Applied Mathematical Sciences, Vol. 8, 2014, no. 163, 8113 - 8124

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.49774

Single Server Queueing Model with Server Delayed

Vacation and Switch over State

R. Sree Parimala

Department of Science and Humanities

Hindusthan Institute of technology

Coimbatore-641 032, Tamil Nadu, India

S. Palaniammal

Department of Science and Humanities

Sri Krishna College of Technology, Kovaipudur

Coimbatore-64 042, Tamil Nadu, India

Copyright © 2014 R. Sree Parimala and S. Palaniammal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper focus on a server delayed vacation of M/M (a, b) / 1queueing

system switch over state. In this model it is assumed that the arrival pattern is

Poisson fashion with parameter λ and service is done in batches which is

exponentially distributed with parameter µ according to the general bulk service

rule introduced by Neuts (11).The batches are served according to FCFS

discipline. The length of the vacations can be controlled by means of the number

of customers ‘kb’ arriving to the system during vacation, and the level ‘k’ may be

chosen according to the arrival rate, the service rate, the cost per unit of waiting

time and the cost of the server being transferred from vacation to work. Secondly

a server allowed for a delay time before a vacation begins. During the delay time,

the server is situated in warm standby state and the service starts immediately if

the batches of ‘a’ customers are present. If server finds (a-1) customers, then

server will stay idle in the system called delay time before he goes for vacation. If

the server finds (a-2) customers in the system the server switch over the system.

So in this system, sever can take only one vacation between two successive

service times.

8114 R. Sree Parimala and S. Palaniammal

The steady state solutions and the system characteristics are derived and analyzed

for this model. Various models studied earlier are discussed as special cases. The

analytical results are numerically illustrated for different values of the parameters

and levels also.

Keywords: Delayed vacation, switch over state, queue size vacation

1. Introduction

In many practical situations the queueing models are used to provide basic

framework for efficient design and analysis including various technical systems

also predictions the behavior of systems such as waiting times of customers,

various vacations for servers and so forth. Queueing systems with 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 general bulk service

Queueing models with server vacation have been analyzed by many authors.

In the last few years, increasing interest in studying queueing systems with

various rules of vacation has led to many extensions of previously existing results.

For example, a batch arrival model with a finite capacity for the buffer size can be

used to model some telecommunications systems using a time division multiple

access (TDMA) scheme. Researchers’ have also done some performance analysis

on systems where, probability distributions of the variables are more general and

closer to reality.

S. Palaniammal (12) has studied M/M (a, b) / (2, 1) queueing model and

derived an analytical solutions for servers repeated and single vacation and

presented the steady state result’s in terms of characteristic equation of a

difference equation. M. I. A fthabbegam (1) has tried analytic solution 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. Medhi. J and Borthakur. A (9) have introduced a general bulk

service rule with two servers. The case of delayed vacation has been analyzed for

a 𝑀𝑥/G1/1/N queueing system by Frey and Takahashi (7) where the term close-

Single server queueing model 8115

down time is used. They have analyzed general service, delay and vacation times

with Poisson (batch) arrivals.

Doshi (6) have analyzed the batch service queueing models without

vacation and with vacation.Takagi. H (13) analyzed the M/G/1/N queues with

vacation and exhaustive service. Also a bulk queueing model M/M (a,b,c)/2 with

servers vacation has been studied by Mishra. S. S and Pandey. N. K (10). Anitha

(2) has analyzed a M/M (a, b)/1 queueing model with multiple exponential

vacations and changeover time and obtained closed form solutions. The Ek / M (a,

b)/1 queueing system and its numerical results are analyzed by Chaudry. M. C and

Easton. G. D (5). The transient of Ek /M (a, b)/1/N derived by Anjanasolanki and

Srivastava. P. N (3).

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 waiting of customers, it may solved by the servers vacation

due to batch criteria. Ke (8) 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.

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 in this paper, an attempt has been made to

analyzethe delayed vacation with server switch over state. The study of this

queueing model is organized as follows. The model is described in Section 2.

Queueing model is formulated mathematically along with notations in Section 3.

