There are a total of 9 projects from which you can select one project that you will
be working on in groups of three.
We will have three sessions in total, with about 10 groups presenting in each
session. This means that each group will have six minutes to present their project
during their assigned presentation time.
1. In the single-server queue, specify the probability that there are n persons in
the queue at time t + δt, denoted as Pn(t + δt), as well as the probability that
there is no person in the queue at time t + δt, denoted as P0(t + δt). Explain
all possibilities and obtain two differential equations.
2. Using the limiting process, and solving the differential equation related to
Project Number One and obtain Pn.
3. Suppose that the density function of the interval between arrivals is
exponential with parameter λ, the service time is a fixed value, say τ, find
Pn(t + τ)
4.
For a general two-state Markov chain, consider the two-state chain with
transition matrix T and find Tn
5. Find the inverse matrix of a 3x3 matrix, with an example.
6. Explain the Gambler's Ruin problem as a Markov chain
7. Incoming telephone calls to an operator are assumed to be a Poisson process
with parameter λ. Find the density function of the length of time for n calls
to be received and find the mean time and variance of the random variable of
the length of time for n calls.
8. Find the exact probability distribution of a random walk.
9. Suppose that A is a gambler and has an initial capital of k units, while B is
the opponent and starts with a-k. From the gambler's point of view, explain
the problem and find P(Ck), which is the probability of the gambler
eventually going broke when starting with an initial capital of k.