This document analyzes a single server queueing system with fixed batch service under multiple vacations with loss and feedback. Customers arrive according to a Poisson process and service times are exponential. The server provides batch service to k customers at a time. After service, batches may rejoin the queue with probability q (feedback) or leave. If fewer than k customers are present, the server takes a vacation. Customers may join the queue upon arrival with probability p or leave (loss). The generating functions for number of customers in the queue during busy and vacation periods are derived. Closed form solutions are obtained for the steady state probabilities and mean number of customers. Numerical studies are conducted to analyze the impact of parameters on mean and variance
HEATED WIND PARTICLE’S BEHAVIOURAL STUDY BY THE CONTINUOUS WAVELET TRANSFORM ...cscpconf
Nowadays Continuous Wavelet Transform (CWT) as well as Fractal analysis is generally used for the Signal and Image processing application purpose. Our current work extends the field of application in case of CWT as well as Fractal analysis by applying it in case of the agitated wind particle’s behavioral study. In this current work in case of the agitated wind particle, we have mathematically showed that the wind particle’s movement exhibits the “Uncorrelated” characteristics during the convectional flow of it. It is also demonstrated here by the Continuous Wavelet Transform (CWT) as well as the Fractal analysis with matlab 7.12 version
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By ranking the octagonal fuzzy numbers it is possible to compare them and using this we convert the fuzzy
valued maximal flow algorithm to a crisp valued algorithm . It is proved that a better solution is obtained when
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New approach for wolfe’s modified simplex method to solve quadratic programmi...eSAT Journals
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In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem (NLPP). In linear programming models, the characteristic assumption is the linearity of the objective function and constraints. Although this assumption holds in numerous practical situations, yet we come across many situations where the objective function and some or all of the constraints are non-linear functions. The non-linearity of the functions makes the solution of the problem much more involved as compared to LPPs and there is no single algorithm like the simplex method, which can be employed to solve efficiently all NPPs.
Keywords: Quadratic programming problem, New approach, Modified simplex method, and Optimal solution.
HEATED WIND PARTICLE’S BEHAVIOURAL STUDY BY THE CONTINUOUS WAVELET TRANSFORM ...cscpconf
Nowadays Continuous Wavelet Transform (CWT) as well as Fractal analysis is generally used for the Signal and Image processing application purpose. Our current work extends the field of application in case of CWT as well as Fractal analysis by applying it in case of the agitated wind particle’s behavioral study. In this current work in case of the agitated wind particle, we have mathematically showed that the wind particle’s movement exhibits the “Uncorrelated” characteristics during the convectional flow of it. It is also demonstrated here by the Continuous Wavelet Transform (CWT) as well as the Fractal analysis with matlab 7.12 version
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Solvability of Fractionl q -Difference Equations of Order 2 3 Involving ...journal ijrtem
In this paper, we study the existence of solutions for non-linear fractional q-difference
equations of order
2 3
involving the p-Laplacian operator with various boundary value conditions. By
using the Banach contraction mapping principle, we prove that, under certain conditions, the suggested
non-linear fractional boundary value problem involving the p-Laplacian operator has a unique solution. Finally,
we illustrate our results with some examples.
Solving Fuzzy Maximal Flow Problem Using Octagonal Fuzzy NumberIJERA Editor
In this paper a general fuzzy maximal flow problem is discussed . A crisp maximal flow problem can be solved
in two methods : linear programming modeling and maximal flow algorithm . Here I tried to fuzzify the
maximal flow algorithm using octagonal fuzzy numbers introduced by S.U Malini and Felbin .C. kennedy [26].
By ranking the octagonal fuzzy numbers it is possible to compare them and using this we convert the fuzzy
valued maximal flow algorithm to a crisp valued algorithm . It is proved that a better solution is obtained when
it is solved using fuzzy octagonal number than when it is solved using trapezoidal fuzzy number . To illustrate
this a numerical example is solved and the obtained result is compared with the existing results . If there is no
uncertainty about the flow between source and sink then the proposed algorithm gives the same result as in crisp
maximal flow problems.
