International Journal of Computer Science Engineering and Information Technology Research (IJCSEITR) ISSN 2249-6831 Vol. 3, Issue 3, Aug 2013, 95-100 © TJPRC Pvt. Ltd.

DETERMINISTIC TRAFFIC MODELS USING ONE DIMENSIONAL CELLULAR AUTOMATA ARAVINDAN ANBARASU Kumaraguru College of Technology, Coimbatore, Tamil Nadu, India

ABSTRACT The paper briefly discusses two deterministic traffic cellular automata models, wolfram rule 184 and deterministic fukui-ishibashi traffic cellular automata. The paper with an introduction to cellular automata which is used for various kinds of modeling also explains one dimensional binary state cellular automata and wolfram numbers in detail. The paper also describes the basics of traffic flow modeling using cellular automata, wolfram rule 184, deterministic fukui-ishibashi traffic cellular automata models and their application to automated freeway traffic situations.

KEYWORDS: Cellular Automata, Wolfram Number, Wolfram Rule 184, Deterministic Fukui-Ishibashi Model INTRODUCTION A Cellular Automaton (CA) is an infinite, regular lattice of simple finite state machines that change their states synchronously, according to a local update rule that specifies the new state of each cell based on the old states of its neighbors. CA's are discrete dynamical systems and are often described as a counterpart to partial differential equations, which have the capability to describe continuous dynamical systems. The meaning of discrete is that space, time and properties of the automaton can have only a finite, countable number of states. The basic idea is not to try to describe a complex system from "above" - to describe it using difficult equations, but simulating this system by interaction of cells following easy rules. It is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modeling. The Essential Properties of a CA

A regular n-dimensional lattice (n is in most cases of one or two dimensions), where each cell of this lattice has a discrete state,

A dynamical behavior, described by so called rules. These rules describe the state of a cell for the next time step, depending on the states of the cells in the neighborhood of the cell.

TERMS Cell: A single element of a cellular space, the smallest unit of the space. Cellular Space: A lattice space made up of cells, each of which is in one of several predefined states. This is what most people nowadays call a Cellular Automaton, but I find the distinction useful. Cellular Automaton: A structure built in a cellular space, an automaton built out of cells. “CA” is an abbreviation that is often used for these terms. Local Rule: The rule governing the transition between states- The definition of a cell’s finite state machine. It’s called

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“local”, because it only uses the Neighborhood as its input. Neighborhood: The surrounding cells that influence its next state. The choice of neighborhood influences the behavior of the cellular space. Configuration: A snapshot of all cell states, representing a single point in time. When talking about a configuration, it’s usually the starting point or a result of running a cellular space. Generation: One step in the evolution of a cellular space, an intermediate configuration. Passage of time in a cellular space is measured in generations. Simple Cellular Automata Cellular automata consist of a regular grid of cells, each in one of a finite number of states, such as "On" and "Off". The grid can be in any finite number of dimensions. For each cell, a set of cells called its neighborhood (usually including the cell itself) is defined relative to the specified cell. An initial state (time t=0) is selected by assigning a state for each cell. A new generation is created (advancing t by 1), according to some fixed rule (generally, a mathematical function) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighborhood. For example, the rule might be that the cell is "On" in the next generation if exactly two of the cells in the neighborhood are "On" in the current generation; otherwise the cell is "Off" in the next generation. Typically, the rule for updating the state of cells is the same for each cell and does not change over time, and is applied to the whole grid simultaneously, though exceptions are known. One Dimensional-Binary State Cellular Automata and Wolfram Number One-dimensional-binary state CA that uses the nearest neighbors to determine their next state is called elementary cellular automata. There are only 2^8 = 256 elementary CA, and it is quite remarkable that one of them is computationally universal. Elementary CA was experimentally investigated in the 1980’s by S.Wolfram. He designed a simple naming scheme that is still used today. The local update rule gets fully specified when one gives the next state for all 8 different contexts: 111 → b7 110 → b6 101 → b5 100 → b4 011 → b3 010 → b2 001 → b1 000 → b0 The Wolfram number of this CA is the integer whose binary expansion is b7b6b5b4b3b2b1b0.

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Elementary CA is known by their Wolfram numbers. For example, elementary CA number 102 has local update rule 111→0 110→1 101→1 100→0 011→0 010→1 001→1 000→0 In this rule, a b c→ b + c (mod 2) and it will be referred to as the xor-CA. Space-Time Diagram for Elementary Cellular Automata A space-time diagram is a pictorial representation of the time evolution of the CA where consecutive configurations are drawn under each other. Horizontal lines represent space, and time increases downwards. For example, the space-time diagram of the xor-CA starting from a single cell in state 1 is the self-similar Rule 102

Figure 1: The Space-Time Diagram of the Xor-CA Starting from a Single Cell in State 1 is the Self-Similar

TRAFFIC MODELS Road Layout and the Physical Environment When applying the cellular automaton analogy to vehicular road traffic flows, the physical environment of the system represents the road on which the vehicles are driving. In a classic single-lane setup for traffic cellular automata, this layout consists of a one-dimensional lattice that is composed of individual cells (our description here thus focuses on unidirectional, single-lane traffic). Each cell can either be empty, or is occupied by exactly one vehicle; the term single-cell models are used to describe these systems. Because vehicles move from one cell to another, TCA models are also called particle-hopping models. Deterministic Traffic Models This traffic model is defined as a one dimensional array with L cells with closed (periodic) boundary conditions. This means that the total number of vehicles N in the system is maintained constant. Each cell (site) may be occupied by one vehicle, or it may be empty. Each cell corresponds to a road segment with a length l equal to the average headway in a

