Discrete-time stochastic processes

# Use this to estimate the total number of busy periods

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Unformatted text preview: backward in time from 2:30 PM. f ) Find the PMF for the number of customers getting on the next bus to arrive after 2:30. (Hint: this is diﬀerent from part (a); look carefully at part e). g) Given that I arrive to wait for a bus at 2:30 PM, ﬁnd the PMF for the number of customers getting on the next bus. Exercise 2.13. a) Show that the arrival epochs of a Poisson process satisfy fS (n) |Sn+1 (s (n) |sn+1 ) = n!/sn+1 . n Hint: This is easy if you use only the results of Section 2.2.2. b) Contrast this with the result of Theorem 2.6 Exercise 2.14. Equation (2.44) gives fSi (x | N (t)=n), which is the density of random variable Si conditional on N (t) = n for n ≥ i. Multiply this expression by Pr {N (t) = n} and sum over n to ﬁnd fSi (x); verify that your answer is indeed the Erlang density. Exercise 2.15. Consider generalizing the bulk arrival process in Figure 2.5. Assume that the epochs at which arrivals occur form a Poisson process {N (t); t ≥ 0} of rate ∏. At each arrival epoch, Sn , the number of arrivals, Zn , satisﬁes Pr {Zn =1} = p, Pr {Zn =2)} = 1 − p. The variables Zn are IID. a) Let {N1 (t); t ≥ 0} be the counting process of the epochs at which single arrivals occur. Find the PMF of N1 (t) as a function of t. Similarly, let {N2 (t); t ≥ 0} be the counting process of the epochs at which double arrivals occur. Find the PMF of N2 (t) as a function of t. b) Let {NB (t); t ≥ 0} be the counting process of the total number of arrivals. Give an expression for the PMF of NB (t) as a function of t. Exercise 2.16. a) For a Poisson counting process of rate ∏, ﬁnd the joint probability density of S1 , S2 , . . . , Sn−1 conditional on Sn = t. b) Find Pr {X1 > τ | Sn =t}. c) Find Pr {Xi > τ | Sn =t} for 1 ≤ i ≤ n. d) Find the density fSi (x|Sn =t) for 1 ≤ i ≤ n − 1. e) Give an explanation for the striking similarity between the condition N (t) = n − 1 and the condition Sn = t. 2.7. EXERCISES 87 Exercise 2.17. a) For a Poisson process of rate ∏, ﬁnd Pr {N (t)=n | S1 =τ } for t > τ and n ≥ 1. b) Using this, ﬁnd fS1 (τ | N (t)=n) c) Check your answer against (2.37). Exercise 2.18. Consider a counting process in which the rate is a rv Λ with probability density fΛ (∏) = αe−α∏ for ∏ > 0. Conditional on a given sample value ∏ for the rate, the counting process is a Poisson process of rate ∏ (i.e., nature ﬁrst chooses a sample value ∏ and then generates a sample function of a Poisson process of that rate ∏). a) What is Pr {N (t)=n | Λ=∏}, where N (t) is the number of arrivals in the interval (0, t] for some given t > 0? b) Show that Pr {N (t)=n}, the unconditional PMF for N (t), is given by Pr {N (t)=n} = αtn . (t + α)n+1 c) Find fΛ (∏ | N (t)=n), the density of ∏ conditional on N (t)=n. d) Find E [Λ | N (t)=n] and interpret your result for very small t with n = 0 and for very large t with n large. e) Find E [Λ | N (t)=n, S1 , S2 , . . . , Sn ]. (Hint: consider the distribution of S1 , . . . , Sn conditional on N (t) and Λ). Find E [Λ | N (t)=n, N (τ )=m] for some τ < t. Exercise 2.19. a) Use Equation (2.44) to ﬁnd E [Si | N (t)=n]. Hint: When you integrate xfSi (x | N (t)=n), compare this integral with fSi+1 (x | N (t)=n + 1) and use the fact that the latter expression is a probability density. b) Find the second moment and the variance of Si conditional on N (t)=n. Hint: Extend the previous hint. c) Assume that n is odd, and consider i = (n + 1)/2. What is the relationship between Si , conditional on N (t)=n, and the sample median of n IID uniform random variables. d) Give a weak law of large numbers for the above median. Exercise 2.20. Suppose cars enter a one-way inﬁnite length, inﬁnite lane highway at a Poisson rate ∏. The ith car to enter chooses a velocity Vi and travels at this velocity. Assume that the Vi ’s are independent positive rv’s having a common distribution F . Derive the distribution of the number of cars that are located in an interval (0, a) at time t. Exercise 2.21. Consider an M/G/1 queue, i.e., a queue with Poisson arrivals of rate ∏ in which each arrival i, independent of other arrivals, remains in the system for a time Xi , where {Xi ; i ≥ 1} is a set of IID rv’s with some given distribution function F (x). 88 CHAPTER 2. POISSON PROCESSES You may assume that the number of arrivals in any interval (t, t + ε) that are still in the system at some later time τ ≥ t + ε is statistically independent of the number of arrivals in that same interval (t, t + ε) that have departed from the system by time τ . a) Let N (τ ) be the number of customers in the system at time τ . Find the mean, m(τ ), of N (τ ) and ﬁnd Pr {N (τ ) = n}. b) Let D(τ ) be the number of customers that have departed from the system by time τ . Find the mean, E [D(τ )], and ﬁnd Pr {D(τ ) = d}. c) Find Pr {N (τ ) = n, D(τ ) = d}. d) Let A(τ ) be the total number of arrivals up to time τ . Find Pr {N (τ ) = n | A(τ ) = a}. e) Find Pr {D(τ + ε) − D(τ ) = d}. Exercise 2.22. The voters...
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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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