Renewal Processes

http://www.sics.se/~aeg/report/node17.html

http://www.math.uah.edu/stat/renewal/index.xhtml

= Discrete Renewal =

find pdf of lifetime function find pdf of remaining lifetime how many events in X years U(s) find probability of event at time t find pgf of Wt and find P(Wt = t)

Continuous renewal

hazard function h(t) find survivor function find mean and median of lifetime of components find U(s) and find u1 u2 and u3 etc find W7 find P(W7 = t) for long running process find E at next t

if you start to observe a continuous renewal process at some random time t find the pdf of time until first component fails find the mean

find count of E after t compare the E[] of new compone to E[] when observing some long running component

what happens if you remove the component and move it elsewhere

When a renewal process has been running for a long time, what reciprocal value is equal to the proportion of time points at which E occurs? The reciprocal of the mean of the random variable T, the time between events When a probability generating function is known F(s), what is the mean of that distribution? "the derivative of the p.g.f at 1 F'(1) gives the mean of the distribution represented by the p.g.f. F(s)" What distinguishes the ordinary renewal process {X(t);t>=0} from general birth processes ? The renewal process is generally not a Markov Process, as it depends on age of the component currently in use. (the poisson process using the exponential distribution for T, time between events is the exception)