Page 177 - Schaum's Outlines - Probability, Random Variables And Random Processes

RANDOM PROCESSES [CHAP 5

0 5 5 ttl- 1
Fig. 5-1

Let Z, = T, - T,-, (5.54)
and To = 0. Then Z, denotes the time between the (n - 1)st and the nth events (Fig. 5-1). The
sequence of ordered r.v.'s {Z,, n 2 1) is sometimes called an interarrival process. If all r.v.'s Z, are
independent and identically distributed, then {Z,, n 2 1) is called a renewal process or a recurrent
process. From Eq. (5.54), we see that

where T, denotes the time from the beginning until the occurrence of the nth event. Thus, (T,, n 2 0)
is sometimes called an arrival process.

B. Counting Processes :
A random process {X(t), t 2 0) is said to be a counting process if X(t) represents the total number
of "events" that have occurred in the interval (0, t). From its definition, we see that for a counting
process, X(t) must satisfy the following conditions:
1. X(t) 2 0 and X(0) = 0.
2. X(t) is integer valued.
3. X(s) ~X(t)ifs < t.
4. X(t) - X(s) equals the number of events that have occurred on the interval (s, t).
A typical sample function (or realization) of X(t) is shown in Fig. 5-2.
A counting process X(t) is said to possess independent increments if the numbers of events which
occur in disjoint time intervals are independent. A counting process X(t) is said to possess stationary
increments if the number of events in the interval (s + h, t + h)--that is, X(t + h) - X(s + hehas the
same distribution as the number of events in the interval (s, t)--that is, X(t) - X(s)--for all s < t and
h > 0.

Fig. 5-2 A sample function of a counting process.

C. Poisson Processes:
One of the most important types of counting processes is the Poisson process (or Poisson counting
process), which is defined as follows: