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Continuous-time Markov chains (CTMCs) are mathematical models used to describe systems that undergo transitions from one state to another in a continuous time domain. They are an extension of Markov chains to the continuous-time case.

Likewise, discrete-time Markov chains have transition matrices that dictate the probabilities of moving between states, CTMCs have transition rates, which represent the rate at which transitions occur between states. The main characteristic of a CTMC is the 'memoryless' property, meaning the probability of transitioning to the next state only depends on the current state, not on the history of past states.

In the context of our problem, both \( \{X(t)\} \) and \( \{Y(t)\} \) are CTMCs. When these two processes are considered together as a joint process \( \{X(t), Y(t)\} \), we are essentially expanding our state space to consider all possible combinations of states from both chains. The challenge is to demonstrate that the joint process itself is also a CTMC and to define its properties, such as the transition rates and the stationary distribution.

In the world of Markov chains, a stationary distribution is a probability distribution that remains unchanged as time progresses, given the system is left to evolve without external interference. It represents a stable state of the Markov process where the proportions of the system in each state do not change over time.

The stationary distribution is crucial in the study of Markov chains because it provides long-term probabilistic information about the states of the system. For a CTMC, the stationary distribution satisfies the following condition:
\[ \sum_{j} \pi(j)q(j, i) = \pi(i) \sum_{j} q(i, j) \]
Where \( \pi(i) \) is the stationary probability of being in state \( i \), and \( q(i, j) \) is the transition rate from state \( i \) to state \( j \). In our exercise, since \( \{X(t)\} \) and \( \{Y(t)\} \) are independent and time reversible, we can express the stationary distribution of the joint process as the product of their individual stationary distributions.

The detailed balance condition is a specific criterion that ensures a Markov chain is time reversible. Time reversibility implies that the process looks the same whether one moves forward or backward in time.

This concept can be quantitatively expressed by the detailed balance equation:
\[ \pi(i)q(i, j) = \pi(j)q(j, i) \]
For each pair of states \( i \) and \( j \), the product of the stationary probability of being in state \( i \) and the transition rate from \( i \) to \( j \) equals the product of the stationary probability of being in state \( j \) and the transition rate from \( j \) to \( i \). Intuitively, this means there is a balance in the flow of probabilities between any two states in both directions. The exercise guides through demonstrating that the joint process \( Z(t) \) satisfies the detailed balance equation, therefore establishing the time reversibility of \( \{X(t), Y(t)\} \).