9. The birth and death process with parameters = 0 and = , >

9. The birth and death process with parameters ëç = 0 and ìç = ì, ç > 0 is called a pure death process. Find Ñ^·(Ï

ÉÏ. Consider two machines. Machine / operates for an exponential time with rate A, and then fails; its repair time is exponential with rate ì,, / = 1, 2.

The machines act independently of each other. Define a four-state continuous-time Markov chain which jointly describes the condition of the two machines. Use the assumed independence to compute the transition probabilities for this chain and then verify that these transition probabilities satisfy the forward and backward equations.

Consider a Yule process starting with a single individual—that is, suppose X(0) = 1. Let Tt denote the time it takes the process to go from a population of size / to one of size / + 1.

(a) Argue that Ti9 / = l , . . . , y , are independent exponentials with respective rates /A.

(b) Let Xx,..., Xj denote independent exponential random variables each having rate A, and interpret Xt as the lifetime of component /. Argue that max (Xx,..., Xj) can be expressed as m a x t A ^ , X j ) = å÷ + å2 + ··· + â,·

(a) What fraction of the attendant's time will be spent servicing cars?

(b) What fraction of potential customers are lost?