This document presents a new solution procedure for the side-constrained traffic assignment problem (SCTAP) based on combining the inner penalty function method with a path-based gradient projection algorithm. The SCTAP incorporates explicit capacity constraints into the standard traffic assignment framework to model bottlenecks and queues. The new algorithm finds solutions at both the path and link level while ensuring all intermediate solutions satisfy the side constraints. The procedure only checks constraints on links belonging to the shortest path, making it efficient.
This document provides an overview and comparison of four types of dynamics used in analytical dynamic traffic assignment models: 1) dynamics based on arc exit-flow functions, 2) dynamics where both entrance and exit flows are controlled, 3) dynamics based on arc exit-time functions, and 4) tatonnement and projective dynamics. It describes the assumptions and constraints of each type of dynamic model and discusses their usefulness and limitations for predicting dynamic network flows.
A Computational Study Of Traffic Assignment AlgorithmsNicole Adams
The document summarizes a study comparing algorithms for solving traffic assignment problems. It classified algorithms as link-based (using link flows), path-based (using path flows), or origin-based (using link flows from origins). It reviewed literature on algorithms like Frank-Wolfe (link-based), path equilibration (path-based), and origin-based algorithm. It chose to implement representative algorithms from each class: Frank-Wolfe, conjugate Frank-Wolfe, bi-conjugate Frank-Wolfe (link-based), path equilibration, gradient projection, projected gradient, improved social pressure (path-based), and Algorithm B (origin-based) to compare their performance on benchmark problems.
A Computational Study Of Traffic Assignment AlgorithmsAlicia Buske
This document summarizes a research study that compares different algorithms for solving traffic assignment problems. The study performs a literature review of prominent traffic assignment algorithms, classifying them based on how the solution is represented (link-based, path-based, origin-based). It then implements representative algorithms from each class and conducts computational tests on benchmark networks of varying sizes. The results are analyzed to compare algorithm performance and identify the impact of different algorithm components on running time.
A Robust Algorithm To Solve The Signal Setting Problem Considering Different ...Joshua Gorinson
This paper presents an algorithm to optimize traffic signal settings that considers the interaction between signal timing and traffic assignment. The algorithm iteratively updates signal timings based on fixed traffic flows, and then updates traffic flows based on the new signal timings, with the goal of minimizing total delay. Two different traffic assignment approaches are considered: user equilibrium assignment and a platoon simulation model. The proposed algorithm is compared to other optimization methods on a real traffic network, demonstrating its robustness in handling different assignment approaches.
Solving bandwidth guaranteed routing problem using routing dataIJCNCJournal
This paper introduces a traffic engineering routing algorithm that aims to accept as many routing demands
as possible on the condition that a certain amount of bandwidth resource is reserved for each accepted
demand. The novel idea is to select routes based on not only network states but also information derived
from routing data such as probabilities of the ingress egress pairs and usage frequencies of the links.
Experiments with respect to acceptance ratio and computation time have been conducted against various
test sets. Results indicate that the proposed algorithm has better performance than the existing popular algorithms including Minimum Interference Routing Algorithm (MIRA) and Random Race based Algorithm for Traffic Engineering (RRATE)
.
Traffic assignment models are used to estimate traffic flows on a transportation network based on origin-destination flows and the network's topology, link characteristics, and performance functions. Traffic is assigned to paths between origin-destination pairs based on travel time or impedance. Traffic assignment is a key part of travel demand forecasting and is used to predict future network flows and performance under different planning scenarios. Common traffic assignment methods include all-or-nothing assignment, user equilibrium assignment, and system optimum assignment.
Optimizing Data Plane Resources for Multipath FlowsIRJET Journal
This document discusses optimizing data plane resources for multipath flows. It introduces the concepts of routing with minimum overhead (RMO) and decomposition with minimum overhead (DMO) to minimize forwarding costs when splitting traffic flows across multiple paths.
The key ideas are:
1) Dividing a traffic flow across multiple paths improves bandwidth utilization but incurs higher forwarding costs due to additional network resources used.
2) RMO and DMO problems are defined to minimize these forwarding costs by reducing the number of paths or nodes used. Efficient algorithms are presented to solve the problems.
3) Simulation results show that algorithms which prefer smaller paths generally perform better at reducing the number of nodes traveled, though they may increase
This document provides an overview and comparison of four types of dynamics used in analytical dynamic traffic assignment models: 1) dynamics based on arc exit-flow functions, 2) dynamics where both entrance and exit flows are controlled, 3) dynamics based on arc exit-time functions, and 4) tatonnement and projective dynamics. It describes the assumptions and constraints of each type of dynamic model and discusses their usefulness and limitations for predicting dynamic network flows.
A Computational Study Of Traffic Assignment AlgorithmsNicole Adams
The document summarizes a study comparing algorithms for solving traffic assignment problems. It classified algorithms as link-based (using link flows), path-based (using path flows), or origin-based (using link flows from origins). It reviewed literature on algorithms like Frank-Wolfe (link-based), path equilibration (path-based), and origin-based algorithm. It chose to implement representative algorithms from each class: Frank-Wolfe, conjugate Frank-Wolfe, bi-conjugate Frank-Wolfe (link-based), path equilibration, gradient projection, projected gradient, improved social pressure (path-based), and Algorithm B (origin-based) to compare their performance on benchmark problems.
A Computational Study Of Traffic Assignment AlgorithmsAlicia Buske
This document summarizes a research study that compares different algorithms for solving traffic assignment problems. The study performs a literature review of prominent traffic assignment algorithms, classifying them based on how the solution is represented (link-based, path-based, origin-based). It then implements representative algorithms from each class and conducts computational tests on benchmark networks of varying sizes. The results are analyzed to compare algorithm performance and identify the impact of different algorithm components on running time.
A Robust Algorithm To Solve The Signal Setting Problem Considering Different ...Joshua Gorinson
This paper presents an algorithm to optimize traffic signal settings that considers the interaction between signal timing and traffic assignment. The algorithm iteratively updates signal timings based on fixed traffic flows, and then updates traffic flows based on the new signal timings, with the goal of minimizing total delay. Two different traffic assignment approaches are considered: user equilibrium assignment and a platoon simulation model. The proposed algorithm is compared to other optimization methods on a real traffic network, demonstrating its robustness in handling different assignment approaches.
Solving bandwidth guaranteed routing problem using routing dataIJCNCJournal
This paper introduces a traffic engineering routing algorithm that aims to accept as many routing demands
as possible on the condition that a certain amount of bandwidth resource is reserved for each accepted
demand. The novel idea is to select routes based on not only network states but also information derived
from routing data such as probabilities of the ingress egress pairs and usage frequencies of the links.
Experiments with respect to acceptance ratio and computation time have been conducted against various
test sets. Results indicate that the proposed algorithm has better performance than the existing popular algorithms including Minimum Interference Routing Algorithm (MIRA) and Random Race based Algorithm for Traffic Engineering (RRATE)
.
Traffic assignment models are used to estimate traffic flows on a transportation network based on origin-destination flows and the network's topology, link characteristics, and performance functions. Traffic is assigned to paths between origin-destination pairs based on travel time or impedance. Traffic assignment is a key part of travel demand forecasting and is used to predict future network flows and performance under different planning scenarios. Common traffic assignment methods include all-or-nothing assignment, user equilibrium assignment, and system optimum assignment.
Optimizing Data Plane Resources for Multipath FlowsIRJET Journal
This document discusses optimizing data plane resources for multipath flows. It introduces the concepts of routing with minimum overhead (RMO) and decomposition with minimum overhead (DMO) to minimize forwarding costs when splitting traffic flows across multiple paths.
The key ideas are:
1) Dividing a traffic flow across multiple paths improves bandwidth utilization but incurs higher forwarding costs due to additional network resources used.
2) RMO and DMO problems are defined to minimize these forwarding costs by reducing the number of paths or nodes used. Efficient algorithms are presented to solve the problems.
