Please answer all a b STOCHASTIC PROCESSES

The n roommates in an apartment, including Charlie and Alex, eat only pizzas. Assume that each roommate orders a pizza independently of the others according to a Poisson process of intensity v and immediately put the box containing the pizza in the common trash. This has a capacity of k boxes, and Charlie has the task of taking down the trash to empty it as soon as it is full. Determine: (a) the expectation and the variance time before Charlie takes down the trash from the moment when it is empty; and | (b) the time expectation that Alex, who replaces Charlie in this task after a very long time, do not take the trash down to first time. Suggestion: You can use the fact that the variance of a exponential law is the square of expectation. The n roommates in an apartment, including Charlie and Alex, eat only pizzas. Assume that each roommate orders a pizza independently of the others according to a Poisson process of intensity v and immediately put the box containing the pizza in the common trash. This has a capacity of k boxes, and Charlie has the task of taking down the trash to empty it as soon as it is full. Determine: (a) the expectation and the variance time before Charlie takes down the trash from the moment when it is empty; and | (b) the time expectation that Alex, who replaces Charlie in this task after a very long time, do not take the trash down to first time. Suggestion: You can use the fact that the variance of a exponential law is the square of expectation