9, includes patrols in the Western, Baltimore, and National Capital regions.
CHART field patrol routes operate based on the following regions [75]:
The following routes within Prince George’s, Montgomery, and Southeastern Howard Counties:
I-95 from Woodrow Wilson Bridge to MD 32 (Exit 38), I-270, I-495, US 50, MD 5, and
The following routes within Baltimore, Anne Arundel Counties and Northeastern Howard Counties:
I-70 from US 29 to Security Blvd, I-83, I-95 from MD 32 to Caton Ave (Exit 50), I-97, I-
The following routes within Carroll, Frederick, Washington and Western Howard Counties:
I-70 from US 29 to the area of Hancock, I-81, I-270, US 15, US 340, and MD 140 from Baltimore/Carroll County line to MD 31
The proposed model is applied to a portion of Maryland's freeway network that is covered by CHART CHART patrol units operate 24/7 in Baltimore and the National Capital regions, while the Western region receives daily coverage from 5:00 AM to 9:00 PM [76].
This study conducts a two-year analysis of 2015 and 2016 In 2016, the Baltimore region network was revised to exclude I-97 and MD100, and the network representation used in the analysis was updated accordingly, as described in section 4.2 Details on the cases examined for each year and the different assumptions applied to each case will be described in the subsequent sections.
Study Area
The proposed model is applied to a portion of Maryland's freeway network and its data Incident data from 2015 and 2016 are analyzed to identify the optimal design for each year.
Two study years provided a dataset of incidents, with each dataset covering incidents that occurred on CHART patrol coverage routes or within a 10‑mile radius of those routes, and excluding incidents outside this radius that CHART units would respond to Incidents that did not occur directly on the patrol routes but lie within 10 miles are assigned to the nearest patrol route, which may increase the count of incidents attributed to boundary routes It is assumed that CHART patrol units detected all of these incidents.
Analysis models the CHART network as a symmetric directed graph in which edges (arcs) correspond to road segments and nodes correspond to major interchanges where re-routing for patrol units is possible Some nodes designate the boundaries of CHART's current coverage area on different routes, and a few additional nodes serve merely to separate different paths This graph-based representation enables routing flexibility for patrol operations and clarifies the geographic extent of CHART coverage.
Based on historic log data, the number of incidents detected and responded to by CHART across the network (not limited to CHART units) is estimated to be more than 11,000 during the studied period.
2015 For 2016, the analysis was carried out based on the number of incidents that were responded by CHART units (not necessarily detected by CHART as well), which was equal to 30,873
In 2015, the analysis was performed on a network of 116 nodes and 119 links For the 2016 analysis, the underlying network graph was adjusted according to CHART officials' indications, resulting in a network of 112 nodes and 115 links.
Each number in Figure 10 and Figure 11 represents one segment Details about the exact location of the nodes and links for 2015 and 2016 are summarized in Appendix A and Appendix
Across these experiments, we assume identical importance factors for all roads, exclude service time from the analysis, and aim to minimize the total patrolling response time (including detection and verification) while accounting for operation costs Each beat can be served by at most two trucks The hourly cost per truck, covering driver wages and vehicle expenses, is estimated at about $50, with the vehicle cost encompassing fuel, maintenance, supplies, and other patrol-related costs The CHART network is modeled with a single depot, so no beat-to-depot assignment is required, and total costs exclude minor deadhead times between depots and beats, which are negligible when depot distances are short.
CHART patrol trucks operate on three weekday shifts—the morning, afternoon, and night shifts—and also run on weekends Night and weekend shifts typically have lower traffic volume and fewer incidents than the morning and afternoon shifts on weekdays, allowing patrol units to travel faster in their assigned beats during those times Consequently, different patrolling speeds can be assumed for each shift, reflecting the varying traffic and incident patterns across weekday mornings, weekday afternoons, weekday nights, and weekends.
CHART patrol trucks provide year-round, continuous coverage of the network by operating in multiple shifts across different times of the day and week Nighttime and weekend periods exhibit similarly low traffic volume and incident density, so they can be managed with the same approach Consequently, the issue is addressed through three distinct cases, described below.
