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Basic theory of one- and two-server ques with Poisson arrivals and exponential servers. Matlab calculations provide numerical examples.

A standard model of a queueing system with a single waiting line and one or more servers assumes that “customers” arrive according to a Poisson process withrate ( λ ) . The customer at the head of the line goes to the first available server, if there are more than one, or to the single server as soonas available, if there is only one. The servers operate independently (of each other and the arrival process), each with exponential service time. We supposeeach server has the same distribution, exponential ( μ ) . Such a system may be analyzed as a Markov birth-death process. An analysis of the long-runprobabilities and expectations of various quantities after the system has settled down to equilibrium yields the results below.

Calculation of these quantities is straightforward, but somewhat tedious if various cases are considered. Matlab procedures for single-server andtwo-server systems are utilized to make these calculations quickly and to present them in a useful way.

Notation

  • N t = number in system (in service and waiting) at time t
  • Q t = number waiting to be served at time t
  • π j = lim t p i j ( t ) = long-run probability of being in state j
  • W t = waiting time for service for customer who arrives at time t
  • D t = waiting time plus service time for customer who arrives at time t
  • A = random variable with distribution of interarrival times
  • S = random variable with distribution of service times

Long-run probabilities π j = P ( N t = j ) for large t , s servers, E [ A ] = 1 / λ , E [ S ] = 1 / μ

For s = 1 ,

  • ρ = E [ S ] / E [ A ] = λ / μ
  • π 0 = 1 - ρ π n = ( 1 - ρ ) ρ n
  • N t is approximately geometric ( 1 - ρ )

For s > 1 ,

  • ρ = E [ S ] / s E [ A ] = λ / s μ
  • π n = π 0 ( s ρ ) n / n ! = π 0 ( λ / μ ) n / n ! 0 n s π 0 ( s s / s ! ) ρ n = π 0 [ ( s ρ ) s / s ! ] ρ n - s s < n

For s = 2

  • π 0 = 1 - ρ 1 + ρ = 2 μ - λ 2 μ + λ

For s = 3

  • π 0 = 1 - ρ 1 + 2 ρ + 3 2 ρ 2

For s = 4

  • π 0 = 1 - ρ 1 + 3 ρ + 8 ρ 2 + 8 3 ρ 3

For large t , with the system in equilibrium

E [ D t ] = E [ A ] E [ N t ] and E [ W t ] = E [ A ] E [ Q t ] E [ S ] E [ N t ]

For s = 1

  • E [ N t ] = ρ 1 - ρ = λ μ - λ
  • E [ Q t ] = ρ E [ N t ] P ( N t > 0 ) = ρ
  • E [ W t ] = E [ S ] E [ N t ] = λ / μ μ - λ
  • D t is approximately exponential ( μ - λ )

For s > 1

  • C = P ( W t > 0 ) = π 0 ( s ρ ) s s ! ( 1 - ρ ) = E [ Q t ] 1 - ρ ρ = s μ ( 1 - ρ ) E [ W t ]
  • P ( W t > v ) = C e - ( μ s - λ ) v v 0
  • P ( D t > v ) = e - μ v 1 + C μ 1 - e - [ μ ( s - 1 ) - λ ] v μ ( s - 1 ) - λ for λ μ ( s - 1 )
  • P ( D t > v ) = e - μ v [ 1 + μ C v ] for λ = μ ( s - 1 )
  • E [ Q t ] = π 0 ( s ρ ) s s ! ρ ( 1 - ρ ) 2
  • E [ N t ] E [ Q t ] + λ μ = E [ Q t ] + s ρ

Matlab calculations for single server queue (in file queue1.m)

L = input('Enter lambda '); % Type desired value, no extra space M = input('Enter mu '); % Type desired value, no extra spacea = [' lambda mu'];b = [L M];disp(a) disp(b)r = L/M; % RhoEN = r/(1 - r); % E[N]EQ = r*EN; % E[Q]EW = EQ/L; % E[W]ED = EN/L; % E[D]A = [' rho EN EQ EW ED']; % Identifies entries in B B = [r EN EQ EW ED]; disp(A)disp(B)v = input('Enter row matrix of values v '); % Type matrix of desired valuesPD = exp(-M*(1 - r)*v); % Calculates P(Dt>v)S = [' v P(D>v)'];s = [v; PD]';disp(S) disp(s)
queue1Enter lambda 0.1 Enter mu 0.2lambda mu 0.1000 0.2000rho EN EQ EW ED0.5000 1.0000 0.5000 5.0000 10.0000Enter row matrix of values v [8 16 24]v P(D>v) 8.0000 0.449316.0000 0.2019 24.0000 0.0907

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
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
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
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 ?
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
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Source:  OpenStax, Topics in applied probability. OpenStax CNX. Sep 04, 2009 Download for free at http://cnx.org/content/col10964/1.2
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