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traffic jam. Traffic density is given by k = N/L. Each vehicle can have a velocity from 0 to vmax. The velocity corresponds to the number of sites that a vehicle advances in one iteration. The movement of vehicles through the cells is determined by a set of updating rules. These rules are applied in a parallel fashion to each vehicle at each iteration. The length of iteration can be arbitrarily chosen to reflect the desired level of simulation detail. The choice of a sufficiently small iteration interval can thus be used to approximate a continuous time system. The state of the s ystem at iteration is determined by the distribution of vehicles among the cells and the speed of each vehicle in each cell. The following notation to characterize each system state: xi: position of the ith vehicle, vi: speed of ith vehicle, and gi: gap between the ith and the ( i+1)th vehicle (i.e., vehicle immediately ahead) and is given by gi = xi+1 - xi – 1. A typical discretisation scheme assumes ΔT = 1 s and ΔX = 7.5 m, corresponding to speed increments of ΔV = ΔX/ ΔT = 27 km/h. Therefore, vehicles assume the discrete speeds v0 = 0km/h, v1 = 27km/h, v2 = 54km/h and so on. The spatial discretisation corresponds to the average length a conventional vehicle occupies in a closely packed jam, i.e-7.5m (and as such, its width is neglected), whereas the time discretisation is based on a typical driver's reaction time and implicitly assume that a driver does not react to events between two consecutive time steps. Wolfram's Rule 184 (CA-184) The simplest deterministic traffic model is wolfram’s rule 184.

Figure 2: Simplest Deterministic Traffic Model is Wolfram’s Rule 184 The shaded portion represents the presence of a vehicle. The un-shaded portion represents the absence of a vehicle This has the physical meaning that a vehicle (Black Square) moves to the right if its neighboring cell is empty. In this model max velocity is 1. For a TCA model, the previous actions as a set of rules that are consecutively applied to all vehicles in the lattice. Rule 1 Acceleration and braking vi (t) ← min {gi (t − 1), 1} Rule 2 Vehicle movement xi (t) ← xi(t − 1) + vi(t) Rule1, sets the speed of the ith vehicle, for the current updated configuration of the system; it states that a vehicle always strives to drive at a speed of 1 cell/time step, unless its impeded by its direct leader, in which case gi(t−1) =0 and

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the vehicle consequently stops in order to avoid a collision. The second rule 2 is not actually a ‘real’ rule; it just allows the vehicles to advance in the system.

Figure 3: Typical Time-Space Diagrams of the CA-184 TCA Model Typical time-space diagrams of the CA-184 TCA model. The shown closed-loop lattices each contain 300 cells, with a visible period of 580 time steps (each vehicle is represented as a single colored dot). Left: vehicles driving a free-flow regime with a global density k = 0.2 vehicles/cell. Right: vehicles driving in a congested regime with k = 0.75 vehicles/cell. The congestion waves can be seen as propagating in the opposite direction of traffic; they have an eternal life time in the system. Both time-space diagrams show a fully deterministic system that continuously repeats itself. Deterministic Fukui-Ishibashi TCA (DFI-TCA) In 1996, Fukui and Ishibashi constructed a generalization of the prototypical CA-184 TCA model. Although their model is essentially a stochastic one, this paper discusses its deterministic version. Fukui and Ishibashi’s idea was twofold: on the one hand, the maximum speed was increased from 1 to vmax cells/time step, and on the other hand, vehicles would accelerate instantaneously to the highest possible speed. Corresponding to the definitions of the rule set of a TCA model; the CA-184’s rule Rule1 changes as follows: Rule1: Acceleration and Braking vi(t) ← min {gi (t − 1), vmax} Just as before, a vehicle will now avoid a collision by taking into account the size of its space gap. To this end, it will apply an instantaneous deceleration: for example, a fast-moving vehicle might have to come to a complete stop when nearing the end of a jam, thereby abruptly dropping its speed from vmax to 0 in one time step.

Figure 4: Left: Several (k, vs) Diagrams for the Deterministic DFI-TCA, Right: Several of the DFI-TCA’s (k, q) Diagrams

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Left: Several (k, vs) diagrams for the deterministic DFI-TCA, each for a different Vmax ∈ {1, 2, 3, 5}. Similarly to the CA-184, the global space-mean speed remains constant, until the critical density is reached, at which point vs. will start to diminish towards zero. Right: several of the DFI-TCA’s (k, q) diagrams, each having a triangular shape (with the slope of the congestion branch invariant for the different vmax). The mean speed remains constant at vmax till critical density after which the mean speed becomes (1-k)/k.

Figure 5: Left: the (k, vs) Diagram for the Deterministic CA-184, Right: the (k, q) Diagram for the Same TCA Model Left: the (k, vs) diagram for the deterministic CA-184, with now vmax → +∞. Right: the (k, q) diagram for the same TCA model, resulting in a critical density k c = 0.

CONCLUSIONS The models discussed here are deterministic models. They are basic models that work in a deterministic way and do not include the effects of sudden braking or idling. There are various stochastic models like Nagel-Schreckenberg TCA to deal with such cases. The basic models discussed here can be applied only to automated freeway systems.

REFERENCES 1.

A deterministic traffic flow model for the two-regime approach –by Ceder, A

2.

Self-organization and a dynamical transition in traffic-flow models- by Ofer Biham and A. Alan Middleton

3.

A cellular automaton model for freeway traffic – by Kai Nagel and Michael Schreckenberg

4.

A set of effective coordination number (12) radii for the -wolfram structure elements –by S. Geller

5.

Traffic Flow Theory -by Sven Maerivoet, Bart De Moor

6.

Statistical mechanics of cellular automata by-Stephen Wolfram