Academic journal article Journal of Digital Information Management

Application of Esscher Transformed Laplace Distribution in Web Server Data

Academic journal article Journal of Digital Information Management

Application of Esscher Transformed Laplace Distribution in Web Server Data

Article excerpt

I. Introduction

The growth of World Wide Web has dramatically changed the way information is accessed and managed, thereby opening the door to exciting new scenarios for the widespread consumption and exchange of information. Its tremendous growth brought huge challenges to Web site designers, content producers and maintainers. The performance of a Web server depends not only on its design and implementation but also on the workload to which it is subjected. There are inter-session and intra-session characteristics which collectively describe session-based Web workload. Goseva-Popstojanova et al. (2006a) introduced number of sessions per user and number of sessions initiated per day/hour as inter-session characteristics and number of requests per session, session length in time and bytes transferred per session as intra-session characteristics and showed that session-based workload and reliability are better indicators of the users perception of the Web quality than the request based matrix. A number of variables in teletraffic engineering such as file sizes, packet arrivals etc. have been shown to possess heavy tailed distributions [Paxon and Floyd (1995)]. Again Goseva-Popstojanova et al. (2006b) found that intra-session characteristics exhibit heavy-tailed behavior.

Heavy tailed distributions (also known as power-law distributions) have been observed in many natural phenomena including both physical and sociological phenomena.

A distribution is said to have a heavy tail if:

P [X > x ] ~ [x.sup.-[alpha]], x [right arrow] [infinity], 0 < [alpha] <2.

This means that regardless of the distribution for small values of the random variable if the asymptotic shape of the distribution is hyperbolic, it is heavy tailed. The simplest heavy tailed distribution is the Pareto distribution which is hyperbolic over its entire range and has the probability density function,

P(x) = [alpha][k.sup.[alpha]] [x.sup.-[alpha]-1], [alpha], k > 0, x [greater than or equal to] k

and its cumulative distribution function is given by:

F(x) = P[X [less than or equal to] x] = 1 - [(k/x).sup.[alpha]], x [greater than or equal to] k

where [alpha] is the shape parameter and tail index and k represents the smallest value, that the random variable can take. As [alpha] decreases, an arbitrarily large portion of the probability mass may be present in the tail of the distribution. This means that a random variable that follows a heavy tailed distribution can give raise to extremely large values with non-negligible probability. As k increases only the tail of the distribution is modeled.

Bestavros and Crovella (1996) have found evidence of heavy tails in the distribution of file sizes and transmission times in their empirical study of WWW. Willinger et al. (1995) also found evidence of heavy tails in activity periods and idle times of individual computers. For a detailed review of heavy tailed distributions, see Sigman (1999). In 2006, Tanenbaun et al. showed the importance of file size distribution in optimizing file system design. Suneetha and Krishnamoorthy (2009) did an in depth analysis of Web log data and their study helped the system administrators and Web designers to improve their system performance. For many years stable Paretian laws are considered for modeling heavy tailed and asymmetric data. Though the theory of stable Paretian distributions are well developed [Samorodnitsky and Taqqu (1994)], because of the lack of analytical form of the densities, infinite second moment, their applications in practical modeling are still rather limited. Again the stable Paretian model does not account for peakedness around the origin which is seen in the Web server data.

Braun and Claffy (1994) suggested Pareto type I distribution for modeling bytes transferred per session (A session is defined as a sequence of requests from the same user during a single visit to the web site). …

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