International Journal of Advancements in Research & Technology, Volume 4, Issue 2, February -2015
ISSN 2278-7763
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AN ANALYSIS OF BULK SERVICE QUEUEING MODEL WITH SERVERS VARIOUS
VACATIONS
R.Sree parimala1, S.Palaniammal2,
1. Research scholar, 2.Professor& Head,
Department of Science and Humanities
1. Hindusthan Institute of technology, Coimbatore-641 032,Tamil Nadu, India.
2. Sri Krishna College of Technology,Kovaipudur,Coimbatore-641 042, Tamil Nadu, India.
E-mail: sree_rsp@rediffmail.com,splvlb@yahoo.com
Abstract
The aim of this paper is focus on M/M (a,b)/(2,1) queueing model with server‟s single and
delayed vacations. In this model it is assumed that the arrival pattern is Poisson fashion with
parameter λ and service is done in batches which are exponentially distributed with parameter µ
according to the general bulk service rule introduced by Neuts (9).The batches are served
according to FCFS discipline. The service starts only when batches of „a‟ customers are present.
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When the queue length is „a‟ but less than or equal to „b‟ then the entire queue is taken up for
service. If there are more than „b‟ customers in the queue then the server accepts first „b‟
customers. In this model the servers takes only one vacation (θ) at a time. (i.e.) on returning from
vacation the server starts serving immediately if there are „a‟ customers waiting in the queue. If
any one of the server finds (a-1) customers in the system and other server is busy or idle, server
will stay idle in the system and wait for the queue size become „a‟. If the server finds (a-2)
customers in the system and other server is busy or idle, the server switch over the system and goes
for vacation. So in this system, sever can take only one vacation between two successive service
times. Any one of the server will always retained in a system. The steady state solutions and the
system characteristics are derived and analyzed for this model. Various models studied earlier are
discussed as special cases of our model. The analytical results are numerically illustrated for
different values of the parameters and levels also.
Keywords: Single vacation, Delayed vacation, switch over state, queue size.
1. Introduction
Queueing theory was born in the early 1900s with the work of A. K. Erlang of the
Copenhagen Telephone Company, who derived several important formulas for Teletraffic
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engineering that today bear his name. The range of applications has grown to include not only
telecommunications and computer science, but also manufacturing, air traffic control, military
logistics, design of theme parks, and many other areas that involve service systems whose
demands are random Queueing models are very useful to provide basic framework for efficient
design and analysis of several practical situations including various technical systems also
predictions the behavior of system such as waiting times of customers, various vacations for
servers and so forth. Queueing systems with server vacations have also found wide applicability in
computer and communication network and several other engineering systems. Such queueing
situations may arise in many real time systems such as telecommunication, data/voice
transmission, manufacturing system, etc.
In computer communication systems, messages which are to be transmitted could consist of
a random number of packets. Vacation models are explained by their scheduling disciplines,
according to which when a service stops, a vacation starts. These predictions help us to anticipate
situations of the system and to take appropriate measures to shorten the queue. In most of the
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queueing models, service begins immediately when the customers arrives. But some of the
physical systems in which idle servers will leave the system for some other uninterrupted task
referred as vacation.
Most of the bulk service Queueing models with server vacation have been analyzed by
many authors. S.Palaniammal (11) has studied M/M(a,b)/(2,1) queueing model and derived
analytic solutions for servers repeated and single vacation and presented the steady state result in
terms of characteristic equation of a difference equation. M.I.Afthabbegam(1) has tried analytic
solutions for M/M(a,b)/1 queues, Ek/M(a,b)/1 queue with servers single and multiple vacation. The
queueing models with vacations have been studied due to their wide applications in flexible
manufacturing or computer communication systems over more than two decades. Several surveys
on server vacation models have been done by Doshi (5), Takagi.H(12) analyzed the M/G/1/N
queues with server vacation and exhaustive service., Medhi.J and Borthakur.A(8) have introduced
a general bulk service rule with two server. Also a bulk queueing model M/M(a,b,c)/2 with servers
vacation has been studied by Mishra.S.SamdPandey.N.K (9).