Robust Fuzzy Data Clustering In An Ordinal Scale Based On A Similarity MeasureIJRES Journal
This paper is devoted to processing data given in an ordinal scale. A new objective function of a
special type is introduced. A group of robust fuzzy clustering algorithms based on the similarity measure is
introduced.
New approach for wolfe’s modified simplex method to solve quadratic programmi...eSAT Journals
Abstract
In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem (NLPP). In linear programming models, the characteristic assumption is the linearity of the objective function and constraints. Although this assumption holds in numerous practical situations, yet we come across many situations where the objective function and some or all of the constraints are non-linear functions. The non-linearity of the functions makes the solution of the problem much more involved as compared to LPPs and there is no single algorithm like the simplex method, which can be employed to solve efficiently all NPPs.
Keywords: Quadratic programming problem, New approach, Modified simplex method, and Optimal solution.
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Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy
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Quantum inspired evolutionary algorithm for solving multiple travelling sales...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
In this paper we present a new iterative method for cryptography, in which we apply Laplace transform of
suitable function for encrypting the plain text and we apply corresponding inverse Laplace transform for
decryption. Finally we developed the results in the generalized form. We also obtained the corresponding
encryption algorithm for this method.
A Novel Design Architecture of Secure Communication System with Reduced-Order...ijtsrd
In this paper, a new concept about secure communication system is introduced and a novel secure communication design with reduced-order linear receiver is developed to guarantee the global exponential stability of the resulting error signals. Besides, the guaranteed exponential convergence rate of the proposed secure communication system can be correctly calculated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "A Novel Design Architecture of Secure Communication System with Reduced-Order Linear Receiver" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20212.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20212/a-novel-design-architecture-of-secure-communication-system-with-reduced-order-linear-receiver/yeong-jeu-sun
International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Supply Chain Coordination with Buy-back and Wholesale-price Contracts under R...IJERA Editor
In order to increase supply chain total profit, we design a supply chain collaboration mechanism between one
supplier and one retailer. Throughout the paper we present one supplier and one retailer set up with random
demand in a decentralized supply chain. An incentive function on the buy-back cost and wholesale price cost is
presented and scheme through the buy-back contract and wholesale-price contract has been developed to enrich
the retailer take part in the collaboration ways. Furthermore, the paper shows that, how corporation between
buy-back contract and wholesale contract could coordinate the supply chain in both a decentralized and
centralized supply chain. Lastly, we set up numerical analyses and the result shows the implied collaboration
mechanism is not permit the decentralized system to achieve the same performance as the centralized decision
however it allows both members in the supply chain gains profit sharing by setting up the contract parameters.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
Balking and Reneging in the Queuing SystemIOSR Journals
In this paper, we have discussed about a steady state solution of the ordered queuing problem with balking and reneging. Here we have taken the waiting line is of chi-square queue with Poisson balking probability which depend not only on the number of customers in the system, but also the rate of services in the system.
A Counterexample to the Forward Recursion in Fuzzy Critical Path Analysis Und...ijfls
Fuzzy logic is an alternate approach for quantifying uncertainty relating to activity duration. The fuzzy
version of the backward recursion has been shown to produce results that incorrectly amplify the level of
uncertainty. However, the fuzzy version of the forward recursion has been widely proposed as an
approach for determining the fuzzy set of critical path lengths. In this paper, the direct application of the
extension principle leads to a proposition that must be satisfied in fuzzy critical path analysis. Using a
counterexample it is demonstrated that the fuzzy forward recursion when discrete fuzzy sets are used to
represent activity durations produces results that are not consistent with the theory presented. The
problem is shown to be the application of the fuzzy maximum. Several methods presented in the literature
are described and shown to provide results that are consistent with the extension principle.
Fast Algorithm for Computing the Discrete Hartley Transform of Type-IIijeei-iaes
The generalized discrete Hartley transforms (GDHTs) have proved to be an efficient alternative to the generalized discrete Fourier transforms (GDFTs) for real-valued data applications. In this paper, the development of direct computation of radix-2 decimation-in-time (DIT) algorithm for the fast calculation of the GDHT of type-II (DHT-II) is presented. The mathematical analysis and the implementation of the developed algorithm are derived, showing that this algorithm possesses a regular structure and can be implemented in-place for efficient memory utilization.The performance of the proposed algorithm is analyzed and the computational complexity is calculated for different transform lengths. A comparison between this algorithm and existing DHT-II algorithms shows that it can be considered as a good compromise between the structural and computational complexities.
