Please Slove this Markov Chain Problems Markov Chains 2) Suppose we have 3 pints

Question

Please Slove this Markov Chain

Problems Markov Chains 2) Suppose we have 3 pints of blood on hand at the end of the day. At tart, we get another pint making 4 pints total During the period we have a demand for 0,1,2, or 3 pints of blood with the following distribution P(x) .3 .2 .1 Assume we receive 1 pint of blood each day at the start of the day. We can hold no more than 5 pints of blood at the end of any given day. Excess blood over 5 pints will be discarded. Construct a one-step transition probability matrix to model this process as a Markov chain.

Explanation / Answer

let the event Yn be the number of pints of blood at the starting of the nth day after receiving 1 pint of blood.

so initially we have 4 pints of blood. demand is 0,1,2 or 3.

we can not hold no more than 5 pints of blood at the end of any given day.

so at the starting of the day after receiving 1 pint of blood, the maximum number of pints of blood available can be 6.

at if any case we have 0 pints of blood at the end of the day then next day starting we would have minimum 1 pint of blood.

so Yn can take the values 1,2,3,4,5,6

and we have the demands as

x: 0 1 2 3

p(x): 0.4 0.3 0.2 0.1

so the transition probability matrix is

1 2 3 4 5 6

1 0.3+0.2+0.1=0.6 0.4 0 0 0 0

2 0.2+0.1=0.3 0.3 0.4 0 0 0

3 0.1 0.2 0.3 0.4 0 0

4 0 0.1 0.2 0.3 0.4 0

5 0 0 0.1 0.2 0.3 0.4

6 0 0 0 0.1 0.2 0.4+0.3=0.7

  

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