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Figure one is a graph labeled, execution time distribution function. The horizontal axis is labeled, Time, and the vertical axis is labeled, probability. The values on the horizontal axis range from 0 to 100 in increments of 10. The values on the vertical axis range from 0 to 1 in increments of 0.1. There is one plotted distribution function on this graph. It begins in the bottom-left corner, at the point (0, 0), and moves right at a strong positive slope. As the plot moves from left to right, the slope decreases as the function increases. About midway across the graph horizontally, the plot is nearly at the top, at a probability value above 0.9. The plot continues to increase at a decreasing rate until it tapers off to a horizontal line by the point (80, 1), at which it continues and terminates at the top-right corner. Figure one is a graph labeled, execution time distribution function. The horizontal axis is labeled, Time, and the vertical axis is labeled, probability. The values on the horizontal axis range from 0 to 100 in increments of 10. The values on the vertical axis range from 0 to 1 in increments of 0.1. There is one plotted distribution function on this graph. It begins in the bottom-left corner, at the point (0, 0), and moves right at a strong positive slope. As the plot moves from left to right, the slope decreases as the function increases. About midway across the graph horizontally, the plot is nearly at the top, at a probability value above 0.9. The plot continues to increase at a decreasing rate until it tapers off to a horizontal line by the point (80, 1), at which it continues and terminates at the top-right corner.
Execution Time Distribution Function F D .

The same results may be achieved with mgd, although at the cost of more computing time. In that case, use g N as in [link] , but use the actual distribution for Y .

Arrival times and counting processes

Suppose we have phenomena which take place at discrete instants of time, separated by random waiting or interarrival times. These may be arrivals of customers in a store,of noise pulses on a communications line, vehicles passing a position on a road, the failures of a system, etc. We refer to these occurrences as arrivals and designate the times of occurrence as arrival times . A stream of arrivals may be described in three equivalent ways.

  • Arrival times : { S n : 0 n } , with 0 = S 0 < S 1 < a . s . (basic sequence)
  • Interarrival times : { W i : 1 i } , with each W i > 0 a . s . (incremental sequence)

The strict inequalities imply that with probability one there are no simultaneous arrivals. The relations between the two sequences are simply

S 0 = 0 , S n = i = 1 n W i and W n = S n - S n - 1 for all n 1

The formulation indicates the essential equivalence of the problem with that of the compound demand . The notation and terminology are changed to correspond to thatcustomarily used in the treatment of arrival and counting processes.

The stream of arrivals may be described in a third way.

  • Counting processes : N t = N ( t ) is the number of arrivals in time period ( 0 , t ] . It should be clear that this is a random quantity for each nonnegative t . For a given t , ω the value is N ( t , ω ) . Such a family of random variables constitutes a random process . In this case the random process is a counting process .

We thus have three equivalent descriptions for the stream of arrivals.

{ S n : 0 n } { W n : 1 n } { N t : 0 t }

Several properties of the counting process N should be noted:

  1. N ( t + h ) - N ( t ) counts the arrivals in the interval ( t , t + h ] , h > 0 , so that N ( t + h ) N ( t ) for h > 0 .
  2. N 0 = 0 and for t > 0 we have
    N t = i = 1 I ( 0 , t ] ( S i ) = max { n : S n t } = min { n : S n + 1 > t }
  3. For any given ω , N ( · , ω ) is a nondecreasing, right-continuous, integer-valued function defined on [ 0 , ) , with N ( 0 , ω ) = 0 .

The essential relationships between the three ways of describing the stream of arrivals is displayed in

W n = S n - S n - 1 , { N t n } = { S n t } , { N t = n } = { S n t < S n + 1 }

This imples

P ( N t = n ) = P ( S n t ) - P ( S n + 1 t ) = P ( S n + 1 > t ) - P ( S n > t )

Although there are many possibilities for the interarrival time distributions, we assume

{ W i : 1 i } is iid, with W i > 0 a . s .

Under such assumptions, the counting process is often referred to as a renewal process and the interrarival times are called renewal times . In the literature on renewal processes, it is common for the random variable to count an arrival at t = 0 . This requires an adjustment of the expressions relating N t and the S i . We use the convention above.

Exponential iid interarrival times

The case of exponential interarrival times is natural in many applications and leads to important mathematical results. We utilize the followingpropositions about the arrival times S n , the interarrival times W i , and the counting process N .

  1. If { W i : 1 i } is iid exponential ( λ ) , then S n gamma ( n , λ ) for all n 1 . This is worked out in the unit on TRANSFORM METHODS, in the discussionof the connection between the gamma distribution and the exponential distribution.
  2. S n gamma ( n , λ ) for all n 1 , and S 0 = 0 , iff N t Poisson ( λ t ) for all t > 0 . This follows the result in the unit DISTRIBUTION APPROXI9MATIONS onthe relationship between the Poisson and gamma distributions, along with the fact that { N t n } = { S n t } .

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
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Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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