Quantum inspired evolutionary algorithm for solving multiple travelling sales...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
In this paper we present a new iterative method for cryptography, in which we apply Laplace transform of
suitable function for encrypting the plain text and we apply corresponding inverse Laplace transform for
decryption. Finally we developed the results in the generalized form. We also obtained the corresponding
encryption algorithm for this method.
A Novel Design Architecture of Secure Communication System with Reduced-Order...ijtsrd
In this paper, a new concept about secure communication system is introduced and a novel secure communication design with reduced-order linear receiver is developed to guarantee the global exponential stability of the resulting error signals. Besides, the guaranteed exponential convergence rate of the proposed secure communication system can be correctly calculated. Finally, some numerical simulations are given to demonstrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "A Novel Design Architecture of Secure Communication System with Reduced-Order Linear Receiver" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-1 , December 2018, URL: http://www.ijtsrd.com/papers/ijtsrd20212.pdf
http://www.ijtsrd.com/engineering/electrical-engineering/20212/a-novel-design-architecture-of-secure-communication-system-with-reduced-order-linear-receiver/yeong-jeu-sun
International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Supply Chain Coordination with Buy-back and Wholesale-price Contracts under R...IJERA Editor
In order to increase supply chain total profit, we design a supply chain collaboration mechanism between one
supplier and one retailer. Throughout the paper we present one supplier and one retailer set up with random
demand in a decentralized supply chain. An incentive function on the buy-back cost and wholesale price cost is
presented and scheme through the buy-back contract and wholesale-price contract has been developed to enrich
the retailer take part in the collaboration ways. Furthermore, the paper shows that, how corporation between
buy-back contract and wholesale contract could coordinate the supply chain in both a decentralized and
centralized supply chain. Lastly, we set up numerical analyses and the result shows the implied collaboration
mechanism is not permit the decentralized system to achieve the same performance as the centralized decision
however it allows both members in the supply chain gains profit sharing by setting up the contract parameters.
These are my note in the class of probabilistic analysis for the "average case" input. The look at:
1.- The use of the indicator function
2.- Enforcing the "Uniform Assumption"
At the end, we look at the application of the insertion sort average case.
Here is the first set of notes for the first class in Analysis of Algorithm. I added a dedicatory for my dear Fabi... she has showed me what real idealism is....
Balking and Reneging in the Queuing SystemIOSR Journals
In this paper, we have discussed about a steady state solution of the ordered queuing problem with balking and reneging. Here we have taken the waiting line is of chi-square queue with Poisson balking probability which depend not only on the number of customers in the system, but also the rate of services in the system.
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In this paper, we study aFM / FM / 1 queue with single working vacation for this fuzzy queueing model. We obtain some system characteristic such as the number of customer in the system in steady state, the virtual time of a customer in the system, the server is in idle period, the server is in regular busy period. Finally, numerical results for all the performance measure are presented to show the effects of system parameters.
Queue with Breakdowns and Interrupted Repairstheijes
This paper considers a queue, consisting of a Poisson input stream and a server. The server is subject to breakdowns. The times to failure of the server follows exponential distribution. The failed server requires repair at a facility, which has an unreliable repair crew. The repair times of the failed server follows exponential distribution, but the repair crew also subjects to breakdown when it is repairing. The times to failure of the repair crew is also assumed to be exponentially distributed. This paper obtains the steady-state performance of the queuewith server breakdowns and interrupted repairs.
A MULTIPLE – CHANNEL QUEUING MODELS ON FUZZY ENVIRONMENTIAEME Publication
In this paper, we investigated a queuing model of fuzzy environment-based a multiple channel queuing model (M/M/C) ( /FCFS) and study its performance under realistic conditions. It applies a nonagonal fuzzy number to analyse the relevant performance of a multiple channel queuing model (M/M/C) ( /FCFS). Based on the sub interval average ranking method for nonagonal fuzzy number, we convert fuzzy number to crisp one. Numerical results reveal that the efficiency of this method. Intuitively, the fuzzy environment adapts well to a multiple channel queuing models (M/M/C) ( /FCFS) are very well.