The Ek/M(a,b)/1 queueing system and its numerical results are analyzed by Chaudry.M.C
and Easton.G.D (4). The transient of Ek/M(a,b)/1/N derived by Anjanasolanki and
Srivastava.P.N(2). In many waiting line systems, the role of server is played by mechanical/
electronic device, such as computer, pallets, ATM, Traffic light, etc., which is subject to accidental
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waiting of customers, it may solved by the servers vacation due to batch criteria. Ke(6) studied the
control policy of the N-Policy M/G/1 queue with server vacations, startup and breakdowns, where
arrival forms a Poisson and service times are generally distributed.
The essence of queueing theory is that it takes into account the randomness of the
arrival process and the randomness of the service process In the literature described above,
customer inter-arrival times and customer service times are required to follow certain probability
distributions with fixed parameters.
The present investigation an attempt has been made to analyze the server‟s delayed and
single vacation. The study of queueing model is organized as follows. The model is described in
Section 2. Section 3 provides the formulation and notations. Steady state behavior of the system
and equation are outlined in Section 4.The steady state solutions have been obtained in Section 5.
The performance measures and mean queue length are derived in Section 6. The cost of our model
are deduced in Section 7.To validate the analytical results and to facilitate the sensitivity analysis,
we present some numerical results for system performance indices in Section 8 and some
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concluding remarks and notable features of investigation done are highlighted in Section 9.
2. Model Description
The study focused on server‟s single and delayed vacation of M/M(a,b)/(2,1) queueing
system with switch over state of server. In this model it is assumed that the arrival pattern is
according Poisson process with parameter λ and service is done in a batch which is exponentially
distributed with parameter µ.The service starts only when batches of „a‟ customers are present.
When the queue length is „a‟ but less than or equal to „b‟ then the entire queue is taken up for
service. If there are more than „b‟ customers in the queue then the server accepts first „b‟
customers. In this model the server takes only one vacation (θ) at a time which is exponentially
distributed. (i.e.) on returning from vacation the server starts serving immediately if there are „a‟
customers waiting in the queue. In this model we make the following assumptions.
(i)
If a server finds the other server is on vacation he will remain in the system, as only one
server is allowed to go on vacation at a time.
(ii)
If any one of the server finds (a-1) customers in the system and other server is busy or
idle, server will stay idle in the system and wait for the queue size become „a‟.
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If the server finds (a-2) customers in the system and other server is busy or idle, the
server switch over the system and goes for a vacation.
So in this system, sever can take only one vacation between two successive service times. Any
one of the server will always retained in a system.
3. Mathematical Formulation
The queueing system can be formulated as a continuous time parameter Markov chain with
states Pjn(n≥0, j = 0,1,2,3) and Qjn ((0 ≤ n ≤ a-2), j = 1,2) denotes the steady state probabilities,
where „n‟ represents the number of customers in the queue and „j‟ signifies the states of the server.
The states of the process
P0n – the probability that one server is idle and the other on vacation,
P1n – the probability that one server is busy and the other on vacation,
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P2n – the probability that both the servers are busy,
P3n – the probability that one server is busy and the other switchover from the system
Q1n – the probability that one server is busy and the other idle,
Q2n – the probability that both are idle in the system respectively.
we define the following limiting probabilities corresponding to different states
P0n= lim𝑛 →∞ 𝑃0𝑛 (𝑡), P1n(t) = lim𝑛 →∞ 𝑃1𝑛 (𝑡)and P2n(t) = lim𝑛→∞ 𝑃2𝑛 (𝑡) exists.