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 numerical results and

graphical illustrations are discussed to facilitate the sensitivity analysis in Section

7 .Concluding remarks and notable features of investigation done are highlighted

in Section 8.

2. Model Description

The queueing system consists of a single server and infinite waiting space for

the customers. The server has a finite capacity ‘b’ and a quorum of size ‘a’. The

customers arrive in single by a Poisson process.In this queueing model there is

one server in the system and we make the following assumptions with intensity.

(i) The queue discipline is FCFS. The customers are served in batches of size [a, b].

(ii) Service times are assumed to be exponentially distributed with mean1 𝜇⁄ , the

traffic intensity is 𝜆 𝑏𝜇⁄ < 1.

8116 R. Sree Parimala and S. Palaniammal

(iii) On completing the service if server finds less than ‘a’ number of customers (i.e. a-1 customers) in the queue, he waits in the system for a period of time

called the delay time before he goes for a vacation. The delay can be interrupted if

the queue size becomes ‘a’, in which case the server resumes service. If the server

finds (a-2) customers in the system switch over the system. Thus the length of the

delay time is min(X, Y), where X is exponentially distributed with mean1 𝜆⁄ and Y is a random variable which is also exponentially distributed with mean1 𝜃1⁄ .

(iv) If the delay time is completed before the queue size becomes ‘a’, the server begins

a vacation whose length is exponentially distributed with mean1 𝜃2⁄ . After completing the vacation, the server resumes service only if ‘kb’ (k ≥ 1) or more customers are in the system otherwise he takes another vacation. The

aforementioned random variables are independent of each other.

3. Mathematical Formulation

The queueing system can be formulated as a continuous time parameter

Markov chain with states’ S(t) = 0,S(t) = 1,S(t) = 2 and S(t) = 3 denotes the

events that the server is on vacation, in delay period, in busy at epoch t and switch

over state respectively.

Let P0n(t) = P(N(t) = n, S(t) = 0) ( n = 0,1,2,3,…)

P1n(t) = P(N(t) = n, S(t) = 1) ( n = 0,1,2,3,…a-1)

P2n(t) = P(N(t) = n, S(t) = 2) ( n = 0,1,2,3,…)

P3n(t) = P(N(t) = n, S(t) = 3) ( n = 0,1,2,3,…a-2)

Where N (t) denotes the number of customers in the system at time t. when

the delay time is exponentially distributed, {N (t), S (t), t ≥ 0} is a standard continuous time Markov chain. From the theory of Markov chain, it follows that

{N (t), S (t), t ≥ 0} has a unique equilibrium distribution which satisfies the following of equations.

The limiting probabilities corresponding to different states are P0n =

lim𝑛→∞

𝑃0𝑛(𝑡), P1n(t) = lim𝑛→∞

𝑃1𝑛(𝑡)and P2n(t) = lim𝑛→∞

𝑃2𝑛(𝑡) exists.

4. Steady State Equations

The steady state equations are given by

𝜆 𝑃00 = 𝜃1𝑃10 (1) 𝜆𝑃0𝑛 = 𝜆𝑃0𝑛−1 + 𝜃1𝑃1𝑛(1 ≤ 𝑛 ≤ 𝑎 − 1) (2) 𝜆𝑃0𝑛 = 𝜆𝑃0𝑛−1(𝑎 ≤ 𝑛 ≤ 𝑘𝑏 − 1) (3) (𝜆 + 𝜃2)𝑃0𝑎−1 = 𝜆𝑃2𝑎−1 + 𝜇𝑃3𝑎−1 (4) (𝜆 + 𝜃2)𝑃0𝑛 = 𝜆𝑃0𝑛−1(𝑛 ≥ 𝑘𝑏) (5)

Single server queueing model 8117 (𝜆 + 𝜃1)𝑃10 = 𝜇𝑃20 (6) (𝜆 + 𝜃1)𝑃1𝑛 = 𝜆𝑃1𝑛−1 + 𝜇𝑃2𝑛(1 ≤ 𝑛 ≤ 𝑎 − 2) (7) (𝜆 + 𝜃1)𝑃1𝑎−1 = 𝜆𝑃3𝑎−1 + 𝜇𝑃2𝑎−1 (8) (𝜆 + 𝜇)𝑃20 = 𝜇 ∑ 𝑃2𝑛