Mx/G(a,b)/1 With Modified Vacation, Variant Arrival Rate With Restricted Admi...IJRES Journal
In this paper, a bulk arrival general bulk service queuing system with modified M-vacation policy, variant arrival rate under a restricted admissibility policy of arriving batches and close down time is considered. During the server is in non- vacation, the arrivals are admitted with probability with ' α ' whereas, with probability 'β' they are admitted when the server is in vacation. The server starts the service only if at least ‘a’ customers are waiting in the queue, and renders the service according to the general bulk service rule with minimum of ‘a’ customers and maximum of ‘b’ customers. At the completion of service, if the number of waiting customers in the queue is less than ‘𝑎’ then the server performs closedown work , then the server will avail of multiple vacations till the queue length reaches a consecutively avail of M number of vacations, After completing the Mth vacation, if the queue length is still less than a then the server remains idle till it reaches a. The server starts the service only if the queue length b ≥ a. It is considered that the variant arrival rate dependent on the state of the server.
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Similar to Analysis of single server queueing system with batch service (20)
Analysis of single server queueing system with batch service
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
79
Analysis Of Single Server Queueing System With Batch Service
Under Multiple Vacations With Loss And Feedback
G. Ayyappan1
G. Devipriya2*
A. Muthu Ganapathi Subramanian3
1. Associate Professor, Pondicherry Engineering College, Pondicherry, India
2. Assistant Professor, Sri Ganesh College of Engineering & Technology ,Pondicherry , India
3. Associate Professor, Kanchi Mamunivar Centre for Post Graduate Studies, Pondicherry, India
*E-mail of the corresponding author: devimou@yahoo.com
Abstract
Consider a single server queueing system with foxed batch service under multiple vacations with loss and
feedback in which the arrival rate λ follows a Poisson process and the service time follows an exponential
distribution with parameter μ. Assume that the system initially contain k customers when the server enters the
system and starts the service in batch. The concept of feedback is incorporated in this model (i.e) after
completion of the service, if this batch of customers dissatisfied then this batch may join the queue with
probability q and with probability (1-q) leaves the system. This q is called a feedback probability. After
completion of the service if he finds more than k customers in the queue then the first k customers will be taken
for service and service will be given as a batch of size k and if he finds less than k customers in the queue then
he leaves for a multiple vacation of exponential length α. The impatient behaviour of customer is also studied in
this model (i.e) the arriving customer may join the queue with probability p when the server is busy or in
vacation. This probability p is called loss probability. This model is completely solved by constructing the
generating function and Rouche’s theorem is applied and we have derived the closed form solutions for
probability of number of customers in the queue during the server busy and in vacation. Further we are providing
the analytical solution for mean number of customers and variance of the system. Numerical studies have been
done for analysis of mean and variance for various values of λ, µ, α, p, q and k and also various particular cases
of this model have been discussed.
Keywords : Single Server , Batch Service, Loss and Feedback, Multiple vacations, Steady state distribution.
1. Introduction
Queueing systems where services are offered in batches instead of personalized service of one at a time.
A service may be of a fixed size (say) k, such a system is called fixed batch size. Bailey (1954), was the first to
consider bulk service. Such models find applications in several situations, such as transportation. Mass transit
vehicles are natural batch servers. There are a great number of numerical and approximations methods are
available, in this paper we will place more emphasis on the solutions by Probability Generating Function.
Formulation of queues with feedback mechanism was first introduced by Takacs. The concepts loss and
feedback are introduced for customers only. After completion of the service, if the customer dissatisfied then he
may join the queue with probability q and with probability (1-q) he leaves the system. This is called feedback in
queueing theory. If the server is busy at the time of the arrival of customer, then due to impatient behaviour of
the customer, customer may or may not join the queue. This is called loss in queueing theory. We assume that p
is the probability that the customer joins the queue and (1-p) is the probability that he leaves the system without
getting service (due to impatient).
