The property of the poisson process in lemma 22 is

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

View Full Document Right Arrow Icon
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 < 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