4. Steady state equations
The steady state equations satisfied by Pjn and Qjnare given by
𝜆 + 𝜇 𝑃00 = 𝜇𝑃10 + 𝜇𝑄10
(1)
𝜆 + 𝜃 𝑃0𝑛 = 𝜆𝑃0𝑛 −1 + 𝜇𝑃1𝑛 + 𝜇𝑄1𝑛 1 ≤ 𝑛 ≤ 𝑎 − 2
(2)
𝜆 + 𝜃 𝑃0𝑎−1 = 𝜆𝑃0𝑎 −2 + 𝜇𝑃1𝑎−1 + 𝜇𝑃3𝑎 −1 + 𝜇𝑄1𝑎−1
(3)
𝜆 + 𝜇 + 𝜃 𝑃10 = 𝜆𝑃0𝑎 −1 + 2 𝜇𝑃20 + 𝜇
𝑏
𝑛=𝑎
𝑃1𝑛
𝜆 + 𝜇 + 𝜃 𝑃1𝑛 = 𝜆𝑃1𝑛−1 + 2 𝜇𝑃2𝑛 + 𝜇𝑃1𝑛+𝑏 1 ≤ 𝑛 ≤ 𝑎 − 2
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(4)
(5)
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𝜆 + 𝜇 + 2𝜃 𝑃1𝑎−1 = 𝜆𝑃1𝑎−2 + 𝛼𝑃3𝑎−1 + 𝜇𝑃1𝑎−1+𝑏
(6)
𝜆 + 𝜇 + 𝜃 𝑃1𝑛 = 𝜆𝑃1𝑛−1 + 𝜇𝑃1𝑛+𝑏 𝑛 ≥ 𝑎
(7)
𝑏
𝑛=𝑎
𝜆 + 2 𝜇 𝑃20 = 𝜆𝑃3𝑎−1 + 𝜃
𝑃1𝑛 + 2𝜇
𝑏
𝑛=𝑎
𝑃2𝑛 + 𝜆 𝑄1𝑎−1
(8)
𝜆 + 2 𝜇 𝑃2𝑛 = 𝜆𝑃2𝑛−1 +𝜃 𝑃1𝑛+𝑏 + 2𝜇𝑃2𝑛 +𝑏 𝑛 ≥ 1
(9)
𝜆 + 𝜇 + 𝛼 𝑃3𝑎−1 = 2𝜇𝑃2𝑎 −1 + 𝜃𝑃1𝑎−1 𝑛 = 𝑎 − 1
(10)
𝜆 𝑄20 = 𝜃𝑃00
(11)
𝜆 𝑄2𝑛 = 𝜃𝑃0𝑛 + 𝜆𝑄2𝑛−1 1 ≤ 𝑛 ≤ 𝑎 − 1
(12)
𝜆 + 𝜇 𝑄1𝑛 = 𝜆 𝑄1𝑛−1 + 𝜃𝑃1𝑛 1 ≤ 𝑛 ≤ 𝑎 − 1
(13)
𝜆 + 𝜇 𝑄10 = 𝜃𝑃10 + 𝜆 𝑄2𝑎−1
(14)
5. Computation of steady state solutions:
Let E denote the forward shifting operator defined by E(P1n) = P1n+1. From equation ( 7)
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(𝜇 Eb+1 – (𝜆 + 𝜇 + 𝜃)E + 𝜆) 𝑃1𝑛 = 0 𝑛 ≥ 1
The characteristic equation of the above equation has only one real root inside the circle |Z| =1 by
Rouche‟s theorem when 𝜌 =
𝜆+𝜃
𝑏𝜇
is less than 1 then 𝑃1𝑛 = 𝑟0𝑛−𝑎+1 𝑃1𝑎−1 𝑛 ≥ 𝑎
(15)
from equation (9), (2𝜇 Eb+1 – (𝜆 + 2𝜇)E + 𝜆) 𝑃2𝑛 = - 𝜃 𝑃1𝑛+𝑏+1 the characteristic equation of this
equation has only one real root by Rouche‟s theorem which lies in the interval (0,1) when 𝜌 =
𝜆
2𝑏𝜇
and using equation (15), after simplification,
𝑃2𝑛 = (𝐴1 𝑟1𝑛 + 𝑘𝑟0𝑛 )𝑃1𝑎−1 𝑛 ≥ 0
where 𝐴1 is a constant and k =
(16)
−𝜃𝑟0𝑏−𝑎 +2