Feedback queues play a vital role in the areas of Computer networks, Production systems subject to
rework, Hospital management, Super markets and Banking business etc. Takacs (1963), introduced the concept
of feedback queues. Disney, McNickle and Simmon (1980), D’Avignon and Disney (1976), Krishnakumar
(2002),Thangaraj and Vanitha . (2009,2010). Ayyappan et al., (2010), Farahmand .K and T. Li(2009) are a few
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
80
to be mentioned for their contribution.
In this paper we are going to concentrate on a very special batch service queue called the fixed size batch
service queue under multiple vacations with loss and feedback. The model under consideration is described in
Section 2. In Section 3 we analyze the model by deriving the system steady state equations and probability
generating functions. Using these generating functions, steady state probabilities are obtained in Section 4. The
operating characteristics are obtained in Section 5. A numerical study is carried out in Section 6 to test the effect
of the system performance measure discussed in Section 5. We are providing the analytical solution for mean
number of customers and variance of the system. Numerical studies have been done for analysis of mean and
variance for various values of λ, µ, α, p, q and k and also various particular cases of this model have been
discussed.
2. DESCRIBE OF THE MODEL
Consider a single server queueing system with foxed batch service under multiple vacations with loss
and feedback in which the arrival rate λ follows a Poisson process and the service time follows an exponential
distribution with parameter μ. Assume that the system initially contain k customers when the server enters the
system and starts the service in batch. The concept of feedback is incorporated in this model (i.e) after
completion of the service, if this batch of customers dissatisfied then this batch may join the queue with
probability q and with probability (1-q) leaves the system. This q is called a feedback probability. After
completion of the service if he finds more than k customers in the queue then the first k customers will be taken
for service and service will be given as a batch of size k and if he finds less than k customers in the queue then
he leaves for a multiple vacation of exponential length α. The impatient behaviour of customer is also studied in
this model (i.e) the arriving customer may join the queue with probability p when the server is busy or in
vacation. This probability p is called loss probability.
Let < N(t),C(t) > be a random process where N(t) be the random variable which represents the number of
customers in queue at time t and C(t) be the random variable which represents the server status (busy/vacation)
at time t.
We define
Pn,1(t) - probability that the server is in busy if there are n customers in the queue at time t.
Pn,2(t) - probability that the server is in vacation if there are n customers in the queue at time t
The Chapman- Kolmogorov equations are
'
0,1 0,1 ,1 ,2( ) ( (1 )) ( ) (1 ) ( ) + P (t)k kP t p q P t q P t (1)
'
,1 ,1 1,1 ,1 ,2( ) ( (1 )) ( ) ( ) (1 ) ( ) + P (t) for n = 1,2,3,...n n n n k n kP t p q P t pP t q P t (2)
'
0,2 0,2 0,1( ) ( ) (1 ) ( )P t pP t q P t (3)
'
,2 ,2 1,2 ,1( ) ( ) ( ) (1 ) ( ) for n = 1,2,3,...,k-1n n n nP t pP t pP t q P t (4)
'
,2 ,2 1,2( ) ( ) ( ) ( ) for n kn n nP t p P t pP t (5)
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.11, 2014
81
3. EVALUATION OF STEADY STATE PROBABILITIES
In this section we are finding the closed form solutions for number of customers in the queue when the server is
busy or in vacation using Generating function technique.
