Suppose that arrivals at a counter come at times of a Poisson process with rate $\lambda$. An arriving particle that finds the counter free gets registered and then locks the counter for an amount of time $\tau$ . Particles that arrive while the counter is locked have no effect.
(a) Find the limiting probability the counter is locked at time $t$.
(b) Compute the limiting fraction of particles that get registered.
Can someone give me a hint where to start?
Let $X(t)=0$ if the counter is open and $1$ if it is closed; we will assume that $X(0)=0$. Then $\{X(t):t\geqslant 0\}$ is a continuous-time Markov chain with generator matrix $$ Q = \begin{pmatrix} -\lambda & \lambda\\ \alpha & -\alpha \end{pmatrix}. $$ It is intuitive that the answers to (a) and (b) are $\frac\lambda{\alpha+\lambda}$ and $\frac\alpha{\alpha+\lambda}$. We can verify this by finding the stationary distribution $\pi$ which is the unique solution to $\pi Q=0$ and $\sum_i \pi_i=1$.