Final answer:
In a queueing model Markov chain, option A (p = 0, q = 1) is null recurrent, option B (p = 1, q = 0) is transient, and options C (p = 0.5, q = 0.5) and D (p = 0.3, q = 0.7) are positive recurrent.
Step-by-step explanation:
The question pertains to the classification of states in a Markov chain as either null recurrent, positive recurrent, or transient, based on the transition probabilities, p and q, where p stands for the probability of success and q for the probability of failure in a Bernoulli trial. The sum of p and q equals 1 for any trial.
Option A describes a scenario where p = 0 and q = 1, meaning there is no possibility of moving forward in the queue (no arrival) and there is a certainty of moving backward or staying in the same state (service occurring). This implies that the state will eventually be reached infinitely often, but the expected time is infinity, classifying the chain as null recurrent.
Option B, with p = 1 and q = 0, results in a chain where there's always advancement in the queue and no service. This condition means the number of items in the queue will continue to grow without bound, leading to a transient state.
Option C has p = q = 0.5, indicating that moving forward or backward in the queue is equally likely. This creates a balanced situation, and the chain is positive recurrent, meaning the expected return time to any state is finite.
Option D, with p = 0.3 and q = 0.7, suggests that there's a higher chance of service than arrivals. This condition still leads to a positive recurrent Markov chain because the queue tends to decrease over time, resulting in a finite expected return time to any state.
Therefore, options C and D are both positive recurrent. Option A is null recurrent, and option B is transient.