𝑏𝑛=𝑎 + 𝜆 𝑄1𝑎−1 (9)

(𝜆 + 𝜇)𝑃2𝑛 = 𝜆𝑃2𝑛−1 +2𝜇𝑃2𝑛+𝑏(1 ≤ 𝑛 ≤ (𝑘 − 1)𝑏 − 1) (10) (𝜆 + 𝜇)𝑃2𝑎−1 = 𝜆𝑃1𝑎−1 + 𝜃1𝑃1𝑎−1 + 𝜆𝑃0𝑎−1 (11) (𝜆 + 𝜇)𝑃2𝑛 = 𝜆𝑃2𝑛−1 + 𝜇 𝑃2𝑛+𝑏 + 𝜃2𝑃0𝑛+𝑏(𝑛 ≥ (𝑘 − 1)𝑏) (12) (𝜆 + 𝜇)𝑃3𝑎−1 =𝜃2𝑃0𝑎−1 (13)

5. Computation of Steady State Solutions

From equation (3), 𝑃0𝑛 = 𝑃0𝑛−1(𝑎 ≤ 𝑛 ≤ 𝑘𝑏 − 1) (14)

Using the result in (5) and solving recursively,

𝑃0𝑛 = 𝑟1𝑛−𝑘𝑏+1𝑃0𝑎−1(𝑛 ≥ 𝑘𝑏) where𝑟1 =

𝜆

𝜆+𝜃2 (15)

From equation (12), (𝜇 Eb+1 – (𝜆 + 𝜇)E + 𝜆) 𝑃2𝑛 = - 𝜃2𝑃0𝑛+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

𝑏𝜇 and using equation (15), after simplification,𝑃2𝑛 =

(𝐴2𝑟2𝑛 + 𝐵𝑟1

𝑛)𝑃0𝑎−1(𝑛 ≥ (𝑘 − 1)𝑏) (16)

where 𝐴1is a constant and B = −𝜃2𝑟1

𝑏−𝑘𝑏+1

𝜇(𝑟1𝑏−1)+𝜃2

From equation (10), substituting n = (k-1)-1, (k-1)b-2, ...(k-2)b and solving

recursively using (15) and (16), it is found that

𝑃2𝑛 = (𝐴2𝑟2𝑛 + 𝐵1𝑟1

𝑛 + 𝐵1𝑟3𝑛) 𝑃0𝑎−1((𝑘 − 2)𝑏 ≤ 𝑛 ≤ (𝑘 − 1)𝑏 − 1)) (17)

where 𝐵1 = 𝜇𝜃2𝑟1

−(𝑘−2)𝑏+1

(𝜃2−𝜇)(𝜇(𝑟1𝑏−1)+𝜃2)

,𝐵2 = −𝜃2𝑟3

−(𝑘−1)𝑏+1

(𝜃2−𝜇) , and 𝑟3 =

𝜆

𝜆+𝜇

By proceeding similarly, we can get the value of 𝑃2𝑛for((0 ≤ 𝑛(𝑘 − 1)𝑏 − 1)). Then the value of 𝑃1𝑛 and 𝑃0𝑛 for(0 ≤ 𝑛 ≤ 𝑎 − 1) can be obtained from equations (2) and (7). Considering the case k = 2 for simplicity, we get the steady

state queue size probabilities as

𝑃0𝑛 = 𝑃0𝑛−1(𝑎 ≤ 𝑛 ≤ 2𝑏 − 1) (18) 𝑃0𝑛 = 𝑟1

𝑛−2𝑏+1𝑃0𝑎−1(𝑛 ≥ 2𝑏) (19) 𝑃2𝑛 = (𝐴2𝑟2

𝑛 + 𝐵𝑟1𝑛)𝑃0𝑎−1(𝑛 ≥ 𝑏) (20)

