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 :
$\{{S}_{n}:0\le n\}$ , 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 :
$\{{W}_{i}:1\le i\}$ , 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
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
$(0,\phantom{\rule{0.166667em}{0ex}}t]$ . 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
$N(t,\phantom{\rule{0.166667em}{0ex}}\omega )$ . 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.
Several properties of the counting process
N should be noted:
$N(t+h)-N\left(t\right)$ counts the arrivals in the interval
$(t,\phantom{\rule{0.166667em}{0ex}}t+h]$ ,
$h>0$ ,
so that
$N(t+h)\ge N\left(t\right)$ for
$h>0$ .
For any given
ω ,
$N(\xb7,\omega )$ is a nondecreasing, right-continuous,
integer-valued function defined on
$[0,\phantom{\rule{0.166667em}{0ex}}\infty )$ , with
$N(0,\phantom{\rule{0.166667em}{0ex}}\omega )=0$ .
The essential relationships between the three ways of describing the stream of arrivals
is displayed in
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 .
If
$\{{W}_{i}:1\le i\}$ is iid exponential
$\left(\lambda \right)$ , then
${S}_{n}\sim $ gamma
$(n,\phantom{\rule{0.166667em}{0ex}}\lambda )$ 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.
${S}_{n}\sim $ gamma
$(n,\phantom{\rule{0.166667em}{0ex}}\lambda )$ 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
$\{{N}_{t}\ge n\}=\{{S}_{n}\le t\}$ .
Questions & Answers
anyone know any internet site where one can find nanotechnology papers?
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
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?