On Markovian Queueing Model as Birth-Death - Global Journals

21 Pages·2014·0.8 MB·English

by  Okoro Otonritse Joshua

Global Journal of Science Frontier Research
Mathematics and Decision Sciences
Volume 13 Issue 11 Version 1.0 Year 2013
Type : Double Blind Peer Reviewed International Research Journal
Publisher: Global Journals Inc. (USA)
Online ISSN: 2249-4626 & Print ISSN: 0975-5896

On Markovian Queueing Model as Birth-Death Process
By Okoro Otonritse Joshua

Landmark University, Nigeria
Abstract- Markovian queueing model has so many application in real life situations. Places where
Markovian queueing model can be applied include, Supermarket, Production system, Post office,
data communication, parking place, assembly of printed circuit boards, call center of an
insurance company, main frame computer, toll booths, traffic lights, e.t.c. Birth-death process
has being markovian foundation on queueing models. This article is an eye opener to novice
researchers, since it explore Markovian queueing model in real life situation. The fundamental of
Markovian Queueing model as birth and death process is hereby reviewed in this article, with
fundamental results applications in M / M / 1, M / M / S, M / M / 1/ K , and M / M / s / K. Here we
reexamined; Average Number of Customers and average number of time in the system, waiting
in the queue, in service respectfully. These summaries of these results are also tabulated.
Keywords: markovian properties, random process, poisson and exponential probability functions,
sum to infinity of a G.P.
GJSFR-F Classification : MSC 2010: 60K25, 65C10

On Markovian Queueing Model as Birth-Death Process

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© 2013. Okoro Otonritse Joshua. This is a research/review paper, distributed under the terms of the Creative Commons
Attribution-Noncommercial 3.0 Unported License http://creativecommons.org/licenses/by-nc/3.0/), permitting all non commercial
use, distribution, and reproduction in any medium, provided the original work is properly cited.
On Markovian Queueing Model as Birth-
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Death Process 130
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Okoro Otonritse Joshua Y
21
Abstract-Markovian queueing model has so many application in real life situations. Places where Markovian queueing
model can b e applied include, Supermarket, Production system, Post office, data communication, parking place, I
n
o
assembly of printed circuit boards, call center of an insurance company, main frame computer, toll booths, traffic lights, si
r
e.t.c. Birth-death process has being markovian foundation on queueing models. This article is an eye opener to novice e
V

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/ M / S,M / M / 1/ K, and M / M / s/K. Here we reexamined; Average Number of Customers and average number of e
u
time in the system, waiting in the queue, in service respectfully. These summaries of these results are also tabulated. s
s
Kinefiynwityo r odfs a: Gm.aPr.kov ian p roperties, random process, poisson and exponential probability functio ns, sum to XIII I

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I. introduction Vol

This article is a review on Markovian queuing model. The general expression for an )) F
explicit markovian queueing model by definition is given as the first is a
h
Poisson rate of arrival with an exponential time distribution and the second rc
a
represent the exponenti al service time. The other dots rep resent other attributes ese
R

similar togeneral queueing model. r
The need for queueing models cannot be ove remphasize bec ause in any service ntie
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r
station, the owner may be interested to know when to increase service points or F
e
number of queues, putting cost into consideration, I n a bank, or a selling out feet, how nc
e
long will one have to wait, and how can we decompose the waiting time during rush ci
S
period. A production manager will want to know the lead time production for an order of
o r for the production in mounting vertical components o n printed circuit boards, how al
n
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c an this lead time be reduc ed and what will be the effect in the production system also ou
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w hen order are prioritized. The information or computer technologist will want to bal
o
estimate th e number of cell delay at the switches, thefraction of cell lost, and thesize of Gl
t he buffer that will be good e nough to accommodate more cells. In air and sea port or

any other out feet, it is important to maximize the available parking space. Managers of

Call centers will want to minimize the waiting time of customers, by increasing call
c enters, operat ors, pooling teams for better efficiency and also traffic light regulators

Auth or: Industrial Mathematics, P hy sica l Science Department C ollege of Science and Engineering Landmark University, Omu-Aran,
Kwara, Nigeria
©2013 Global Journals Inc. (US)
On Markovian Queueing Model as Birth-Death Process
and to llbooth managers wi ll need to give acceptable waiting time and acceptable
amount to pay to motorist respectively. There may be need for Server managers to
increase efficiency and capacity of their servers in order to handle more transactions.

