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Unformatted text preview: = 1 9 (c) What is the pmf of X ? p k = P [ X = k ] = Î» Î» 1 ...Î» k1 Î¼ 1 Î¼ 2 ...Î¼ k p k 1 2 3 4 p k = P ( X = k ) 1 9 2 9 2 9 2 9 2 9 1 (d) What is the (large t ) expected # of jobs in the system? E [ X ] = 0 Â· p + 1 Â· p 1 + 2 Â· p 2 + 3 Â· p 3 + 4 Â· p 4 = 0 Â· 1 9 + 2 9 Â· (1 + 2 + 3 + 4) = 20 / 9 (d) What is the probability that an incoming in message is turned away? An incoming message is lost when system is full i.e., X = 4 P (message lost) = P (system full) = P ( X = 4) = p 4 = 2 / 9 (e) What is the expected number of messages in the buffer? Let Y = # of messages in the buï¬€er; note that Y is a function of X X 1 2 3 4 Y 1 2 p k 1 9 2 9 2 9 2 9 2 9 E [ Y ] = 0 Â· p + 1 Â· p 1 + 2 Â· p 2 + 3 Â· p 3 + 4 Â· p 4 = 0 Â· 1 9 + 0 Â· 2 9 + 0 Â· 2 9 + 1 Â· 2 9 + 2 Â· 2 9 = 5 / 9 2...
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 Spring '11
 Zhou
 Probability theory, Exponential distribution, #, AirTrain Newark, 0 1 1 2 3 4 1 9 k

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