3 Weekday Nights (9 PM – 5 AM) and Weekends
Assuming 52 weeks per year, the annual working hours for weekday morning and afternoon shifts during standard hours are estimated at 2,080 hours for one year of operation, and the annual working hours for night and weekend shifts are also estimated.
Patrol operations total 4576 hours per year Travel times are calculated using the average patrolling speed of 40 mph for morning and afternoon shifts on weekdays, while night and weekend shifts use a standard patrolling speed of 55 mph to estimate travel time.
The input for the model, including the travel time and the number of incidents for each
Incidents Sub- network Link Travel
Number of Incidents Sub- network Link Travel
Number of Incidents Sub- network
Incidents Sub-network Link Travel Time
Incidents Sub- network Link Travel Tim
Table 6 – Input: Night and Weekend
Link Travel Tim (min) No of
Incidents Sub-network Link Travel Tim
(min) No of Inciden Sub-network
Number of Incidents Sub- network Link Travel
Number of Incidents Sub- network
Analysis for 2015 Data
Incident Duration Reduction Savings
the parameter for the CHART network, we need to determine the value of time and estimate the average response time reduction caused by the CHART patrol program
Value of time varies by source, and the Department of Transportation (DOT) provides travel time values (VOTT) for 2009 and 2012, based on two trip types on surface modes: intercity and local trips The intercity VOTT are presented in Table 7, while the local-trip VOTT appear in Table 8 According to these DOT recommendations for 2009 and 2012, travel-time values can be used for both intercity and local surface-trip analyses.
2015 are extrapolated and added up to the tables, too
Table 7 - Recommended Hourly Values of Travel Time Savings for Intercity Trips
Table 8 - Recommended Hourly Values of Travel Time Savings for Local Trips
According to a U.S Department of Transportation (US DOT) report, the value of travel time for the All-Purpose category is estimated using weighted averages that reflect trip-purpose distributions across modes The intercity travel by conventional surface modes is distributed as 78.6% personal and 21.4% business, while local travel by surface modes is 95.4% personal and 4.6% business [78].
A study by the Center for Advanced Transportation Technology (CATT) at the University of Maryland refines the travel-time value for Maryland freeway users by analyzing major high-volume corridors around Baltimore and the National Capital region Analyzing segments of I-95, I-495, I-270, MD 295, and US 29 (as shown in Figure 12), the research proposes travel-time values of $29.82 per hour for passengers, $45.40 per hour for cargo, and $20.21 per hour for truck drivers.
Value-of-time estimates vary by trip purpose, trip mode, vehicle type, and trip type, along with other pertinent factors According to CHART officials, the study uses an average value of time of $20 per hour for the subject network, providing a key input for cost-benefit analyses and travel-time valuations.
To estimate the average response time reduction attributable to the CHART patrol program, we refer to the CHART evaluation studies The CHART evaluation reports [80]-[82] show that the average incident duration with CHART response is about 10 minutes shorter than incidents without CHART assistance Therefore, we estimate that the CHART patrol program reduces the average response time by roughly 5 to less than 10 minutes Following the same approach as in the numerical example, the parameter is estimated from plausible response-time-reduction scenarios, with the results listed in Table 9 Based on this analysis, the parameter α is estimated to be about 15 for the subject network.
Table 9 - Parameter α Estimated for the CHART Network
VEH-HR Saving Per one min RTR
VEH-HR Saving Per Incident Per 1 min
Avg Cost Saving Per 1 min RTR (α)
Value of Time = 20 & Number of Incidents = 693
Results
To solve the problem on the subject network, the approach combines network decomposition with a neighbor search algorithm First, guided by the network decomposition algorithm, the model is solved on three dense sub-networks, and these results are used to obtain a viable solution for the full network Next, this solution is refined by a neighbor search procedure: for each beat, all neighboring links are examined to determine whether adding a neighboring link to the beat and removing it from its current beat can yield a better overall solution The process repeats until no further improvement is possible.
