Suppose you work for NASA. Starting at time 0, you launch rockets into space according to a Poisson
process with rate $\lambda$ rockets per hour; i.e. the times of rocket launches are exactly the times of events in
a Poisson process with rate $\lambda$. Suppose that each rocket launched has a random speed (given in miles per
hour), independent of all else. Further suppose that this random speed is always drawn i.i.d. according to
a r.v. $S$, and that there exist two strictly positive constants $L < H$ such that $P(S = L) = P(S = H) = \frac{1}{2}$.
Assume that the rockets travel according to the following simplified linear model of physics: if a rocket is
launched at a time $u$, and has initial random speed $s$ miles/hour, then for all $w \ge 0$, its position at time
$u + w$ will equal $ws$. Assume that all times are given in hours, and all positions are given in miles.
(1) Suppose that each rocket, in addition to being assigned a random speed, is also assigned a random
“target height”, drawn i.i.d. from an exponential distribution with mean 1 mile (drawn independently
for each rocket). Let $N(L, t)$ denote the number of rockets which were launched before time $t$, assigned
initial random speed $L$, but have not yet reached their target height by time $t$. Model $\{N(L, t), t \ge 0\}$ as a
birth-death process, giving all birth and death rates.
(2) Consider again the setup of part (1). Let $N(t)$ denote the total number of rockets (both low and
high speed) which were launched before time $t$, but have not yet reached their target height. Prove that
$N(t)$ converges in distribution to a limiting r.v. as $t \to\infty$.
For problem (1), I try to figure out what is the transition probability of the underlying DTMC. I think the state space should be the number of $N(L,t)$. It seems related to the property of Poisson process that conditional on the number of coming events in $[0,t]$ of a Poisson process, the unordered coming events distribute as a uniform distribution over $[0,t].$ But I don't see how to model it a continuous time Markov chain. And it seems problem (2) related to problem (1).