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
I only see partial conversation and what's the question here!
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
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?