Discrete-time stochastic processes

This proves the following strong law for renewal

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Unformatted text preview: ) Pr {X > t} = exp(−∏π t2 ) √ b) E [X ] = 1/(2 ∏). Exercise 2.27. This problem is intended to show that one can analyze the long term behavior of queueing problems by using just notions of means and variances, but that such analysis is awkward, justifying understanding the strong law of large numbers. Consider an M/G/1 queue. The arrival process is Poisson with ∏ = 1. The expected service time, E [Y ], is 1/2 and the variance of the service time is given to be 1. a) Consider Sn , the time of the nth arrival, for n = 1012 . With high probability, Sn will lie within 3 standard derivations of its mean. Find and compare this mean and the 3σ range. b) Let Vn be the total amount of time during which the server is busy with these n arrivals (i.e., the sum of 1012 service times). Find the mean and 3σ range of Vn . c) Find the mean and 3σ range of In , the total amount of time the server is idle up until Sn (take In as Sn − Vn , thus ignoring any service time after Sn ). d) An idle period starts when the server completes a service and there are no waiting arrivals; it ends on the next arrival. Find the mean and variance of an idle period. Are successive idle periods IID? e) Combine (c) and (d) to estimate the total number of idle periods up to time Sn . Use this to estimate the total number of busy periods. f ) Combine (e) and (b) to estimate the expected length of a busy period. Exercise 2.28. The purpose of this problem is to illustrate that for an arrival process with independent but not identically distributed interarrival intervals, X1 , X2 , . . . ,, the number of arrivals N (t) in the interval (0, t] can be a defective rv. In other words, the ‘counting process’ is not a stochastic process according to our definitions. This illustrates that it is necessary to prove that the counting rv’s for a renewal process are actually rv’s . a) Let the distribution function of the ith interarrival interval for an arrival process be FXi (xi ) = 1 − exp(−αi xi ) for some fixed α ∈ (0, 1). Let Sn = X1 + · · · Xn and show that E [Sn ] = 1 − αn−1 . 1−α b) Sketch a ‘reasonable’ sample function for N (t). 2 c) Find σSn . 2.7. EXERCISES 91 d) Use the Chebyshev inequality on Pr {Sn ≥ t} to find an upper bound on Pr {N (t) ≤ n} that is smaller than 1 for all n and for large enough t. Use this to show that N (t) is defective for large enough t. Chapter 3 RENEWAL PROCESSES 3.1 Introduction Recall that a renewal process is an arrival process in which the interarrival intervals are positive,1 independent and identically distributed (IID) random variables (rv’s). Renewal processes (since they are arrival processes) can be specified in three standard ways, first, by the joint distributions of the arrival epochs, second, by the joint distributions of the interrarival times, and third, by the joint distributions of the counting process {N (t); t ≥ 0} in which N (t) represents the number of arrivals to a system in the interval (0, t]. The simplest characterization is through the interarrival times, since they are IID. Perhaps the most useful characterization, however, is through the counting process. These processes are called renewal processes because the process probabilistically starts over at each arrival epoch, Sn = X1 + · · · + Xn . That is, if the nth arrival occurs at Sn = τ , then, counting from Sn = τ , the j th subsequent arrival epoch is at Sn+j − Sn = Xn+1 + · · · + Xn+j . Thus, given Sn = τ , {N (τ + t) − N (τ ); t ≥ 0} is a renewal counting process with IID interarrival intervals of the same distribution as the original renewal process. Because of this renewal property, we shall usually refer to arrivals as renewals. The ma jor reason for studying renewal processes is that many complicated processes have randomly occurring instants at which the system returns to a state probabilistically equivalent to the starting state. These embedded renewal epochs allow us to separate the long term behavior of the process (which can be studied through renewal theory) from the behavior of the actual process within a renewal period. Example 3.1.1. Consider a G/G/m queue. The arrival counting process to a G/G/m queue is a renewal counting process, {N (t); t ≥ 0}. Each arriving customer waits in the queue until one of m identical servers is free to serve it. The service time required by each 1 Renewal processes are often defined in a slightly more general way, allowing the interarrival intervals Xi to include the possibility 1 > Pr {Xi = 0} > 0. All of the theorems in this chapter are valid under this more general assumption, as can be verified by complicating the proofs somewhat. Allowing Pr {Xi = 0} > 0 allows multiple arrivals at the same instant, which makes it necessary to allow N (0) to take on positive values, and appears to inhibit intuition about renewals. Exercise 3.2 shows how to view these more general renewal processes while using the definition here, thus show...
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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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