 Holding rate of a continuous-time Markov chain

ID: holding-rate-of-a-continuous-time-markov-chain

The holding rate at state

i

Q = (q_{ij})

q_{i} = - q_{ii} = \sum_{j \neq = i} q_{ij} .

For a conservative chain with finite

q_{i} > 0

T_{i}

Pr (T_{i} > t) = e^{- q_{i} t}

and

E T_{i} = 1/ q_{i}

. Conditional on a jump, the next state is

j

q_{ij} / q_{i}

. Thus the rate determines when departure occurs, while normalized outgoing rates determine where it goes.

If the outgoing rates are

q_{i 1} = 2

and

q_{i 2} = 3

for two distinct other states, the total rate is five, the mean holding time is

1/5

, and departure chooses them with probabilities

2/5

and

3/5

. A zero rate gives an absorbing state with infinite holding time. Statewise finite rates alone do not rule out explosion of a continuous-time Markov chain.

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