All these and more can be modeled using Markov queueing processing. Ivo Adan and

Jacques in 2002 givesome application of queue model.
Birth and death process has been regarded as an important subclass of Markov

Chains and is frequently used to model growth of biological population, Zhong Li 2013. N
otes
3 He also compute the expected extinction time of birth-death chain. In 2010 on the
1
0 application queueing theory to epidemic model Carlos M. H. and others give an
2

r expression that relates basic reproductive number, and the server utilization, . also
a
e
Y
they derive new approximations to quasistationary distribution (QSD) of SIS
22 (Susceptible- Infected- Susceptible) and SEIS (Susceptible- Latent- Infected-
Susceptible) stochastic epidemic models. In their work they considered all individual in
I
n
o a close population to be server of which this individual may either be busy (infected) or
si
r
e idle (susceptible). Research work on epidemiology continuous markov chain in
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queueing model is just too few. In 1971 phase generalization of the typical
XI

queueing model, were considered, where the queueing-type birth-and-death process is

e
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s defined on a continuous-time n-state Markov chain. It was conclude that the phase
s
I
XIII generalization of the steady-state queue will not yield, in general closed-form
solution. Hence there will be need to employ numerical method to solve any specific
e
m
case. Some applications to classical birth-death Markov process are given by Carlos M.
u
ol
V and Carlos C. 1999. John Willey in 2006 and son give a thorough treatment of queue

) F system and queueing network; among other method used, continuous Markov chain
) was employed. Forrest and Marc, 2011 used the continuous-time Markov chain that
h
c
r counts particles in a system over a time as a birth and death proceces to obtain
a
e
s
e expressions for Laplace transforms of transition probabilities in a general birth-death
R
r process with arbitrary birth and death rates and make explicit important derivation
e
ti
on connecting transition probabilities and continued fraction. Markovian model from
r
F
Markov chain where used in application and examples to illustrate key points. Solution
e
c
n techniques of Markovaregeneration processes where investigated.
e
Sci It surprising to note that, no so much research has been done using
of queuetheory and model inEpidemiologyanalysis which has to do with study of disease
nal origin and spread pattern of disease development. In this article our focus is on
r
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o Markovian queueing model as a birth-death process with emphasis on epidemiological
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al analysis.
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II. definitions from queue model
a) Memoryless property of the exponential distribution
Memoryless means that the probability of time of occurrence of the event no matter
how long since the last event occur is the same. That is,
in real world situation this not always true. In most cases it is applicable to
©2013 Global Journals Inc (US)
On Markovian Queueing Model as Birth-Death Process
phenomena that follow random variable and random processes. For example, the longer
a real traffic light has been red, the greater the probability that it will turn green in the
next, say, 10 seconds, this situation is not a random process. If the probability of a traffic
light turning green in the next 10 seconds does not change independent of how long it
has been red, then the distributionof the red light is memoryless. Only two distributions
are memoryless - the exponential (continuous) and geometric (discrete). Here is the
N memoryless proof for the exponential distribution…
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[ ] [ ] 130
r 2
r r
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( ) ( ) ( )
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. This memoryless property state that XI

b) The Little’s Formula e
u
s
s
Assume that entering customers are required to pay an entrance fee (according to some I
rule) to the system. Then we have Average rate at which the system earns XIII

e
average amount an entering customer pays where is the average arrival rate of m

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entering customers ol
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lim and denotes the number of customer arrivals by time .( see [1] )) F
h
and [2]) c
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c) Birth-Death Process of Markov Chain
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The birth-Death process is a case of Markov time continuous process. The current size ti
n
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of the population represent the state. For a birth-death Markov time continuous process r
F
the movement from one state to another; known as transition is limited to birth and e
c
n
death. Let represent each state such that the state can move from to by birth cie
S
and by death, we assume that the movement from one state to another is
of
independent from each other. Let and for … represent birth and death al
n
process respectively. We define pure death process as such that and pure birth ur
o
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process as such that for all . the probability transition from state to and
al
to , is [ ] and [ ], respectively where ob
Gl

[ ] and [ ] are the probability of the time until a birth is
less than the time until a death andprobability of the time until a birth is less
than the time until a death respectively. The process remain in state with
exponential distribution . For a death to occur there must be a birth, for there to

be any first noticeable change in the system, the process must move from state to state
which implies one birth and no death, of which its probability is given by
©2013 Global Journals Inc. (US)
On Markovian Queueing Model as Birth-Death Process

