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By J.A. White (Auth.)
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Additional resources for Analysis of Queueing Systems
As we have already seen, the exponential random variable falls within the family of Erlang random variables ; that is, the Erlang density function reduces to the Probability Theory 37 exponential when k = 1. Similarly, the exponential random variable can be considered to fall within the family of hyperexponential random variables in that the hyperexponential density reduces to the exponential density when ρ = 0 or ρ — 1. Thus, the hyperexponential random variable can be considered a mixture of exponential random variables, while the Erlang can be considered a sum of exponential random variables.
110) S 2. Probability Theory and Transform Methods 58 Dirac delta function. 113) does exist and is equal to unity. This and the following property make δ(ή useful in deriving waiting-time distributions for many queueing systems. 114) t >a+8 0, is Then the convolution of δ(ί) and g(t) f ô(t - a)g(t) dt = lim f •Ό ε-0 Κ ^ dt = lim ε G(a + ε) - G(a) ε-0 where g(x) dx, G(t) = and •Ό ô(t - a)g(t) dt = g(a) ^0 since r G(a + ε) - G(a) h m ^ - ^ — U d . 115) 59 Transform Methods The Laplace transform of ô(t) is given by ^[δ(ή] 1 dt = lim f -e~ = f ô(t)e~ st st •'Ο £ - 0 ^0 .
Let us now turn our attention to the problem of finding the Laplace transform of an integral of the form Jo f(x) dx. 102) 2. 103) J5? f ' / W dx\ Convolutions. 105) defines the convolution of the functions/^) and g(y ) and can be extended to define the convolution of an arbitrary number of functions. Although the analysis required to determine the convolution of functions is not conceptually complex, it may prove quite tedious. To simplify this problem we will use the Laplace transform. Assume that h(t) and g(t) are functions for which <&[h(t)] and [g(t)] exist.
Analysis of Queueing Systems by J.A. White (Auth.)