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see Section 12 The best general-purpose approach for the computation of the state probabilities is the discrete FFT method An explicit expression for the generating functionCreating QR In Visual Basic .NETUsing Barcode drawer for VS .NET Control to generate, create QR Code image in .NET framework applications.P (z) =Barcode Maker In VS .NETUsing Barcode creator for .NET Control to generate, create bar code image in .NET framework applications.j =0Encoding Code 39 Full ASCII In VS .NETUsing Barcode creation for .NET Control to generate, create Code 39 Extended image in VS .NET applications.pj zj ,Make Bar Code In .NETUsing Barcode encoder for VS .NET Control to generate, create barcode image in VS .NET applications.|z| 1 Generate Delivery Point Barcode (DPBC) In .NET FrameworkUsing Barcode printer for Visual Studio .NET Control to generate, create Postnet 3 of 5 image in .NET applications.can be given It is a matter of tedious algebra to derive from (932) that P (z) = (1 ) where Generating Data Matrix In JavaUsing Barcode printer for Java Control to generate, create ECC200 image in Java applications.1 (z){1 G(z)} , 1 (z){1 G(z)}/(1 z)Code 39 Full ASCII Creation In VB.NETUsing Barcode creation for .NET Control to generate, create Code39 image in .NET framework applications.(933)Bar Code Drawer In .NET FrameworkUsing Barcode generator for ASP.NET Control to generate, create barcode image in ASP.NET applications.G(z) =GS1 - 12 Encoder In VS .NETUsing Barcode drawer for ASP.NET Control to generate, create Universal Product Code version A image in ASP.NET applications.j =1Painting USS Code 39 In Visual C#Using Barcode printer for .NET Control to generate, create Code-39 image in .NET applications. j zj Decoding UPC-A Supplement 2 In .NETUsing Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications.and (z) =Printing Barcode In JavaUsing Barcode maker for Java Control to generate, create barcode image in Java applications.e {1 G(z)}t (1 B(t)) dt Create Bar Code In .NET FrameworkUsing Barcode maker for ASP.NET Control to generate, create bar code image in ASP.NET applications.The derivation uses that e {1 G(z)}t is the generating function of the compound Poisson probabilities rn (t); see Theorem 121 Moreover, the derivation uses that the generating function of the convolution of two discrete probability distributions is the product of the generating functions of the two probability distributions The other details of the derivation of (933) are left to the reader For constant and phase-type services, no numerical integration is required to evaluate the function (z) in the discrete FFT method Asymptotic expansion The state probabilities allow for an asymptotic expansion when it is assumed that the batch-size distribution and the service-time distribution are not heavy-tailed Let us make the following assumptionTHE M X /G/1 QUEUE Assumption 931 (a) The convergence radius R of G(z) = j =1 j zj is larger st than 1 Moreover, 0 e {1 B(t)} dt < for some s > 0 (b) lims B 0 e st {1 B(t)} dt = , where B is the supremum over all s with 0e st {1 B(t)} dt] < (c) limx R0 G(x) = 1 + B/ for some number R0 with 1 < R0 R Under this assumption we obtain from Theorem C1 in Appendix C that pj j as j , (934)where is the unique solution to the equation ( ){1 G( )} = 1 on (1, R0 ) and the constant is given by = (1 )(1 ) ( ){1 G( )} A formula for the average queue size The long-run average number of customers in queue is Lq = j =1 (j 1)pj Using the relation P (1) = j =1 jpj , we obtain after some algebra from (933) that(935)(1 )G ( ) +1 1 G( )(936)Lq = E(X2 ) 1 2 2 (1 + cS ) + 1 , 2 1 2(1 ) E(X)where X denotes the batch size Note that the rst part of the expression for Lq gives the average queue size in the standard M/G/1 queue, while the second part re ects the additional effect of the batch size The formula for Lq implies directly a formula for the long-run average delay in queue per customer By Little s formula Lq = Wq 932 The Waiting-Time Probabilities The concept of waiting-time distribution is more subtle for the case of batch arrivals than for the case of single arrivals Let us assume that customers from each arrival group are numbered as 1, 2, Service to customers from the same arrival group is given in the order in which those customers are numbered For customers from different batches the service is in order of arrival De ne the random variable Dn as the delay in queue of the customer who receives the nth service In the batch-arrival queue, limn P {Dn x} need not exist To see this, consider the particular case