Download e-book for kindle: Analysis of Queueing Systems by J.A. White (Auth.)

By J.A. White (Auth.)

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Now rS(t)p(0)e- 'dt = s (i- ) P since Furthermore, 00 J o 00 jc= 00 ι o 00 J À \x / \ x=i x + 1 — A J ! "(1- Hence

Since p(x) = 0, χ < 1, we have F(x) = Σ y= — POO = o. 6, X X X * < 1 26 2. 0, χ >4 The graph of F(x) is shown in Fig. 1. 2h J_ 0 Fig. 4. There are several discrete probability distributions that are of general interest in operations research and probability theory. Some of these distributions are of particular interest in queueing theory. For example, as we shall see in later chapters, the Poisson distribution is a fundamental building block in the theory of queues. Specifically, the initial treatment of most queueing systems usually entails the assumption that the number of arrivals to the system in a fixed interval of time is Poisson distributed.

The conditional density function of X given X ,f(*i | x ), is given by x /(xj I x ) 2 = 2 2 ^2) /(*i> and the conditional probability mass function of X given X , p(x \ x ), is defined as 1 2 x 2 Example 2JS The time Τ required to provide a given service is an exponential random variable with parameter A. However, the parameter A is also a random variable with density function 0 < A < oo g(k) = μβ~ \ μ Find the joint density function of Τ and A and the conditional distribution of A given T. 67), f(t, A) =f(t I k)g(k) = μ Α ^ - ^ , 0<ί<οο, +ί) 0 < A < oo, μ>0 The conditional density function of A given Τ is defined by where h(t) is the marginal density function of T.

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