SYST 611 Homework 7

Issued: Thursday, 10/29/2009

Due: Thursday, 11/5/2009

The class on Thursday 10/29 covered Luenberger, pages 225-239.

Two questions, fifty points each.

Problem 1.

Each American family is classified as living in an urban, rural, or a suburban location. During a given year, 15% of all urban families move to the suburbs, while 5% move to a rural location (the other 80% stay urban).  Also, 6% of all suburban families move to an urban location and 4% move to a rural location. 4% of all rural families move to an urban location while 6% move to a suburban location.

(a)    Write out the probability transition matrix (the P-matrix) for the one-step Markov chain representing this problem.

(b)   If a family now lives in an urban location, what is the probability that it will live in an urban area two years from now?  A suburban area two years from now?  A rural area two years from now?

(c)    Suppose that at present 40% of all families live in an urban area, 35% live in a suburban area, and 25% live in a rural area.  Two years from now, what percentage of American families will live in an urban area?

(d)   Over a very long period of time (hundreds of years), what percentage of American families will live in a rural location?

Problem 2.  Diffusion model.

Consider a container consisting of two compartments, A and B, separated by a membrane.  There is a total of N molecules in the container.  Individual molecules occasionally pass through the membrane from one compartment to the other.  If at any time there are j molecules in compartment A, and N−j molecules in compartment B, then there is a probability of j/N that the next molecule to cross the membrane will be from A to B, and a probability of (N−j)/N that the next crossing is in the opposite direction.  Let y(k) represent the state where k molecules are in compartment A, and therefore N−k molecules are in compartment B.

(a)  Set up a Markov Chain model for this process showing  y(0), y(1), y(2), and y(3), if N = 1,000.

(b) Is the Markov Chain ergodic?

(c) Write out the flow balance equations for state y(1) if N = 1,000.