8118 R. Sree Parimala and S. Palaniammal

where B = −𝜃2𝑟1

𝑏−2𝑏+1

𝜇(𝑟1𝑏−1)+𝜃2

, 𝑟1 = 𝜆

𝜆+𝜃2 and

𝑃2𝑛 = (𝐴2𝑟2𝑛 + 𝐵1𝑟1

𝑛 + 𝐵2𝑟3𝑛) 𝑃0𝑎−1( 0 ≤ 𝑛 ≤ 𝑏 − 1) (21)

where 𝐵1 = 𝜇𝜃2𝑟1

(𝜃2−𝜇)(𝜇(𝑟1𝑏−1)+𝜃2)

,𝐵2 = −𝜃2𝑟3

−𝑏+1

(𝜃2−𝜇) ,and 𝑟3 =

𝜆

𝜆+𝜇

solving the equation (7) using difference equation technique, can be obtained as

𝑃1𝑛 = 𝐴3𝑟4𝑛 + 𝐴2𝑓(𝑟2)𝑟2

𝑛 + 𝐵1𝑓(𝑟1)𝑟1𝑛 + 𝐵2𝑓(𝑟3)𝑟3

𝑛) 𝑃0𝑎−1(0 ≤ 𝑛 ≤ 𝑏 − 1) (22)

Here𝑓(𝑥) =𝜇𝑥

(𝜆+𝜃1)𝑥−𝜆 and 𝑟4 =

𝜆

𝜆+𝜃1

Taking the summation over k = 1, 2, 3…n in equation (2) and adding equation (1),

we have 𝜆𝑃0𝑛 = 𝜃1

𝜆∑ 𝑃1𝑘

𝑛𝑘=1 ( 0 ≤ 𝑛 ≤ 𝑎 − 1) and substituting for 𝑃1𝑛 from (22),

simplifying,

𝑃0𝑛=𝜃1

𝜆 [𝐴3

1−𝑟4𝑛+1

1−𝑟4+ 𝐴2𝑓(𝑟2)

1−𝑟2𝑛+1

1−𝑟2 + 𝐵1 𝑓(𝑟1)

1−𝑟1𝑛+1

1−𝑟1+ 𝐵2𝑓(𝑟3)

1−𝑟3𝑛+1

1−𝑟3]𝑃0𝑎−1

𝑓𝑜𝑟 (0 ≤ 𝑛 ≤ 𝑎 − 1) (23)

From equation (13), 𝑃3𝑎−1 =𝜃2

𝜆+ 𝜇𝑃0𝑎−1

𝑃3𝑎−1=𝜃1𝜃2

𝜆(𝜆+ 𝜇)[ 𝐴3

1−𝑟4𝑎

1−𝑟4+ 𝐴2𝑓(𝑟2)

1−𝑟2𝑎

1−𝑟2+ 𝐵1 𝑓(𝑟1)

1−𝑟1𝑎

1−𝑟1+ 𝐵2𝑓(𝑟3)

1−𝑟3𝑎

1−𝑟3]𝑃0𝑎−1 (24)

From equation (6) we get the value of the constant 𝐴3 as

𝐴3 = 𝐴2[𝜇

𝜆+𝜃1- 𝑓(𝑟2)] + 𝐵1[

𝜇

𝜆+𝜃1 - 𝑓(𝑟1)] + 𝐵2[

𝜇

𝜆+𝜃1 - 𝑓(𝑟3)] (25)

Similarly using (9) and (25) the constant 𝐴2 can be obtained as

𝐴2 = [

𝑟41−𝑟4

𝑎 − 𝐵1 𝑔(𝑟1) − 𝐵2𝑔(𝑟3) ]

𝑔(𝑟2) here 𝑔(𝑥) = 𝑓(𝑥)[

1−𝑥𝑎

1−𝑥

𝑟4

1−𝑟4𝑎 +

𝜇

𝜆+𝜃1 – 1] (26)

Thus we have obtained all the steady state probabilities in terms of

𝑃0𝑎−1.Using the normalizing condition,

∑ 𝑃0𝑛∞𝑛=0 + ∑ 𝑃1𝑛

𝑎−2𝑛=0 + ∑ 𝑃2𝑛

∞𝑛=0 + 𝑃3 𝑎−1 = 1 (27)

