AN ANALYSIS OF BULK SERVICE QUEUEING MODEL WITH … · 2015-02-26 · Introduction Queueing theory was born in the early 1900s with the work of A. K. Erlang of the ... situations

AN ANALYSIS OF BULK SERVICE QUEUEING MODEL WITH SERVERS VARIOUS

VACATIONS

R.Sree parimala1, S.Palaniammal

2,

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: [email protected],[email protected]

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.

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

International Journal of Advancements in Research & Technology, Volume 4, Issue 2, February -2015 ISSN 2278-7763

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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

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

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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(iii) 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,

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 𝑃1𝑛𝑏𝑛=𝑎 (4)

𝜆 + 𝜇 + 𝜃 𝑃1𝑛 = 𝜆𝑃1𝑛−1 + 2 𝜇𝑃2𝑛 + 𝜇𝑃1𝑛+𝑏 1 ≤ 𝑛 ≤ 𝑎 − 2 (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)

𝜆 + 2 𝜇 𝑃20 = 𝜆𝑃3𝑎−1 + 𝜃 𝑃1𝑛𝑏𝑛=𝑎 + 2𝜇 𝑃2𝑛

𝑏𝑛=𝑎 + 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)

(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 = 𝑟0

𝑛−𝑎+1𝑃1𝑎−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 = (𝐴1𝑟1𝑛 + 𝑘𝑟0

𝑛)𝑃1𝑎−1 𝑛 ≥ 0 (16)

where 𝐴1is a constant and k = −𝜃𝑟0

𝑏−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 = (𝐴1𝐵𝑛 (𝑟1) + k 𝐵𝑛 (𝑟0) + 𝑟0𝑛−𝑎 ) 𝑃1𝑎−1 1 ≤ 𝑛 ≤ 𝑎 − 2 (17)

where 𝐵𝑛 𝑥 = 2𝜇𝑅

𝜆(𝑥−𝑅)( 𝑥𝑛 –

𝑥

𝑅 𝑎−1

𝑅𝑛 ) , R = 𝜆

𝜆+𝜇+𝜃 and k =

−𝜃𝑟0𝑏−𝑎+2

𝜆+2𝜃 𝑟0−𝜆


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Similarly solving equation (13) recursively using (17)

𝑄1𝑛 = ( 𝐴2𝑟2𝑛 + 𝐴1𝑔𝑛 (𝑟1) + k 𝑔𝑛 (𝑟0) + 𝑘0𝑟0

𝑛−𝑎 ) 𝑃1𝑎−1, 1 ≤ 𝑛 ≤ 𝑎 − 1 (18)

where 𝐴2is a constant, 𝑟2 = 𝜆

𝜆+𝜇 , 𝑔𝑛 (x) =

2𝜇𝜃𝑅

𝜆(𝑥−𝑅)(

𝑥𝑛+1

𝜆+𝜇 𝑥− 𝜆+

𝑥

𝑅 𝑎−1 𝑅𝑛

𝜃 ) and 𝑘0 =

𝜃 𝑟0

𝜆+𝜇 𝑟0−𝜆

By adding (2) , (12) and using the equations (1), (11)