Patrol planning is solved for three cases, with the weekday morning beat configuration assigning one truck to each of the remaining beats The weekday afternoon shift has 13 beats, similar to the weekday morning shift, but requires 17 patrol trucks; four beats are allocated double trucks, and the remaining beats receive a single truck Night and weekend shifts require fewer patrol trucks than the weekday morning and afternoon shifts, with eight trucks patrolling ten designed beats for these shifts, where two beats use double trucks and six beats use a single truck.
Figure 13 Beat Configuration for the Weekday Morning Shift
Table 10 – Fleet Size and Allocation for the Weekday Morning Shift
Figure 14 Beat Configuration for the Weekday Afternoon Shift
Table 11 – Fleet Size and Allocation for the Weekday Afternoon Shift
Figure 15 Beat Configuration for the Night and Weekend Shift
Table 12 – Fleet Size and Allocation for the Night and Weekend Shift
Table 13 presents the major characteristics and performance metrics of the designed program, with results shown for each shift and for the overall totals The analysis estimates an annual full-time operating cost of $5,616,000 The designed network is projected to require 5,898 hours of patrolling response time to address 11,805 incidents over a year, producing an average response time of under 32 minutes per incident, including detection and verification, so incidents are addressed in about 30 minutes from the time they occur on the network.
Optimal beat configuration, fleet size, and fleet allocation can vary significantly with the time of day due to changing incident densities and travel times To optimize performance while minimizing operating costs, it is advantageous to design different configurations for each part of the day The same logic supports creating separate networks for weekdays and weekends to reflect distinct demand patterns Additionally, since incident density and traffic volume can vary throughout the year, seasonal or monthly designs can deliver more precise, period-specific solutions.
New findings reaffirm that accurately determining fleet size and the number of beats is more effective than relying on preset assumptions The patrol program's efficiency depends heavily on beat configuration and fleet allocation For optimal performance, the network should be designed within a comprehensive model that simultaneously considers all relevant factors, rather than addressing each issue in isolation.
Shift Duration (hours per year) 2080 2080 4576 8736
[including detection and verification times] (min) 31.7 28.1 36.4 31.9
[including detection and verification times] (hours) 1810 1929 2159 5898
Sensitivity Analysis
Designing the network for patrol programs starts by specifying inputs and making informed assumptions about the program When data are incomplete or input values vary, the exact parameter values may be uncertain Therefore, this section conducts sensitivity analysis to quantify how changes in these inputs influence the beat configuration, the required fleet size, and the allocation of fleets.
This section investigates a set of influential parameters and their impact on the optimal design A study by the Center for Advanced Transportation Technology (CATT) at the University of Maryland recommends a specific travel-time value for Maryland freeway users, derived from an analysis of major high-volume corridors The recommended value is about $30 per hour for travel on several major Maryland freeways Consequently, additional scenarios are solved assuming a travel-time value of $30 per hour The best configuration for weekday morning and weekday afternoon shifts, based on this value, is shown in Figures 16 and 17, respectively In addition, the corresponding fleet size and fleet allocation results for the weekday morning and weekday afternoon shifts—based on the $30 per hour value—are presented in Tables 14 and 15.
Results indicate that increasing the value of time from $20 per hour to $30 per hour leads to a larger fleet size The weekday morning shift expands from 15 to 18 patrol units, and the weekday afternoon shift increases from 17 to 22 patrol units This occurs because a higher value of time reduces acceptable incident duration, necessitating more patrol units to maintain coverage.
Figure 16 Beat Configuration for the Weekday Morning Shift - VOT0$/hr
Table 14 – Fleet Size and Allocation for the Weekday Morning Shift - VOT0$/hr
Figure 17 Beat Configuration for the Weekday Afternoon Shift - VOT0$/hr
Table 15 - Fleet Size and Allocation for the Weekday Afternoon Shift - VOT0$/hr
B Maximum Number of Trucks per Beat
In the main analysis, the maximum number of patrol units per beat is assumed to be two
Generally, assigning a large number of patrol units to a single beat is impractical because maintaining a relatively constant headway between all patrols is difficult, a condition assumed for calculating the average response time In this section, two scenarios for the maximum number of patrol units per beat are considered to inform network design: one truck per beat and three trucks per beat.
