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Remark . The counting process is a Poisson process in the sense that N t Poisson ( λ t ) for all t > 0 . More advanced treatments show that the process has independent, stationary increments. That is

  1. N ( t + h ) - N ( t ) = N ( h ) for all t , h > 0 , and
  2. For t 1 < t 2 t 3 < t 4 t m - 1 < t m , the class { N ( t 2 ) - N ( N 1 ) , N ( t 4 ) - N ( t 3 ) , , N ( t m ) - N ( t m - 1 ) } is independent.

In words, the number of arrivals in any time interval depends upon the length of the interval and not its location in time, and the numbers of arrivals in nonoverlappingtime intervals are independent.

Emergency calls

Emergency calls arrive at a police switchboard with interarrival times (in hours) exponential (15). Thus, the average interarrival time is 1/15 hour (four minutes).What is the probability the number of calls in an eight hour shift is no more than 100, 120, 140?

p = 1 - cpoisson(8*15,[101 121 141])p = 0.0347 0.5243 0.9669
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We develop next a simple computational result for arrival processes for which S n gamma ( n , λ ) .

Gamma arrival times

Suppose the arrival times S n gamma ( n , λ ) and g is such that

0 | g | < and E n = 1 | g ( S n ) | <

Then

E n = 1 g ( S n ) = λ 0 g

VERIFICATION

We use the countable sums property (E8b) for expectation and the corresponding property for integrals to assert

E n = 1 g ( S n ) = n = 1 E [ g ( S n ) ] = n = 1 0 g ( t ) f n ( t ) d t where f n ( t ) = λ e - λ t ( λ t ) n - 1 ( n - 1 ) !

We may apply (E8b) to assert

n = 1 0 g f n = 0 g n = 1 f n

Since

n = 1 f n ( t ) = λ e - λ t n = 1 ( λ t ) n - 1 ( n - 1 ) ! = λ e - λ t e λ t = λ

the proposition is established.

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Discounted replacement costs

A critical unit in a production system has life duration exponential ( λ ) . Upon failure the unit is replaced immediately by a similar unit. Units fail independently. Cost of replacement of a unit is c dollars. If money is discounted at a rate α , then a dollar spent t units of time in the future has a current value e - α t . If S n is the time of replacement of the n th unit, then S n gamma ( n , λ ) and the present value of all future replacements is

C = n = 1 c e - α S n

The expected replacement cost is

E [ C ] = E n = 1 g ( S n ) where g ( t ) = c e - α t

Hence

E [ C ] = λ 0 c e - α t d t = λ c α

Suppose unit replacement cost c = 1200 , average time (in years) to failure 1 / λ = 1 / 4 , and the discount rate per year α = 0 . 08 (eight percent). Then

E [ C ] = 1200 · 4 0 . 08 = 60 , 000
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Random costs

Suppose the cost of the n th replacement in [link] is a random quantity C n , with { C n , S n } independent and E [ C n ] = c , invariant with n . Then

E [ C ] = E n = 1 C n e - α S n = n = 1 E [ C n ] E [ e - α S n ] = n = 1 c E [ e - α S n ] = λ c α
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The analysis to this point assumes the process will continue endlessly into the future. Often, it is desirable to plan for a specific, finite period. The result of [link] may be modified easily to account for a finite period, often referred to as a finite horizon .

Finite horizon

Under the conditions assumed in [link] , above, let N t be the counting random variable for arrivals in the interval ( 0 , t ] .

If Z t = n = 1 N t g ( S n ) , then E [ Z t ] = λ 0 t g ( u ) d u

VERIFICATION

Since N t n iff S n t , n = 1 N t g ( S n ) = n = 0 I ( 0 , t ] ( S n ) g ( S n ) . In the result of [link] , replace g by I ( 0 , t ] g and note that

0 I ( 0 , t ] ( u ) g ( u ) d u = 0 t g ( u ) d u
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Replacement costs, finite horizon

Under the conditions of [link] , consider the replacement costs over a two-year period.

SOLUTION

E [ C ] = λ c 0 t e - α u d u = λ c α ( 1 - e - α t )

Thus, the expected cost for the infinite horizon λ c / α is reduced by the factor 1 - e - α t . For t = 2 and the numbers in [link] , the reduction factor is 1 - e - 0 . 16 = 0 . 1479 to give E [ C ] = 60000 · 0 . 1479 = 8 , 871 . 37 .

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In the important special case that g ( u ) = c e - α u , the expression for E n = 1 g ( S n ) may be put into a form which does not require the interarrival times to be exponential.

General interarrival, exponential g

Suppose S 0 = 0 and S n = i = 1 n W i , where { W i : 1 i } is iid. Let { V n : 1 n } be a class such that each E [ V n ] = c and each pair { V n , S n } is independent. Then for α > 0

E [ C ] = E n = 1 V n e - α S n = c · M W ( - α ) 1 - M W ( - α )

where M W is the moment generating function for W .

DERIVATION

First we note that

E [ V n e - α S n ] = c M S n ( - α ) = c M W n ( - α )

Hence, by properties of expectation and the geometric series

E [ C ] = c n = 1 M W n ( - α ) = M W ( - α ) 1 - M W ( - α ) , provided | M W ( - α ) | < 1

Since α > 0 and W > 0 , we have 0 < e - α W < 1 , so that M W ( - α ) = E [ e - α W ] < 1 .

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Uniformly distributed interarrival times

Suppose each W i uniform ( a , b ) . Then (see Appendix C ),

M W ( - α ) = e - a α - e - b α α ( b - a ) so that E [ C ] = c · e - a α - e - b α α ( b - a ) - [ e - a α - e - b α ]

Let a = 1 , b = 5 , c = 100 and α = 0 . 08 . Then,

a = 1; b = 5;c = 100; A = 0.08;MW = (exp(-a*A) - exp(-b*A))/(A*(b - a)) MW = 0.7900EC = c*MW/(1 - MW) EC = 376.1643
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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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RAW Reply
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Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
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Hafiz
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Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Damian Reply
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Kyle
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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