The property of the poisson process in lemma 22 is

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , which will be our o cial deﬁnition, is nice because it allows us to construct the process easily. Deﬁnition. Let ⌧1 , ⌧2 , . . . be independent exponential( ) random variables. Let Tn = ⌧1 + · · · + ⌧n for n 1, T0 = 0, and deﬁne N (s) = max{n : Tn s}. We think of the ⌧n as times between arrivals of customers at a bank, so Tn = ⌧1 + · · · + ⌧n is the arrival time of the nth customer, and N (s) is the number of arrivals by time s. To check the last interpretation, consider the following example: ⌧1 0 ⇥ T1 ⌧2 ⇥ T2 ⌧3 ⇥ T3 ⌧4 ⌧5 ⇥ T4 s ⇥ T5 Figure 2.1: Poisson process deﬁnitions. and note that N (s) = 4 when T4 s &lt; T5 , that is, the 4th customer has arrived by time s but the 5th has not. Recall that X has a Poisson distribution with mean , or X = Poisson( ), for short, if P (X = n) = e n n! for n = 0, 1, 2, . . . 81 2.2. DEFINING THE POISSON PROCESS To explain why N (s) is called the Poisson process rather than the exponential process, we will compu...
View Full Document

Ask a homework question - tutors are online