Figure 18 and Figure 19 present the weekday morning and weekday afternoon beat configurations with one truck per beat, while Figure 20 shows the weekday afternoon beat configuration under a maximum of three trucks per beat; Table 16 provides the corresponding fleet size and fleet allocation for the weekday afternoon shift under the three-truck-per-beat scenario The beat configuration, fleet size, and fleet allocation under the maximum three trucks per beat do not change for the weekday morning shift compared with the main results Table 17 reports objective values for three scenarios of the maximum number of trucks per beat; increasing the maximum allows the model to select a larger fleet size for a given beat if it yields a better solution, but the improvement is not substantially significant, since the model can instead create additional, smaller beats rather than one large beat with more units, although a sufficiently fine breakdown into small links is required.
Figure 18 Beat Configuration for the Weekday Morning Shift – One Truck per Beat
Figure 19 Beat Configuration for the Weekday Afternoon Shift – One Truck per Beat
Figure 20 Beat Configuration for the Weekday Afternoon Shift – Maximum Three Trucks per Beat
Table 16 - Fleet Size and Allocation for the Weekday Afternoon Shift – Maximum Three Trucks per Beat
Table 17 - Maximum Number of Trucks per Beat
One of the most influential parameters in designing the network for freeway service patrol programs is the standard patrolling speed of patrol units, so an additional scenario with a 55 mph standard patrolling speed is considered for the weekday morning and weekday afternoon shifts The beat configurations for these shifts, based on the 55 mph standard patrolling speed, are shown in Figure 21 and Figure 22, respectively The fleet size and fleet allocation for the weekday morning and weekday afternoon shifts are listed in Table 18 and Table 19, respectively.
Raising the standard patrolling speed from 40 MPH to 55 MPH reduces the required patrol fleet size on weekday shifts: morning shifts decrease from 15 to 14 units, while afternoon shifts decrease from 17 to 15 units This shows that faster emergency response patrols on their assigned beats enable a smaller fleet to maintain effective coverage.
Figure 21 Beat Configuration for the Weekday Morning Shift - 55 MPH
Table 18 - Beat Configuration for the Weekday Morning Shift - 55 MPH
Figure 22 Beat Configuration for the Weekday Afternoon Shift - 55 MPH
Table 19 - Beat Configuration for the Weekday Afternoon Shift - 55 MPH
Non-Patrolling Detection: Result
Based on historic log data for 2015, the number of incidents that were both detected and responded to by CHART (not limited to CHART patrol units) is estimated at more than 11,000, with an initial assumption that CHART patrol units detected all of these incidents; a larger incident dataset is also available, encompassing all incidents CHART units responded to but did not necessarily detect, which indicates more than 30,000 incidents in 2015 occurring on CHART patrol coverage routes or within about 10 miles of patrol routes For this larger dataset, because a significant majority of incidents are not detected by CHART and detection details by CHART patrol units are unavailable, we assume incidents are detected by other sources and apply a non-patrolling detection response time method Additionally, per CHART officials, we assume only one response unit is assigned to each beat for this dataset, with all other assumptions remaining consistent with those used for the previous dataset.
Based on the non-patrolling detection dataset, the problem is resolved across three cases, with beat configurations for the weekday morning, weekday afternoon, and night/weekend shifts shown in Figures 23–25 For each shift, the details of links covered by each beat are presented in Tables 20–22, with additional information available in the Appendix.
