Consider the M/M/1 scheme with a capacity of a.

(a) Make sure that the limiting probabilities in (2.4.6) are positive for all ρ ≠ 1.

(b) Figure out for which ρ the probability π_k, as a function of k, is increasing (certainly, for k = 0, ...,a); for which ρ it is decreasing; and for which ρ the probability π_k does not depend on k for k = 0, ...,a. Interpret the result proceeding from the significance of the arrival/service ratio ρ.

(c) How do the probabilities π₀, ..., π_a look for large ρ; formally, for ρ→∞? Interpret the answer.