Modeling and Analysis of Stochastic Systems, Second Edition by Vidyadhar G. Kulkarni

By Vidyadhar G. Kulkarni

Based at the author's greater than 25 years of educating event, Modeling and research of Stochastic structures, moment variation covers crucial sessions of stochastic strategies utilized in the modeling of numerous platforms, from provide chains and stock structures to genetics and organic structures. for every type of stochastic strategy, the textual content comprises its definition, characterization, purposes, temporary and restricting habit, first passage occasions, and cost/reward types. besides reorganizing the cloth, this variation revises and provides new routines and examples. New to the second one version: a brand new bankruptcy on diffusion methods that offers an available and non-measure-theoretic therapy with purposes to finance; a extra streamlined, application-oriented method of renewal, regenerative, and Markov regenerative methods; and, appendices that acquire correct effects from research and differential and distinction equations. instead of provide distinctive tips that paintings in particular difficulties, this ebook offers thorough assurance of common instruments that permit the answer and research of stochastic versions. After studying the cloth within the textual content, scholars can be well-equipped to construct and examine beneficial stochastic versions for numerous events. a suite of MATLAB[registered]-based courses will be downloaded from the author's web site and a options handbook is accessible for qualifying instructors.

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Extra resources for Modeling and Analysis of Stochastic Systems, Second Edition (Chapman & Hall/CRC Texts in Statistical Science)

Example text

3 . Clearly, the DTMC has a higher tendency to move to state 1, thus implying that this is a model of the weather at a sunny place! 5 Clinical Trials. Suppose two drugs are available to treat a particular disease, and we need to determine which of the two drugs is more effective. This is generally accomplished by conducting clinical trials of the two drugs on actual patients. ” Suppose drug i is effective with probability pi , i = 1, 2. In practice the values of p1 and p2 are unknown, and the aim is to determine if p1 ≥ p2 or p2 ≥ p1 .

17) = aj is the initial distribution. 18) ai pi,j , i∈S where (n) pi,j = P(Xn = j|X0 = i), i, j ∈ S, n ≥ 0. 19) It is called the n-step transition probability, since it is the probability of going from state i to state j in n transitions. 20) where δi,j is one if i = j and zero otherwise, and (1) pi,j = P(X1 = j|X0 = i) = pi,j , i, j ∈ S. 21) (n) If we can compute the n-step transition probabilities pi,j , we can compute the marginal distribution of Xn . Intuitively, the event of going from state i to state j involves going from state i to some intermediate state r at time k ≤ n, followed by a trajectory from state r to state j in the remaining n − k steps.

Thus the next generation again consists of two individuals. This process is repeated indefinitely. Let Xn be the state of the population in the n-th generation. Then it can be seen that {Xn , n ≥ 0} is a DTMC on state-space S with transition probability matrix given below (where the rows and columns are indexed in the same order as in S): ⎡ ⎤ 1 0 0 0 0 0 ⎢ 1/4 1/2 0 1/4 0 0 ⎥ ⎢ ⎥ ⎢ 0 0 0 1 0 0 ⎥ ⎢ ⎥. 14) ⎥ ⎢ 1/16 1/4 1/8 1/4 1/4 1/16 ⎥ ⎣ 0 0 0 1/4 1/2 1/4 ⎦ 0 0 0 0 0 1 DTMCS IN OTHER FIELDS 25 Thus once the population state reaches (dd, dd) or (rr, rr), it stays that way forever.

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