<< Chapter < Page Chapter >> Page >

Message routing

A junction point in a network has two incoming lines and two outgoing lines. The number of incoming messages N 1 on line one in one hour is Poisson (50); on line 2 the number is N 2 Poisson (45). On incoming line 1 the messages have probability p 1 a = 0 . 33 of leaving on outgoing line a and 1 - p 1 a of leaving on line b. The messages coming in on line 2 have probability p 2 a = 0 . 47 of leaving on line a. Under the usual independence assumptions, what is the distribution of outgoing messages on line a?What are the probabilities of at least 30, 35, 40 outgoing messages on line a?

SOLUTION

By the Poisson decomposition, N a Poisson ( 50 · 0 . 33 + 45 · 0 . 47 = 37 . 65 ) .

ma = 50*0.33 + 45*0.47 ma = 37.6500Pa = cpoisson(ma,30:5:40) Pa = 0.9119 0.6890 0.3722
Got questions? Get instant answers now!

VERIFICATION of the Poisson decomposition

  1. N k = i = 1 N I E k i .
    This is composite demand with Y k = I E k i , so that g Y k ( s ) = q k + s p k = 1 + p k ( s - 1 ) . Therefore,
    g N k ( s ) = g N [ g Y k ( s ) ] = e μ ( 1 + p k ( s - 1 ) - 1 ) = e μ p k ( s - 1 )
    which is the generating function for N k Poisson ( μ p k ) .
  2. For any n 1 , n 2 , , n m , let n = n 1 + n 2 + + n m , and consider
    A = { N 1 = n 1 , N 2 = n 2 , , N m = n m } = { N = n } { N 1 n = n 1 , N 2 n = n 2 , , N m n = n m }
    Since N is independent of the class of I E k i , the class
    { { N = n } , { N 1 n = n 1 , N 2 n = n 2 , , N m n = n m } }
    is independent. By the product rule and the multinomial distribution
    P ( A ) = e - μ μ n n ! · n ! k = 1 m p k n k ( n k ) ! = k = 1 m e - μ p k p k n k n k ! = k = 1 m P ( N k = n k )
    The second product uses the fact that
    e μ = e μ ( p 1 + p 2 + + p m ) = k = 1 m e μ p k
    Thus, the product rule holds for the class { N k : 1 k m } , so that it is independent.

Extreme values

Consider an iid class { Y i : 1 i } of nonnegative random variables. For any positive integer n we let

V n = min { Y 1 , Y 2 , , Y n } and W n = max { Y 1 , Y 2 , , Y n }

Then

P ( V n > t ) = P n ( Y > t ) and P ( W n t ) = P n ( Y t )

Now consider a random number N of the Y i . The minimum and maximum random variables are

V N = n = 0 I { N = n } V n and W N = n = 0 I { N = n } W n

Computational formulas

If we set V 0 = W 0 = 0 , then

  1. F V ( t ) = P ( V t ) = 1 + P ( N = 0 ) - g N [ P ( Y > t ) ]
  2. F W ( t ) = g N [ P ( Y t ) ]

These results are easily established as follows. { V N > t } = n = 0 { N = n } { V n > t } . By additivity and independence of { N , V n } for each n

P ( V N > t ) = n = 0 P ( N = n ) P ( V n > t ) = n = 1 P ( N = n ) P n ( Y > t ) , since P ( V 0 > t ) = 0

If we add into the last sum the term P ( N = 0 ) P 0 ( Y > t ) = P ( N = 0 ) then subtract it, we have

P ( V N > t ) = n = 0 P ( N = n ) P n ( Y > t ) - P ( N = 0 ) = g N [ P ( Y > t ) ] - P ( N = 0 )

A similar argument holds for proposition (b). In this case, we do not have the extra term for { N = 0 } , since P ( W 0 t ) = 1 .

Special case . In some cases, N = 0 does not correspond to an admissible outcome (see [link] , below, on lowest bidder and [link] ). In that case

F V ( t ) = n = 1 P ( V n t ) P ( N = n ) = n = 1 [ 1 - P n ( Y > t ) ] P ( N = n ) = n = 1 P ( N = n ) - n = 1 P n ( Y > t ) P ( N = n )

Add P ( N = 0 ) = P 0 ( Y > t ) P ( N = 0 ) to each of the sums to get

F V ( t ) = 1 - n = 0 P n ( Y > t ) P ( N = n ) = 1 - g N [ P ( Y > t ) ]

Maximum service time

The number N of jobs coming into a service center in a week is a random quantity having a Poisson (20) distribution. Suppose the service times (in hours) for individualunits are iid, with common distribution exponential (1/3). What is the probability the maximum service time for the units is no greater than 6, 9, 12, 15, 18 hours?SOLUTION

Solution

P ( W N t ) = g N [ P ( Y t ) ] = e 20 [ F Y ( t ) - 1 ] = exp ( - 20 e - t / 3 )
t = 6:3:18; PW = exp(-20*exp(-t/3));disp([t;PW]')6.0000 0.0668 9.0000 0.369412.0000 0.6933 15.0000 0.873918.0000 0.9516
Got questions? Get instant answers now!

Questions & Answers

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
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
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
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
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