Question

Markov Chains:

A small town purchases salt by railroad-car loads to be used for melting ice and snow on the roads during the winter. One railroad car holds twelve tons of salt. The amount of salt used in any one storm depends upon the severity and duration of the storm. Past experience shows that, of the storms which are serious enough to call for any salt, some will require only one pass of the salt trucks, some will require two passes, and a few will require three passes. Each pass (a complete coverage of all of the streets of the town) consumes five tons of salt. Also from past experience, it has been estimated that 50 percent of the storms are one-pass storms, 40 percent are two-pass, and 10 percent are three-pass. The initial supply of salt at the beginning of winter is five railroad cars, or sixty tons. Construct a Markov chain model to show the consumption of salt over time, where time is measured discretely in the number of storms since the beginning of winter.

Please give the answer in either Matrix Form or as a Transition Diagram for a thumbs up. Thank you

Answer 1

Transition matrix

Salt consumed ( In tonnes )	0	5	10	15	20	25	30	35	40	45	50	55	60	......
0	0	0.50	0.40	0.10	0	0	0	0	0	0	0	0	0	.....
5	0	0	0.50	0.40	0.10	0	0	0	0	0	0	0	0	.....
10	0	0	0	0.50	0.40	0.10	0	0	0	0	0	0	0	.....
15	0	0	0	0	0.50	0.40	0.10	0	0	0	0	0	0	....
20	0	0	0	0	0	0.50	0.40	0.10	0	0	0	0	0	......
25	0	0	0	0	0	0	0.50	0.40	0.10	0	0	0	0	.....
30	0	0	0	0	0	0	0	0.50	0.40	0.10	0	0	0	......
35	0	0	0	0	0	0	0	0	0.50	0.40	0.10	0	0	......
40	0	0	0	0	0	0	0	0	0	0.50	0.40	0.10	0	.....
45	0	0	0	0	0	0	0	0	0	0	0.50	0.40	0.10	.....
50	0	0	0	0	0	0	0	0	0	0	0	0.50	0.40	......
55	0	0	0	0	0	0	0	0	0	0	0	0	0.50	.....
60	0	0	0	0	0	0	0	0	0	0	0	0	0	.....
......	......	......	.....	.....	......	.....	.......	....	.......	.....	.....	......	......	.....

It is assumed here that the initial stock of salt if exhausts then the city brings in more salt for consumption and hence the transition matrix continues in the same pattern.

Also it is assumed that time domain is discrete and consists of points of time at which storms occur in the city and therefore salt passes are made