Holding times, jump chains and transition probabilities for continuous time Markov processes Flashcards
(18 cards)
What is a continuous-time Markov chain?
A continuous-time Markov chain is a stochastic process where the process moves between states continuously over time, with the probability of transitioning depending only on the current state.
What is the distribution of the holding time in state i in a continuous-time Markov chain?
The holding time in state i is exponentially distributed with rate parameter q_i, meaning the expected time spent in state i is 1/q_i.
Why is the distribution of holding times exponential?
The holding time is exponential because the process exhibits the memoryless property: the probability of staying in a state for additional time is independent of how long the process has already been in that state.
What is the jump chain in a continuous-time Markov process?
The jump chain is a discrete-time Markov chain that captures the sequence of states visited by the continuous-time Markov chain, ignoring the times spent in each state.
How is the jump chain related to the holding times in a continuous-time Markov chain?
The jump chain tracks the sequence of states visited, while the holding times between jumps are exponentially distributed, with each state having its own rate of leaving.
What is the alarm clock analogy in the context of continuous-time Markov chains?
In the alarm clock analogy, each state has an ‘alarm clock’ with an exponentially distributed ringing time. The chain transitions to the state whose alarm clock rings first.
Why is the time at which an alarm clock rings exponential?
The time at which the alarm clock rings must be exponential because if it were not, the holding time would not be memoryless, contradicting the fundamental assumption of a continuous-time Markov process.
What is the probability of transitioning from state i to state j in a continuous-time Markov process?
The probability of transitioning from state i to state j in a small time interval h is p_ij(h) = q_ij * h + o(h), where q_ij is the rate at which the process moves from state i to state j.
What happens when we condition on moving to a particular state in a continuous-time Markov process?
Even when conditioning on moving to a particular state, the holding time in the current state remains exponential with the same rate parameter q_i, ensuring the memoryless property is preserved.
How is the rate q_i related to the rates q_ij for a continuous-time Markov chain?
The rate q_i at which the chain leaves state i is the sum of the rates q_ij for all j ≠ i, where q_ij is the rate of transition from state i to state j.
What is the Chapman-Kolmogorov equation for continuous-time Markov chains?
The Chapman-Kolmogorov equation in continuous time is P(s+t) = P(s)P(t), which expresses the probability of transitioning from one state to another over two time intervals s and t.
How do the Chapman-Kolmogorov equations help in continuous-time Markov processes?
The Chapman-Kolmogorov equations allow us to compute transition probabilities over longer periods by breaking them into smaller intervals, using the property that the transition probabilities over time are multiplicative.
What is the transition probability matrix P(t) for continuous-time Markov chains?
The transition probability matrix P(t) describes the probabilities of moving from one state to another over a time interval t, and can be used to compute the future state distribution from the current state.
What is the significance of the generator matrix in continuous-time Markov processes?
The generator matrix is a key tool that simplifies the study of transition probabilities in continuous-time Markov chains. It is easier to work with than directly computing transition probabilities.
What is the probability of two transitions occurring in a very small time interval h in a continuous-time Markov chain?
The probability of two transitions occurring in a very small time interval h is o(h), meaning it is negligible and can be ignored when considering the probability of only one transition in such a small interval.
What is the probability that the chain stays in state i over a very small time interval h?
The probability that the chain stays in state i over a small time interval h is p_ii(h) = 1 - q_i * h + o(h), where q_i is the rate at which the chain leaves state i.
What does the formula p_ij(h) = q_ij * h + o(h) represent in a continuous-time Markov chain?
The formula p_ij(h) = q_ij * h + o(h) represents the probability of transitioning from state i to state j in a very small time interval h, where q_ij is the rate of leaving state i to go to state j.
How is the transition probability matrix for the jump chain related to the continuous-time Markov chain?
The transition probability matrix for the jump chain describes the probabilities of moving between states in discrete time, and it can be derived from the rates of transitioning between states in the continuous-time Markov chain.