When steady state prevails, the equations (1) to (5) becomes
0,1 ,1 ,2( (1 )) (1 ) + Pk kp q P q P (6)
,1 1,1 ,1 ,2( (1 )) (1 ) + P for n = 1,2,3,....n n n k n kp q P pP q P (7)
0,2 0,1(1 )pP q P (8)
,2 1,2 ,1(1 ) for n = 1,2,3,...,k-1n n npP pP q P (9)
,2 1,2( ) for n kn np P pP (10)
Generating functions for the number of customers in the queue when the server is busy or in the vacation are
defined as
,1
0
( ) n
n
n
G z P z
and ,2
0
( ) n
n
n
H z P z
Multiply the equation (6) with 1 and (7) with zn
on both sides and summing over n = 0 to ∞, we get
1 1
1
,1 ,2
0 0
( )[ ( (1 )) ] ( ) (1 )
k k
k k n n
n n
n n
G z pz p q z H z q P z P z
(11)
Adding equation (8), (9) and (10) after multiply with 1, zn
(n = 1, 2, 3, . . ., k-1) and zn
( n = k, k+1, k+2, . . . )
respectively, we get
1 1
,1 ,2
0 0
( )[ (1 )] (1 )
k k
n n
n n
n n
H z p z q P z P z
(12)
From the equations (11) and (12), we get
1
( ) (1 )
( )
( (1 )) (1 )k k
H z p z
G z
pz p q z q
(13)
From equation (12), we get
1 1
,1 ,2
0 0
(1 )
( )
(1 )
k k
n n
n n
n n
q P z P z
H z
p z
(14)
Equation (14) represents the probability generating function for number of customers in the queue when the
server is in vacation.
From the equations (13) and (14), we get
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1 1
,1 ,2
0 0
1
( (1 )) (1 )
( )
[ ( (1 )) ][ (1 )]
k k
n n
n n
n n
k k
p z q P z P z
G z
pz p q z p z
(15)
Equation (15) represents the probability generating for number of customers in the queue when the server is
busy.
Put z = 1 in equation (13), we get
(1) (1)
(1 )
G
p
H
k q p
(16)
The normalized condition is (1) (1) 1G H (17)
Using the equation (1) in (17), we get
(1 )
(1)
(1 )
k q p
H
k q
(18)
From the equations (16) and (18) , we get
Steady state probability that the server is busy = (1)
(1 )
G
p
k q
Steady state probability that the server in vacation =
(1 )
(1)
(1 )
k q p
H
k q
The generating function G(z) has the property that it must converge inside the unit circle. We notice that the
denominator of G(z),
1
( (1 )) (1 )k k
pz p q z q
has k+1 zero's. Applying Rouche's theorem,
we notice that k zeros of this expression lies inside the circle 1z and must coincide with k zeros of
numerator of G(z) and one zero lies outside the circle 1z . Let z0 be a zero which lies outside the
circle 1z .
As G(z) converges, k zeros of numerator and denominator will be cancelled, we get
0
( )
( (1 )) ( )
A
G z
p z z z
(19)
Put z = 1 in the equation (19), we get
0
(1)
(1 )
A
G
z
(20)
Using (16) and (18) in (20), we obtain
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2
0(1 )
(1 )
p
A z
k q
(21)
From the equation (19) and (21), we get
0
0
(1 )
( )
( (1 ) ) ( )( )
zp
G z
k q p z z p pz
(22)
By applying partial fractions, we get
1
0
1
0 00 0
(1 )
( )
( (1 ) ) ( )
n n
n
n
n n
z p p z
G z z
k q p p pz p z
(23)
Comparing the coefficient of zn
on both sides of the equation (23), we get
1 1
,1
(1 )
(s - ) for n = 0,1,2,...
( (1 ) ) ( )
n n
n
r s
P r
k q p s r
(24)
where
0
1
and s =
p
r
z p
Using equation (24) in (8) and (9) , apply recursive for n = 1,2,3,. . . ,k-1, we get
,2 ,1
0
(1 ) n
n t
t
q
P P
p
for n = 0,1,2,3,. . . ,k-1 (25)
1
,2 1,2
n k
n k
p
P P
p
for n ≥ k (26)
Equations (24), (25) and (26) represent the steady state probabilities for number of customers in the queue when
the server is busy / in vacation.
4. STABILITY CONDITION
The necessary and sufficient condition for the system to the stable is
(1 )
p
k q
< 1.
5. PARTICULAR CASE
If we take p =1 and q = 0, the results coincides with the results of the model single server batch service under
multiple vacation.
6. SYSTEM PERFORMANCE MEASURES
In this section, we will list some important performance measures along with their formulas. These measures are
used to bring out the qualitative behaviour of the queueing model under study. Numerical study has been dealt in
very large scale to study the following measures
a. Probability that there are n customers in the queue when the server is busy
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1 1
,1
1
0
(1 )
(s - ) for n = 0,1,2,...