∑

( ) N
otes
3
1
20

r
a
Ye The probability for moving from state to is given by

. The probability of having any other moves other
24
than this two is non-zero instead is given by ,
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n for …. This also implies that
o
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er . Generally we can represent
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the birth and death process by
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e {
m
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Vol This can be

)
F
) { the Kronecker’s delta
h
c
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es
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er {
nti
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r
F
e
c
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cie Then the matrix is the infinitesimal generator of the process define by [ ],
S
of where are called transition rate.
al
n
r
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al ‖ ‖
b
o
Gl

‖ ‖

Note that since the process remain in the same state in zero step with

probability one and move to another state in zero step with probability one. We have
©2013 Global Journals Inc (US)
On Markovian Queueing Model as Birth-Death Process

hence differentiating term by term and setting

∑ this implies that

otherwise

Also∑
N
otes
d) Probability Transition of Birth-Death Process and Differential Equation from 130
Kolmogorov 2
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Kolmogorov backward differential equationdescribe the transition probabilities in their a
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dependence on the initial point
25
Basically
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∑ ( ) e
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XI

e
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( ) ( ) ( ) ∑ ( ) s
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The last summation is for . m
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)) F
( ) ( )

h
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∑ ( ) ea
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But ∑ ( ) ∑ ( ) er
ti
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( )
e
c
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of
hence we have al
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Jo
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( ) Glo
and so;

©2013 Global Journals Inc. (US)
On Markovian Queueing Model as Birth-Death Process
also we have

,
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3
1
0 we then derive the differential equation knowing there is no birth without death that
2

ar
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26
I
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e and again
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XI

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ss we know that
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XIII
∑ ( )
e
m
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)
F
) Differentiating with respect to we have

h
c
r
a
Rese ∑ ( )
r
e
ti
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o Setting gives
r
F
e
nc ∑ ( )
e
ci
S
of
al ∑ ( )
n
r
u
o
J
bal Therefore
o
Gl

While the forward Kolmogorov differential equation describes the probability
distribution of a state in time keeping the initial point fixed, decomposing the interval
into and
©2013 Global Journals Inc (US)
On Markovian Queueing Model as Birth-Death Process

∑ ( )

( ) ( ) ( ) ∑ ( )

N The last summation is for .
otes
( ) 130
2
r
a
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Similar apply here as in Kolmogorov back differential equation Y
27
( )

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With the same initial condition XI

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u
We know that s
s
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XIII
∑ ( )
e
m
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ol
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by Chapman-Kolmogorov

Differentiating with respect to we have )) F

h
c
r
a
∑ ( ) se
Re
r
e
ti
n
Setting gives o
r
F
∑ ( ) e
nc
e
ci
S
of
∑
al
n
r
u
o
J
therefore al
b
o
Gl
For
e
Where e ∑ ∑

©2013 Global Journals Inc. (US)
On Markovian Queueing Model as Birth-Death Process
Also changes or number of event over a period of time of let for a Poisson pure
birth process with the infinitesimal transition rate

{ follows a Bernoulli trial performed times

with the probability of success hence we have N
otes
3
1
0 ( )
2 {

r
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28 ( ) ( )

I
n
o
si From the formula
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XI e ∑ ∑

e
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s
s
XIII I

e
m
olu
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∑ ∑ ( )
)
F
)

rch
ea ∑
s
Re ( )

r
e
nti
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Fr ∑
( )
e
c
n
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ci
S

of ∑
al ( )
rn
u
o
J
al
b ∑
Glo

But hence cannot start from zero, but from and letting
we have

∑

©2013 Global Journals Inc (US)
On Markovian Queueing Model as Birth-Death Process

∑

Which is Poisson process
e) The law of Rare Events
N
This law holds when the probability of success occurrence from large number of
otes
independent Bernoulli trails is small and constant from one occurrence to another. Let 130
follows the binomial distribution, such that is the total number of success in 2

r
trails for … . a
e
Y
29
( )

I
n
( ) ( ) … … ersioV

XI
Multiplying and dividing the right-hand side by , we have

e
u
s
s
I
( ) ( )( )…( ) ( ) XIII
e
m
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ol
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If we let in such a way that remains constant, then
)) F

h
( )( )…( )→ c
r
a
e
s
e
R
r
e
( ) ( ) ( ) → ti
n
o
r
F
e
Where we need the fact that c
n
e
ci
S

lim ( ) of
al
n
r
u
Hence, in the limit as with (and as Jo
al
b
o
( ) → Gl

Thus, in the case of large and small
( )

©2013 Global Journals Inc. (US)