STAT2005/7004 Introduction to Stochastic Processes
1. A factory has produced N televisions for sale. Here N is a Poisson random variable
with mean λ. These televisions can fail to function at some time. Assume each tele-
vision malfunctions independently. Let Xi denote the life time of the ith television
produced by this factory, which is a positive random variable that follows an exponen-
tial distribution with a rate parameter µ. Let N(t) be the number of televisions which
have failed to work by time t. [16 marks]
(a) Find the mean of N(t) for each t > 0. [3 marks]
(b) Identify the distribution of N(t) for each t > 0. [4 marks]
(c) DoesN(t) have independent increment? Explain your answer. Does your conclusion
still hold if Xi follows a general distribution with a proper density function f , rather
than an exponential distribution? [2 marks]
(d) Is (N(t), t ≥ 0) a nonhomogeneous Poisson process? Explain your answer. Does
your conclusion still hold if Xi follows a general distribution with a proper density
function f , rather than an exponential distribution? [3 marks]
Let Yi be a Gamma random variable with parameter (α, β). Assume Yi are independent.
Define a compound Poisson process
Y (t) =
N(t)∑
i=1
Yi.
(e) Find E(Y (t)) and cov(Y (s), Y (s+ t)) for each s, t > 0. [4 marks]
2. Consider a job shop that consists of 3 identical machines and 2 technicians. Suppose
that, the amount of time each machine operates before breaking down is exponentially
distributed with parameter 0.2 and, a technician takes to fix a machine is exponentially
distributed with parameter 0.3. Suppose that all the times to breakdown and times
to repair are independent random variables and let X(t) be the number of operating
machines at time t. [10 marks]
(a) Write the Kolmogorov’s forward differential equations in terms of pj(t) := P0j(t) =
P(X(t) = j|X(0) = 0), for j = 0, 1, 2, 3. [4 marks]
(b) Obtain the equilibrium probabilities pj = limt→∞ pj(t). [4 marks]
(c) What is the average number of busy technicians in the long-run? [2 marks]
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3. Passengers arrive to a passport control area in a very small airport according to a
Poisson process having rate equal to 40 passengers per hour. The passport control has
a sole officer who completes checks after independent and exponentially distributed
times with expected value equal to 1 minutes. [14 marks]
(a) Obtain the probability that the server is idle. [2 marks]
(b) Calculate the expected number of passengers in the passport control area. [3 marks]
(c) What is the probability that passengers form a queue and what is the expected
size? [3 marks]
(d) Determine the expected time an arriving passenger spends in the passport control
area and the expected time this passenger waits to be served. [2 marks]
(e) What is the probability that a passenger waits at least 5 minutes until his/her
passport starts to be checked by the officer? [4 marks]
