TSP3251

Applied Probability and Stochastic Process



Mini-Project




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Student Admission Model

Question 1

The aim of this project is to determine how Markov Chain could be used to model the process of student admission for the course of Communication offered by the Faculty of Broadcasting of Oxford University in UK.

A model is thus being designed to describe a system which at any given point of time, is in one, and only one, of the defined sets of states and most importantly the future state of the system depends only on the current state and not on any of the past states. For this reason the Markov Chain is often referred to as a “memoryless” model.

This model approximates the process of student admission as a series of transitions between four possible states where a potential student would be in. They are:
• State 0: has not applied to OXFORD UNIVERSITY
• State 1: has applied to OXFORD UNIVERSITY but an accept/reject decision has not yet been made
• State 2: has applied to OXFORD UNIVERSITY and has been rejected
• State 3: has applied to OXFORD UNIVERSITY and has been accepted (a successful placement has been made)


The following diagram shows how the process of admission takes place and identifies the states involved at a given time. The states can be identified by the Bolded lines.



As illustrated in the diagram above, a potential student of OXFORD UNIVERSITY can be classified under one, and only one, of the four states. When a potential student applies for the Communication course, his state could change between the given four states. For example, after one month of his application, he may well be in State 1, 2 or 3, i.e. the decision to accept/reject his application may not have been decided yet; or his application could have been rejected or; his application could have been accepted.

At the start of the year (month 1 in the admissions year) all potential students are in state 1.

A probability could be assigned to any of the transitions by analyzing the admission statistics of the course for a quite considerable period of time. The specific probabilities and factors which need to be considered would be discussed later in Question 2 of this project.

State 0(E0) State 1(E1) State 2(E2) State 3(E3)
State 0: has not applied to OXFORD UNIVERSITY P00 P01 0 0
State 1: has applied to OXFORD UNIVERSITY but an accept/reject decision has not yet been made 0 P11 P12 P13
State 2: has applied to OXFORD UNIVERSITY and has been rejected 0 0 1 0
State 3: has applied to OXFORD UNIVERSITY and has been accepted 0 0 0 1

A transition matrix could be obtained from the transition table and could be summarized as below.






T is called the transition matrix of the above system. In our example, a state is the stage at which any potential student of OXFORD UNIVERSITY is in the system at a particular time. The entry Pij in the above matrix represents the probability of transition from the state corresponding to i to the state corresponding to j.

In the model some of the transitional probabilities would always remain constant. For example, it would have been the final result if a potential student did apply to do Communication in OXFORD UNIVERSITY and was rejected. Thus even after a month, two and always the probability of transition from State 2 to State 2 would always be 100%, i.e. 1.

Furthermore, when we look at our matrix in detail we would understand that the Admissions manager has control over the elements in one row of the above transition matrix, namely row 1. The elements in this row reflect:
• Transition from 1 to1: the speed with which applications are processed each month.
• Transition from 1 to 2: the proportion of applicants who are rejected each month.
• Transition from 1 to 3: the proportion of applicants who are accepted each month.

To be more specific, the Admissions manager has to decide the proportion of applications that should be (i) handled within that month (ii) accepted that month, and (iii) rejected that month.

Thus, for example, the transition probability P12 (a potential student who have applied for placement to be rejected) or P13 (a potential student who have applied for placement to be accepted) depends mainly on the capability of the Admissions manager to evaluate the applications.

The objective of the model is not just to analyze the history of how the admissions to the course have been made, but more importantly it could be used in order to predict future admissions to the Communication course will take place.
The model would inform the management how long it would take to process an average application in the future, what factors needs to be considered in order to improve the situation, etc.

Furthermore the model could be compared with newly designed models in order to predict and discover an ideal time period for an application to complete given the competitive environment in which OXFORD UNIVERSITY operates. Even though this would not be so relevant in the current context, as OXFORD UNIVERSITY is the only university offering a course in Communication in UK, it would, however, be necessary in the future for them to be informed on ways to tackle the competition faced by OXFORD UNIVERSITY in offering Communication as a Undergraduate degree in UK.



Question 2

As seen in the Transition Matrix in Question 1 above, there are five transitions for which we need to define the probability. They include the probability of P00, P01, P11, P12, and P13. The remaining 11 probabilities would remain constant regardless of the duration of the transition as shown in the transition matrix in Question 1.

As discussed before, these probabilities could be assigned after analyzing the admission statistics of the course for a considerable period of time. Suppose for example after analyzing the data for the past 3 years for the course of Communication, and by discussion sessions held by experienced persons in this field, the probabilities were decided as follows for each month:
P00 = 0.97
P01 = 0.03
P11 = 0.10
P12 = 0.15
P13 = 0.75


Thus the new transition matrix is as follows:






Since each potential student could be categorized to one and only one of the four states, each row sums to 1. Because we are dealing with probabilities, each entry must be between 0 and 1, inclusive. The most important fact that lets us model this situation as a Markov chain is that the next state of the potential student depends only on the current state, not previous history.

The resulting transition diagram could be drawn as follows:














Checking whether the chain is ergodic

Since f0 is < 1, therefore E0 is a Transient State

Since f1 is < 1, therefore E1 is a Transient State

Since f2 is = 1, therefore E2 is a Persistent State

Since f3 is = 1, therefore E3 is a Persistent State

As all of the states are not Persistent, the chain is therefore not ergodic.

Calculation of Steady-State Distribution

from this we get:

we also know that

which solves to

this indicates that this Markov Chain does not have a steady-state distribution as the transition matrix is not regular.

Proof that the transition matrix is not regular
A Square matrix is called regular if for some integer n all entries of Tn are positive. For example when n =2,


Thus, transition matrix T is not a regular matrix, because for all positive integer n, all the entries in Tn are not positive.
The steady-state distribution is used to depict a long-term conclusion using a Markov Chain Model. In this case, the steady-state vector does not exist; therefore, it is not possible to make predictions on the longer term. For example it is not possible to correctly make assumptions using the Student Admission Model for long periods, for example for 3 years.