Endless Line and Extinction Probability - Pillai

Based on https://drive.google.com/file/d/1ljJwnkpANoN43WcVuiVAjx_8R3IiDo_H/view?usp=drivesdk The content described here comes from Lecture#20 by Prof.Pillai. The extinction probability for queues and related stochastic processes is explored here, specifically drawing parallels with population models (branching processes). It defines a busy period in a queue and questions its termination, characterizing queues based on traffic intensity (rho)—where slow traffic has an extinction probability of one, but heavy, unstable traffic may still have a finite probability of termination. A significant portion of the source focuses on Martingales, defining them as stable stochastic processes where the conditional expectation of future values equals the present value, and illustrating their application through examples like the gambler's ruin problem and deriving key inequalities such as Hoeffding’s inequality. Finally, the text introduces the concept of a stopping time as a random instant where a specific condition is met, showing how this concept extends the Martingale stability property (E{X_n} = E{X_0}) to random time instants (E{X_T} = E{X_0}).