By Aliakbar Montazer Haghighi


A helpful consultant to the Interrelated components of Differential Equations, distinction Equations, and Queueing Models

Difference and Differential Equations with functions in Queueing Theory provides the original connections among the tools and purposes of differential equations, distinction equations, and Markovian queues. that includes a entire selection of themes which are utilized in stochastic techniques, fairly in queueing idea, the booklet completely discusses the connection to structures of linear differential distinction equations.

The booklet demonstrates the applicability that queueing conception has in quite a few fields together with telecommunications, site visitors engineering, computing, and the layout of factories, outlets, places of work, and hospitals. besides the wanted prerequisite basics in likelihood, statistics, and Laplace rework, Difference and Differential Equations with functions in Queueing Theory provides:

  • A dialogue on splitting, delayed-service, and not on time suggestions for single-server, multiple-server, parallel, and sequence queue models
  • Applications in queue types whose recommendations require differential distinction equations and producing functionality methods
  • Exercises on the finish of every bankruptcy in addition to opt for answers

The booklet is a superb source for researchers and practitioners in utilized arithmetic, operations learn, engineering, and business engineering, in addition to an invaluable textual content for upper-undergraduate and graduate-level classes in utilized arithmetic, differential and distinction equations, queueing conception, chance, and stochastic processes.

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Extra resources for Difference and Differential Equations with Applications in Queueing Theory

Example text

1), for a case for an arbitrary (continuous measurable) function of X, say g(X), where X is a bounded random variable with continuous pdf fX(x), will be: E ( g ( X )) = ∫ Ω g ( X ) dF = ∫ ∞ −∞ g ( x ) f ( x ) dx, provided that the integral converges absolutely. 2 The kth moments of the continuous random variable X with pdf fX(x), denoted by E[Xk], k = 1, 2, . . 3) x f ( x ) dx < +∞. 4), the kth of X exists if and only if the kth absolute moment of X, E(|X|k), is finite. Notes: (1) It can be shown that if the kth, k = 1, 2, .

X =1  x =1 ∞ ∞ x−2 x −1   λ λ +λ = e − λ λ 2 ( x − 1)!   x = 2 ( x − 2 )! x =1 ∑ ∑ x ∑ x ∑ = e − λ [λ 2 e λ + λ e λ ] = λ 2 + λ , Var ( X ) = λ 2 + λ − λ 2 = λ . 4. CONTINUOUS RANDOM VARIABLES So far we have been discussing discrete random variables, discrete distribution functions, and some of their properties. We now discuss continuous cases. , an interval or a union of intervals). The set consisting of all subsets of real numbers R is extremely large and it will be impossible to assign probabilities to all of them.

2. Markov’s Inequality Let X be a nonnegative discrete random variable with a finite mean, E(X). Let a be a fixed positive number. Hence, P {X ≥ a} ≤ E (X ) . 3) Proof: By definition, E (X ) = ∑ xf ( x) x = ∑ xf ( x ) + 0≤ x < a ∞ ∑ xf ( x ). 4) is positive. Hence, E (X ) ≥ ∞ ∑ xf ( x). 5) we have: E (X ) ≥ ∞ ∞ x=a x=a ∑ af ( x) = a∑ f ( x). 6) functionS of a random Variable 51 Hence, E ( X ) ≥ aP {X ≥ a} . 3) follows. 3. Chebyshev’s Inequality Let X be a nonnegative random variable with finite mean μ and variance σ2.

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