3) Simulation results show that algorithms which prefer smaller paths generally perform better at reducing the number of nodes traveled, though they may increase
A new approach in position-based routing Protocol using learning automata for...ijasa
This document summarizes a research paper that proposes a new position-based routing protocol called PBLA (Position-Based routing protocol using Learning Automata) for vehicular ad hoc networks (VANETs) in urban scenarios. PBLA uses learning automata and traffic information to initially find the best and shortest path to a mobile destination. It has two phases: a learning phase where vehicles learn traffic patterns on streets to assign costs; and a routing phase where the shortest path is found using Dijkstra's algorithm and packets are forwarded between intersections greedily. The performance of PBLA is evaluated against GPSR and GPCR protocols in a simulated urban road network, showing it can efficiently route packets in high mobility V
The document discusses a proposed congestion control method called ELFIQM for wireless networks. ELFIQM is inspired by the Engset loss formula queue model and aims to distribute network load evenly across multiple routes. It does this by estimating key parameters like minimum queue length, blocking probability, and congestion rate. Simulation results showed ELFIQM improved packet delivery ratio, throughput, and reduced network overhead by up to 25% compared to other queue-based load balancing techniques.
Joint Routing and Congestion Control in Multipath Channel based on Signal to ...IJECEIAES
Routing protocol and congestion control in Transmission Control Protocol (TCP) have important roles in wireless mobile network performance. In wireless communication, the stability of the path and successful data transmission will be influenced by the channel condition. This channel condition constraints come from path loss and the multipath channel fading. With these constraints, the algorithm in the routing protocol and congestion control is confronted with the uncertainty of connection quality and probability of successful packet transmission, respectively. It is important to investigate the reliability and robustness of routing protocol and congestion control algorithms in dealing with such situation. In this paper, we develop a detailed approach and analytical throughput performance with a cross layer scheme (CLS) between routing and congestion control mechanism based on signal to noise ratio (SNR) in Rician and Rayleigh as multipath fading channel. We proposed joint routing and congestion control TCP with a cross layer scheme model based on SNR (RTCP-SNR). We compare the performance of RTCP-SNR with conventional routing-TCP and routing-TCP that used CLS with routing aware (RTCP-RA) model. The analyses and the simulation results showed that RTCP-SNR in a multipath channel outperforms conventional routing-TCP and RTCP-RA.
A Model For The Dynamic System Optimum Traffic Assignment ProblemSean Flores
This document describes a model for the dynamic system optimum traffic assignment problem. The model seeks to reduce total travel delays in a road network by routing drivers along routes with the lowest marginal delay. It is applicable to networks with many origin-destination pairs and bottlenecks. The model discretizes time and simulates vehicle movement to determine link flows and queues while respecting the first-in, first-out queue discipline at bottlenecks. Numerical results are provided for two test networks to demonstrate the model.
An Analytical Review of the Algorithms Controlling Congestion in Vehicular Ne...iosrjce
This document provides an analytical review of algorithms that control congestion in vehicular networks (VANETs). It begins with an introduction to VANETs and the issues they face with congestion due to high volumes of broadcast safety and non-safety messages. It then reviews several proposed hop-by-hop congestion control algorithms, including one that uses a utility-based approach where packets are prioritized based on encoded utility information. The document analyzes the characteristics and limitations of different algorithms to maintain efficient network operation while preventing overloads during congestion.
This document provides an analytical review of algorithms that control congestion in vehicular networks (VANETs). It begins with an introduction to VANETs and the issues they face with congestion due to frequent broadcasting of safety and non-safety messages. It then discusses several proposed congestion control algorithms for VANETs, including a utility-based approach that encodes utility information in packets to prioritize them. The document analyzes the characteristics of different hop-by-hop congestion control approaches and highlights their advantages over end-to-end approaches for VANETs. Finally, it provides a comparison of various proposed algorithms for VANET congestion control.
Newton-raphson method to solve systems of non-linear equations in VANET perfo...journalBEEI
This document discusses using the Newton-Raphson method to solve systems of non-linear equations to optimize VANET performance. It selects three MAC protocol parameters - transmission power, bitrate, and contention window - as variables. Simulations are run with different values of these parameters. The results for energy consumption, packet delivery ratio, and delay are used to derive three non-linear equations. The Newton-Raphson method is then applied to find the coefficients a, b, and c to optimize the objective function of minimizing energy*delay/PDR. Tables 4-6 show the objective function results for different parameter configurations.
Channel Aware Mac Protocol for Maximizing Throughput and FairnessIJORCS
The proper channel utilization and the queue length aware routing protocol is a challenging task in MANET. To overcome this drawback we are extending the previous work by improving the MAC protocol to maximize the Throughput and Fairness. In this work we are estimating the channel condition and Contention for a channel aware packet scheduling and the queue length is also calculated for the routing protocol which is aware of the queue length. The channel is scheduled based on the channel condition and the routing is carried out by considering the queue length. This queue length will provide a measurement of traffic load at the mobile node itself. Depending upon this load the node with the lesser load will be selected for the routing; this will effectively balance the load and improve the throughput of the ad hoc network.
CROSS LAYER DESIGN APPROACH FOR EFFICIENT DATA DELIVERY BASED ON IEEE 802.11P...pijans
Intelligent Transportation Systems (ITS) have been one of the promising technology that has a great
interest attention from many researchers over the world. Vehicular Ad-hoc Network (VANET)
communications environment as a part of ITS opens the way for a wide range of applications such as safety
applications, mobility and connectivity for both driver and passengers to exploit the transport systems in a
smoothly, efficiently and safer way. Several challenging tasks facing adopting VANET functionality for ITS
such as modelling of wireless transmission and routing issues. These research issues have become more
critical due to the high mobility of vehicles nodes (transmitters and receivers) and unexpected network
topology due to the high speed of nodes. In fact, modelling radio propagation channel in VANET
environment which considers as one of a stringent communications environment is a challenging task. The
selection of a suitable transmission model plays a key role in the routing decisions for VANET. Different
propagation models allow calculating the Received Signal Strength (RSS) based on key environmental
properties such as the distance between transmitter vehicle and a receiver vehicle, the gain and antenna
height of transmitter and a receiver vehicles. Hence, it is useful to calculate RSS and SNR values for a
specific propagation model and then these values can be used later for routing decision in order to find the
best path with high SNR. This paper evaluates the performance of different transmission models (free-
space, two-ray and log-normal) in terms of Receive Signal Strength (RSS). In addition, the performance of
such wireless transmission models for vehicular communication in terms of PDR, throughput and delay is
evaluated by applying the proposed cross layer routing approach based on IEEE 802.11p. By using
MATLAB, the obtained results confirm the best packet delivery ratio for our proposed approach, where it
indicates poor quality of DSSS PHY with high number vehicles. The minimum delay achieved when traffic
density is decreased
CROSS LAYER DESIGN APPROACH FOR EFFICIENT DATA DELIVERY BASED ON IEEE 802.11P...pijans
Intelligent Transportation Systems (ITS) have been one of the promising technology that has a great interest attention from many researchers over the world. Vehicular Ad-hoc Network (VANET) communications environment as a part of ITS opens the way for a wide range of applications such as safety applications, mobility and connectivity for both driver and passengers to exploit the transport systems in a smoothly, efficiently and safer way. Several challenging tasks facing adopting VANET functionality for ITS such as modelling of wireless transmission and routing issues. These research issues have become more critical due to the high mobility of vehicles nodes (transmitters and receivers) and unexpected network topology due to the high speed of nodes. In fact, modelling radio propagation channel in VANET environment which considers as one of a stringent communications environment is a challenging task. The selection of a suitable transmission model plays a key role in the routing decisions for VANET. Different propagation models allow calculating the Received Signal Strength (RSS) based on key environmental properties such as the distance between transmitter vehicle and a receiver vehicle, the gain and antenna height of transmitter and a receiver vehicles. Hence, it is useful to calculate RSS and SNR values for a specific propagation model and then these values can be used later for routing decision in order to find the best path with high SNR. This paper evaluates the performance of different transmission models (freespace, two-ray and log-normal) in terms of Receive Signal Strength (RSS). In addition, the performance of such wireless transmission models for vehicular communication in terms of PDR, throughput and delay is evaluated by applying the proposed cross layer routing approach based on IEEE 802.11p. By using MATLAB, the obtained results confirm the best packet delivery ratio for our proposed approach, where it indicates poor quality of DSSS PHY with high number vehicles. The minimum delay achieved when traffic density is decreased.
