CONTINUOUS-TIME MARKOV CHAINS In the Markov chain models that we have con�idered so far. we have assumed

qij = 0,

for - j I > 1.

In a birth-death process, the long-term expected frequencies of transitions from

to j and of transitions from j to must be the same, leading to the local

balance equations

for all j.

The local balance equations have the same structure as in the discrete-time case, leading to closed-form formulas for the steady-state probabilities.

Example 7.15 (continued) . The local balance equations take the form

0, 1 , . . . , m -1,

and we obtain 1I"i+1 = p7ri , where p = )../J-t. Thus, we have 1I"i = pi1l"0 for all i. The normalization equation 1 = E�o 1I"i yields

1 = 11"0 L /,

i=O

and the steady-state probabilities are

1I" pt

i = 0, 1 , . . . , m.

1 + p + ' " + pm

378 Markov Chains Chap. 7