𝜆+2𝜃 𝑟0 −𝜆
from equation (5), substituting n = a-2, a-3, a-4,...1 and solving recursively using (15) and (16),
𝑃1𝑛 = (𝐴1 𝐵𝑛 (𝑟1 ) + k 𝐵𝑛 (𝑟0 ) + 𝑟0𝑛−𝑎 ) 𝑃1𝑎−1 1 ≤ 𝑛 ≤ 𝑎 − 2
where 𝐵𝑛 𝑥 =
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2𝜇𝑅
( 𝑥𝑛 –
𝜆(𝑥−𝑅)
𝑥 𝑎−1
𝑅
𝜆
𝑅 𝑛 ) , R = 𝜆+𝜇 +𝜃 and k =
(17)
−𝜃𝑟0𝑏 −𝑎 +2
𝜆+2𝜃 𝑟0 −𝜆
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Similarly solving equation (13) recursively using (17)
𝑄1𝑛 = ( 𝐴2 𝑟2𝑛 + 𝐴1 𝑔𝑛 (𝑟1 ) + k 𝑔𝑛 (𝑟0 ) + 𝑘0 𝑟0𝑛−𝑎 ) 𝑃1𝑎−1 ,
𝜆
2𝜇𝜃𝑅
where 𝐴2 is a constant, 𝑟2 = 𝜆+𝜇 , 𝑔𝑛 (x) = 𝜆(𝑥−𝑅)(
1 ≤𝑛 ≤𝑎−1
𝑥 𝑛 +1
+
𝑥− 𝜆
𝜆+𝜇
𝑥 𝑎−1 𝑅 𝑛
𝑅
𝜃
) and 𝑘0 =
(18)
𝜃 𝑟0
𝜆+𝜇 𝑟0 −𝜆
By adding (2) , (12) and using the equations (1), (11)
𝜇
𝑛
𝑘=0( 𝑃1𝑛
𝑃0𝑛 + 𝑄2𝑛 = 𝜆
+ 𝑄1𝑛 ), 0 ≤ 𝑛 ≤ 𝑎 − 2
from equations (17) and (18) substituting the values of 𝑃1𝑛 𝑎𝑛𝑑 𝑄1𝑛
𝜇
𝑛
𝑘=0
𝑃0𝑛 + 𝑄2𝑛 = 𝜆
( 𝐴2 𝑟2𝑛 + 𝐴1 [𝐵𝑛 (𝑟1 ) + 𝑔𝑛 (𝑟1 )] + k [𝐵𝑛 (𝑟1 ) + 𝑔𝑛 (𝑟1 )] + (1+ 𝑘0 )𝑟0𝑛−𝑎 ) 𝑃1𝑎−1
After simplification 𝐵𝑛 (𝑥) + 𝑔𝑛 (𝑥) =𝐶𝑛 𝑥 =
2𝜇 𝑥 𝑛
𝜆+𝜇 𝑥− 𝜆
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𝜇
𝑛
𝑘=0
𝑃0𝑛 + 𝑄2𝑛 = 𝜆
( 𝐴2 𝑟2𝑛 + 𝐴1 𝐶𝑛 𝑟1 + 𝑘 𝐶𝑛 𝑟0 + (1+ 𝑘0 )𝑟0𝑛−𝑎 ) 𝑃1𝑎−1
Further giving an expansion and simplifying the above equation,
𝜇
𝑃0𝑛 + 𝑄2𝑛 = 𝜆 [𝐴2
here𝐷(𝑟1 ) =
1−𝑟2𝑛 +1
1−𝑟2
2𝜇
𝜆+𝜇 𝑟1 − 𝜆
+ 𝐴1 𝐷(𝑟1 )
and F (𝑟0 ) =
1−𝑟1𝑛 +1
1−𝑟1
+ F(𝑟0 )
1
𝜆+𝜇 𝑟0 − 𝜆
1−𝑟0𝑛 +1
1−𝑟0
[2𝜇𝑘 −
𝜆
𝑟0𝑎
] 𝑃1𝑎−1
( 1-
𝑟0
𝑅
(19)
)]
The probability for one of the server is busy and the other switchover from the system is solved by
using (10)
𝑃3 𝑎−1 = [𝐴1 𝑇 𝑟1 + k 𝐴1 𝑇 𝑟0 + 𝑅1 ] 𝑃1𝑎−1
where𝑇 𝑥 =
2𝜇 𝑥 𝑎 −1
𝜆+𝜇 +𝛼
and 𝑅1 =
(20)
𝜃
𝜆+𝜇 +𝛼