B for the exact location of the links According to the results, 17 patrol units are needed to patrol during the weekday morning shift, and 19 units are needed to patrol during the weekend afternoon shift As expected, the night and weekend shift require less number of patrol units, compared to the weekday morning and weekday afternoon shifts, because of lower incident frequencies Eleven patrol units are needed to patrol during the night and weekend shift Please note that, for each shift, the number of incidents per each beat is provided in Table 20 through Table 22 Details regarding the number of incidents per each link, during each shift, are also presented in Appendix C This information could be useful to determine where to assign additional units during each shift
Major characteristics and performance measures of the designed program are summarized in Table 23 The result for each shift including fleet size, shift duration and number of incidents during one year, average response time, total response time, and operations costs are provided in Table 23 According to the result, the total operating cost is estimated to be $6,261,000 for one year of full-time operation Also, the total response time for the designed network is estimated the weekday morning and weekday afternoon shifts, and 55 MPH for the night and weekend shifts) and obviously will decrease if patrol units can drive faster
Figure 23 Non-Patrolling Detection: Beat Configuration for the Weekday Morning Shift
Table 20 - Non-Patrolling Detection: Beat Configuration for the Weekday Morning Shift
Beat Covered Links Number of
Figure 24 Non-Patrolling Detection: Beat Configuration for the Weekday Afternoon Shift
Table 21 - Non-Patrolling Detection: Beat Configuration for the Weekday Afternoon Shift
Beat Covered Links Number of Incidents
Figure 25 Non-Patrolling Detection: Beat Configuration for the Night and Weekend Shift
Table 22 - Non-Patrolling Detection: Beat Configuration for the Night and Weekend Shift
Beat Covered Links Number of
Table 23 - Non-Patrolling Detection: Performance Measures
Avg Response Time (min) - 40 MPH 13.7 12.4 -
Avg Response Time (min) - 55 MPH - - 15.4
Total Response Time (hours) - 40 MPH 2267 2220 -
Total Response Time (hours) - 55 MPH - - 2443
Non-Patrolling Detection: Sensitivity Analysis
In this study, sensitivity analysis is conducted on the non-patrolling detection dataset to assess how varying parameters influence beat configuration and fleet size A key parameter examined is the average response speed of emergency units to arrive at the incident location after being informed of the incident occurrence This response speed may differ from the standard patrolling speed discussed for the previous dataset, since units are already alerted and can drive faster.
Two additional scenarios for average response speed, 55 MPH and 65 MPH, are considered for weekday morning and weekday afternoon shifts, and one extra scenario at 65 MPH is analyzed for night and weekend shifts The beat configurations for the shifts—weekday morning at 55 MPH, weekday afternoon at 55 MPH, weekday morning at 65 MPH, weekday afternoon at 65 MPH, and night and weekend at 65 MPH—are illustrated in Figures 26 through 30.
Table 24 summarizes the performance measures from the sensitivity analysis results using the specified average response speeds for each shift, capturing fleet size, average response time, and total response time across speed scenarios The findings show that for both weekday morning and weekday afternoon shifts, increasing speed from 40 MPH to 55 MPH and from 55 MPH to 65 MPH reduces the required patrol units while also shortening the average response time Although a higher response speed is desirable from a safety perspective, practical constraints such as safety concerns and traffic volumes—especially during peak hours—may compel patrol units to slow down.
Figure 26 Non-Patrolling Detection: Beat Configuration for the Weekday Morning Shift – 55 MPH
Figure 27 Non-Patrolling Detection: Beat Configuration for the Weekday Afternoon Shift – 55 MPH
Figure 28 Non-Patrolling Detection: Beat Configuration for the Weekday Morning Shift – 65 MPH
Figure 29 Non-Patrolling Detection: Beat Configuration for the Weekday Afternoon Shift – 65 MPH
Figure 30 Non-Patrolling Detection: Beat Configuration for the Night and Weekend Shift – 65 MPH
Table 24 - Non-Patrolling Detection Sensitivity Analysis: Performance Measures
Avg Response Time (min) - 55 MPH 11.5 11.7 -
Avg Response Time (min) - 65 MPH 10.6 10.5 16.2
Total Response Time (hours) - 55 MPH 1901 2088 -
Total Response Time (hours) - 65 MPH 1756 1874 2572
Although the proposed model can determine the optimal beat configuration and the total number of beats, it can also be configured to use a pre-specified beat count This approach is useful when resources are limited and the network must be designed around the maximum number of patrol units available In this case, the number of beats is adjusted to match the fleet size, with the maximum number of beats capped by the number of patrol units For example, if there are at most ten patrol units available, the system should support up to ten beats, since each beat requires at least one patrol unit.
To conduct sensitivity analysis for the non-patrolling detection dataset, we fix the number of beats and design the network around 11 beats The beat configuration for the weekday morning and weekday afternoon shifts, each based on 11 beats, is shown in Figure 31 and Figure 32, respectively.
