Discrete-time stochastic processes

Such stochastic processes are generally called

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Unformatted text preview: w that Pi ≤ P1 [FX (δ )]i−1 . c) Show that E [N (t + δ ) − N (t)] ≤ P1 [1 − FX (δ )]−2 . d) Show that E [N (t + δ ) − N (t)] ≤ P1 [1 − o(δ )]. e) Show that E [N (t + δ ) − N (t)] ≥ P1 . f ) Use parts d) and e), along with Blackwell’s theorem, to verify (3.45) and the three equations of (3.19). Exercise 3.16. Let Y (t) = SN (t)+1 − t be the residual life at time t of a renewal process. First consider a renewal process in which the interarrival time has density fX (x) = e−x ; x ≥ 0, and next consider a renewal process with density fX (x) = 3 ; (x + 1)4 x ≥ 0. For each of the above densities, use renewal-reward theory to ﬁnd: i) the time-average of Y (t) ii) the second moment in time of Y (t) (i.e., limT →1 1 T RT 0 Y 2 (t)dt) £ § For the exponential density, verify your answers by ﬁnding E [Y (t)] and E Y 2 (t) directly. Exercise 3.17. Consider a variation of an M/G/1 queueing system in which there is no facility to save waiting customers. Assume customers arrive according to a Poisson process of rate ∏. If the server is busy, the customer departs and is lost forever; if the server is not busy, the customer enters service with a service time distribution function denoted by FY (y ). Successive service times (for those customers that are served) are IID and independent of arrival times. Assume that customer number 0 arrives and enters service at time t = 0. a) Show that the sequence of times S1 , S2 , . . . at which successive customers enter service are the renewal times of a renewal process. Show that each inter-renewal interval Xi = Si − Si−1 (where S0 = 0) is the sum of two independent random variables, Yi + Ui where Yi is the ith service time; ﬁnd the probability density of Ui . b) Assume that a reward (actually a cost in this case) of one unit is incurred for each customer turned away. Sketch the expected reward function as a function of time for the sample function of inter-renewal intervals and service intervals shown below; the expectation is to be taken over those (unshown) arrivals of customers that must be turned away. Rt c) Let 0 R(τ )dτ denote the accumulated reward (i.e., cost) from 0 to t and ﬁnd the limit Rt as t → 1 of (1/t) 0 R(τ )dτ . Explain (without any attempt to be rigorous or formal) why this limit exists with probability 1. 132 CHAPTER 3. RENEWAL PROCESSES ✛ ❄ Y1 ✲ S0 = 0 ❄ ✛ ❄ Y1 S1 ✲ ❄ ✛ ❄ S2 Y1 ✲ ❄ d) In the limit of large t, ﬁnd the expected reward from time t until the next renewal. Hint: Sketch this expected reward as a function of t for a given sample of inter-renewal intervals and service intervals; then ﬁnd the time-average. e) Now assume that the arrivals are deterministic, with the ﬁrst arrival at time 0 and the nth arrival at time n − 1. Does the sequence of times S1 , S2 , . . . at which subsequent customers start service still constitute the renewal times of a renewal process? Draw a sketch ≥R ¥ t of arrivals, departures, and service time intervals. Again ﬁnd limt→1 0 R(τ ) dτ /t. Exercise 3.18. Let Z (t) = t − SN (t) be the age of a renewal process and Y (t) = SN (t)+1 − t be the residual life. Let FX (x) be the distribution function of the inter-renewal interval and ﬁnd the following as a function of FX (x): a) Pr {Y (t)>x | Z (t)=s} b) Pr {Y (t)>x | Z (t+x/2)=s} c) Pr {Y (t)>x | Z (t+x)>s} for a Poisson process. Exercise 3.19. Let Z (t), Y (t), X (t) denote the age, residual life, and duration at time t for a renewal counting process {N (t); t ≥ 0} in which the interarrival time has a density given by f (x). Find the following probability densities; assume steady-state. a) fY (t) (y | Z (t+s/2)=s) for given s > 0. b) fY (t),Z (t) (y , z ). c) fY (t) (y | X (t)=x). d) fZ (t) (z | Y (t−s/2)=s) for given s > 0. e) fY (t) (y | Z (t+s/2)≥s) for given s > 0. Exercise 3.20. a) Find limt→1 {E [N (t)] − t/X } for a renewal counting process {N (t); t ≥ 0} with inter-renewal times {Xi ; i ≥ 1}. Hint: use Wald’s equation. b) Evaluate your result for the case in which X is an exponential random variable (you already know what the result should be in this case). £§ c) Evaluate your result for a case in which E [X ] < 1 and E X 2 = 1. Explain (very brieﬂy) why this does not contradict the elementary renewal theorem. Exercise 3.21. Customers arrive at a bus stop according to a Poisson process of rate ∏. Independently, buses arrive according to a renewal process with the inter-renewal interval 3.9. EXERCISES 133 distribution FX (x). At the epoch of a bus arrival, all waiting passengers enter the bus and the bus leaves immediately. Let R(t) be the number of customers waiting at time t. a) Draw a sketch of a sample function of R(t). b) Given that the ﬁrst£bus arrives at time X1 = x, ﬁnd the expected number of customers § Rx picked up; then ﬁnd E 0 R(t)dt , again given the ﬁrst bus arrival at X1 = x. Rt c) Find limt→1 1 0 R(τ )dτ (with probability 1). Assuming that FX...
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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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