Stochastic Analysis Assignment

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SCHOOL OF ELECTRICAL ENGINEERING
UNIVERSITY OF THE WITWATERSRAND, JOHANNESBURG
ELEN 3007: PROBABILISTIC SYSTEM ANALYSIS
STOCHASTIC PROCESSES ASSIGNMENT
LAONE K. MATSHEDISO, 722441
14 OCTOBER 2016

INTRODUCTION
A stochastic process is a model for a time dependent random phenomenon or simply a collection of variables ordered in time [1]. The analysis of how systems behave in a random environment is called Stochastic modelling and has many applications, such as clinical research optimization, graphical derivations and data modelling in economics as well as information modulation and control transmission through telecommunication channels. In terms of Electrical Engineering, probability theory is used in the analysis of random signals and
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These inputs are fed into the cars AI and which will then plan a route and give constant directions to the vehicle command such as speed up, turn left etc. Figure 1 below shows a simplified flow diagram of the algorithm. The information given to the AI is constantly changing thus the directions to the vehicle command are updated frequently.

Figure 1: Simplified Algorithm for Self-driving car
These cars rely on sensors such as the Laser Illuminating Detection and Ranging – or LIDAR which is a 3D sensor that gathers real-time information on the location of the car and its surroundings. These sensors include lasers, radars and cameras detect objects in all direction and create a 3D map that must be interpreted by the on board computers AI. Figure 2 below shows an example of a 3D map created by information from the real time sensors. Figure 2: 3D map rendered to AI from sensors Lane Tracking
One of the duties of the AI is to work out the probability of another on the road switching lanes. This process is known as lane tracking
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This can be done using a neural network model based on the theory of conditional probability and Bayes’ rule [4]. A neural network is a computer system modelled on the human brain and nervous system. Queueing theory which is another discipline within the mathematical theory of probability can also be used to calculate the flow of traffic.
Calculating traffic flow can be considered as a point process as it consists of single arrivals of discrete entities i.e. individual cars.
The model uses three parameters to characterize traffic states: velocity (v), flow (q), and density (k). These parameters change with time. As a result, traffic states change continuously as a function of time. If S is defined as a variable to represent a traffic state, S can be written as S=(q,k,v). Additionally, velocity (v) and flow (q) can be represented as a function of density (k) by the traffic flow fundamental relationship q=kv. Thus the state of traffic can be represented by one variable S(t)=k(t).
Traffic speed/velocity was selected as the indicative traffic parameter defining traffic conditions at a specific location and time. This means that the traffic state is measured by vt the speed at

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