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# ch4_add_1 - = 1 9(c What is the pmf of X p k = P X = k = Î...

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Birth and Death Processes Example: Communication System A communication system has two processers that decode messages and a buffer that holds at most two messages. A message that arrives when the buffer is full is lost. Each processer needs an average of 2 minutes to decode a message. Messages arrive at a rate of 1 per minute. Assume that inter-arrival times and processing times are independent exponential random variables. (a) Draw a state diagram with possible states and corresponding birth/death rates. Solution: 1 1 0 1 2 3 4 1 1 .5 1 1 1 Let T 1 = time until processer 1 finishes and T 2 = time until processer 2 finishes. E [ T i ] = 2 i = 1 , 2 implies that rate parameter for both distributions are 1/2 i.e., T 1 , T 2 iid Exp (1 / 2) For k = 1, implying that μ 1 = 1 2 = . 5 For k 2, D = min { T 1 , T 2 } giving D Exp ( 1 2 + 1 2 = 1); thus μ k = 1 , k = 2 , 3 , 4 (b) What is the (large t ) probability that the system is empty? S = 1 + 1 . 5 + 1 · 1 1 · . 5 + 1 · 1 · 1 1 · 1 · . 5 + 1 · 1 · 1 · 1 1 · 1 · 1 · . 5 = 9 implying that p 0 = 1 S = 1 9 (c)

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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» k-1 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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ch4_add_1 - = 1 9(c What is the pmf of X p k = P X = k = Î...

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