Probability and Random Processes Second Edition

Introduction In the last two chapters we will discuss two applications for probability modeling. In this chapter Probability modeling in Traffic Engineering will be addressed and in the next chapter Probability modeling in Medical Imaging will be addressed. The design of telephone switching system and networks in the early 1900's took into consideration the random arrival pattern of customers initiating the calls and the random durations that the telephone lines were occupied. The resources required to satisfy a prescribed quality of service to the customers were determined by the application of queuing theory, a sub-field of applied probability. Teletraffic engineering is the application of this theory to the sizing and building of telephone systems and networks [1]. Traffic engineering has since been extended to size and evaluate digital voice and data communication networks [2]. This problem continues to be relevant as nearly 3 billion people around the world, about 40% of the world's population will have access to the Internet and its diverse applications by the end of 2014 [14]. The random pattern of the Internet user's access to applications on the data network can be quite different from that of the telephone customers. The characterization of Internet traffic has been a challenging problem in the twenty years since the commercialization of the network infrastructure and the distribution of information through the World Wide Web. The identification and application of stochastic models for traffic generated from various voice, video and data applications continues to be an important engineering problem for the current and future generations of communications networks. The performance of telephone networks has however been well managed and controlled based on a class of queueing models that capture fairly accurately the random arrival times and call holding times of telephone users. The probabilistic descriptions of two such models, both assuming Poisson arrivals but distinctly different holding times, one exponentially distributed and the other being deterministic are presented in detail in this chapter. The public switched telephone network, engineered to perform with exemplary precision, is a classic example of how an appropriate stochastic model of customer behavior can lead to robust probabilistic metrics that guarantee the required grade of service (GOS). In 1909 Erlang [3,4] proposed that a Poisson process described well the random arrival pattern of telephone calls. He also determined that the call holding time was an exponentially distributed random variable. His models were based on the assumption …