( (1 ) ) ( )
1
Where and s =
n n
n
r s
P r
k q p s r
p
r
z p
b. Probability that there are n customers in the queue when the server in vacation
,2 ,1
0
(1 ) n
n t
t
q
P P
p
for n = 0,1,2,...,k-1
1
,2 1,2
n k
n k
p
P P
p
for n ≥ k
c. Mean number of customers in the queue
,1 ,2
0
( )q n n
n
L n P P
d.Variance of the number of customers in the queue
2 2 2
,1 ,2
0 0
( ) ( )n n q
n n
V x n P n P L
.
7. NUMERICAL STUDIES
The values of parameters λ, µ, α, p, q and k are chosen so that they satisfy the stability condition discussed in
section 4. The system performance measures of this model have been done and expressed in the form of tables
for various values λ, µ, α and k.
Tables 3, 6 and 9 show the impact of arrival rate λ and k over Mean number of customers in the queue.
Tables 10 show the impact of p and k over Mean number of customers in the queue.
Further we infer the following
Mean number of customers in the queue increases as arrival rate λ increases.
P1 and P2 are probabilities of server in busy and in vacation
Tables 1, 4and 7 show the steady probabilities distribution when server is busy for various values of λ and
different batch size k.
Tables 2, 5 and 8 show the steady state probabilities distribution when the server is in vacation for various values
of λ and different batch size k.
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Table 9: Average number of customers in the queue and Variance for various values
of λ, p = 0.8, q =0 .2 and µ = 10, α = 5 and batch size is K = 6
Λ µ Α P1 P2 Mean Variance
1 10 5 0.0167 0.9833 2.6600 3.1023
1.5 10 5 0.0250 0.9750 2.7400 3.2143
2 10 5 0.0333 0.9667 2.8200 3.3391
2.5 10 5 0.0417 0.9583 2.9000 3.4767
3 10 5 0.0500 0.9500 2.9800 3.6271
3.5 10 5 0.0583 0.9417 3.0601 3.7904
4 10 5 0.0667 0.9333 3.1402 3.9667
4.5 10 5 0.0750 0.9250 3.2204 4.1558
5 10 5 0.0833 0.9167 3.3007 4.3580
5.5 10 5 0.0917 0.9083 3.3811 4.5734
6 10 5 0.1000 0.9000 3.4617 4.8020
6.5 10 5 0.1083 0.8917 3.5425 5.0440
7 10 5 0.1167 0.8833 3.6235 5.2995
7.5 10 5 0.1250 0.8750 3.7048 5.5686
8 10 5 0.1333 0.8667 3.7864 5.8517
8.5 10 5 0.1417 0.8583 3.8683 6.1488
9 10 5 0.1500 0.8500 3.9507 6.4603
9.5 10 5 0.1583 0.8417 4.0334 6.7862
Table 10: Average number of customers in the queue and Variance for various values
of p, λ = 5, q = 0.2 and µ = 10, α = 5 and batch size is K = 2
p µ λ P1 P2 Mean Variance
0.1 10 5 0.0313 0.9688 0.6002 0.3602
0.2 10 5 0.0625 0.9375 0.7016 0.4920
0.3 10 5 0.0938 0.9063 0.8050 0.6469
0.4 10 5 0.1250 0.8750 0.9112 0.8269
0.5 10 5 0.1563 0.8438 1.0208 1.0343
0.6 10 5 0.1875 0.8125 1.1346 1.2717
0.7 10 5 0.2188 0.7813 1.2532 1.5425
0.8 10 5 0.2500 0.7500 1.3774 1.8507
0.9 10 5 0.2813 0.7188 1.5080 2.2012
1 10 5 0.3125 0.6875 1.6462 2.6004
8. Conclusion:
The Numerical studies show the changes in the system due to impact of batch size, vacation rate , arrival
rate . The mean number of customers in the system increases as batch size and arrival rate increase . The mean
number of customers in the system decreases as vacation rate increases. Various special cases have been
discussed, which are particular cases of this research work. This research work can be extended further by
introducing various concepts like bulk arrival , interruption etc.
References
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