CPCRT: Crosslayered and Power Conserved Routing Topology for congestion Cont...IOSR Journals
The document describes a proposed Crosslayered and Power Conserved Routing Topology (CPCRT) for congestion control in mobile ad hoc networks. The CPCRT aims to improve transmission performance by distinguishing between packet loss due to link failure versus other causes, while also conserving power used for packet transmission. It builds upon an earlier Crosslayered Routing Topology (CRT) approach by incorporating power conservation. The CPCRT is intended to identify the root cause of packet loss, avoid unnecessary congestion handling from link failures, allow congestion handling at specific high-traffic nodes rather than all nodes, and optimize resource and power usage for packet routing in mobile ad hoc networks.
A Cell-Based Variational Inequality Formulation Of The Dynamic User Optimal A...Joe Osborn
This summarizes a research paper that develops a cell-based dynamic traffic assignment formulation using a variational inequality approach. The formulation encapsulates the Cell Transmission Model to capture traffic dynamics like shockwaves and queues. It aims to precisely follow the ideal dynamic user optimal principle where all used routes between an origin-destination pair have equal travel times. An alternating direction method is used to solve the variational inequality problem. The paper evaluates the formulation using two scenarios to demonstrate traffic dynamics, interactions across links, and adherence to the dynamic user optimal principle.
Optimizing IP Networks for Uncertain Demands Using Outbound Traffic ConstraintsEM Legacy
This document summarizes an approach for optimizing routing in IP networks with uncertain traffic demands. The approach uses simple outbound traffic constraints at each node to bound the maximum traffic originating from that node. It formulates the traffic engineering problem to find optimal link weights that minimize maximum link utilization under these demand uncertainties. Computational results on a sample network show the impact of the proposed uncertainty models on performance measures like link utilization.
IMPACT OF PARTIAL DEMAND INCREASE ON THE PERFORMANCE OF IP NETWORKS AND RE-OP...EM Legacy
12th GI/ITG CONFERENCE ON MEASURING, MODELLING AND EVALUATION OF COMPUTER AND COMMUNICATION SYSTEMS 3rd POLISH-GERMAN TELETRAFFIC SYMPOSIUM
PGTS 2004
Eueung Mulyana, Ulrich Killat
IRJET-Mitigation of Harmonic Power Flow in Unbalanced and Polluted Radial Dis...IRJET Journal
This document presents a novel algorithm to compute harmonic power flow in radial distribution systems accounting for unbalanced conditions and harmonic pollution. It proposes using a dynamic data structure (DDS) to model each branch and store relevant parameters. A recursive solution technique is used to calculate voltages starting from the farthest branches and moving upstream. The method is tested on IEEE 13, 37, and 123 bus test systems. Results show the total harmonic distortion at each node to evaluate the proposed approach. The algorithm aims to efficiently solve for harmonic power flow in unbalanced and non-sinusoidal radial distribution systems.
In this paper, we propose a new traffic flow model of the Long Term Evaluation (LTE) network for the Evolved Universal Terrestrial Radio Access Network (E-UTRAN). Here only one Evolve Node B (eNB) nearest to the Mobility Management Entity (MME) and Serving Gateway (S-GW) will use the S1 link to bridge the E-UTRAN and Evolved Packet Core (EPC). All the eNBs of a tracking area will be connected to each other by the X2 link. Determination of capacity of a links of such a network is a challenging job since each node offers its own traffic and at the same time conveys traffic of other nodes. In this paper, we apply maximum flow algorithm including superposition theorem to solve the traffic flow of radio network. Using the total flow per subcarrier, a new traffic model is also developed in the paper. The relation among the traffic parameters: ‘blocking probability’, ‘offered traffic’, ‘instantaneous capacity’, ‘average holding time’, and ‘number of users’ are shown graphically under both QPSK and 16-QAM. The concept of the network will be helpful to improve the SINR of the received signal ofeNBslocated long distance relative to MME/S-GW.
PERFORMANCE EVALUATION OF LTE NETWORK USING MAXIMUM FLOW ALGORITHMijcsit
In this paper, we propose a new traffic flow model of the Long Term Evaluation (LTE) network for the Evolved Universal Terrestrial Radio Access Network (E-UTRAN). Here only one Evolve Node B (eNB) nearest to the Mobility Management Entity (MME) and Serving Gateway (S-GW) will use the S1 link to
bridge the E-UTRAN and Evolved Packet Core (EPC). All the eNBs of a tracking area will be connected to each other by the X2 link. Determination of capacity of a links of such a network is a challenging job since each node offers its own traffic and at the same time conveys traffic of other nodes. In this paper, we apply maximum flow algorithm including superposition theorem to solve the traffic flow of radio network. Using the total flow per subcarrier, a new traffic model is also developed in the paper. The relation among the traffic parameters: ‘blocking probability’, ‘offered traffic’, ‘instantaneous capacity’, ‘average holding
time’, and ‘number of users’ are shown graphically under both QPSK and 16-QAM. The concept of the network will be helpful to improve the SINR of the received signal ofeNBslocated long distance relative to MME/S-GW.
CONGESTION CONTROL USING FUZZY BASED LSPS IN MULTIPROTOCOL LABEL SWITCHING NE...ijfcstjournal
In this paper, we have proposed a fuzzy based decision making component for high volume traffic MPLS
networks, by implementing Traffic Engineering, Quality of Service and Multipath routing. The approach
explicitly proves to be successful in solving the issues and challenges pertaining to stability, scalability in
high volume and dynamic traffic. Furthermore, it will work to handle congestion by higher link utilization
and provides efficient rerouting of traffic along with fault tolerance in the network. In this research work,
fuzzy calculations are done for fixing the attributes of the MPLS label(s), which is put on particular packet
representing the Forwarding Equivalence Class. Fuzzy controller consists of two sub fuzzy systems- Label
Switched Path setup System (LsS) and Traffic Splitting System (TSS). The computation of dynamic status of
Load and Delay is utilized by LsS to arrange the paths in preference order. The attained Link Capacity and
Utilization Rate are employing by TSS for maintaining congestion free path. The impact of this is to
facilitate, we have better decision making for splitting the traffic for different promising paths. This was
apparent from the series of traffic scenarios. Observations are obtained using this realization.
CONGESTION CONTROL USING FUZZY BASED LSPS IN MULTIPROTOCOL LABEL SWITCHING NE...ijfcstjournal
In this paper, we have proposed a fuzzy based decision making component for high volume traffic MPLS
networks, by implementing Traffic Engineering, Quality of Service and Multipath routing. The approach
explicitly proves to be successful in solving the issues and challenges pertaining to stability, scalability in
high volume and dynamic traffic. Furthermore, it will work to handle congestion by higher link utilization
and provides efficient rerouting of traffic along with fault tolerance in the network. In this research work,
fuzzy calculations are done for fixing the attributes of the MPLS label(s), which is put on particular packet
representing the Forwarding Equivalence Class. Fuzzy controller consists of two sub fuzzy systems- Label
Switched Path setup System (LsS) and Traffic Splitting System (TSS). The computation of dynamic status of
Load and Delay is utilized by LsS to arrange the paths in preference order. The attained Link Capacity and
Utilization Rate are employing by TSS for maintaining congestion free path. The impact of this is to
facilitate, we have better decision making for splitting the traffic for different promising paths. This was
apparent from the series of traffic scenarios. Observations are obtained using this realization.
How To Research For A Research Paper. Write A ReLori Moore
The document provides instructions for completing a 5-step process to request an assignment be written by a writer on the HelpWriting.net platform. It explains that users must first create an account, then complete an order form with instructions and deadline. Writers will then bid on the request and the user can choose a writer to complete the assignment, making revisions until satisfied with the final product.
Best Custom Research Writing Service Paper WLori Moore
The document discusses the steps to get writing assistance from HelpWriting.net, including creating an account, submitting a request form with instructions and sources, and choosing a writer to complete the assignment. Clients can then review and approve the work before authorizing final payment. HelpWriting.net offers revisions and refunds to ensure clients' needs and expectations are met.
