Probabilistic techniques in the analysis of dynamic processes

In this thesis we study some stochastic problems related to networks and search engines. We study systems in a dynamic setting where input is continually injected into the system and the algorithm (protocol) processes it. The goal is the design and analysis of protocols that are stable and efficient in the long run. We first consider the problem of routing calls on telephone or ATM networks, for which we present a routing protocol and we compare it with the one that is commonly used (Dynamic Alternative Routing). We prove that under a standard input model (Poisson arrivals, exponential durations) our protocol has exponentially smaller bandwidth requirement than the traditional approach. Next we study the problem of load balancing on networks, where requests are continually created and serviced. We analyze a protocol under a variety of input models and we prove bounds on the expected load and the expected waiting time of a new request as time passes. Subsequently, we address a problem related to search engines. We introduce the problem of sampling results from search-engine queries, which has applications in data mining the results and offering services to the user. We present algorithms for the problem and we analyze their running time and the quality of results that we obtain. We supplement the analysis with several experiments. In modeling of dynamic phenomena as stochastic processes we place some stochastic assumption on the stream of inputs to the system. The stochastic process that generates the stream of inputs might be stationary, periodic, or even bursty. The goal is to obtain results that are valid under the weakest set of assumptions. To this end, we have to develop and apply various mathematical tools. In our analyses we apply tools from Markov processes, queueing theory, renewal theory and martingale or martingale-like processes, that enable us to handle the dependencies between the quantities that appear in our systems.