𝑃0𝑛 + 𝑄2𝑛 = 𝜇

𝜆 (𝑛

𝑘=0 𝑃1𝑛 + 𝑄1𝑛), 0 ≤ 𝑛 ≤ 𝑎 − 2

from equations (17) and (18) substituting the values of 𝑃1𝑛 𝑎𝑛𝑑 𝑄1𝑛

𝑃0𝑛 + 𝑄2𝑛 = 𝜇

𝜆 (𝑛

𝑘=0 𝐴2𝑟2𝑛 + 𝐴1[𝐵𝑛(𝑟1) + 𝑔𝑛 (𝑟1)] + k [𝐵𝑛 (𝑟1) + 𝑔𝑛 (𝑟1)] + (1+ 𝑘0)𝑟0

𝑛−𝑎 ) 𝑃1𝑎−1

After simplification 𝐵𝑛 (𝑥) + 𝑔𝑛 (𝑥) =𝐶𝑛 𝑥 = 2𝜇𝑥𝑛

𝜆+𝜇 𝑥− 𝜆

𝑃0𝑛 + 𝑄2𝑛 = 𝜇

𝜆 (𝑛

𝑘=0 𝐴2𝑟2𝑛 + 𝐴1𝐶𝑛 𝑟1 + 𝑘 𝐶𝑛 𝑟0 + (1+ 𝑘0)𝑟0

𝑛−𝑎 ) 𝑃1𝑎−1

Further giving an expansion and simplifying the above equation,

𝑃0𝑛 + 𝑄2𝑛 = 𝜇

𝜆 [𝐴2

1−𝑟2𝑛+1

1−𝑟2 + 𝐴1𝐷(𝑟1)

1−𝑟1𝑛+1

1−𝑟1 + F(𝑟0)

1−𝑟0𝑛+1

1−𝑟0 ] 𝑃1𝑎−1 (19)

here𝐷(𝑟1) = 2𝜇

𝜆+𝜇 𝑟1− 𝜆 and F (𝑟0) =

1

𝜆+𝜇 𝑟0− 𝜆 [2𝜇𝑘 −

𝜆

𝑟0𝑎 ( 1-

𝑟0

𝑅 ) ]

The probability for one of the server is busy and the other switchover from the system is solved by

using (10)

𝑃3 𝑎−1 = [𝐴1𝑇 𝑟1 + k 𝐴1𝑇 𝑟0 + 𝑅1 ] 𝑃1𝑎−1 (20)

where𝑇 𝑥 = 2𝜇𝑥𝑎−1

𝜆+𝜇+𝛼 and 𝑅1 =

𝜃

𝜆+𝜇+𝛼

Also by adding (17), (18) and using the results of equation (19), we obtain

𝑃1𝑛 + 𝑄1𝑛 = (𝐴2𝑟2𝑛 + 𝐴1𝐷 𝑟1 𝑟1

𝑛 + 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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𝜆 + 2 𝜇 𝐴1 + 𝑘 = 𝜆 [ 𝐴1𝑇 𝑟1 + k 𝑇 𝑟0 + 𝑅1] + 𝜃𝑟0−𝑎+1

𝑟0𝑎−𝑟0

𝑏+1

1−𝑟0 +

2𝜇[ 𝐴1 𝑟1𝑎−𝑟1

𝑏+1

1−𝑟1 + k

𝑟0𝑎−𝑟0

𝑏+1

1−𝑟0 ] + 𝜆 [( 𝐴2𝑟2

𝑛 + 𝐴1𝑔𝑛 (𝑟1) + k 𝑔𝑛 (𝑟0) + 𝑘0𝑟0𝑛−𝑎 )]

By simplifying and using 𝑔𝑎−1(x) = 2𝜇𝑥𝑎−1

𝜆+𝜇 𝑥− 𝜆 we obtain the value of constant 𝐴2 as follows

𝐴2 = 1

𝑟2𝑎−1 [𝐴1𝑆 𝑟1 + 𝑘 𝑆 𝑟0 -

𝜃𝑟0

𝜆 (

1−𝑟0𝑏−𝑎+1

1−𝑟0) - (𝑅1 + 𝑘0𝑟0

𝑛−1)] (22)

here𝑆 𝑥 = 𝜆+2 𝜇

𝜆 -

2𝜇𝑅1𝑟1𝑎−1

𝜃 -

2 𝜇

𝜆 𝑥𝑎−𝑥𝑏+1

1−𝑥 -

2𝜇𝑥𝑎−1

𝜆+𝜇+𝛼

Also to obtain the value of 𝐴1, by adding (4) and (14),

𝐴1 = 1

𝑧(𝑟1) [𝜆{ 𝑟0 -

𝜃𝑟0

𝜆 (

1−𝑟0𝑏−𝑎+1

1−𝑟0) - (𝑅1 + 𝑘0𝑟0

𝑛−1) } + F (𝑟0) [𝜇 𝑟0−𝑟0

𝑎

1−𝑟0 - 𝜆 ] +

2𝜇[ k + 𝜇

2 𝑟0−𝑟0

𝑏−𝑎+2

1−𝑟0 ] (23)

