Introduction to Matrix Analytic Methods in Queues 1

1Introduction

Stochastic (also referred to as random) models play an important role in our day-to-day activities. Just look around your living or working space and see the items, are needed on a daily basis. Dairy products, appliances, electronic devices, to name a few among several dozens of items, are manufactured/grown/cultivated and shipped from the factories to nearby stores for consumers. How are these delivered? A supply chain plays a major part in all of these, and as is well-known most things involved in this chain are random. How do we study randomness? Through probability. Whether someone likes probability or is allergic to it, it is there to rescue us and provide a good quality of life. While this book is being written, the whole planet has been affected by Covid-19 disrupting almost everyone in one form or the other. The containers used in shipping items via cargo ships are scarce, resulting in prohibitively high costs; trucks needed to haul away the filled containers after dropping the empty ones are in short supply, making all the cargo ships anchor close to the ports where they arrive. According to the news, the bottleneck is expected to last well into 2022. The point of this discussion is to illustrate the havoc played by uncertainty and why it is important to have several scenarios analyzed through stochastic modeling.

In this chapter, we present key concepts and results on probability, random variables, renewal processes including Poisson, and matrix analysis. ...