Question: Consider an M/M/1 system with parameters λ and µin which customers are impatient. Specifically, a customer that finds k other customers already in the system joins the queue with probability exp(-αk/µ) or leaves with complementary probability 1 - exp(-αk/µ), where α ≥ 0:
a) Find the asymptotic probability px(k) in terms of px(0).
b) Give an expression for px(0) in terms of the system parameters.
c) Find the stability condition assuming µ > 0 and α > 0.
) Find the asymptotic distribution of x and its mean when α → ∞.