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Routing protocol and congestion control in Transmission Control Protocol (TCP) have important roles in wireless mobile network performance. In wireless communication, the stability of the path and successful data transmission will be influenced by the channel condition. This channel condition constraints come from path loss and the multipath channel fading. With these constraints, the algorithm in the routing protocol and congestion control is confronted with the uncertainty of connection quality and probability of successful packet transmission, respectively. It is important to investigate the reliability and robustness of routing protocol and congestion control algorithms in dealing with such situation. In this paper, we develop a detailed approach and analytical throughput performance with a cross layer scheme (CLS) between routing and congestion control mechanism based on signal to noise ratio (SNR) in Rician and Rayleigh as multipath fading channel. We proposed joint routing and congestion control TCP with a cross layer scheme model based on SNR (RTCP-SNR). We compare the performance of RTCP-SNR with conventional routing-TCP and routing-TCP that used CLS with routing aware (RTCP-RA) model. The analyses and the simulation results showed that RTCP-SNR in a multipath channel outperforms conventional routing-TCP and RTCP-RA.
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This document describes a model for the dynamic system optimum traffic assignment problem. The model seeks to reduce total travel delays in a road network by routing drivers along routes with the lowest marginal delay. It is applicable to networks with many origin-destination pairs and bottlenecks. The model discretizes time and simulates vehicle movement to determine link flows and queues while respecting the first-in, first-out queue discipline at bottlenecks. Numerical results are provided for two test networks to demonstrate the model.
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This document provides an analytical review of algorithms that control congestion in vehicular networks (VANETs). It begins with an introduction to VANETs and the issues they face with congestion due to high volumes of broadcast safety and non-safety messages. It then reviews several proposed hop-by-hop congestion control algorithms, including one that uses a utility-based approach where packets are prioritized based on encoded utility information. The document analyzes the characteristics and limitations of different algorithms to maintain efficient network operation while preventing overloads during congestion.
This document provides an analytical review of algorithms that control congestion in vehicular networks (VANETs). It begins with an introduction to VANETs and the issues they face with congestion due to frequent broadcasting of safety and non-safety messages. It then discusses several proposed congestion control algorithms for VANETs, including a utility-based approach that encodes utility information in packets to prioritize them. The document analyzes the characteristics of different hop-by-hop congestion control approaches and highlights their advantages over end-to-end approaches for VANETs. Finally, it provides a comparison of various proposed algorithms for VANET congestion control.
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The proper channel utilization and the queue length aware routing protocol is a challenging task in MANET. To overcome this drawback we are extending the previous work by improving the MAC protocol to maximize the Throughput and Fairness. In this work we are estimating the channel condition and Contention for a channel aware packet scheduling and the queue length is also calculated for the routing protocol which is aware of the queue length. The channel is scheduled based on the channel condition and the routing is carried out by considering the queue length. This queue length will provide a measurement of traffic load at the mobile node itself. Depending upon this load the node with the lesser load will be selected for the routing; this will effectively balance the load and improve the throughput of the ad hoc network.
CROSS LAYER DESIGN APPROACH FOR EFFICIENT DATA DELIVERY BASED ON IEEE 802.11P...pijans
Intelligent Transportation Systems (ITS) have been one of the promising technology that has a great
interest attention from many researchers over the world. Vehicular Ad-hoc Network (VANET)
communications environment as a part of ITS opens the way for a wide range of applications such as safety
applications, mobility and connectivity for both driver and passengers to exploit the transport systems in a
smoothly, efficiently and safer way. Several challenging tasks facing adopting VANET functionality for ITS
such as modelling of wireless transmission and routing issues. These research issues have become more
critical due to the high mobility of vehicles nodes (transmitters and receivers) and unexpected network
topology due to the high speed of nodes. In fact, modelling radio propagation channel in VANET
environment which considers as one of a stringent communications environment is a challenging task. The
selection of a suitable transmission model plays a key role in the routing decisions for VANET. Different
propagation models allow calculating the Received Signal Strength (RSS) based on key environmental
properties such as the distance between transmitter vehicle and a receiver vehicle, the gain and antenna
height of transmitter and a receiver vehicles. Hence, it is useful to calculate RSS and SNR values for a
specific propagation model and then these values can be used later for routing decision in order to find the
best path with high SNR. This paper evaluates the performance of different transmission models (free-
space, two-ray and log-normal) in terms of Receive Signal Strength (RSS). In addition, the performance of
such wireless transmission models for vehicular communication in terms of PDR, throughput and delay is
evaluated by applying the proposed cross layer routing approach based on IEEE 802.11p. By using
MATLAB, the obtained results confirm the best packet delivery ratio for our proposed approach, where it
indicates poor quality of DSSS PHY with high number vehicles. The minimum delay achieved when traffic
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PERFORMANCE EVALUATION OF LTE NETWORK USING MAXIMUM FLOW ALGORITHMijcsit
In this paper, we propose a new traffic flow model of the Long Term Evaluation (LTE) network for the Evolved Universal Terrestrial Radio Access Network (E-UTRAN). Here only one Evolve Node B (eNB) nearest to the Mobility Management Entity (MME) and Serving Gateway (S-GW) will use the S1 link to
bridge the E-UTRAN and Evolved Packet Core (EPC). All the eNBs of a tracking area will be connected to each other by the X2 link. Determination of capacity of a links of such a network is a challenging job since each node offers its own traffic and at the same time conveys traffic of other nodes. In this paper, we apply maximum flow algorithm including superposition theorem to solve the traffic flow of radio network. Using the total flow per subcarrier, a new traffic model is also developed in the paper. The relation among the traffic parameters: ‘blocking probability’, ‘offered traffic’, ‘instantaneous capacity’, ‘average holding
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CONGESTION CONTROL USING FUZZY BASED LSPS IN MULTIPROTOCOL LABEL SWITCHING NE...ijfcstjournal
In this paper, we have proposed a fuzzy based decision making component for high volume traffic MPLS
networks, by implementing Traffic Engineering, Quality of Service and Multipath routing. The approach
explicitly proves to be successful in solving the issues and challenges pertaining to stability, scalability in
high volume and dynamic traffic. Furthermore, it will work to handle congestion by higher link utilization
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fuzzy calculations are done for fixing the attributes of the MPLS label(s), which is put on particular packet
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CONGESTION CONTROL USING FUZZY BASED LSPS IN MULTIPROTOCOL LABEL SWITCHING NE...ijfcstjournal
In this paper, we have proposed a fuzzy based decision making component for high volume traffic MPLS
networks, by implementing Traffic Engineering, Quality of Service and Multipath routing. The approach
explicitly proves to be successful in solving the issues and challenges pertaining to stability, scalability in
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Switched Path setup System (LsS) and Traffic Splitting System (TSS). The computation of dynamic status of
Load and Delay is utilized by LsS to arrange the paths in preference order. The attained Link Capacity and
Utilization Rate are employing by TSS for maintaining congestion free path. The impact of this is to
facilitate, we have better decision making for splitting the traffic for different promising paths. This was
apparent from the series of traffic scenarios. Observations are obtained using this realization.
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تتميز هذهِ الملزمة بعِدة مُميزات :
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A Gradient Projection Algorithm For Side-Constrained Traffic Assignment
1. A Gradient Projection Algorithm for Side-constrained
Traffic Assignment
Joseph N. Prashker* and Tomer Toledo**
*Technion - Israel Institute of Technology
Department of Civil Engineering
Haifa
Israel
Email: prashker@netvision.net.il
**Massachusetts Institute of Technology
Center for Transportation and Logistics
Cambridge, MA
USA
Email: toledo@mit.edu
EJTIR, 4, no. 2 (2004), pp. 177-193
Received: October 2003
Accepted: May 2004
Standard static traffic assignment models do not take into account the direct effects of
capacities on network flows. Separable link performance functions cannot represent
bottleneck and intersection delays, and thus might load links with traffic volumes, which far
exceed their capacity. This work focuses on the side-constrained traffic assignment problem
(SCTAP), which incorporates explicit capacity constraints into the traffic assignment
framework to create a model that deals with capacities and queues. Assigned volumes are
bounded by capacities, and queues are formed when capacity is reached. Delay values at
these queues are closely related to Lagrange multipliers values, which are readily found in
the solution. The equilibrium state is defined by total path travel times, which combine link
travel times and delays at bottlenecks and intersections for which explicit capacity
constraints have been introduced.
