By J.A. White (Auth.)

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**Extra resources for Analysis of Queueing Systems**

**Sample text**

Y are the service times of the η units already in the system and X is waiting time for the arriving unit, then u 2 n E(X) = E{Y + y + - + Y ) l 2 n 2. Probability Theory and Transform Methods 46 Let h(X X ,X ) be a function of the random variables X X . , X ) is given by u 2 n n l9 (j E[h{X 2 Xh X X l9 9 dx x '~> n)f( l> ~'> n)dX! 70) X ... X )]={ u 2 n ( l> ' " j X ,. ,X i9 E{a X 1 + aX 1 2 2 and constants n a a ,... 72) = E(X )E(X ) X 2 Conditional expectation. Let X and X be dependent random variables.

As we shall show later, the exponential random variable is integrally related to the Poisson random variable and this relationship will prove to be of paramount importance throughout our discussion of queueing theory. Y be the times between the occurGamma distribution. 43) where Γ(η) = (η - 1)! when η is an integer. If, however, η is not an integer, the gamma random variable cannot be represented by the sum of identically distributed exponential random variables. When η is unity the gamma density function reduces to the exponential density function.

88), we should be able to define/(i) given L(s). 91) defines the inverse Laplace transform, sometimes denoted by i f " (5). 91) can be used to determine the inverse Laplace transform, its use requires a knowledge of contour integration (LePage, 1961; Schmidt, 1974). To circumvent this problem, extensive tables of Laplace and inverse Laplace transforms have been developed. 88). b dt st Jb ~st e b dt= -j. b —a - = ct )e~ dt ~ase _ -bs e s(b — a) s(b — a) For (d), WW] {a + bt + 2 . ~ b ,r si Ä + fc Γ 00 te~ st + β ί f V 2c s* 5 Laplace Transform of Special Functions In subsequent chapters we will encounter situations requiring the development of the Laplace transform of special functions.