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It follows quite directly from the classical theory of semi-Markov process that explicit expressions can be given for the Laplace transform of the conditional mean More recently, it has been shown that numerical inversion can be an effective algorithm; see Abate et al [1] For related discussions of transient analysis, design and control, see s 13, 16, and 18 in this volume 173 OUTLINE OF THE CHAPTER Unlike the more elementary M =G=I model considered in Duf eld and Whitt [14], however, the conditional mean here is not available in closed form In order to rapidly compute the time-dependent conditional mean aggregate demand, we exploit numerical inversion of Laplace transforms It follows quite directly from the classical theory of semi-Markov process that explicit expressions can be given for the Laplace transform of the conditional mean.

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1213 A random current Even if sources can be individually described in a detailed, or even exact way (in principle), we generally take a stochastic point of view, and thus models proposed for traf c and traf c superpositions are random processes The main reasons for this choice are the impossibility of managing within a deterministic framework the innumerable factors and details which intervene in practice, and to take into account the unpredictability of traf c For example, the fact that the length of calls and the precise arrival times of new connections are not known in advance demands the adoption of a probabilistic approach Such models can be de ned in discrete or continuous time, according to the interpretation which is given to the ow, the time scale considered, and the questions that are of interest The choice also depends on the analytic tools available, which can favor one approach or other for technical reasons For example, let us take a process X(k) that models audio traf c in discrete time, giving the number of bytes to be transmitted in interval k of width 40 ms This time series corresponds to a traf c rate on a discrete time grid whose resolution is close to an established mechanism of audio transmission, namely the packet based TCP/IP (we revisit this shortly) It also constitutes a traf c representation based on a fairly ne time scale Beginning from a rate series X(k), the nature of traf c is described by 2 statistical quantities, starting with the mean X (k) = E[X(k)], variance X (k) = 2 E[X(k) k ] (assuming it exists, which is not always the core), or even the entire distribution function X(k) for every k = 1, 2, 3, etc Regarding its temporal structure, the most important quantity is the auto-correlation function, de ned by cX (k, l) = E[(X(k) (k))(X(l) (l))]/ X (k) X (l) Typically, we assume that a ow is stationary, that is its statistics do not depend on the time of origin, 2 2 we therefore obtain (k) = , X (k) = X and cX (k, l) = c (( = k l)) = 2 E[(X(k) )(X(l) )]/ X In this context, let us now examine the nature of a traf c superposition as mentioned earlier Let two processes X1 and X2 have the same (statistical) characteristics as X, in other words, they are identical but independent copies of X Assuming that their superposition reduces to a simple addition, from the superposition Y (k) = X1 (k) + X2 (k) we can easily show that the 2 2 2 2 mean Y = X1 + X2 = 2 X and variance Y (k) = X1 (k) + X2 (k) = 2 X behave in a linear fashion, but on the other hand that the structure of covariances remains constant: c= (( )) cX ( ) Although the covariance amplitude (the variance) increased, the ratio of the size of variations to the mean, / , is reduced by a factor of 2 The simple fact that multiplexing two independent ows results in another ow of reduced variability explains why larger links are more effective; for the same probability of traf c congestion, they can transport traf c with a mean rate which is closer to their capacity: this is the well known multiplexing gain However, when it comes to peak rates, if X1 and X2 are transported by links of capacity C which supply a concentrator, the latter s output must have a capacity of at east 2C to ensure a zero loss rate.

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EXTREME MODELING (XM) Encode QR In Visual C#NET Using Barcode generator for Visual It makes a lot of sense to have a simple and lightweight method of escribing the system or parts of a system in terms of what it does and the order in which it does it If we have such a picture, we will be able to use it to de ne test sets that do more than just test one simple function The extreme X-machine (XXM) is a suitable candidate These were introduced by Thomson (2003, 2005), and we have been using these for a number of years with great success They are popular with programmers, and customers can also make sense of them The simple machine of the type in Fig 66 is called an X-machine by Eilenberg (1974) and is discussed at length in Holcombe (1998).

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