Instructions:

Suppose people arrive to an area in accordance to a homogeneous Poisson process, having

rate . Whenever a person arrives to the area, he/she stays in that area for an exponentially distributed amount of time having rate , independently of everything else.

Let Q(t) denote the number of people currently in the area at time t, and let D(t) denote the number of people that have departed from the area by time t.

3. Write R code that can be used to simulate Q(s) and Q(t), when 0 < s < t.

Warning:

Q(s) and Q(t) will be dependent random variables! What you will want to do is create an R function whose input is s, t, , and .

Hint:

Q(s) can be interpreted as the number of customers arriving in (0,s] that are still present at time s, but not at time t, plus the number of customers arriving in (0,s] that are still present at time t. Likewise, Q(t) can be interpreted as the number of customers arriving in (0,s] that are still present at time t, plus the number of customers arriving in (s,t] that are still present at time t.

4. Find, for 0< s < t, E[Q(s)Q(t)].

Hint: The hint from Problem 3 can be used to find this expected value as well. What we are essentially doing is writing Q(s) and Q(t) as functions of independent random variables.

5. Finally, use Monte Carlo simulation to justify that the formula you found in Problem 4 is correct. Try verifying it for the case where s= 1,

t = 3, = 2, and = 1 (there's nothing special about these numbers, they are random).