An Introduction to Queueing Theory: Modeling and Analysis in by U. Narayan Bhat

By U. Narayan Bhat

This introductory textbook is designed for a one-semester path on queueing conception that doesn't require a path in stochastic techniques as a prerequisite. via integrating the mandatory heritage on stochastic methods with the research of versions, the paintings presents a legitimate foundational creation to the modeling and research of queueing structures for a wide interdisciplinary viewers of scholars in arithmetic, information, and utilized disciplines equivalent to machine technological know-how, operations examine, and engineering.

Key features:

* An introductory bankruptcy together with a historic account of the expansion of queueing thought within the final a hundred years.

* A modeling-based technique with emphasis on id of types utilizing subject matters similar to selection of info and exams for stationarity and independence of observations.

* Rigorous therapy of the principles of uncomplicated versions standard in functions with applicable references for complicated topics.

* A bankruptcy on modeling and research utilizing computational tools.

* A accomplished therapy of statistical inference for queueing systems.

* A dialogue of operational and determination problems.

* Modeling workouts as a motivational instrument, and evaluation routines protecting heritage fabric on statistical distributions.

An creation to Queueing Theory can be used as a textbook via first-year graduate scholars in fields resembling computing device technological know-how, operations study, commercial and platforms engineering, in addition to similar fields comparable to production and communications engineering. Upper-level undergraduate scholars in arithmetic, facts, and engineering can also use the e-book in an optionally available introductory direction on queueing conception. With its rigorous insurance of uncomplicated fabric and huge bibliography of the queueing literature, the paintings can also be priceless to utilized scientists and practitioners as a self-study reference for purposes and additional research.

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Extra info for An Introduction to Queueing Theory: Modeling and Analysis in Applications

Example text

Suppose f (x) is the probability density of a random variable X and φ(θ ) its Laplace transform. ) ∞ φ(θ ) = e−θx f (x)dx, Re(θ ) > 0. 0 Two easily established properties of φ(θ) are E(X) = −φ (0), E(X ) = φ (0). 23) Let B represent the length of the busy period. 19), we get 1 , µ−λ 1+ρ V [B] = 2 . 25) There may be occasions when a busy period starts out with an initial number of i customers in the system. Because of the Markovian properties of the arrival process we can show that the transition of the underlying Markov process from i to 0 can be considered to be made up of i intervals with the same distribution representing the transitions from i → i − 1, i − 1 → i − 2, .

The corresponding forward Kolmogorov equations for Pn (t) (n = 0, 1, 2, . . ) are P0 (t) = µP1 (t), P1 (t) = −(λ + µ)P1 (t) + µP2 (t), Pn (t) = −(λ + µ)Pn (t) + λPn−1 (t) + µPn+1 (t), n = 2, 3, . . 18) with the initial condition P1 (0) = 1, Pn (0) = 0 for n = 1. Solving these differencedifferential equations requires the use of PGFs and Laplace transforms. ) Let π0 (θ ) be the Laplace transform of the busy period defined as ∞ π0 (θ ) = 0 e−θt P0 (t)dt, Re(θ) > 0.

K − 1, 52 4 Simple Markovian Queueing Systems µn = nµ, n = 1, 2, . . , s − 1, = sµ, n = s, s + 1, . . , K. 1) Note that we assume the arrivals to be denied entry to the system (or the arrival process stops) once the number in the system reaches K. 3) in the first K rows: 0 1 .. A= . K −1 K ⎡ −λ λ ⎢ µ −(λ + µ) ⎢ .. ⎢ . ⎢ ⎢ ⎢ ⎢ ⎣ ⎤ λ .. sµ −(µ + sµ) λ sµ −sµ ⎥ ⎥ ⎥ ⎥ ⎥. 2) For the limiting probabilities {pn }, n = 0, 1, 2, . . 4). 6) can be given as Writing λ sµ 1 n! λ µ n pn = 1 s! λ µ s = = ρ and λ µ 0 ≤ n ≤ s, p0 , λ sµ n−s s ≤ n ≤ K.

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