4.1 Introduction

We describe the structure of a simple point process on the line. Consider a strictly increasing sequence of random variables { T ^P _n} ^? _{n = ? ?} defined on a probability space . The { T ^P _n} ^? _{n = ? ?} represent the arrival times of certain events say an incoming signal to a network measured in seconds before or after some fixed time which we take to be 0. We suppose that T ^P ₀ ? 0 < T ^P ₁. The sojourn times or interarrival times relative to 0 are denoted:

Except for n = 0 and n = 1, the X ^P _n represents the interarrival time between the n ? 1 ^th and the n ^th arrivals. X ^P ₀ represents the time since the last arrival before 0, and X ^P ₁ represents the time until the first arrival after time 0. If multiple arrivals may occur at the same { T ^P _n}, we call this a multiple point process. We shall give a description of these point processes under various dependence structures.

Figure 4.1: A trajectory of a simple point process.

Throughout we shall assume that time is measured in seconds with nanosecond precision; hence precise to nine decimal places. The operation of converting any time t measured in seconds to nanoseconds and rounding up to the next...