Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Tutorial 8 November 4/5, 2010 1. Type A, B, and C items are placed in a common buffer, each type arriving as part of an inde-pendent Poisson process with average arrival rates, respectively, of a, b, and c items per minute. For the first four parts of this problem, assume the b uffer is discharged immediately whenever it contains a total of ten items. (a) What is the probability that, of the fir s t ten items to arrive at the buffer, only the first and one other are typ e A? (b) What is the probability that any particular discharge of the buffer contains five times as many typ e A items as type B items? (c) Determine the PDF, expectation, and variance for the total time between consecutive dis-charges of the buffer. (d) Determine the probability that exactly two of each of the three item types arrive at th e buffer input during any particular five minute interval. 2. A store opens at t = 0 and potential customers arrive in a Poisson manner at an average arrival rate of λ potential customers per hour. As long as the store is open, and independently of all other events, each particular potential customer becomes an actual cus tomer with probability p. The store closes as soon as ten actual customers have arrived. (a) What is the probability that exactly three of the first five potential customers become actual customers? (b) What is the probability that the fifth potential customer to arrive becomes the third actual customer? (c) What is the PDF and expected value f or L, the duration of the interval from store opening to store closing? (d) Given only that exactly three of the first five potential customers became actual customers, what is the conditional expected value of the total time the store is open? (e) Considering only customers arriving between t = 0 and the closing of the store, what is th e probability that no two actual customers arrive within τ time units of each other? 3. Problem 6.24, page 335 in text. Consider a Poisson process with parameter λ, and an independent random variable T , which is exponential with parameter ν. Find the PMF of the number of Poisson arrivals during the time interval [0, T ]. Page 1 of 1 Textbook problems are courtesy of Athena Scientific, and are used with permission.MIT OpenCourseWare http://ocw.mit.edu 6.041 / 6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit:
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