Discrete-time stochastic processes

5 assume that the epochs at which arrivals occur form

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Unformatted text preview: ns form independent Poisson processes. This ability to view independent Poisson processes either independently or as a splitting of a combined process is a powerful technique for finding almost trivial solutions to many problems. It was next shown that a non-homogeneous Poisson process could be viewed as a (homogeneous) Poisson process on a non-linear time scale. This allows all the properties of (homogeneous) Poisson properties to be applied directly to the non-homogeneous case. The simplest and most useful result from this is (2.31), showing that the number of arrivals in any interval has a Poisson PMF. This result was used to show that the number of customers in service at any given time τ in an M/G/1 queue has a Poisson PMF with a mean approaching ∏ times the expected service time in the limit as τ → 1. Finally we looked at the distribution of arrivals conditional on n arrivals in the interval (0, t]. It was found that these arrivals had the same joint distribution as the order statistics of n uniform IID rv’s in (0, t]. By using symmetry and going back and forth between the uniform variables and the Poisson process arrivals, we found the distribution of the interarrival times, the arrival epochs, and various conditional distributions. 2.7 Exercises Exercise 2.1. a) Find the Erlang density fSn (t) by convolving fX (x) = ∏ exp(−∏x) with itself n times. b) Find the moment generating function of X (or find the Laplace transform of fX (x)), and use this to find the moment generating function (or Laplace transform) of Sn = X1 + X2 + · · · + Xn . Invert your result to find fSn (t). c) Find the Erlang density by starting with (2.14) and then calculating the marginal density for Sn . Exercise 2.2. a) Find the mean, variance, and moment generating function of N (t), as given by (2.15). b) Show by discrete convolution that the sum of two independent Poisson rv’s is again Poisson. c) Show by using the properties of the Poisson process that the sum of two independent Poisson rv’s must be Poisson. Exercise 2.3. The purpose of this exercise is to give an alternate derivation of the Poisson distribution for N (t), the number of arrivals in a Poisson process up to time t; let ∏ be the rate of the process. a) Find the conditional probability Pr {N (t) = n | Sn = τ } for all τ ≤ t. b) Using the Erlang density for Sn , use (a) to find Pr {N (t) = n}. 2.7. EXERCISES 83 Exercise 2.4. Assume that a counting process {N (t); t≥0} has the independent and stationary increment properties and satisfies (2.15) (for all t > 0). Let X1 be the epoch of the first arrival and Xn be the interarrival time between the n − 1st and the nth arrival. a) Show that Pr {X1 > x} = e−∏x . b) Let Sn−1 be the epoch of the n − 1st arrival. Show that Pr {Xn > x | Sn−1 = τ } = e−∏x . c) For each n > 1, show that Pr {Xn > x} = e−∏x and that Xn is independent of Sn−1 . d) Argue that Xn is independent of X1 , X2 , . . . Xn−1 . Exercise 2.5. The point of this exercise is to show that the sequence of PMF’s for the counting process of a Bernoulli process does not specify the process. In other words, knowing that N (t) satisfies the binomial distribution for all t does not mean that the process is Bernoulli. This helps us understand why the second definition of a Poisson process requires stationary and independent increments as well as the Poisson distribution for N (t). a) For a sequence of binary rv’s Y1 , Y2 , Y3 , . . . , in which each rv is 0 or 1 with equal probability,° find a joint distribution for Y1 , Y2 , Y3 that satisfies the binomial distribution, t¢ pN (t) (k) = k 2−k for t = 1, 2, 3 and 0 ≤ k ≤, but for which Y1 , Y2 , Y3 are not independent. Your solution should contain four 3-tuples with probability 1/8 each, two 3-tuples with probability 1/4 each, and two 3-tuples with probability 0. Note that by making the subsequent arrivals IID and equiprobable, you have an example where N (t) is binomial for all t but the process is not Bernoulli. Hint: Use the binomial for t = 3 to find two 3-tuples that must have probability 1/8. Combine this with the binomial for t = 2 to find two other 3-tuples with probability 1/8. Finally look at the constraints imposed by the binomial distribution on the remaining four 3-tuples. b) Generalize part a) to the case where Y1 , Y2 , Y3 satisfy Pr {Yi = 1} = q and Pr {Yi = 0} = 1 − q . Assume q < 1/2 and find a joint distribution on Y1 , Y2 , Y3 that satisfies the binomial distribution, but for which the 3-tuple (0, 1, 1) has zero probability. c) More generally yet, view a joint PMF on binary t-tuples as a ° ¢ non-negative vector in a 2t dimensional vector space. Each binomial probability pN (τ ) (k) = τ q k (1 − q )τ −k constitutes k a linear constraint on this vector. For each τ , show that one of these constraints may be replaced by the constraint that the components of the vector sum to 1. d) Using part c), show that at...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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