Also by adding (17), (18) and using the results of equation (19), we obtain
𝑃1𝑛 + 𝑄1𝑛 = (𝐴2 𝑟2𝑛 + 𝐴1 𝐷 𝑟1 𝑟1𝑛 + F (𝑟0 ) 𝑟0𝑛 ) 𝑃1𝑎−1 0 ≤ 𝑛 ≤ 𝑎 − 1
(21)
To find the value of constants, from equation (8), using the results of𝑃3 𝑎−1 , 𝑃2𝑛 , 𝑄1𝑛
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𝑟0𝑎 −𝑟0𝑏+1
𝜆 + 2 𝜇 𝐴1 + 𝑘 = 𝜆 [ 𝐴1 𝑇 𝑟1 + k 𝑇 𝑟0 + 𝑅1 ] + 𝜃𝑟0−𝑎+1
𝑟1𝑎 −𝑟1𝑏+1
2𝜇[ 𝐴1
1−𝑟1
1
𝑟2𝑎 −1
[𝐴1 𝑆 𝑟1 + 𝑘 𝑆 𝑟0 -
here𝑆 𝑥 =
𝜆+2 𝜇
𝜆
-
2𝜇 𝑅1 𝑟1𝑎 −1
𝜃
-
𝜆+𝜇 𝑥− 𝜆
𝜃𝑟0 1−𝑟0𝑏−𝑎 +1
𝜆
(
1−𝑟0
2𝜇
𝑥 𝑎 −𝑥 𝑏 +1
𝜆
1−𝑥
+
] + 𝜆 [( 𝐴2 𝑟2𝑛 + 𝐴1 𝑔𝑛 (𝑟1 ) + k 𝑔𝑛 (𝑟0 ) + 𝑘0 𝑟0𝑛−𝑎 )]
1−𝑟0
2𝜇 𝑥 𝑎 −1
By simplifying and using 𝑔𝑎−1 (x) =
𝐴2 =
𝑟0𝑎 −𝑟0𝑏+1
+k
1−𝑟0
28
we obtain the value of constant 𝐴2 as follows
) - (𝑅1 + 𝑘0 𝑟0𝑛−1 )]
(22)
2𝜇 𝑥 𝑎 −1
-
𝜆+𝜇 +𝛼
Also to obtain the value of 𝐴1 , by adding (4) and (14),
𝐴1 =
1
𝑧(𝑟1
[𝜆{ 𝑟0 )
𝜃𝑟0 1−𝑟0𝑏−𝑎 +1
(
𝜆
1−𝑟0
𝑟0 −𝑟0𝑎
) - (𝑅1 + 𝑘0 𝑟0𝑛−1 ) } + F (𝑟0 ) [𝜇
-𝜆]+
1−𝑟0
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𝜇
2𝜇[ k + 2
𝑟1𝑎 −𝑟1
Where 𝑧(𝑟1 ) = [𝐷 𝑟1 { 𝜆 + 𝜇
1−𝑟1
𝑟0 −𝑟0𝑏−𝑎 +2
1−𝑟0
]
(23)
𝑆 𝑟
1
} - 2 𝜇 - 𝑟 𝑎 −1
]
2
Thus we obtained all the steady state probabilities in terms of 𝑃1𝑎−1 which it may now be
determined by using the normalizing condition. Hence all the probabilities are completely in terms
of the queue parameters.
To obtain the value of𝑃1𝑎−1 , by using the normalizing condition
𝑎−1
𝑛=0( 𝑃1𝑛
+ 𝑄1𝑛 +𝑃0𝑛 + 𝑄2𝑛 ) +
∞
𝑛=𝑎
𝑃1𝑛 +
∞
𝑛=0 𝑃2𝑛
+ 𝑃3 𝑎−1 = 1
(24)
Substituting the results from the equations (19), (22), (15), (16) and (20) we obtain
𝑃1−1𝑎−1 = 𝐴2 [ 𝐻(𝑟2 ) +
1−𝑟2𝑎
1−𝑟2
1
] + 𝐴1 𝐷(𝑟1 ) [𝐻(𝑟1 ) +
1
𝐴1 (1−𝑟 + 𝑇 𝑟1 ) + k (1−𝑟 + 𝑇 𝑟0 ) + 𝑅1 .
