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# hw2 - traﬃc with rate 30 calls/min The durations of calls...

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CSE-ECE 861: Homework 2 Due: Wed. Feb. 22, 2012 1. Let { S ( t ) , t 0 } be the sum of a random number of independent Poisson processes, each with rate λ . Is S ( t ) a Poisson process? Carefully prove or disprove. 2. Use Little’s Law to verify PASTA for an M/M/1/N queue and an M/M/N/N queueing system 3. Let τ 1 and τ 2 be two exponentially distributed, independent random variables with means 1 1 and 1 2 , respectively. Show that the random variable min( τ 1 , τ 2 ) is exponentially distributed with mean 1 / ( λ 1 + λ 2 ). Also show that P ( { τ 1 < τ 2 } ) = λ 1 / ( λ 1 + λ 2 ). Use these facts to show that the queue can be described by a continuous-time Markov chain with transition rates q n,n +1 = λ and q n,n - 1 = μ, n = 0 , 1 , 2 , . . . . 4. A telephone company establishes a direct connection between two cities expecting Poisson
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Unformatted text preview: traﬃc with rate 30 calls/min. The durations of calls are independent and exponentially distributed with mean 3 min. Interarrival times are independent of call durations. How many circuits should the company provide to ensure that an attempted call is blocked (because all circuits are busy) with probability less than 0.01? It is assumed that blocked calls are lost (i.e., a blocked call is not attempted again). 5. Find the mean, variance, and moment generating function of a Poisson, Erlang, exponential, and Gaussian random variables. 6. Solve problems 2-3, 2-5, 2-9, 2-11, 2-17, 2-18, 2-20, 2-23 in the Schwartz book....
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