<< Chapter < Page Chapter >> Page >
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

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
Samson Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied probability' conversation and receive update notifications?

Ask