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In Section 6.1 we have already mentioned a result of Cohen [13] that relates the (regularly varying) tail behavior of service and waiting times in the GI =G=1 queue with the FCFS discipline; it shows that the waiting time is ``one degree heavier'' than the service time tail, in the case of regular variation. 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 I

Again using the bounds from equations (12.21) and (12.22), we can rewrite equation (12.28)

(a) SAX strong anion exchanger, WAX weak anion exchanger, SCX strong cation exchanger, WCX weak cation exchanger. (b) Correlation coef cient for the linearity of ln p k0 versus 1= I plots.13 [Reprinted, by permission, from J. Stahlberg, B. Jonsson, and C. Horvath, Anal. Chem. 63, 1991, 1867 1874. Theory for Electrostatic Interaction Chromatography of Proteins . # 1991 by American Chemical Society.]

P B > u du;

Again noting that max{|di d j | | i, j} = , the worst case is Z >1+ f + f + TL + TF (12.32)

x ! 0:

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Lineweaver Burk method yields signi cantly poorer precision than any of the other three methods. The Cornish Bowden Eisenthal method clearly yields the most precise values for Km, in the presence of either absolute or relative error. 2.5.4. One-Substrate, Two-Product Enzyme Kinetics Hydrolytic enzymes that release two products in a de ned sequence require a slightly more complicated model for their kinetic behavior. These enzymes pass through two intermediate stages in the catalytic cycle, E S and E S0 , and the formation of products involves two steps with associated rate constants k2 and k3 : ! ! E S$ E S E S0 P1 E P2 P1

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

Another key performance metric of a VoD service is system response time, de ned as the time from initiating a new request to the time video playback starts. Ignoring system administration and network delay issues, system response time consists of two components: scheduling delay and pre ll delay. Scheduling delay is the delay incurred at the admission scheduler plus the delay incurred at the server scheduler, as derived in Section 12.3. For pre ll delay, we note that the client pre lls the rst Y video blocks before starting video playback. Hence, the average pre ll delay can be obtained from DP = YQ RV (12.33)

Lemma 6.2.1. lent:

In this section, we evaluate the performance of the staggered-push architecture using numerical results. All results are computed using the derivations in Sections 12.3 to 12.5 with the system parameters listed in Table 12.1.

2:24

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

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 Queue

To illustrate performance and resource requirements of the architecture, we rst consider a design example in this section. We assume that there are 8 servers in the system, with a client server ratio of 50 (i.e., up to 400 concurrent streams). Using the parameters in Table 12.1, the server buffer requirement is calculated to be 6.25MB. Compared to the amount of memory in today s PC, this buffer requirement is relatively small. Moreover, as conventional PCs can be expanded to 1GB or more memory, in theory a client server ratio of over 8,000

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