1
0
𝜇
𝑎
where H(x) = 𝜆 [ 1−𝑥 -
Copyright © 2015 SciResPub.
𝑥(1−𝑥 𝑎 )
(1−𝑥)2
1−𝑟1𝑎
1−𝑟1
] + F (𝑟0 )[𝐻(𝑟0 ) +
1−𝑟0𝑎
1−𝑟0
𝑟 +𝑘
0
] + (1−𝑟
)+
0
(25)
]
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6. Performance measures
Performance measures are important features of queueing systems as they reflect the
efficiency of the queueing system under consideration. The steady-state probabilities at service
completion, vacation termination, departure, and arbitrary epochs are known, various performance
measures of the queue can be easily obtained such as the average number of customers in the
queue at any arbitrary epoch (Lq), probability of the servers busy period (𝑃2𝐵 ), Probability of one
of the servers busy and vacation or idle period (𝑃1𝐵 ), Probability of both the servers vacation or
idle period (𝑃0𝐵 ), and Probability of the switch over state to any one of server (𝑃3𝑎−1 ).
Mean queue length
The results of our model are listed below.
Let 𝐿𝑞 be the expected number of customers in the queue then
𝐿𝑞 =
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𝑎−1
𝑛=0 𝑛( 𝑃1𝑛 +
𝑎−1
𝑛=0 𝑛( 𝑃0𝑛
𝑄1𝑛 )+
+ 𝑄2𝑛 ) +
∞
𝑛=𝑎
𝑛𝑃1𝑛 +
∞
𝑛=0 𝑛𝑃2𝑛
+ 𝑃3 𝑎−1
(26)
Using equations (19),(16),(20),(21) and (15) ,
𝐿𝑞 = [𝐴2 𝐻1 𝑟2 + 𝐴1 𝐷(𝑟1 )𝐻1 𝑟1 + F (𝑟0 )𝐻1 𝑟0 +
𝑟0
1−𝑟0
{a +
𝑟0
1−𝑟0
}+
𝐴1 𝑟1
(1−𝑟1
)2
+
(1−𝑟0 )2
+𝐴1 𝑇 𝑟1 + k 𝑇 𝑟0 + 𝑅1 ] 𝑃1𝑎−1
𝜇𝑎 (𝑎−1)
here𝐻1 𝑥 = 2𝜆(1−𝑥) + [
𝑥 1−𝑥 𝑎 −𝑎𝑥 𝑎 (1−𝑥)
(1−𝑥)𝑎
](1−
𝜇𝑥
(1−𝑥)𝜆
𝑘𝑟0
(27)
)
Probability that both servers are busy (𝑷𝟐𝑩 )
1
1
𝑃2𝐵 = ( 𝐴1 1−𝑟 + 𝑘 1−𝑟 )𝑃1𝑎−1
1
(28)
0
Probability that one server is busy and the other is idle or on vacation (𝑷𝟏𝑩 )
1−𝑟 𝑎
1−𝑟1𝑎
2
1−𝑟1
𝑃1𝐵 = (𝐴2 1−𝑟2 + 𝐴1 𝐷 𝑟1
+ F (𝑟0 )
1−𝑟0𝑎
1−𝑟0
𝑟
+ 1−𝑟0 ) 𝑃1𝑎−1
(29)
0
Probability that the servers are either idle or on vacation (𝑷𝟎𝑩 )
𝜇
𝑎
𝑃0𝐵 = [𝐴2 {
−
𝜆
1−𝑟
2
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𝑟2 (1−𝑟2𝑎 )
(1−𝑟2
)2
𝑎
}+𝐴1 𝐷(𝑟1 ){1−𝑟 −
1
𝑟1 (1−𝑟1𝑎 )
(1−𝑟1
)2
𝑎
} + F(𝑟0 ){1−𝑟 −
0
𝑟0 (1−𝑟0𝑎 )
(1−𝑟0 )2
}]𝑃1𝑎−1 (30)
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30
Probability that the server switch over the system (𝑷𝟑𝒂−𝟏 )
𝑃3𝑎−1 = [𝐴1 𝑇 𝑟1 + k 𝐴1 𝑇 𝑟0 + 𝑅1 ] 𝑃1𝑎−1
(31)
This completes analytic analysis of M/M (a,b)/(2,1) queueing model.