This paper presents a new solution procedure for the SCTAP based on the inner penalty
function method combined with a path-based adaptation of the gradient projection algorithm.
This procedure finds a solution at the path level as well as at the link level. All intermediate
solutions produced by the algorithm are strictly feasible. The procedure used to ensure that
side-constraints are not violated is efficient since it is only performed on constrained links
that belong to the shortest path.
2. 178 A radient Projection Algorithm for Side-constrained Traffic Assignment
1. Introduction
The static traffic assignment problem (TAP) deals with predicting traffic flows on the links of
a transportation network, given the travel demand between origins and destinations. The
effects of traffic flows on travel times are represented by link performance functions. User
equilibrium (UE) traffic assignment is found by solving a mathematical minimization
problem (Beckmann et al. 1956).
The solution of the standard TAP formulation has been thoroughly investigated in the
literature. The Frank-Wolfe (F-W) algorithm is widely used due to its simplicity and minimal
computer memory requirements. However, the method suffers from slow convergence to the
optimum, especially in the vicinity of the optimal solution.
In recent years, considerable attention has been given to path-based solution algorithms for
the TAP. Path-based algorithms were previously considered infeasible for large networks.
However, Jayakrishnan et al. (1994) and Chen and Lee (1999) showed that large-scale
problems could be efficiently solved using path-based algorithms with existing computing
capabilities. Moreover, the path-based solution is useful in cases that require knowledge of
used paths and path flows, such as ATMSATIS applications. In particular, two path-based
solution algorithms were introduced in the literature: Larsson and Patriksson (1992) proposed
the Disaggregate Simplicial Decomposition algorithm (DSD) and Jayakrishnan et al. (1994)
used the Gradient Projection (GP) algorithm. Both algorithms were shown to be superior to
the F-W algorithm in terms of convergence and computing time (Jayakrishnan et al. 1994,
Tatineni et al. 1998). Chen and Lee (1999) compared DSD and GP on several realistic
networks. They found that GP performed better in most aspects.
A class of problems closely related to TAP is the side-constrained traffic assignment problem
(SCTAP). A side-constrained problem is created when additional constraints are added to the
TAP. These constraints may represent capacity limitations at bottlenecks and intersections or
external constraints on the transportation system.
The purpose of this paper is two-fold: first, we discuss the SCTAP formulation and highlight
its advantages in terms of modeling realism compared to standard traffic assignment.
Secondly, we present a new development of the GP algorithm to efficiently solve the SCTAP.
The new algorithm exploits the characteristics of the GP algorithm to combine it with a
penalty function method.
The rest of the paper is structured as follows: The next section motivates the use of SCTAP.
The extension of the TAP to include side-constraints and solution algorithms to the SCTAP
are presented in section 3. A new path-based solution algorithm for the SCTAP and
assignment results of the proposed algorithm are presented in section 4. A summary of the
results is presented in section 5.
2. Motivation
Static traffic assignment has been the main tool for transportation network analysis within the
four-step planning paradigm. Over the years, many deficiencies of TAP have been raised,
pointing at the over-simplicity and unrealistic aspects of the results produced by this model.
Research efforts in recent years have led to the development of dynamic assignment models
that relax some of the simplifying assumptions made in static TAP. However, dynamic
3. Joseph N. Prashker and Tomer Toledo 179
assignment models are still far from being able to replace static models in practice. One
important reason for that is the ability of TAP models to efficiently handle large-scale
applications that may be beyond the capabilities of more complex and detailed models.
Furthermore, given the long-term focus of some of the applications with the associated
uncertainty in inputs and the data intensive and computation-heavy nature of dynamic models
it is likely that static models will continue to be an important transportation modeling tool in
the foreseeable future. Thus, efforts to improve the realism of static models are very
important (FHWA 2002). As we will discuss next, the addition of side-constraints to the
standard TAP model has a potential to significantly improve the realism of the solution, while
making relatively simple modifications in the TAP that do not overly complicate the solution
process.
TAP solutions, especially in congested networks, often assign links with flows that exceed
their capacity. For example, approximately 15% of the links in the ADVANCE network
solution (Chen and Lee 1999) are over-saturated. Such assignment results cannot be used for
realistic engineering analysis: Not only that traffic flows on over-saturated links are
unrealistically high, but the assignment on the alternative paths is also distorted since these
links will often be under-utilized. This behavior is typical to models that use link performance
functions that do not provide an upper bound on flows and so tend to under-estimate queuing
delays, such as the BPR function (BPR 1964). Daganzo (1977) proposed to use link
performance functions that are asymptotic to the capacity, such as Davidson’s function
(Davidson 1966) in order to restrict assigned flows. However, application of these functions
produces very high travel times in saturated links (Boyce et al. 1981), and so does not
improve the realism of the solution. The introduction of explicit capacity constraints to the
TAP model is a simple and intuitive alternative to the use of asymptotic link performance
functions to prohibit over-saturated links.
Another shortcoming of standard static assignment is that separable link performance
functions (in which the cost of a link is a function of the flow on that link only) cannot
capture delays caused by the interaction between flows on two or more links. Non-separable
cost functions, which may capture these interactions, require complicated and less efficient
solution techniques (Patriksson 1994). Moreover, existence and uniqueness of equilibrium are
not always ensured (Smith 1979, Sheffy 1985). Similar to capacities, it may be simpler to
represent the limitations imposed by link interactions by introducing side-constraints in the
problem formulation, rather than through the link performance functions. For example, in the
next section we will discuss a simple constraint that can guarantee that the total flow entering
a merging area is smaller or equal to the saturation flow of the merged link. However, it may
be much more difficult to calibrate a link performance function that captures the merging
effect.
3. Side-constrained traffic assignment (SCTAP)
3.1 Formulation
We begin by briefly reviewing some of the basic definitions of the assignment model. Let
G=(N, A) be a directed graph, where N and A are the sets of nodes and links, respectively.
4. 180 A radient Projection Algorithm for Side-constrained Traffic Assignment
Denote rs
q the demand for trips from origin r R
∈ to destination s S
∈ . R and S are subsets
of N. A set, rs
K , of paths that connect r to s is defined for each OD pair. The user equilibrium
(UE) principle states that for each OD pair, all used paths have equal and minimal travel
times:
min
min
0
,
0
rs rs rs
k k
rs rs rs
k k
f t t
k rs
f t t
⎧ > ⇒ =
⎪
∀ ∀
⎨
= ⇒ ≥
⎪
⎩
(1)
rs
k
f is the flow on path rs
k K
∈ , rs
k
t and min
rs
t are travel times on path k and on the shortest
path from r to s, respectively. Path travel times are the sum of travel times on all the links that
comprise the path.
Assuming separable link performance functions, the solution of the following optimization
problem, defined in the space of path flows variables, corresponds to the UE principle:
( )
0
min
a
V
a
a
Z t w dw
= ∑∫ (2)
Subject to:
rs rs
k
k
f q rs
= ∀
∑ (3)
0 ,
rs
k
f k rs
≥ ∀ ∀
(4)
rs rs
a k ak
rs k
V f a
δ
= ∀
∑∑
(5)
a
t and a
V are travel time and flow on link a, respectively. rs
ak
δ is an indicator variable, which
takes a value of 1 if link a is on path k for OD pair rs and 0 otherwise.
We consider the addition of explicit side-constraints to the TAP formulation in Equations (2)-
(5) to represent various conditions and control measures in the network. The extended
formulation was first introduced in Thompson and Payne (1975), which considered capacity
constraints and flow-independent link travel times. Smith (1987) proved their results.
Patriksson (1994) extended them to flow-dependent travel times and general side-constraints.
The theoretical development below considers a set of general side-constraints:
( ) 0
i
g V i I
≤ ∀ ∈ (6)
( )
i
g V is some function of the vector of link flows in the network, V. The subscript i denotes
the constraint index within the set I.