Where 𝑧(𝑟1) = [𝐷 𝑟1 { 𝜆 + 𝜇 𝑟1𝑎−𝑟1

1−𝑟1 } - 2 𝜇 -

𝑆 𝑟1

𝑟2𝑎−1 ]

Thus we obtained all the steady state probabilities in terms of 𝑃1𝑎−1which 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−1𝑛=0 𝑃1𝑛 + 𝑄1𝑛 +𝑃0𝑛+ 𝑄2𝑛 ) + 𝑃1𝑛

∞𝑛=𝑎 + 𝑃2𝑛

∞𝑛=0 + 𝑃3 𝑎−1 = 1 (24)

Substituting the results from the equations (19), (22), (15), (16) and (20) we obtain

𝑃1 𝑎−1−1 = 𝐴2 [ 𝐻(𝑟2) +

1−𝑟2𝑎

1−𝑟2 ] + 𝐴1𝐷(𝑟1) [𝐻(𝑟1) +

1−𝑟1𝑎

1−𝑟1 ] + F (𝑟0)[𝐻(𝑟0) +

1−𝑟0𝑎

1−𝑟0 ] + (

𝑟0+𝑘

1−𝑟0) +

𝐴1(1

1−𝑟1 + 𝑇 𝑟1 ) + k (

1

1−𝑟0 + 𝑇 𝑟0 ) + 𝑅1. (25)

where H(x) = 𝜇

𝜆 [

𝑎

1−𝑥 -

𝑥(1−𝑥𝑎 )

(1−𝑥)2 ]


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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 = 𝑛(𝑎−1𝑛=0 𝑃1𝑛+ 𝑄1𝑛 )+ 𝑛(𝑎−1

𝑛=0 𝑃0𝑛 + 𝑄2𝑛 ) + 𝑛𝑃1𝑛∞𝑛=𝑎 + 𝑛𝑃2𝑛

∞𝑛=0 + 𝑃3 𝑎−1 (26)

Using equations (19),(16),(20),(21) and (15) ,

𝐿𝑞 = [𝐴2𝐻1 𝑟2 + 𝐴1 𝐷(𝑟1)𝐻1 𝑟1 + F (𝑟0)𝐻1 𝑟0 + 𝑟0

1−𝑟0 {a +

𝑟0

1−𝑟0 } +

𝐴1𝑟1

(1−𝑟1)2 + 𝑘𝑟0

(1−𝑟0)2

+𝐴1𝑇 𝑟1 + k 𝑇 𝑟0 + 𝑅1 ] 𝑃1𝑎−1 (27)

here𝐻1 𝑥 = 𝜇𝑎 (𝑎−1)

2𝜆(1−𝑥) + [

𝑥 1−𝑥𝑎 −𝑎𝑥𝑎 (1−𝑥)

(1−𝑥)𝑎 ] ( 1 −

𝜇𝑥

(1−𝑥)𝜆 )

Probability that both servers are busy (𝑷𝟐𝑩)

𝑃2𝐵 = ( 𝐴11

1−𝑟1+ 𝑘

1

1−𝑟0)𝑃1𝑎−1 (28)

Probability that one server is busy and the other is idle or on vacation (𝑷𝟏𝑩)

𝑃1𝐵 = (𝐴21−𝑟2

𝑎

1−𝑟2 + 𝐴1𝐷 𝑟1

1−𝑟1𝑎

1−𝑟1 + F (𝑟0)

