ABSTRACT
A point process is a sequence of real numbers with properties{t1, t2, ...} (3.1)t1 < t2 < . .. and limi→ ∞
ti = +∞ .
That means, a point process is a strictly increasing sequence of real numbers, which does not have a finite limit point. In practice, point processes occur in numerous situations: arrival time points of customers at service stations (workshops, filling stations, supermarkets, ...), failure time points of machines, time points of traffic accidents, occurrence of nature catastrophies, occurrence of supernovas,... Generally, at time point a certain event happens. Hence, the are called event times. With re-ti ti gard to the arrival of customers at service stations, the are also called arrival times.ti If not stated otherwise, the assumption is made.t1 ≥ 0 Although the majority of applications of point processes refer to sequences of time points, there are other interpretations as well. For instance, sequences can{t1, t2, ...} be generated by the location of potholes in a road. Then denotes the distance of theti
pothole from the beginning of the road. Or, the location is measured, at which ani th imaginary straight line, which runs through a forest stand, hits trees. (This is the base of the well-known Bitterlich method for estimating the total number of trees in a forest stand.) Strictly speaking, since both road and straight line through a forest stand have finite lengths, to meet assumption (3.1), they have to be considered finite samples from a point process. A point process can equivalently be represented by the sequen of its{t1, t2, ...} ce interevent (interarrival) times
{ y1, y2, ...} with yi = ti − ti−1; i = 1, 2, ...; t0 = 0. Counting Process Frequently, the event times are of less interest than the number of events, which occur in an interval This number is denoted as :(0, t], t > 0. n(t)
n(t) = max {n, tn ≤ t}. For obvious reasons, is said to be the counting process belonging to the{n(t), t ≥ 0} point process Here and in what follows, it is assumed that more than one{t1, t2, ...}. event cannot occur at a time. Point processes with this property are called simple. The numberof events, which occur in an interval , is(s, t] s < t,
n(s, t) = n(t) − n(s).
