Consider a machine that works for an exponential amount of time having mean 1/λ before bre...

Consider a machine that works for an exponential amount of time having mean 1/λ before breaking down. Suppose that it takes an exponential amount of time having mean 1/ω to repair the machine. Let Y(t) be the state of the machine at time t for t ≥ 0, where

Then {Y(t), t ≥ 0} is a continuous-time stochastic process. Furthermore, it can be shown that [see Ross (2003, pp. 364-366)]

Thus, the distribution of Y(t) depends on both t and Y(0). By letting t → ∞ in these equations, compute the steady-state distribution of Y(t). Does it depend on Y(0)?