Markov Chains Driven by Clock Times. Let {Xn : n ≥ 0} be astochastic process on S whose dynamics are as follows. Whenever the processenters a state i, a set of independent clock times τij , j ∈ Si are...

Markov Chains Driven by Clock Times. Let {Xn : n ≥ 0} be astochastic process on S whose dynamics are as follows. Whenever the processenters a state i, a set of independent clock times τij , j ∈ Si are started,where Si is the subset of states in S that can be reached from state i inone step. The times τij are geometrically distributed with parameters γij(P{τij > m} = γmij ). Then the sojourn time in state i is the minimumτi = minj∈Si τ_ij
, and, at the end of the sojourn, the process jumps to thestate j ∈ Si for which τij = τi. Find the distribution of τi