To obtain the value of 𝑃0𝑎−1 by substituting for 𝑃0𝑛−1, 𝑃1𝑛 and 𝑃2𝑛 from (18) to (24)

Single server queueing model 8119

∑ 𝑃0𝑛∞𝑛=0 = [𝐴3𝐻(𝑟4)+𝐴2𝑓(𝑟2)𝐻(𝑟2)+𝐵1𝑓(𝑟1)𝐻(𝑟1) +𝐵2𝑓(𝑟3)𝐻(𝑟3)+

𝑟1

1−𝑟1 +

(2b – a)]𝑃0𝑎−1 here 𝐻(𝑥) = 𝜃1

𝜆 [

𝑎

1−𝑥 -

𝑥(1−𝑥𝑎)

(1−𝑥)2 ] (28)

∑ 𝑃2𝑛∞𝑛=0 = [

𝐴2

1−𝑟2 + B

𝑟1𝑏

1−𝑟1+ 𝐵1

1−𝑟1𝑏

1−𝑟1+ 𝐵2

1−𝑟3𝑏

1−𝑟3 ] 𝑃0𝑎−1 (29)

Also∑ 𝑃1𝑛𝑎−2𝑛=0 =[𝐴3

1−𝑟4𝑎−1

1−𝑟4+ 𝐴2𝑓(𝑟2)

1−𝑟2𝑎−1

1−𝑟2 + 𝐵1 𝑓(𝑟1)

1−𝑟1𝑎−1

1−𝑟1+ 𝐵2𝑓(𝑟3)

1−𝑟3𝑎−1

1−𝑟3]𝑃0𝑎−1 (30)

Substituting the equations (24), (28), (29) and (30) in (27) gives

𝑃0𝑎−1−1 =𝐴3 (𝐻(𝑟4) + 𝑘

1−𝑟4𝑎−1

1−𝑟4) + 𝐴2 {𝑓(𝑟2) (𝐻(𝑟2) + 𝑘

1−𝑟2𝑎−1

1−𝑟2) +

1

1−𝑟2} +

𝐵1 { 𝑓(𝑟1) (𝐻(𝑟1) + 𝑘1−𝑟1

𝑎−1

1−𝑟1) +

1−𝑟1𝑏

1−𝑟1} + 𝐵2 {𝑓(𝑟3) (𝐻(𝑟3) + 𝑘

1−𝑟3𝑎−1

1−𝑟3) +

1−𝑟3𝑏

1−𝑟3} +

𝐵𝑟1𝑏+𝑟1

1−𝑟1+ (2𝑏 − 𝑎) (31)

here k =𝜃1𝜃2

𝜆(𝜆+ 𝜇)

6. Some 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 (𝑃𝐵), Probability when the server is idle (𝑃1), Probability of the server in warm standby position (𝑃𝑤) are derived.

6.1 Mean Queue Length

The results of our model are listed below.

Let 𝐿𝑞 be the expected number of customers in the queue then

𝐿𝑞 = ∑ 𝑛𝑎−2𝑛=0 𝑃1𝑛+∑ 𝑛

∞𝑛=0 𝑃0𝑛 + ∑ 𝑛𝑃2𝑛

∞𝑛=1 + (a-1)𝑃3 𝑎−1 (32)

Substituting for 𝑃0𝑛 ,𝑃2𝑛 (0 < n

8120 R. Sree Parimala and S. Palaniammal

𝐿𝑞 = [𝐴3{𝐻1(𝑟4) + 𝐺(𝑟4) +𝑘(𝑎−1)(1−𝑟4

𝑎)

