(a) Given that an arrival occurs in the interval (nδ, (n + 1)δ) for the sampled-time M/M/1 model in Figure 6.5, find the conditional PMF of the state of the system at time nδ (assume n is arbitrarily large and assume positive recurrence).

(b) For the same model, again in steady state but not conditioned on an arrival in (nδ, (n + 1)δ), find the probabilityQ(i, j) (i ≥ j > 0) that the system is in state i at nδ and that i - j departures occur before the next arrival.

(c) Find the expected number of customers seen in the system by the first arrival after time nδ. Note: The purpose of this exercise is to make you cautious about the meaning of 'the state seen by a random arrival'.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.