Summary: Examination Introduction to Stochastic Processes
(LNMB/Dutch Master Program)
Monday December 19, 2005, 12.00 - 15.00 hours
This exam consists of 5 problems
Use of book is not permitted
Motivate your answers!
1. Suppose that print jobs arrive at a network printer with independent and
exponentially distributed inter-arrival times with parameter 10 per hour. It
takes the printer exactly 6 seconds to print each page. The sizes of the jobs
are i.i.d. and have a Poisson distribution with a mean of 2 pages.
a. What is the probability that precisely 20 jobs arrive between 8.30 hours and
b. What is the probability that a print job arrives while the previous job is
not finished, when this previous job consists of 6 pages? (Assume that the
printing of this previous job started immediately after its arrival).
c. What is the expected arrival time of the first job after 12.00 hours, when
the previous job arrived at 11.58 hours?
d. Let M(t) be the number of print jobs consisting of more than 3 pages that
arrive in a time interval (0, t]. Give an expression for P(M(t) = m).
2. In a simple model, the state of a car dynamo at the beginning of year n is