hwk7 - E3106, Solutions to Homework 7 Columbia University...

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Columbia University Exercise 14 . Letting the number of cars in the station be the state variable, we have a birth-death process with λ 0 = λ 1 = λ 2 =20 1 = μ 2 =12 λ i =0 ,i> 2 . Hence the limiting probabilities must satisfy, as the equations on p. 380-371 12 P 1 =2 0 P 0 12 P 2 =2 0 P 1 12 P 3 =2 0 P 2 P 0 + P 1 + P 2 + P 3 =1 which leads to P 0 = Ã 1+ 5 3 + 5 3 ¸ 2 + 5 3 ¸ 3 ! 1 = 27 272 . (1) The fraction of the attendant’s time spent servicing cars is equal to the fraction of time there are cars in the system and is therefore 1 P 0 = 245 / 272 . (2) The fraction of potential customers that are lost is equal to the fraction of customers that arrive when there are 3 cars in the station, and is therefore P 3 = μ 5 3 2 P 0 = 125 272 . Exercise 15 .L e t X ( t ) denote the number of customers in the service center at time t ,th en { X ( t ) ,t 0 } is a birth and death process with state space { 0 , 1 , 2 , 3 } and rates λ 0 = λ 1 = λ 2 =3 μ 1 =2 2 = μ 3 =4
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hwk7 - E3106, Solutions to Homework 7 Columbia University...

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