Question

A computer device can be either in a busy mode (state 1) processing a task, or in an idle mode (state 2), when there are no tasks to process. Being in a busy mode, it can finish a task and enter an idle mode any minute with the probability 0.2. Thus, with the probability 0.8 it stays another minute in a busy mode. Being in an idle mode, it receives a new task any minute with the probability 0.1 and enters a busy mode. Thus, it stays another minute in an idle mode with the probability 0.9. The initial state X0 is idle. Let Xn be the state of the device after n minutes.

(a) Find the distribution of X2.

Answer 1

To find the distribution of X2, we analyze the probabilities of transitioning between states. The probability of being in state 2 after 2 minutes is 0.08.

Step-by-step explanation:

To find the distribution of X2, we need to analyze the probabilities of the device transitioning between states.

Given that X0 is idle, we start in state 2.

To move to state 1 (busy), the device must receive a new task, which happens with probability 0.1. Therefore, the probability of transitioning from state 2 to state 1 is 0.1.

Once in state 1, there is a 0.2 probability of finishing the task and transitioning to state 2 (idle), and a 0.8 probability of staying in state 1 for another minute.

Therefore, the probability of transitioning from state 1 to state 2 is 0.2, and the probability of staying in state 1 is 0.8.

Using this transition information, we can calculate the distribution of X2.

1. Starting in state 2, there's a 0.1 probability of transitioning to state 1.
2. Once in state 1, there's a 0.8 probability of staying in state 1.
3. Therefore, the probability of being in state 2 after 2 minutes is (0.1)*(0.8) = 0.08.