# 15.1 Some random selection problems  (Page 5/6)

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The same results may be achieved with mgd, although at the cost of more computing time. In that case, use $gN$ 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 : $\left\{{S}_{n}:0\le n\right\}$ , with $0={S}_{0}<{S}_{1}<\phantom{\rule{0.277778em}{0ex}}\cdots \phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ (basic sequence)
• Interarrival times : $\left\{{W}_{i}:1\le i\right\}$ , with each ${W}_{i}>0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$ (incremental sequence)

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

${S}_{0}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{S}_{n}=\sum _{i=1}^{n}{W}_{i}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{W}_{n}={S}_{n}-{S}_{n-1}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{0.277778em}{0ex}}n\ge 1$

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 : ${N}_{t}=N\left(t\right)$ is the number of arrivals in time period $\left(0,\phantom{\rule{0.166667em}{0ex}}t\right]$ . It should be clear that this is a random quantity for each nonnegative t . For a given $t,\phantom{\rule{0.277778em}{0ex}}\omega$ the value is $N\left(t,\phantom{\rule{0.166667em}{0ex}}\omega \right)$ . 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.

$\left\{{S}_{n}:0\le n\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{{W}_{n}:1\le n\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{{N}_{t}:0\le t\right\}$

Several properties of the counting process N should be noted:

1. $N\left(t+h\right)-N\left(t\right)$ counts the arrivals in the interval $\left(t,\phantom{\rule{0.166667em}{0ex}}t+h\right]$ , $h>0$ , so that $N\left(t+h\right)\ge N\left(t\right)$ for $h>0$ .
2. ${N}_{0}=0$ and for $t>0$ we have
${N}_{t}=\sum _{i=1}^{\infty }{I}_{\left(0,t\right]}\left({S}_{i}\right)=max\left\{n:{S}_{n}\le t\right\}=min\left\{n:{S}_{n+1}>t\right\}$
3. For any given ω , $N\left(·,\omega \right)$ is a nondecreasing, right-continuous, integer-valued function defined on $\left[0,\phantom{\rule{0.166667em}{0ex}}\infty \right)$ , with $N\left(0,\phantom{\rule{0.166667em}{0ex}}\omega \right)=0$ .

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

${W}_{n}={S}_{n}-{S}_{n-1},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{{N}_{t}\ge n\right\}=\left\{{S}_{n}\le t\right\},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left\{{N}_{t}=n\right\}=\left\{{S}_{n}\le t<{S}_{n+1}\right\}$

This imples

$P\left({N}_{t}=n\right)=P\left({S}_{n}\le t\right)-P\left({S}_{n+1}\le t\right)=P\left({S}_{n+1}>t\right)-P\left({S}_{n}>t\right)$

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

$\left\{{W}_{i}:1\le i\right\}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{iid,}\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{W}_{i}>0\phantom{\rule{4pt}{0ex}}\mathrm{a}.\mathrm{s}.\phantom{\rule{0.166667em}{0ex}}$

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 $\left\{{W}_{i}:1\le i\right\}$ is iid exponential $\left(\lambda \right)$ , then ${S}_{n}\sim$ gamma $\left(n,\phantom{\rule{0.166667em}{0ex}}\lambda \right)$ for all $n\ge 1$ . This is worked out in the unit on TRANSFORM METHODS, in the discussionof the connection between the gamma distribution and the exponential distribution.
2. ${S}_{n}\sim$ gamma $\left(n,\phantom{\rule{0.166667em}{0ex}}\lambda \right)$ for all $n\ge 1$ , and ${S}_{0}=0$ , iff ${N}_{t}\sim$ Poisson $\left(\lambda t\right)$ 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 $\left\{{N}_{t}\ge n\right\}=\left\{{S}_{n}\le t\right\}$ .

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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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 By Mahee Boo By John Gabrieli By Jill Zerressen By Yasser Ibrahim By OpenStax By Richley Crapo By Marion Cabalfin By OpenStax By Brooke Delaney By OpenStax