An open air rock concert is taking place in beautiful Pine Valley, and enthusiastic fans from the entire state of Alifornia are heading there long before the much anticipated event. The arriving cars have to be directed to one of three large (practically unlimited) parking lots, $a, b$ and $c$ situated near the valley entrance. The traffic cop at the entrance to the valley decides to direct every third car (in the order of their arrival) to a particular lot. Thus, cars $1,4,7,10$ and so on are directed to lot $a$, cars $2,5,8,11$ to lot $b$ and cars $3,6,9,12$ to lot $c$.

Suppose that the total arrival process $N(t), t \geqslant 0$, at the valley entrance is Poisson, of rate $\lambda>0$ (the initial time $t=0$ is taken to be considerably ahead of the actual event). Consider the processes $X^{a}(t), X^{b}(t)$ and $X^{c}(t)$ where $X^{i}(t)$ is the number of cars arrived in lot $i$ by time $t, i=a, b, c$. Assume for simplicity that the time to reach a parking lot from the entrance is negligible so that the car enters its specified lot at the time it crosses the valley entrance.

(a) Give the probability density function of the time of the first arrival in each of the processes $X^{a}(t), X^{b}(t), X^{c}(t)$.

(b) Describe the distribution of the time between two subsequent arrivals in each of these processes. Are these times independent? Justify your answer.

(c) Which of these processes are delayed renewal processes (where the distribution of the first arrival time differs from that of the inter-arrival time)?

(d) What are the corresponding equilibrium renewal processes?

(e) Describe how the direction rule should be changed for $X^{a}(t), X^{b}(t)$ and $X^{c}(t)$ to become Poisson processes, of rate $\lambda / 3$. Will these Poisson processes be independent? Justify your answer.