1−𝑟0𝑎

1−𝑟0+

𝑟0

1−𝑟0 ) 𝑃1𝑎−1 (29)

Probability that the servers are either idle or on vacation (𝑷𝟎𝑩)

𝑃0𝐵 = 𝜇

𝜆 [𝐴2{

𝑎

1−𝑟2−

𝑟2(1−𝑟2𝑎 )

(1−𝑟2)2}+𝐴1𝐷(𝑟1){

𝑎

1−𝑟1−

𝑟1(1−𝑟1𝑎 )

(1−𝑟1)2 } + F(𝑟0){

𝑎

1−𝑟0−

𝑟0(1−𝑟0𝑎 )

(1−𝑟0)2 }]𝑃1𝑎−1(30)


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Probability that the server switch over the system (𝑷𝟑𝒂−𝟏)

𝑃3𝑎−1 = [𝐴1𝑇 𝑟1 + k 𝐴1𝑇 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

M = 2𝐶0 + 𝑊0𝐿𝑞 + 𝐶1 𝜇 (2𝑃2𝐵 +𝑃3𝑎−1 +𝑃1𝐵 ), where 𝐿𝑞 is the mean queue length and𝑃2𝐵,𝑃1𝐵

denotes the probability that the servers busy and 𝑃3𝑎−1represents 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

a = 10

b = 25

𝐿𝑞 𝑃0𝐵 𝑃1𝐵 𝑃2𝐵 𝑃3𝑎−1

5 5.16056 0.5117 0.3115 0.00066 0.000041

10 8.81293 0.2423 0.6943 0.0158 0.006710

15 12.1141 0.1048 0.6796 0.1685 0.088760

20 23.5442 0.0497 0.9205 0.2292 0.014329

6

a = 20

b = 30

9.1025 0.7774 0.2701 0.0009 0.000080

12 11.2353 0.4747 0.5455 0.0117 0.003421

18 16.1302 0.2239 0.6485 0.0778 0.004560

24 26.5239 0.1488 0.7287 0.1224 0.098789

10

a = 30

b = 50

14.4438 0.7224 0.3231 0.0007 0.000049

20 19.6429 0.5106 0.5756 0.0121 0.009878

30 28.6061 0.3238 0.7076 0.0624 0.023140

40 45.5809 0.1434 0.7477 0.1109 0.094531


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Table 8.2 𝐿𝑞 for various values of𝜆, a when b= 50, 𝜃= 0.5 and 𝜇= 1

𝜆 a=10 a=20 a=30 a=40 a=45

5 5.2399 9.0993 15.2845 19.2809 21.0005

10 7.5620 11.6587 15.6054 19.9154 22.0534

15 12.0918 12.7643 15.9769 20.1236 22.4333

20 16.4365 15.8790 17.9896 21.4732 23.2970

25 18.9994 19.4367 20.9076 22.8553 24.9982

Table 8.3: Comparison of 𝐿𝑞 for M/M (a,b)/1 and M/M(a,b)/(2,1) models

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 M/M(a,b)/1Repeated

vacation

M/M(a,b)/(2,1)

Repeated vacation

M/M(a,b)/(2,1)