7. Cost Model
In this section, the cost analysis for the models analyzed by considering different costs
associated with the servers and customers waiting time. Let

𝐶0 = fixed cost per unit time for each server



𝑊0 = waiting cost per unit service by each server
𝐶1 = cost per unit service by each server
𝐵𝑠 = size of the waiting batch in the system
If M denotes the expected total cost per unit time for operating the system, then
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M = 2𝐶0 + 𝑊0 𝐿𝑞 + 𝐶1 𝜇 (2𝑃2𝐵 +𝑃3𝑎−1 +𝑃1𝐵 ), where 𝐿𝑞 is the mean queue length and𝑃2𝐵 ,𝑃1𝐵
denotes the probability that the servers busy and 𝑃3𝑎−1 represents the probability of server switch
over from the system.
8. Numerical Analysis
Now we present Computational procedures and discussion of numerical results in this
Section. The numerical values of the performance measures for the various values of the
parameters a, b, 𝜃, 𝜇, 𝜆 are given in the tables (8.1) to (8.4).
Table 8.1The Performance measures for 𝜃= 0.2 and 𝜇= 1
𝜆
5
10
15
20
6
12
18
24
10
20
30
40
a = 10
b = 25
a = 20
b = 30
a = 30
b = 50
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𝐿𝑞
5.16056
8.81293
12.1141
23.5442
9.1025
11.2353
16.1302
26.5239
14.4438
19.6429
28.6061
45.5809
𝑃0𝐵
0.5117
0.2423
0.1048
0.0497
0.7774
0.4747
0.2239
0.1488
0.7224
0.5106
0.3238
0.1434
𝑃1𝐵
0.3115
0.6943
0.6796
0.9205
0.2701
0.5455
0.6485
0.7287
0.3231
0.5756
0.7076
0.7477
𝑃2𝐵
0.00066
0.0158
0.1685
0.2292
0.0009
0.0117
0.0778
0.1224
0.0007
0.0121
0.0624
0.1109
𝑃3𝑎−1
0.000041
0.006710
0.088760
0.014329
0.000080
0.003421
0.004560
0.098789
0.000049
0.009878
0.023140
0.094531
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31
Table 8.2 𝐿𝑞 for various values of𝜆, a when b= 50, 𝜃= 0.5 and 𝜇= 1
𝜆
5
10
15
20
25
a=10
5.2399
7.5620
12.0918
16.4365
18.9994
a=20
9.0993
11.6587
12.7643
15.8790
19.4367
a=30
15.2845
15.6054
15.9769
17.9896
20.9076
a=40
19.2809
19.9154
20.1236
21.4732
22.8553
a=45
21.0005
22.0534
22.4333
23.2970
24.9982
Table 8.3: Comparison of 𝐿𝑞 for M/M (a,b)/1 and M/M(a,b)/(2,1) models
𝜆
𝜃=5
5.5
11.5
17.5
23.5
5.5
11.5
17.5
23.5
9.5
19.5
29.5
39.5
a = 10
b= 30
a = 25
b= 30
a = 40
b= 50
Repeated
M/M(a,b)/1
𝐿𝑞
13.665
25.025
42.313
85.086
22.403
35.695
35.085
100.128
26.984
38.860
60.941
120.300
Bs
1
1
2
3
0
1
1
3
0
0
1
2
Repeated
M/M(a,b)/(2,1)
𝐿𝑞
5.119
8.129
12.625
18.453
12.044
13.016
16.021
21.516
19.587
21.346
26.559
35.874
Bs
0
0
1
1
1
0
0
1
0
0
0
0
Single
M/M(a,b)/1
𝐿𝑞
11.643
23.546
42.087
83.987
15.438
30.012
51.089
97.333
20.343
30.176
53.418
119.332
Bs
1
1
2
3
0
1
1
3
0
0
1
2
Single
M/M(a,b)/(2,1)
𝐿𝑞
5.353
8.826
13.347
18.635
12.041
12.732
14.555
17.524
19.589
20.925
24.261
29.537
Bs
0
0
1
1
1
0
0
0
0
0
0
0
Single and
delayed
M/M(a,b)/(2,1)
Bs
𝐿𝑞
4.999
0
8.098
0
13.009
1
18.323
1
12.009
0
12.756
0
14.112
0
19.5034
0
19.6734
0
20.7903
0
24.2644
0
28.8760
0
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The expected total cost per unit time for the operating system M is compared with single
and repeated vacation of M/M(a,b)/(2,1) for various values of a, b when 𝜃= 0.1 and 𝜇= 1
Table 8.4 𝐿𝑞 and M for various values of a, b where𝜃= 0.1 and 𝜇= 1
𝜆
5
10
15
20
8
16
24
32
10
20
30
40
a = 10
b= 25
a = 25
b= 40
a = 40
b= 50
Copyright © 2015 SciResPub.
