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In Section 6.2.2 this result, and an extension, will be discussed in some more detail for the M =G=1 FCFS queue. A similar result for the M =G=1 queue with the PS discipline (due to Zwart and Boxma [42]) will be discussed in Section 6.2.3. That result shows that, under processor sharing, the waiting time tail is just as heavy as the service time tail. In Section 6.2.4 we prove that the latter phenomenon also occurs in the M =G=1 queue with the LCFS-PR discipline. In the present section we rst introduce some notation, and we present a very useful lemma that relates the tail behavior of a regularly varying probability distribution and the behavior of its Laplace Stieltjes transform (LST) near the origin. Consider the M =G=1 queue. Customers arrive according to a Poisson process with rate l; their service times B1 ; B2 ; . . . are i.i.d. (independent, identically distributed) random variables with nite mean b and LST bfsg. A generic service time is denoted by B. By B* we denote a random variable of which the distribution is that of a residual service time: P B* > x 1 b IBarcode Generator In .NET FrameworkUsing Barcode drawer for .NET Control to generate, create bar code image in .NET framework applications.P B > u du;Decoding Bar Code In Visual Studio .NETUsing Barcode scanner for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications.x ! 0:Code 3/9 Creation In C#Using Barcode generation for VS .NET Control to generate, create Code39 image in .NET applications.Its LST is given by b*fsg : 1 bfsg =bs. The traf c load r : lb of the M=G=1 queue is assumed to be less than one, so that the steady-state waiting time distribution exists. A very useful property of probability distributions with regularly varying tails is a characterization of the behavior of its LST near the origin. Let F be the distribution of a nonnegative random variable, with LST ffsg and nite rst n moments m1 ; . . . ; mn (and m0 1). De ne " fn fsg : 1 Encoding Code 39 Extended In Visual Studio .NETUsing Barcode creator for ASP.NET Control to generate, create Code 3/9 image in ASP.NET applications.# s j ffsg mj : j! j 0 Code 3 Of 9 Creation In VB.NETUsing Barcode creator for VS .NET Control to generate, create Code-39 image in VS .NET applications.Lemma 6.2.1. lent:ECC200 Creation In .NETUsing Barcode creator for Visual Studio .NET Control to generate, create Data Matrix 2d barcode image in VS .NET applications.Let n < n < n 1, C ! 0. The following statements are equivafn fsg C o 1 sn L 1=s ; 1 F t C o 1 s 5 0; s real; t 3 I: 6:3 6:4 Generate USS Code 39 In Visual Studio .NETUsing Barcode encoder for .NET framework Control to generate, create Code 39 image in VS .NET applications. 1 n n t L t ; G 1 n Bar Code Generator In .NET FrameworkUsing Barcode creation for .NET Control to generate, create bar code image in Visual Studio .NET applications.6.2 WAITING TIME TAIL BEHAVIOR Generate GTIN - 12 In VS .NETUsing Barcode printer for Visual Studio .NET Control to generate, create UPC E image in Visual Studio .NET applications.The case C > 0 is due to Bingham and Doney [4]. The case C 0 is treated in Boxma and Dumas [12, Lemma 2.2]. The case of an integer n is more complicated; see Bingham et al. [5, Theorem 8.1.6 and Chap. 3]. 6.2.2 The M=G=1 FCFS QueueBar Code Recognizer In Visual Studio .NETUsing Barcode decoder for .NET framework Control to read, scan read, scan image in VS .NET applications.We rst formulate the main result of Cohen [13] for the GI =G=1 queue with FCFS discipline in full generality. There is no need to specify the interarrival time distribution; the mean interarrival time and traf c load are denoted by 1=l and r lb (as before). In what follows, W denotes the steady-state waiting time. Theorem 6.2.2.Paint EAN 128 In VB.NETUsing Barcode creation for VS .NET Control to generate, create USS-128 image in .NET applications.For r < 1 and n > 1,   n  1 n x r x x3I L x ( P W > x \$ A L x : b 1 r b 6:5 Painting Bar Code In JavaUsing Barcode encoder for Java Control to generate, create bar code image in Java applications.P B > x \$ n 1 Printing Barcode In VS .NETUsing Barcode creator for ASP.NET Control to generate, create bar code image in ASP.NET applications.Pakes [29] has extended this result to the larger class of subexponential distributions (i.i.d. stochastic variables X1 and X2 have a subexponential tail if t3I P X1 X2 > t =P X1 > t \$ 2 . 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