1−𝑟4} + 𝐴2 {𝑓(𝑟2)[𝐻1(𝑟2) + 𝐺(𝑟2) + 𝐺1(𝑟2) +

1−𝑟2𝑎

1−𝑟2 ] +

𝑏𝑟2𝑏

1−𝑟2+

𝑟2𝑏+1

(1−𝑟2)2 }+ 𝐵1 { 𝑓(𝑟1)[ 𝐻1(𝑟1) + 𝐺(𝑟1) + 𝐺1(𝑟1) +

1−𝑟1𝑎

1−𝑟1]} +

𝐵2 {𝑓(𝑟3)[(𝐻1(𝑟3) + 𝐺(𝑟3) + 𝐺1(𝑟3))1−𝑟3

𝑎

1−𝑟3}+

2𝑏𝑟1

1−𝑟1 +

𝑟12

(1−𝑟1)2+

(2𝑏−1)2𝑏−𝑎(𝑎−1)

2 +

B[𝑏𝑟1

𝑏

1−𝑟1+

𝑟1𝑏+1

(1−𝑟1)2 ] ]𝑃0 𝑎−1 (33)

Here 𝐻1(𝑥) = 𝜃1

𝜆[

𝑎(𝑎−1)

2(1−𝑥) +

𝑥

(1−𝑥){𝐺(𝑥)}],

𝐺(𝑥) =𝑥(1−𝑥𝑎)−𝑎𝑥𝑎(1−𝑥)

(1−𝑥)2 and

𝐺1(𝑥) = 𝑥(1−𝑥𝑏)−𝑏𝑥𝑏(1−𝑥)

(1−𝑥)2

6.2 Probability that the server is busy (𝑷𝑩)

𝑃𝐵 = ( 𝐴21

1−𝑟2+ B

𝑟1𝑏

1−𝑟1+ 𝐵1

1−𝑟1𝑏

1−𝑟1+ 𝐵2

1−𝑟3𝑏

1−𝑟3)𝑃0𝑎−1

6.3 Probability that server is idle (𝑷𝟏)

𝑃1 = [𝐴3𝐻(𝑟4) + 𝐴2𝑓(𝑟2)𝐻(𝑟2) + 𝐵1𝑓(𝑟1)𝐻(𝑟1) + 𝐵2𝑓(𝑟3)𝐻(𝑟3) + 𝑟1

1 − 𝑟1+ (2𝑏 − 𝑎)]𝑃0𝑎−1

6.4 Probability that the server is in warm standby position (𝑷𝑾)

𝑃𝑊 = [𝐴3(1−𝑟4

𝑎−1)

1−𝑟4 + 𝐴2𝑓(𝑟2)

(1−𝑟2𝑎−1)

1−𝑟2+ 𝐵1𝑓(𝑟1)

(1−𝑟1𝑎−1)

1−𝑟1+ 𝐵2𝑓(𝑟3)

(1−𝑟3𝑎−1)

1−𝑟3]𝑃0𝑎−1

This completes analytic analysis of M/M (a,b)/1 queueing model.

7. Numerical Calculations

Numerical values of the expected queue length, the probability that the

server is busy, on vacation and in warm standby position are calculation for

various values of the parameters and that for ρ = λ 𝑏μ⁄ > 0.5, the mean queue

length increases rapidly.

Single server queueing model 8121

Table: 7.1 Mean Queue length for various values of a, 𝜃2, λ, 𝜃1 = 10, μ = 1 and b=50

