ENGG7302, Assignment 3 (Stochastic Process)
Due date : Noon (12pm), Friday, 18 November 2016.
Submit : Through turnitin
UQ Shopping Center (UQSC) has requested your company to assess the safety of UQSC in the case of a fire breakout. The main task for your company is to assess if the emergency exit door in UQSC is well placed, in the sense that the position of the door maximizes the likelihood that customers are able to escape in the case of a fire.
As a preliminary study, your company will analyze the position of the emergency exit door only at the top level of UQSC. The map of this floor is shown in Figure 1a. For simplicity of the safety analysis, UQSC¡¯s management divides this top level into 9 regions, as shown in Figure 1b. Shoppers can only enter or leave the top level of UQSC through the elevator at Region 5. The existing emergency exit door is at Region 6.
(a) (b) Fig. 1. Top level of UQ Shopping Center (UQSC).
Furthermore, UQSC has provided two information to your company, i.e.: 1. The flow of shoppers in the top level of UQSC (in terms of region).
Shoppers can only move from one region to its neighboring regions in its North, South, East, and West, i.e.,
o From region-1 to region-2 and region-4.
o From region-2 to region-1, region-3, and region-5. o From region-3 to region-2 and region-6.
o From region-4 to region-1 and region-7.
o From region-5 to region-2 and region-8.
o From region-6 to region-3 and region-9.
o From region-7 to region-4 and region-8.
o From region-8 to region-7, region-9, and region-5. o From region-9 to region-6 and region-8.
Shoppers¡¯ movements from region to region depend only on where they are at the moment. To understand the shoppers¡¯ behavior better, the management of UQSC recorded the movement of shoppers from region to region over two weeks (flowData.txt). However, they ignore the flow of people entering and
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leaving the floor. For simplification, one can assume that all shoppers have the same behavior. Furthermore, their behaviors are independent of which time or day they are in.
File format of flowData.txt:
Each line represents the captured transition of a shopper from one region to another, with the first number indicating the region where the shopper is at the moment, and the second number indicating the region where the shopper is at the next time step.
2. Data on whether shoppers can successfully escape the building, based on real and simulated fire drills (escapeData.txt). Again for simplification, one can assume that all shoppers have the same behavior.
File format of escapeData.txt:
Each line represents whether a shopper at a particular region manages to escape the building if the emergency exit door is at a particular region. The first number in a line is the region where the shopper is, the second region is the region where the emergency exit door is, while the third number indicates whether the shopper manages to escape or not, with 0 means fail and 1 otherwise.
Does the existing emergency exit door provide the highest likelihood for shoppers to escape the building in the case of a fire? If it does not, where should the emergency exit door be (in terms of region), such that the shoppers will have the highest likelihood of escaping the building in the case of a fire?
Solving the above problem requires solving five main steps. Each step is worth equal mark (20/100). For each step, please explain clearly:
1. What is the goal of the step.
2. Why the step is needed for solving the problem.
3. How you achieve your goal for the particular step, e.g., how you model the problem, what assumptions do you use, how do you compute the solution, are there any computational simplifications you made due to problem structures, the program (if any).
To help you get started, here are the first two steps:
1. Model the shoppers¡¯ movement between regions as a Markov Chain.
2. Compute the transition probability matrix of the Markov Chain from previous
step from the given data set. For this purpose, please use Maximum Likelihood Estimation (MLE).
Note on grading:JThe journey is worth more than the destination.J
Logical steps and explanation are valued more than the final answer, i.e., you will not get a full mark if you only provide the final answer without any logical explanation.