Figure 31 Non-Patrolling Detection: Beat Configuration for the Weekday Morning Shift – Pre-Specified 11
Figure 32 Non-Patrolling Detection: Beat Configuration for the Weekday Afternoon Shift: Pre-Specified 11
Recent results show that, with a fixed fleet size, we can determine the optimal beat configurations for each shift Conversely, there are scenarios where the aim is to determine the required fleet size and the allocation of resources to fit a given beat configuration This dual perspective supports two complementary optimization problems: configuring beat patterns under resource constraints and sizing and allocating fleets to realize a predefined beat layout, enabling more efficient planning and improved utilization of assets.
The problem is addressed using the current CHART operating beat configuration, which includes 11 beats as shown in Figure 33, to determine the optimal fleet size and fleet allocation across the beats The resulting fleet size and allocation by shift are presented in Table 25, based on the assumption of a constant headway between patrol units on the same beat.
Figure 33 CHART Current Beat Configuration
Table 25 – Fleet Size and Allocation Based on the Current Beat Configuration
Beat Weekday Morning Weekday Afternoon Night and Weekend
Conclusions
Our analysis shows that accounting for all relevant factors in the model can elevate the performance of the patrol program In particular, the number of beats, beat configuration, fleet size, and fleet allocation, along with other model elements, must be determined rather than simply assumed.
To optimize program performance while minimizing operating costs, the study shows that configurations should reflect time-varying incident densities across different times of day, week, and year While there’s no need to design the network for every moment, data processing and statistical analysis of incident data can reveal periods that require individual design In this work, the network is designed around official CHART shifts: weekday morning, weekday afternoon, night, and weekend shifts Additional scenarios could explore peak and non-peak hours, as well as seasonal or monthly designs that may prove helpful.
Sensitivity analysis reveals that varying parameters such as value of time, emergency trucks' average response speed, and standard patrolling speed significantly influence the optimal beat configuration, fleet size, and fleet allocation, so these values must be carefully selected and embedded into the model; when uncertainty exists, using a range of plausible values to design the network and assessing the resulting impact on the solution is advisable, and increasing the maximum number of patrol units per beat also affects the optimal solution, though the difference in objective values is not pronounced because the model can form additional beats with fewer units per beat rather than a single large beat with more units, so for this approach the network should be decomposed into sufficiently small links.
Results show that increasing the value of time from $20 per hour to $30 per hour significantly increases fleet size per shift This happens because a higher time value pushes the optimization model to shorten total incident duration, leading to the deployment of additional patrol units.
Results indicate that as patrol units’ average response speed, i.e., standard patrolling speed, increases, fewer units are needed to cover the network, even though the average response time may also decrease However, pushing speed higher is often constrained by safety concerns Additionally, high traffic volumes, especially during peak morning and afternoon hours, can force patrols to slow down, reducing overall patrol efficiency despite the potential gains from faster speeds.
The proposed model can determine the optimal beat configuration and the number of beats, and it also supports designing the beat layout based on a pre-specified number of beats This approach is particularly valuable when the available fleet is limited; for example, beat configurations can be determined with a pre-specified 11 beats In addition, both fleet size and fleet allocation can be derived for any given beat configuration under the assumption of a constant headway between patrol units within the same beat.
Analysis across all shifts shows that the Baltimore region and the National Capital region require more patrol units than the Western region This aligns with incident activity, since the Western region records fewer incidents than both Baltimore and the National Capital region Additionally, the Baltimore region may require one or two more patrol units than the National Capital region during certain shifts.
During planning, when data is available, incidents should be classified by detection method and the network should be designed to accommodate both classes within a single model This approach matters because the average response time differs between patrolling and non-patrolling detection methods, so considering both detection methods in the model can optimize responsiveness and overall performance.
Agencies operating patrol programs should tailor fleet size and beat configuration to each shift according to incident frequencies, with the total fleet allocated across shifts in proportion to the incident load per shift For existing layouts, simple sensitivity analyses—such as swapping patrol links between beats or transferring a link to a neighboring beat—and evaluating the resulting configuration can reveal performance changes Likewise, testing small adjustments to fleet size for individual beats (increasing or decreasing) helps assess the benefit or cost of those changes.