Single vacation

M/M(a,b)/(2,1) single

and delayed vacation

𝐿𝑞 M 𝐿𝑞 M 𝐿𝑞 M 𝐿𝑞 M

5 a = 10

b= 25

51.020 182.490 5.132 74.770 5.26245 75.0636 3.6723 72.076

10 99.760 332.725 8.404 90.556 8.9852 92.0953 4.0011 73.2368

15 153.028 496.448 14.237 111.643 16.9276 118.7974 5.2123 82.6394

20 227.314 723.154 23.113 140.984 30.2383 161.1403 5.9021 90.4263

8 a = 25

b= 40

87.517 291.698 12.289 93.188 12.3472 93.3515 11.4987 91.0007

16 168.171 537.657 15.453 108.081 16.0342 109.7927 11.8127 97.0012

24 256.541 806.717 23.387 136.061 26.1690 144.1992 11.9921 101.9390

32 378.552 1177.000 37.406 181.535 46.0609 207.2089 12.2231 108.2140

10 a = 40

b= 50

115.778 376.377 19.653 113.943 19.6701 114.0156 18.1739 112.3721

20 217.798 686.235 22.337 126.705 22.6798 127.7262 18.2645 116.7685

30 329.634 1026.000 30.752 156.153 32.8542 162.454 19.0305 120.6432

40 483.482 1491.000 47.529 210.265 55.4393 233.9873 20.4235 121.2786

𝜆 𝜃 = 5

Repeated

M/M(a,b)/1

Repeated

M/M(a,b)/(2,1)

Single

M/M(a,b)/1

Single

M/M(a,b)/(2,1)

Single and

delayed

M/M(a,b)/(2,1)

𝐿𝑞 Bs 𝐿𝑞 Bs 𝐿𝑞 Bs 𝐿𝑞 Bs 𝐿𝑞 Bs

5.5 a = 10

b= 30

13.665 1 5.119 0 11.643 1 5.353 0 4.999 0 11.5 25.025 1 8.129 0 23.546 1 8.826 0 8.098 0 17.5 42.313 2 12.625 1 42.087 2 13.347 1 13.009 1 23.5 85.086 3 18.453 1 83.987 3 18.635 1 18.323 1 5.5

a = 25

b= 30

22.403 0 12.044 1 15.438 0 12.041 1 12.009 0 11.5 35.695 1 13.016 0 30.012 1 12.732 0 12.756 0 17.5 35.085 1 16.021 0 51.089 1 14.555 0 14.112 0 23.5 100.128 3 21.516 1 97.333 3 17.524 0 19.5034 0 9.5

a = 40

b= 50

26.984 0 19.587 0 20.343 0 19.589 0 19.6734 0 19.5 38.860 0 21.346 0 30.176 0 20.925 0 20.7903 0 29.5 60.941 1 26.559 0 53.418 1 24.261 0 24.2644 0 39.5 120.300 2 35.874 0 119.332 2 29.537 0 28.8760 0


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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

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.

0

5

10

15

20

25

30

35

1 2 3 4

Repeated vacationSingle vacation

Single and Delayed vacation

Lq

λ


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1. AfthabBegum.M.I,(1996), “Queueing models with bulk service and vacation”, Ph.D,

Dissertation, Bharathiar university, Coimbatore, Tamil nadu, India.

2. AnjanaSolanki and Srivastava.P.N(1998), “Transient state analysis of the queueing system

Ek/ M(a,b)/1/N”, Operations research, Vol.35,No.4,353-359.

3. Choudhury.G and Paul.M(2005),”A two phase queueing system with Bernoulli Feedback”,

Information and management science,Vol.16,35-52

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numerical analysis”, Computer and operations research,Vol.9,197-205.

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29-66.

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startup and breakdown,” Comput. Indust.Engg.,44: 567 -579.

7. Madan.K.C and AI-Rawwash.M(2005),”On the Mx/G/1 queue with feedback and optional

server vacations based on a single vacation policy”, Applied mathematical and

computations, Vol 160, 909 -919.

8. Medhi.J.H and Borthakur.A (1972), “On a two server Markovian queue with a general bulk

service rule”, Cahiers duecentre d‟ Etudes de Rechercheoperationnelle, Vol.21, 183-189.

9. Mishra.S.S and Pandey.N.K (2002), “A Bulk queueing model M/ M(a,b,c)/2 for non-

Identical servers with vacation”, International journal of Management and systems, Vol.18,

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10. Neuts.M.F(1967),”A general class of bulk queues with Poisson input”, Applied

Mathematical and Statistics,Vol.38, 759 – 770.

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vacation”, Ph.D, Dissertation, Bharathiar university, Coimbatore, Tamil nadu, India.

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