𝜆 𝜃1= 10

Lq P1

a=10 a=20 a=30 a=40 a=1

0 a=20 a=30

a=4

0

10

𝜃2 = 1

45.76

5

50.89

0

55.98

0

57.00

9

0.27

9

0.253 0.232 0.20

1

20 58.36

4

61.63

8

64.98

7

67.87

9

0.43

3

0.410 0.384 0.33

2

30 71.87

6

79.12

1

80.00

3

84.09

9

0.67

7

0.600 0.581 0.56

7

40 127.0

8

131.6

9

139.7

6

145.8

6

0.80

0

0.791 0.788 0.76

1

10

𝜃2 =0.75

51.89

0

55.18

0

59.58

0

61.23

1

0.27

7

0.232 0.200 0.19

9

20 63.63

8

70.68

7

76.43

4

80.98

0

0.43

3

0.420 0.400 0.38

2

30 87.12

1

100.6

0

99.81

3

103.6

4

0.61

7

0.601 0.591 0.58

7

40 151.6

6

143.7

2

174.2

6

169.0

1

0.82

0

0.801 0.788 0.76

1

10

𝜃2 =0.5

55.98

0

59.58

0

63.98

7

67.90

8

0.24

7

0.238 0.210 0.19

9

20 74.98

7

64.43

4

85.00

3

90.87

6

0.42

3

0.420 0.390 0.38

0

30 103.0

3

109.0

3

119.7

6

122.6

1

0.63

7

0.622 0.590 0.56

5

40 175.7

6

184.2

0

187.5

6

191.0

5

0.82

0

0.817 0.788 0.76

1

10

𝜃2 =0.25

70.54

3

74.34

3

82.54

3

87.54

3

0.22

2

0.202 0.190 0.17

9

20 105.0

9

115.3

9

105.0

0

105.0

9

0.44

3

0.430 0.410 0.39

2

30 155.9

7

165.8

2

155.9

8

155.9

7

0.62

7

0.611 0.596 0.58

2

40 245.3

1

255.3

2

245.3

2

245.3

1

0.81

2

0.801 0.788 0.76

1

8122 R. Sree Parimala and S. Palaniammal

Table: 7.2 𝐿𝑞 for various values of a and λ when 𝜃1 = 2, 𝜃2 = 0.5, 𝑏 = 50 and μ = 1

𝜆 a=10 a=20 a=30 a=40

5 52.399 55.0993 58.2845 62.2809

10 60.620 61.6587 64.6054 72.9154

15 70.0918 72.7643 75.9769 80.1236

20 80.4365 85.8790 87.9896 91.4732

25 98.9994 99.4367 100.9076 112.8553

30 105.8493 112.9998 134.9080 141.0987

35 134.7202 136.8765 140.8761 145.3027

40 181.0023 183.0998 189.0087 191.2341

Table: 7.3 Performance measures for various values of the parameters

ρ 𝐿𝑞 PB PW PI

a =5

b=10 𝜃1 = 2

𝜃2 = 0.5 μ = 1

0.1 7.098 0.188 0.042 0.776

0.2 10.627 0.234 0.039 0.719

0.3 13.665 0.377 0.024 0.660

0.4 16.111 0.421 0.012 0.543

0.5 19.009 0.499 0.011 0.448

0.6 23.121 0.546 0.009 0.332

0.7 29.008 0.609 0.004 0.301

0.8 38.908 0.770 0.0001 0.201

0.9 61.202 0.812 0.027 0.099

a =10

b=20 𝜃1 = 5 𝜃2 = 2 μ = 1

0.1 12.009 0.179 0.023 0.887

0.2 20.565 0.224 0.021 0.764

0.3 21.998 0.321 0.018 0.643

0.4 24.408 0.465 0.014 0.594

0.5 26.009 0.504 0.012 0.483

0.6 31.142 0.599 0.009 0.359

0.7 38.999 0.623 0.007 0.216

0.8 55.868 0.755 0.004 0.211

0.9 99.435 0.845 0.001 0.112

Figure 1 LqVs λ for various values of θ2 when a = 20, b = 50, µ = 1 and θ1 = 10,

it displays the effect of expected queue length Lq. It is noted that the expected

queue length Lq increases with the increase in batch size a.

Single server queueing model 8123

Fig 1

8. Conclusion

In this present study, a M/M(a,b)/1 vacation queueing models with servers

delayed vacation and the state of switch over are considered. In general,

analytical solution of bulk service queueing models are extremely complicated.

An attempt has been made to study the analytical solution of single server

queueing model in which 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. The explicit expressions for expected queue length may

be helpful in setting traffic management strategies based on performance indices.

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Received: October 2, 2014; Published: November 19, 2014

http://dx.doi.org/10.1016/0305-0548%2882%2990018-1http://dx.doi.org/10.1007/bf01149327http://dx.doi.org/10.1016/s0360-8352%2802%2900235-8http://dx.doi.org/10.1214/aoms/1177698869http://dx.doi.org/10.1287/opre.42.5.926