By U. Narayan Bhat

ISBN-10: 0817684212

ISBN-13: 9780817684211

This introductory textbook is designed for a one-semester path on queueing conception that doesn't require a path in stochastic strategies as a prerequisite. through integrating the mandatory heritage on stochastic approaches with the research of types, this e-book offers a foundational creation to the modeling and research of queueing structures for a vast interdisciplinary viewers of scholars. Containing routines and examples, this quantity can be used as a textbook by way of first-year graduate and upper-level undergraduate scholars. The paintings can also be valuable as a self-study reference for functions and extra learn.

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Extra info for An Introduction to Queueing Theory: Modeling and Analysis in Applications (2nd Edition)

Example text

1. 1) without going through the generator matrix as illustrated below. Considering the transitions of the process Q(t) during (t, t + Δt], we have P0 (t + Δt) Pn (t + Δt) = P0 (t)[1 − λ0 Δt + o(Δt)] + P1 (t)[μ1 Δt + o(Δt)] = Pn (t)[1 − λn Δt − μn Δt + o(Δt)] +Pn−1 (t)[λn−1 Δt + o(Δt)] +Pn+1 (t)[μn+1 Δt + o(Δt)] +o(Δt) n = 1, 2, . . 4) Subtracting Pn (t) (n = 0, 1, 2 . 4) and dividing by Δt, we get P0 (t + Δt) − P0 (t) Δt Pn (t + Δt) − Pn (t) Δt = −λ0 P0 (t) + μ1 P1 (t) + = −(λn + μn )Pn (t) o(Δt) Δt +λn−1 Pn−1 (t) + μn+1 Pn+1 (t) o(Δt) .

If we are interested in an exact model for the early or late occurrence of events, we may consider the displacement from the deterministic epoch as a random variable with some distribution like the uniform or the normal. Under these conditions, it is possible to have the kth scheduled event occurring later than the occurrence of the (k + 1)th scheduled event. 1. PROBABILITY DISTRIBUTIONS AS MODELS 17 Exponential Distribution, Poisson Process (M) Let F (x) = 1 − e−λx , x ≥ 0, λ > 0. 2) Then we get, f (x) = E[Zn ] = d F (x) = λe−λx dx 1 λ and ψ(θ) = λ .

Zn . Using F (x), the distribution of Sn can be obtained as the n-fold convolution of F (x) with itself, which we denote as Fn (x). 2) 0 as the Laplace–Stieltjes transform of F (x). We then have ∞ e−θx dFn (x) = [φ(θ)] . 3) 0 The distribution of the renewal counting process N (t) for a specific value of t can be derived as follows. Let Pn (t) = P [N (t) = n] . 4) Consider two events {N (t) ≥ n} and {Sn ≤ t}. These are equivalent events. By equating their probabilities, we get P [N (t) ≥ n] P [Sn ≤ t] Fn (t).

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An Introduction to Queueing Theory: Modeling and Analysis in Applications (2nd Edition) by U. Narayan Bhat


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