The optimal solution to the SCTAP is characterized by the Karush-Kuhn-Tucker (KKT) first
order optimality conditions (see, for example, Ravindran et al. 1987):
( )
( ) ,
i
rs rs rs
a a ak i ak
a i a a
g V
t V k rs
V
δ λ δ τ
∂
+ ≥ ∀ ∀
∂
∑ ∑ ∑
(7)
5. Joseph N. Prashker and Tomer Toledo 181
rs rs
k
k
f q rs
∗
= ∀
∑
(8)
0 ,
rs
k
f k rs
∗
≥ ∀ ∀ (9)
( ) 0
i
g V i
≤ ∀ (10)
0
i i
λ ≥ ∀ (11)
0
rs
rs
τ ≥ ∀ (12)
( )
( ) 0 ,
i
rs rs rs rs
k a a ak i ak
a i a a
g V
f t V k rs
V
δ λ δ τ
∗ ∂
⎡ ⎤
+ − = ∀ ∀
⎢ ⎥
∂
⎣ ⎦
∑ ∑ ∑
(13)
( ) 0
i i
g V i
λ = ∀ (14)
i
λ and rs
τ are Lagrange multipliers associated with the side-constraints and trip demands,
respectively.
Equations (10), (11) and (14), and can be summarized jointly by:
( )
( )
0 0
0 0
i i
i i
g V i
g V i
λ
λ
< ⇒ = ∀
= ⇒ ≥ ∀
(15)
Assuming that side-constraints capture limited capacities of various road facilities, the above
equations can be interpreted as defining the delays caused by these limitations. When flows
are such that capacities are not reached, there are no delays associated with the corresponding
constraint. When flows reach the capacity, queues form and drivers experience delays.
From Equations (7), (9) and (13) we get:
( )
( )
( )
( )
0
,
0
i
rs rs rs rs
k a a ak i ak
a i a a
i
rs rs rs rs
k a a ak i ak
a i a a
g V
f t V
V
k rs
g V
f t V
V
δ λ δ τ
δ λ δ τ
∗
∗
∂
⎧
> ⇒ + =
⎪
∂
⎪
∀ ∀
⎨
∂
⎪ = ⇒ + ≥
⎪ ∂
⎩
∑ ∑ ∑
∑ ∑ ∑
(16)
This is a generalization of the UE principle stated in Equation (1). The network equilibration
is defined over generalized path travel times that include link travel times and delays at
queues. The delays are captured by the magnitude of the Lagrange multiplier of the side-
constraint:
( ) ,
i
ai i
a
g V
d a i
V
λ
∂
= ∀ ∀
∂
(17)
ai
d is the delay on link a caused by constraint m.
6. 182 A radient Projection Algorithm for Side-constrained Traffic Assignment
It is important to note that, unlike the link flows solution, the path flows and the Lagrange
multipliers are generally not unique. An optimal solution to the SCTAP problem will yield
one out of the possibly infinite such solutions. It is therefore important to use path flow
solutions and interpret Lagrange multipliers with caution. Larsson and Patriksson (1999)
discuss this issue in detail and provide sufficient conditions for uniqueness.
3.2 Side-constraints
Different facilities in the network may be represented in the assignment model through a
simplification of the mechanism that operates them to a single (or set) of constraints. The
most common side-constraints are link capacity constraints:
j j
V C
≤ (18)
j
C is the capacity of link j. The subscript j denotes links in the subset J A
⊆ for which
capacity constraints are defined.
Capacity constraints represent link geometry and bottleneck capacities created by lane drops,
lane closure, or road incidents. Facilities that limit usage time of links (e.g. fixed-time traffic
signals, portals and ramp controls) can also be modeled with capacity constraints. Other
constraints may also be used. Bell (1995) modeled the operations of traffic-actuated signals in
which green proportions change according to the relative demands on all links approaching
the intersection with the constraint:
1
j
j IN j
V W
S c
∈
≤ −
∑ (19)
IN is the set of all links approaching the intersection. j
S is the saturation flow of link j
approaching the intersection. c is the cycle time and W is the lost time. Thus,
W
c
is the
proportion of lost time.
In merging situations, the total flow entering the merging area is limited by the capacity of the
common outgoing link. This can be represented by:
j out
j IN
V C
∈
≤
∑ (20)
IN denotes the set of all links entering the merge. out
C is the capacity of the outgoing link.
The above constraint assumes non-priority merging. However, it may also be used as an
approximation for priority merges, such as an on-ramp merging into a freeway or two-way
stop or yield controlled intersections. While in these situations the major stream is supposed
to have absolute priority over the minor stream in the allocation of available capacity, there is
ample empirical evidence (e.g. Bunker and Troutbeck 2003, Bonneson and Fitts 1999) that
behaviors such as yielding, gap forcing and intersection blockages by minor stream vehicles
result in capacity allocation that is more similar to that of a non-priority merge. To further
simplify the constraints, the merging constraint can be replaced by simple capacity constraints
for each merging link under the assumption that all the merging links are saturated in
congested conditions. In this case the exit capacity from each one of the links entering the
merge will be proportional to its saturation flow:
7. Joseph N. Prashker and Tomer Toledo 183
j
j j out
l
l IN
S
V C C
S
∈
≤ =
∑
(21)
j
S and l
S are the saturation flows of links j and l that enter the merge area, respectively.
3.3 A simple example
The following example illustrates the ability of the SCTAP model to produce more realistic
assignment results compared to standard traffic assignment. Consider the network presented
in Figure 1Figure1, which consists of a central intersection that may represent a city center
surrounded by two ring roads. Travel times on each link are given by the BPR formula with
parameters 0.15
α = and 4
β = . The free-flow travel times ( 0
t ) and capacities of the links in
this network are presented inTable 1. Trip demands are for passing traffic (1→2, 2→1) and
for trips to the center (1→7, 2→7). These demands are presented inTable 2.
6
2
9
3
7
1
8
5
4
10
11
Figure1. Example network
User equilibrium flows on this network were found using two models: the standard model
and the side-constrained model. In the latter, constraints on intersection flows, given by
Equation Fout! Verwijzingsbron niet gevonden., were imposed on intersections 3, 4, 5 and
6. Both solutions are presented in Figure 2. The unconstrained solution shows high levels of
traffic passing through the center and the inner ring. The outer ring is not used at all. As a
result the flows on four links exceed capacities. The flows through intersections 3, 4, 5 and 6
are more than double the capacity of these intersections. It is more realistic to assume that
limited capacities and delays near the center will divert traffic to other paths bypassing the
center. This is apparent in the SCTAP solution: only center-bounded traffic goes into the
center. Passing traffic is diverted from the center to the two ring roads. The resulting link
8. 184 A radient Projection Algorithm for Side-constrained Traffic Assignment
flows are such that the flows through all intersections satisfy the capacity constraints of these
facilities. This example shows that the introduction of side-constraints can significantly alter
traffic assignment patterns and provide more realistic and plausible predictions. The impact
on decision making may not only be in terms of the conclusions that will be drown from the
assignment results, but also in terms of the fidelity decision-makers associate with the
assignment results.