M/M(a,b)/1Repeated
vacation
M
𝐿𝑞
51.020
182.490
99.760
332.725
153.028
496.448
227.314
723.154
87.517
291.698
168.171
537.657
256.541
806.717
378.552
1177.000
115.778
376.377
217.798
686.235
329.634
1026.000
483.482
1491.000
M/M(a,b)/(2,1)
Repeated vacation
M
𝐿𝑞
5.132
74.770
8.404
90.556
14.237
111.643
23.113
140.984
12.289
93.188
15.453
108.081
23.387
136.061
37.406
181.535
19.653
113.943
22.337
126.705
30.752
156.153
47.529
210.265
M/M(a,b)/(2,1)
Single vacation
M
𝐿𝑞
5.26245 75.0636
8.9852
92.0953
16.9276 118.7974
30.2383 161.1403
12.3472 93.3515
16.0342 109.7927
26.1690 144.1992
46.0609 207.2089
19.6701 114.0156
22.6798 127.7262
32.8542 162.454
55.4393 233.9873
M/M(a,b)/(2,1) single
and delayed vacation
M
𝐿𝑞
3.6723
72.076
4.0011
73.2368
5.2123
82.6394
5.9021
90.4263
11.4987
91.0007
11.8127
97.0012
11.9921
101.9390
12.2231
108.2140
18.1739
112.3721
18.2645
116.7685
19.0305
120.6432
20.4235
121.2786
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32
From the table (8.3) we infer that Lq in single and repeated vacation is less compared to Lq
in this single and delayed vacation only when the difference between the batch size a and b is less.
When the difference between the batch size a and b is more, the waiting queue in the system is less
in the model with single and delayed vacation compared to the model with single vacation and
repeated vacation this may be, because of the fact that one server is always retained in the system.
The table values of (8.4) shows that the number of batches (of size a and b) waiting in the
queue is less by comparing the other vacation models. It is also seen that Lq and M are
significantly more in M/M(a,b)/1 model compared to M/M(a,b)/(2,1) queueing model.
Figure 1: Comparison of M/M(a,b)/(2,1) queueing model
35
30
25
Lq
Repeated
vacation
Single vacation
20
15
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Single and
Delayed vacation
10
5
0
1
λ
2
3
4
To appreciate the research effectiveness of the presented method in comparison with the
graphical approaches, the sample under study was examined 𝐿𝑞 (expected number of customers in
the queue) is very less than other M/M(a,b)/(2,1) queueing models.
9. Conclusion
In this present study, a M/M(a,b)/(2,1) queueing models with servers vacation depends on
the batch sizes and the state of switch over are considered. In general, analytical solution of bulk
service queueing models are extremely complicated in the two server‟s case. We have made
attempt to study the analytical solution of two servers bulk service queueing models in which only
one server is allowed for vacation at a time to avoid the inconvenience to the customers. This
model is applicable to a variety of real world stochastic service system.
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33
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