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A distinction is made between the cases in which the interarrival time distribution has a heavier tail, the service time distribution has a heavier tail, and both tails are ``just as heavy'' In Section 66 the input process and the workload process are studied for the GI =G=1 queue with heavy-tailed interarrival and=or service time distribution The heavy traf c behavior of those processes is characterized Section 67 contains conclusions and some topics for further research (Note: Several of the results in this chapter have appeared in recent reports of the authors and their colleagues; in those cases, proofs are generally omitted Also, several chapters in this volume are related to the present work We mention in particular the contributions of Jelenkovic ( 10), of Likhanov ( 8), of Makowski and Parulekar ( 9), of Norros ( 4), of Brichet et al.



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Byzantine in this context refers to a theoretical computing problem known as the Byzantine generals. In this exercise, an army commander sends out his generals to scout the enemy. Some don t return, and some return compromised by the enemy as spies. The commander s task is to devise an algorithm that allows him to determine which generals are lying and which generals are telling the truth. Practical applications of this idea aren t plentiful, but voting algorithms for reliability are based on the same ideas.

( 5), and of Resnick and Samorodnitsky ( 7)).

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Tesseract is an optical character recognition engine for various operating systems. It is free software, released under the Apache License, Version 2.0, and development has been sponsored by Google since 2006. In 2006, Tesseract was considered one of the most accurate open-source OCR engines then available.

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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

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business continuity is a strategic planning element. Business continuity is related to disaster recovery: DR describes the IT mechanisms for bringing a functioning system back online, while business continuity describes the policies and processes for keeping things running while you re executing the DR plan. Business continuity covers a wide range of topics, such as the following:

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Delegation of responsibility and decision-making authority in case key executives are affected by a disaster. Regulatory requirements for reporting, data preservation, and auditing capabilities while operating from a DR site. Putting bounds on the recovery times and systems capabilities for a DR site; business continuity becomes the requirements analysis for a DR plan. Relationships with suppliers, distributors, and other partners, such that the business is able to function (even at a reduced level of output).

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Business continuity makes sure the enterprise is capable of running after any number of large-scale failures. Disaster recovery ensures there are IT systems to run that business. And availability planning keeps the IT systems running through smaller-scale failures. As we become adept at solving problems at one level of complexity, the issues get larger and more abstract.

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

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