Table 1. Link parameters
Up
Node
Down
Node
t0 C
Up
Node
Down
Node
t0 C
1 3 20 3000 6 2 20 3000
1 4 20 3000 6 3 15 3000
1 8 30 4000 6 7 6 2000
1 9 30 4000 6 10 20 4000
2 5 20 3000 7 3 6 2000
2 6 20 3000 7 4 6 2000
2 10 30 4000 7 5 6 2000
2 11 30 4000 7 6 6 2000
3 1 20 3000 8 1 30 4000
3 6 15 3000 8 4 20 4000
3 7 6 2000 8 11 30 4000
3 9 20 4000 9 1 30 4000
4 1 20 3000 9 3 20 4000
4 5 15 3000 9 10 30 4000
4 7 6 2000 10 2 30 4000
4 8 20 4000 10 6 20 4000
5 2 20 3000 10 9 30 4000
5 4 15 3000 11 2 30 4000
5 7 6 2000 11 5 20 4000
5 11 20 4000 11 8 30 4000
Table 1 Trip demands
Origin Destination Demand
1 2 4000
1 7 3000
2 1 4000
2 7 3000
9. Joseph N. Prashker and Tomer Toledo 185
UE not considering constraints
2680
1180
1180
1180
1180
2680 820
820
820
820
2680
2680
3500
2000
3500
3500
3500
2000
2000
2000
UE considering constraints
750
1250
1250
750
2250
750
2250
1250
1250
1250
1250
1500
750
750
1500
1500
1500
750
750
1250
1250
1250
750
1250
2250
2250
1250
1250
Legend
V/C<0.5
0.5<V/C<1
1<V/C<1.5
Figure 2. Equilibrium traffic flows with and without capacity constraints
3.4 Solution algorithms
Several solution algorithms for the SCTAP were proposed in the literature. Inouye (1986), for
the case of capacity side-constraints, and Yang and Yagar (1994, 1995) for linear side-
constrains, used the inner penalty function method to generate a sequence of standard traffic
10. 186 A radient Projection Algorithm for Side-constrained Traffic Assignment
assignment sub-problems, with penalty terms added to the objective function for each side-
constraint. Penalty factors were decreased in consecutive iterations. The sequence of sub-
problem solutions approaches the optimal solution of the original problem. The F-W
algorithm was used to solve assignment sub-problems. However, these methods exhibit the
same slow convergence properties of the F-W algorithm. Larsson and Patriksson (1995)
considered a simpler version of the problem with only simple capacity constraints. They used
an augmented Lagrangean dual algorithm to solve the problem. The objective function is
augmented to include a Lagrangean exterior penalty term expressing capacity constraints. At
every iteration, a new solution is found and the Lagrange multipliers estimates are updated.
The unconstrained dual problems are solved with the Disaggregated Simplicial
Decomposition (DSD) method proposed by Larsson and Patriksson (1992). Lawphongpanich
(2000) and Larsson et al. (2004) propose modified variants of the basic algorithm that
improve convergence and stability. These methods are applicable to general side-constraints.
Other Lagrangean dualization approaches were proposed by Inouye (1986) and Hearn and
Lawphongpanich (1990).
4. The GPCAP algorithm
As discussed earlier the GP algorithm (Jayakrishnan et al. 1994) is an efficient method to
solve the standard TAP problem. We propose a solution algorithm adapted from the GP
algorithm for the SCTAP. The algorithm is a path-based penalization method. Our algorithm
consists of two levels: an inner penalization step and a gradient projection step. In the former,
the SCTAP is replaced with an unconstrained sub-problem. The sub-problem is similar in
structure to TAP with an additional penalty term in the objective function for each side-
constraint. The sub-problem is solved using the GP algorithm. Next, a new sub-problem is
generated with smaller penalty terms. The optimal solution of the previous sub-problem is
used as an initial solution to the new one. The sequence of sub-problems solutions converges
to the optimal solution of the original problem when the penalty terms approach zero. The
algorithm is named GPCAP (Gradient Projection with CAPacity constraints). Its details are
described next.
4.1 Inner penalty function
In the inner penalty function method, a penalty term for every side-constraint is added to the
objective function. The penalty term is such that when the solution nears a feasibility border
its value rises rapidly. This ensures feasibility of the solution at every iteration. The penalty
term contains a parameter . This parameter is decreased in consecutive iterations through
multiplication by a scaling factor σ(0<σ<1) thus creating the sequence of sub-problems to be
solved. A suitable penalty function is (Yang and Yagar 1994):
( )
( )
ln
n i
n
i
i
g V
P V i
H
γ
−
⎛ ⎞
= − ∀
⎜ ⎟
⎝ ⎠
(22)
( )
n
i
P V is the penalty term associated with side constraint i at iteration n of the inner penalty
function. n
γ is the penalty parameter at iteration n. i
H is a scaling factor given by:
11. Joseph N. Prashker and Tomer Toledo 187
( )
( )
sup
i i
V
H g V m
= − ∀ (23)
Hence, at the nth
iteration, the sub-problem is given by:
( )
0
min
a
V
n
rs rs
a k ak i
a rs k i
t f dV P V
δ
⎛ ⎞
+
⎜ ⎟
⎝ ⎠
∑ ∑∑ ∑
∫ (24)
Subject to the constraints in Equations (3), (4) and(5).
An important feature of this method is that the optimal values of Lagrange multipliers
associated with the penalized constraints ( i
λ ) are given by the derivative of the penalty
function when 0
γ → . Equation (17) relates i
λ to delays in the network. Hence, an explicit
expression for delays is given by:
0
( )
,
( )
n
n
i
ai
i a
g V
d a i
g V V
γ
γ
+
→
∂
= ∀ ∀
− ∂
(25)
4.2 Gradient projection
The sub-problems created through inner penalization are solved by the GP method. In this
method, given a feasible solution, a gradient related search direction for the new solution is
chosen. The step size in this direction can be determined in various ways. When the resulting
new solution is not feasible (i.e. the step size was too large) a projection function transforms
the solution into the feasible area.
This method is very efficient for problems with many simple constraints. It allows the
solution to advance along arcs defining the feasible area rather than along a straight line
inside the feasible region as does the F-W algorithm. Feasibility borders do not slow down
the progress of the solution, since the projection function will ensure feasibility. This allows
for faster convergence, especially in cases (such as TAP) in which the optimal solution lays
on the boundaries with many binding constraints.
Adaptation of the GP algorithm to this problem is based on ensuring the validity of the
demand constraints [Equation (3)]. Flow-carrying paths are separated in two groups: shortest
paths at the current solution, one for each OD pair, and all other used paths, which are non-
optimal at the current solution. The problem is defined in terms of the non-optimal paths
only. Flows on the shortest paths are calculated such that trip demands will be satisfied:
rs rs rs
k
k
k k
f q f rs
≠
= − ∀
∑ % (26)
rs
k
f and rs
k
f
% are the flows on the optimal and non-optimal paths, respectively.
According to the UE principle, the optimal solution will be reached when travel times on all
paths carrying flow will be equal, and not greater than the travel time on any unused path. In
order to advance towards such a solution, flows are transferred from currently non-optimal
paths to the shortest one for every OD pair. The gradient of the objective function in the space
of non-optimal path flows is given by:
12. 188 A radient Projection Algorithm for Side-constrained Traffic Assignment
( )
( ) ( ) ( ) 1
( ) ( )
( )
,
n rs rs
i
a a ak ak
rs rs
rs
a i
k i a
k k
rs rs rs rs rs rs
k k k
k k k
g V
Z f Z f Z f
t V
f f g V V
f
t t d d c c k rs
γ δ δ
⎡ ⎤
∂
∂ ∂ ∂
= − = + −
⎢ ⎥
∂ ∂ − ∂
∂ ⎣ ⎦
= − + − = − ∀ ∀
∑ ∑
%
%
% (27)
Several choices of search direction and step size choices may be used. The simplest option is
to use the gradient direction with a predetermined step size. The step size may also be based
on exact or inexact (e.g. Armijo rule) line search. Bertsekas and Gallager (1982) and
Jayakrishnan et al. (1994) found that for network problems, a variation of the GP algorithm
that uses a diagonally scaled Newton’s search direction gives best results. The Hessian matrix
is approximated by its diagonal elements only:
2
2
( )
rs
k rs
k
Z f
H
f
∂
∂
⎡ ⎤
= ⎢ ⎥
⎣ ⎦
%
%
(28)
The resulting GP iteration is given by:
( )
1
, 1 , ,
,
m
rs m rs m rs m
k k k rs
k
Z f
f f H k rs
f
−
+
∂
⎡ ⎤
= − ∀ ∀
⎣ ⎦ ∂
%
% %
%
(29)
The superscript m denotes the GP iteration counter.
,
rs m
k
H can be explicitly written as:
,
2
,
2 2
( ) ( ) ( )
1 1
( ) ( )
rs m
k
rs m n
a a i i
k
i
a L a i a i a m
V
t V g V g V
H
V g V V g V V
γ
∈
⎡ ⎤
∂ ∂ ∂
= + +
⎢ ⎥
∂ ∂ − ∂
⎣ ⎦
∑ ∑ (30)
,
rs m
k
L is the set of all links that belong to either path k or the shortest path k but not to both.
