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Unformatted text preview: CS112  Homework #8 1. A network router has buffer capacity for at most two packets (including the one being served) Packets that arrive when the buffer is full, leave without being served. Potential packets arrive at a Poisson rate of three per millisecond, and the successive service times are independent exponential random variables with mean 0.25 milliseconds. What is: (a) The average number of packets in the buffer? (b) The fraction of arriving packets that actually get served? (c) If the router could work twice as fast, what fraction of arriving packets would actually get served? (d) If another router was installed (so now there are two servers and a shared buffer space for 4 packets) that serves packets at the same rate, what fraction of packets would be served? 2. Consider a system with 3 machines. Each machine has a time to failure which is exponential with mean 1 / . When a machine fails, there is a single repairman and the time to repair a machine is modeled as the sum of two exponential random variables, X 1 and X 2 . X i has parameter i . The repairman serves the failed machines in FirstComeFirstServe order. The system is considered to fail (crash) if the number of operational machines falls to 1. When this happens, the system remains down until both machines are repaired. Assume that when the system is down (only one machine is operational) the operational machine is idle and there are no failures when idle. (a) Define the state space and show the state transition rate diagram. (b) Explain in detail the procedure you would use to determine the mean time for the system to fail given it starts in the state with all machines are operational....
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 Spring '08
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