Consider a CTMC X(t) on the state space S = No. If the CTMC is in state i € N, then а it stays in state i for an exponen

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Consider a CTMC X(t) on the state space S = No. If the CTMC is in state i € N, then а it stays in state i for an exponentially distributed length of time with mean i-!, at which point it moves to state i + 1 with probability p and state i - 1 with probability 1 - (where 0 < p < 1). State 0 is an absorbing state. (a) Explain why this is a BDP(1. Hi) (birth-death process) and give the values of ; and Hi for i ES. (b) (i) Let h?n denote the probability of hitting state 2n from state i (where i < 2n). Show that han = A р where A and B are to be determined and given in terms of p and n. (ii) Suppose X(0) = n for some ne N. Show that p" han p" + (1 - p) a(':' +B.