Validity of the non-negativity constraints [Equation Fout! Verwijzingsbron niet gevonden.]
is ensured by a projection function:
( )
{ }
1
, 1 , , , ,
max 0, ,
rs m rs m rs m rs m rs m
k k k k k
f f H c c k rs
−
+
⎡ ⎤
= − − ∀ ∀
⎣ ⎦
% %
(31)
After performing the projection, flows on all non-optimal paths are smaller or equal to the
previous iteration. Shortest paths flows are increased according to Equation (26).
4.3 Overall structure
The GPCAP algorithm requires efficient combination of the GP algorithm with the inner
penalty function steps. In doing so, it is necessary to ensure that side-constraints are not
violated when performing the GP iteration. Each iteration of the GP algorithm diverts flows
from non-optimal paths to currently shortest paths. Therefore, only side-constraints that
involve links that belong to the optimal path k (and do not belong to some non-optimal path
k) may be violated. To ensure that the new flows on these links do not violate constraints, the
flow diverted in each iteration is bounded by the smallest slack in side-constraints:
13. Joseph N. Prashker and Tomer Toledo 189
( ) ( )
-1
, ( )
inf ( ) - ,
rs m rs rs
i
k ak i
ak
i
m
a a V
g V
f g V k rs
V
δ δ ω
⎡ ⎤
∂
∆ = − − ∀ ∀
⎢ ⎥
∂
⎢ ⎥
⎣ ⎦
∑
% (32)
,
rs m
k
f
∆% is an upper bound on the flow diverted from non-optimal path k to the shortest path
k . ω is a small positive constant that ensures a strictly feasible solution at each iteration.
The number of checks required to perform this step is relatively small. For example, in the
case of link capacity constraints [Equation (18)] it is at the most equal to the number of
constrained links on the shortest path. The bound on the flow that can be diverted would in
this case simplify to:
( )
,
inf - ,
rs m m
k j j
j k
j k
f C V k rs
ω
∈
∉
∆ = − ∀ ∀
%
(33)
The GP algorithm is easily adapted for these checks, hence allowing it to efficiently cope with
side-constraints. The only modification required is the addition of a term to the projection
function corresponding to Equation (32). The modified projection function is given by:
( )
{ }
1
, 1 , , , , , ,
max 0, , ,
rs m rs m rs m rs m rs m rs m rs m
k k k k k k
k
f f H c c f f k rs
−
+
⎡ ⎤
= − − − ∆ ∀ ∀
⎣ ⎦
% % % %
(34)
Given a feasible initial solution, the GPCAP algorithm is summarized in these steps:
1. Initialization: Set σ and 0
γ values and iteration counters m:=0, n:=0.
2. Inner penalty function iteration: Set n:=n+1, 1
n n
γ σγ −
=
3. GP iteration:
• Update link traffic flows and costs. m:=m+1.
• Calculate the shortest path for every OD pair. If it is a new path, add it to the path list.
• Update path flows according to equations (34) and (26).
4. GP convergence: If GP converged go to step 5. Otherwise, go to step 3.
5. Overall convergence: If overall convergence holds calculate link flows and travel times,
stop. Otherwise, set m:=0, go to step 2.
Next, we present a procedure for finding a feasible initial solution based on a technique
proposed by Daganzo (1977). The procedure is based on the GPCAP algorithm itself and
retains the same structure. Given an infeasible solution, we define a temporary feasible area
that contains this solution. Based on this feasibility area, a new solution is calculated by
performing an iteration of the GPCAP algorithm. The procedure is repeated until the
temporary feasibility area is contained in the original one. The feasible area is narrowed by
temporarily changing the right-hand side of side-constraints to ˆ
( )
i i
g V C
≤ :
0 ( ) 0
ˆ
( )(1 ) ( ) 0
i
i
i i
g V
C
g V g V
ε
<
⎧
= ⎨
+ ≥
⎩ (35)
ˆ
i
C is the temporary right-hand side value of constraint i. ε is a small positive constant.
At each iteration constraints that are violated will have very high travel costs (due to the
penalty value). This will cause flows on links associated with these constraints to be smaller
14. 190 A radient Projection Algorithm for Side-constrained Traffic Assignment
in the next iteration. The use of the inner penalty function in the procedure ensures that once a
constraint is satisfied it cannot be violated again. The starting solution for this process can be
an all-or-nothing (AON) assignment or an incremental assignment.
Given an infeasible starting solution, the initializing algorithm proceeds as follows:
1. Initialization: Set σ and 0
γ values and iteration counters m:=0, n:=0.
2. Inner penalty function iteration: Set n:=n+1, 1
n n
γ σγ −
= .
3. GP iteration:
• Update link flows.
• If all side-constraints are satisfied, stop. Otherwise, calculate temporary right-hand
side values according to equation (35) and costs. m:=m+1.
• Calculate the shortest path for every OD pair. If it is a new path, add it to the path list.
• Update path flows according to equations (34) and (26).
4. GP convergence: If GP converged go to step 2. Otherwise, go to step 3.
4.4 Numerical results
The convergence of GPCAP algorithm was compared to Yang and Yagar’s algorithm. Their
algorithm is based on the F-W algorithm. The two algorithms were tested on the Sioux Falls
network (Leblanc 1973). This network contains 24 nodes, 76 links and 550 OD pairs.
Capacity constraints were imposed on 12 links. The algorithm was run to achieve a very
accurate solution. The progress of the algorithms was measured by the deviation from this
solution. Convergence is measured by the maximum absolute percentage error (MAPE)
between the optimal and current flows over all links:
max
n opt
a a
opt
a A
a
V V
MAPE
V
∈
−
=
(36)
opt
a
V and n
a
V are the flows on link a in the optimal and nth
iteration solutions, respectively.
Convergence results are presented in Figure . GPCAP iterations are more complex because of
the need to calculate second derivatives of the objective function and perform path
comparisons to identify common links. However, the additional information used allows the
solution to converge after many fewer iterations. The GPCAP algorithm converged faster and
in a more stable manner. The gap between the algorithms widens as they approach the
optimum. Overall, Yang and Yagar’s algorithm took approximately 27%, 108% and 7820%
time longer than GPCAP to get to levels of accuracy within 10%, 5% and 1% of the optimal
solution, respectively.
15. Joseph N. Prashker and Tomer Toledo 191
0
10
20
30
40
50
60
10 100 1000 10000 100000
Running Time (sec.)
Deviation
from
optimal
solution
(%)
GPCAP Yang
Figure 3. Convergence in Sioux-Falls network
5. Summary
This paper presented a model and solution algorithm for the static traffic assignment problem
with side-constraints. The SCTAP is created by adding side-constraints to the standard static
TAP. Side-constraints may be used to represent various bottlenecks in the network as well as
external factors that affect traffic flow. Several such constraints were described. Queuing
delays are created when side-constraints are satisfied as equalities, i.e., when traffic flows
equal the capacity of the facilities the constraints represent. These delays are closely related to
the optimal values of the corresponding Lagrange multipliers. The resulting UE state is
defined by total path travel times comprised of link travel times and queuing delays.
The SCTAP solution may significantly enhance the realism of the assignment, as the
demonstrated by a simple example. This may not only improve the reliability of long-term
planning decisions, but also allow assignment results to be used for short-term policy and
operational analysis, such as evaluation of demand management and congestion pricing
strategies, estimation of environmental impacts and design of traffic plans for special events.
We introduced the GPCAP algorithm to solve the SCTAP. The algorithm is based on
combining an internal penalty function method with a path-based adaptation of the GP
algorithm. All intermediate solutions produced by the algorithm are strictly feasible. The
procedure used to ensure that side-constraints are not violated is efficient since it is only
performed on constrained links that belong to the shortest path. The algorithm was tested
against Yang and Yagar’s algorithm with favorable convergence results.
16. 192 A radient Projection Algorithm for Side-constrained Traffic Assignment
The GPCAP algorithm provides a solution that includes path flows. These are useful for
estimating turning flows and for route identification. While the path flow solution is not
unique and therefore needs to be interpreted with caution, it is useful for practical analysis, as
in the case of link of interest problems.
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