WEBVTT
Kind: captions
Language: en
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Now we are moving into the one type of renewal
process that is called alternating renewal
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process. Let Xi's are iid random variable
which constitute on times and Yi's are iid
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random variable which constitutes off times.
Assume that mean is the existent it is finite
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and the X plus Y has the distribution F. Suppose
that the renewal occurs at the end of every
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Xi's whereas no arrivals at the end of every
Yi's. Assume that Xi's and Yi's are independent
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random variables also. Then the Xi plus Yi
are called the alternative renewal process.
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See the example for this situation. A machine
works for the time X1 and then breaks down
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and has to be repaired which takes a time
Y1. Then works for the time X2 then it is
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down for a time Y2. That's a second repair
time and so on. So Xi's are nothing but the
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machine works and the Yi's are nothing but
the repair time. If you suppose that the machine
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is good as new after each repair then this
constitute alternative renewal process. So
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this is the example of alternative renewal
process with the proper assumptions.
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Now we are moving into the next renewal processes
that is delayed renewal process. It is not
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always reasonable to insist that the first
renewal occurs at time S naught that is equal
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to 0 in the origin itself, at the time 0 itself.
For instant in applications where the occurrence
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are not or at times when a component of the
system must be replaced. One might well be
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interested in situations where there is already
a working component in place at time 0. For
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this reason we defined a delayed renewal process
to be a sequence S naught, S1, S2 and
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so on where Sn is nothing but S naught plus
the summation of first n Xi's and the inter
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arrival times Xi's are positive and iid random
variables as in the ordinary renewal process.
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And the initial delay S naught which is great
or equal to 0 that is independent of inter
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arrival times Xi's.
Notice that the distribution of initial delay
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random variable X naught is not recurred to
be the same as that of the inter arrival time
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random variables Xi's. Hence a delayed renewal
process is a renewal process in which the
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first arrival time X1 is independently and
is allowed to have a different distribution
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that is F1. The distribution of all remaining
iid random variables that distribution is
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F. So the F1 is different from F. The X1 is
equal to t1 that is nothing but that delay
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and there is no such delay then X1 is also
distributed in the same way so the distribution
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of X1 is F as usual. Then the renewal process
is said to be a non delayed version.
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Now we are discussing the central limit theorem
on the renewal process. As n tends to infinity
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the random variable S suffix n that is nothing
but the nth time renewal minus n times mu
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divided by Sigma times square root of n will
be normal distribution with the mean 0 and
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variance 1. So this convergence takes place
in a distribution. So this is a weak distribution,
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weak convergence.
So as n tends to infinity the random variable
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Sn and the mean of Sn is n times mu and the
variance of Sn is Sigma square n and the random
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variable minus the mean divided by the standard
deviation is normal distributed with the mean
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0 and variance 1 as n tends to infinity as
n tends to infinity the n of t the counting
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process it's a renewal process becomes a normal
distribution in the mean t divided by mu and
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the variance is Sigma square t divided by
mu cube where mu is the mean of inter arrival
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time as a t tends to infinity the random variable
n of t minus t by mu divided by Sigma times
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square root of t by mu cube will be normal
distribution with the mean 0 and the variance
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1. So this is also a convergence in distribution.
Now we are going to discuss the long-run properties
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of renewal process. There are two types of
long run properties. One is
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to obtain the long-run average of the quantity
of interest. The other one is to obtain the
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pointwise limit. For example the long-run
average of age that is limit t tends to infinity
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the integration 0 to t A of s, A of s is the
age divided by t. while the pointwise limit
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of the expected age that
is the limit t tends to infinity expectation
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of A of t.
One can study the average number of renewals
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per unit time in the long run. It is called
the long run renewal rate. For a renewal process
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having a distribution F for the inter arrival
times the long run renewal rate is nothing
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but limit t tends to infinity n of
t divided by t. That will be 1 divided by
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mu with the probability 1 where mu is nothing
but the mean of inter arrival time.
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Since S suffix n of t plus last renewal time
prior to t and S suffix n
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of t plus 1 is
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the first renewal time of t we know the relation
of S of n of t with the S of n of t plus 1
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and that lies, the t lies between those two
renewal times.
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You can divide by t you can divide by n of
t. Therefore S of n of t divided by n of t
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less than or equal to t divided by n of t
that is less than or equal to S of n plus
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t plus 1 divided by n of t. Now we can evaluate
the first one the
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S of n of t divided by n of t limit t tends
to infinity. That
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is nothing but the numerator is nothing but
the summation of Xi's t n of t denominator
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is n of t and that is nothing but expectation
of X in a long run -
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in the long run the summation of X1, X2 till
X of n t divided by n of T
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that is nothing but the expectation of X that
is same as the mu with the probability 1.
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And similarly one can evaluate the last value
that is S of n of t plus 1 divided by n of
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t that is also will be mu with the probability
1. Since t lies between S of n of t and S
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of n of t plus 1 and the throughout divided
by n
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of t in all three therefore as limit t tends
to infinity n of t divided by t will be 1
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divided by mu with the probability 1.