5 - Discrete Distributions, Queues, Trees, and Search Codes

Abstract

This chapter discusses probability models related to random walks and, in particular, derive waiting time distributions in the simple random walk, a correlated random walk, and some queuing processes. The waiting time distributions provide discrete probability models—the simplest of which is the binomial distribution—for many practical situations. In addition, the parameters of random walk models are estimated. Among the probability distributions related to lattice paths, the simplest and probably the oldest one is the binomial distribution. Using the Lagrange formula approach, Consul and Shenton have generated a class of distributions called Lagrange distributions and studied their properties. Various theorems are proven in the chapter. The chapter presents a few examples of certain aspects of queues involving batches, dealing mainly with the number of customers served in a busy period, which are directly related to lattice path combinatorics.