By U. Narayan Bhat
This introductory textbook is designed for a one-semester direction on queueing thought that doesn't require a path in stochastic techniques as a prerequisite. via integrating the required heritage on stochastic tactics with the research of types, this booklet presents a foundational creation to the modeling and research of queueing platforms for a vast interdisciplinary viewers of scholars. Containing workouts and examples, this quantity can be used as a textbook via first-year graduate and upper-level undergraduate scholars. The paintings can also be important as a self-study reference for functions and additional learn.
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Additional resources for An Introduction to Queueing Theory: Modeling and Analysis in Applications (2nd Edition)
5 Spaces of Bounded Functions 39 principle the index set T need not have a semimetric on it when it comes to us. FUrthermore, if it does, this semi metric may not be the right one to use. The next few results of this section are somewhat technical, but they lead to the important result that for Gaussian limits one can always use a certain variance metric. A possible limit X of a net Xa. can be identified from marginal convergence. Next, according to the addendum, we should look for a semimetric p that makes T totally bounded and the paths of X uniformly continuous.
J) and lE, )(a -v-+ L1 32 1 Stochastic Convergence To see this, assume without loss of generality that [j) and lE are complete, so that the marginals of (X"" Y",) are asymptotically tight. By Prohorov's theorem the net of joint variables (X"" Y",) is relatively compact. Every limit point L must have f(x)g(y) dL(x, y) = f dL 1 g dL 2 · These numbers uniquely identify the (one) limit point. 7 Example (Slutsky's lemma). If X", "'" X and Y", "'" c with X separable and c a constant, then (X"" Y",) "'" (X, c).
They can also be extended to countable products, with slightly more effort. If [j)i is a metric space with metric di for every natural number i, then the Cartesian product [j)l x [j)2 X . can be metrized by the metrics 1 d((Xl,X2, .. )) = sup -;- (di(Xi, Zi) i d((Xl,X2,,,,),(Zl,Z2,''')) = Z 1\ LTi (di(Xi,Zi) 1) , 1\ 1). These generate the same topology; the product topology. The results of this section hold for [j)l x [j)2 X .... In particular, coordinatewise asymptotic tightness is the same as joint asymptotic tightness, and given asymptotic tightness, coordinatewise asymptotic measurability is the same as joint asymptotic measurability.