1. VARIATION OF PROCESS GENERALIZATION AND RENEWAL
Delays Renewal Process
It is assumed that {Xk} are all independent positive random variables, but only X2,
X3, … is distributed identically with distribution function F, X1 is likely to have a
different distribution with distribution function G. Delay renewal process described when
we have all the ingredients for an ordinary renewal process, at a beginning of time until
the first renewal has a different distribution of the time in another incident.
Delay the process of renewal will occur if the components in operation at time t=0
is not new, but all subsequent sequence is a new replacement the next. For example,
suppose the first is y units of time after the start of an ordinary renewal process. Then the
time until the first renewal after the origin in the process of delay will have a distribution
of excess lifetime y of ordinary renewal process.
If W0 = 0 and Wn = X1 + …… + Xn’ and if N(t) the number of renewal up to time t.
But now is essential to between the average number of delays in the process of renewal.
MD(t) = E[N(t)]
And the renewal function with distribution F,
M(t) = Fk (t )
k 1
To delay the process of renewal is a fundamental theorem
M d (t ) 1
lim which = E(X2)
t t
And the renewal the theorem states that
h
lim [ M D (t ) M D (t h)
t
with X2, X3, …… is a continuous random variable.

2. Stationary Process Renewal
Delay the process of renewal in the first state to have the distribution function
x
1
G ( x) {1 F ( y )}dy
0
called a stationary renewal process. We seek a renewal process model start with an
unlimited period of time in the past, so the lifetime of this state is maintained on the
origin of the excess of the limit distribution has lived in the ordinary renewal process. We
recognize G as the limit distribution.
Anticipation that such a process that shows the number of stationary, or the
properties of time-invariant. To stationary renewal process
t
M D (t ) E[ N (t )]
And for all t
Pr{Yt D x} G ( x)
So, what is generally only a renewal asymptotic relation to identified, valid for all, in
stationary renewal process.
Cumulative and Relationship Process
Suppose associated with the i-th unit, or a lifetime interval, is a random Yi
(distribution identified {Yi}) in addition to lifetime Xi. We follow the Xi and Yi are
dependent, but is assumed that the pair (Xi,Yi), (X2,Y1), …….. are independent. We use
notation F(x) = Pr{Xi x}, G(y) = Pr{Yi y}, = E(Xi), and v = E(Yi).

3. Renewal Process Involving Two Components for Each Interval renewal
Suppose Yi describe part of the duration of the Xi as shown bellow :
Y1 Y2 Y3
X1 X2 X3
We can describe the Y occurred at the beginning of the interval, but this assumption is
not essential for the following results :
If p(t) is the probability that the fall in the interval Y of some of renewal interval.
When X1, X2, …… is a continuous random variable, then the renewal theorem implies
asymptotic critical evaluation as follows,
E (Yi )
lim p (t )
t E( X i )
Some examples of concrete
Replacement Model
In this model the replacement does not happen on its own (instant). Suppose Yi is
the time of surgery and Zi lag period that preceded installment of the operating unit to the
(i+1). We assume that the sequence of time between the successful replacement of Xk+Zk
, k=1,2, …… there is a renewal process. Then p(t), the probability that the system is an
operation at time t, the convergence of E[Yi]/E[Xi].

4. A Queuing Model
Queue process is a process where costumer come at some designated place where
the service of sorts some refundable, litterateur example, the payment counter at the bank
or the supermarket at payment caser. It is assumed that the time between the arrival or
inter-arrival time, and time spent in the provision of service provided in costumer follow
probabilistic rules.
If the arrival of the queue follow a Poisson process of intensity , then the
successful Xk from the beginning of the k-th busy period for the start of the next busy
period shaped the process of renewal. Each Xk drawn from the busy parts of Zk and Yk is
not busy. Then p(t), the probability that the queue is empty at time t, the convergence to
E(Y1)/E(X1).
The Peter Principle
“Peter Principle” states that employees will be promoted until finally reaching the
position where the so called qualified person (competent). If this happens, the person will
remain in employment until retirement. Based on the model of a single job from the
“Peter Principle” is as follows: The person selected at random from the population and
placed in jobs. If the person is competent, he will stay on the job to have a random time
cumulative distribution function F and mean µ and promoted. If not competent, people
live for a random time which has cumulative distribution function G and mean v > µ and
dismissed. If you work in a state of vacant, them the others will be selected at random
and the process repeated. It is assumed that the infinite population contains a fraction p of
the good and q = 1 – p incompetent.
Renewal occurs every time when filled E[Xk] = p + (1 p)v.
Cumulative Process
Interpret Y is a cost value associated with the i-th renewal process. Class of
problem is naturally set in the context of the general of the pair {Xi,Yi} where Xi is due
process of renewal. Attention here is focused on the so called cumulative processes
N (t ) 1
W (t ) Yk
k 1

5. Accumulated cost or value until time t (assumed that the transaction was made at the
beginning of the cycle of renewal). The fundamental theorem of renewal set forth in this
question are:
E{Y1 ]
lim 1 E{W (t )}
t
t