Analysis of a Single Server Markovian Queueing System with Working Vacations and Vacation Interruptions with Setup Times

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Analysis of a Single Server Markovian Queueing System with Working Vacations and Vacation Interruptions

with Setup Times

P. Manoharan

*

and T. Jeeva Department of Mathematics,

Annamalai University, Annamalai Nagar, Tamilnadu, INDIA.

email:manomaths.hari@gmail.com, jeevasamarmaths@gmail.com

(Received on: September 28, 2018) ABSTRACT

In this paper, we study an Analysis of a single server Markovian queueing system with single working vacations and vacation interruptions with setup times.

The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy: the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. such policy is called vacation interruption. When a vacation ends, the system becomes empty, the server going to be set-up period. During a closed-down period, an arriving customer cannot be served immediately and experiences a period of set-up time. We connect the above mentioned two polices and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the set-up period. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems.

Keywords: Markovian queue, Working vacation, Vacation Interruption, Set up time, Matrix-geometric method, Stochastic decomposition.

1. INTRODUCTION

Working vacation is a kind of semi-vacation policy was introduced by Servi and Finn

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a customer is served at a lower rate rather than completely stopping the service during a

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vacation. In the classical vacation queueing models, during the vacation period the server doesn't continue on the original work and such policy may cause the loss or dissatisfaction of the customers. For the working vacation policy, the server can still work during the vacation and may accomplish other assistant work simultaneously. So, the working vacation is more reasonable than the classical vacation in some cases. The details can be seen in the monograpraphs of Takagi

12

and Tian and Zhang

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the survey of Doshi

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. However, in these models, the server stops the original work in the vacation period and cannot come back to the regular busy period until the vacation period ends. Subsequently, Wu and Takagi

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generalized the model in to an M/G/1 queue with general working vacations. Baba

1

studied a GI/M/1 queue with working vacations by using the matrix analytic method. Banik et al.

2

analyzed the GI/M/1/N queue with working vacations. Liu et al.

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established a stochastic decomposition result in the M/M/1 queue with working vacations. Li et al.

8

established the conditional stochastic decomposition result in the M/G/1 queue with exponentially working vacations using matrix analytic approach. Shakir Majid and P. Manoharan

10

analyzed the M/M/1 queue with single working vacation and vacation interruption using matrix geometric method.For example, when the number of customers exceeds the special value and if the server continues to take the vacation, the cost of waiting customers and providing service in the vacation period will be large. In 2007, Li and Tian

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first introduced vacation interruption policy and studied an M/M/1 queue. Next, Li et al.

6

analyzed the GI/M/1 queue. Using the method of a supplementary variable, Zhang and Hou

16

considered an M/G/1 queue with working vacations and vacation interruption. Sreenivasan et al.

11

studied an MAP/PH/1 queue with working vacations, vacation and N-Policy.

Power-Saving in ICT systems is an important issue because ICT devices consume a large amount of energy. One simple method is to turn off on idle device and to switch it on again when some jobs arrive. This is because in the current technology idle devices still consume of peak processing a job

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on the other hand, a quick response in crucial for delay sensitive applications. An off server needs some setup time in order to be active during which the server consumes energy but cannot process a job.Thus, there is a trade-off can be analyzed using single server queueing models with setup times which are extensively studied in the literature

4,13

. Our models are suitable for a down link of a mobile station with a power saving mode. A mobile station receives data from a base station. Arriving messages are stored in the base station and the mobile station downloads these messages from the base station upon the completion of a download, if there are no messages in the base station the mobile station is turned off in order to save energy. However, when a message arrives, the base station sends a signal in order to wake up the mobile station. The mobile station needs some random setup time to be active so as to receive waiting messages.

In this paper, we consider an M/M/1 queue with vacation interruptions and working

vacations with setup times. The server can stop the vacation ones some indices of the system,

such as the number of customers, achieve a certain value in the vacation period. Certainly, it

is possible for the server to take an interrupted vacation, so we call this policy vacation

interruption. When a vacation ends, the server begins a closed-down period. The working

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vacation, vacation interruption and set-up times are connected in this paper and the server enters into vacation when there are no customers and he can take service at a lower rate in the vacation period, the server will come back to the set-up period. In terms of the quasi birth and death process and matrix-geometric solution method, we provide expressions for the distributions of the queue length and the waiting time. Furthermore, we indicate the stochastic decomposition structures.There is the specific relationship between the results of this model and the results of the classical M/M/1 queues without vacations.

The rest of this paper is organized as follows. In section 2, we describe the quasi birth and death process of the model. In section 3 obtains the distribution of the queue length and the state probabilities of the server in steady state. Finally in section 4, we indicate the stochastic decomposition structures of the queue length and the waiting time such a system.

2. QUASI BIRTH AND DEATH PROCESS MODEL

Consider a classical Markovian queue with an arrival rate ๐œ† and a service rate ๐œ‡

๐‘

. Upon the completion of a service, if there are no customers in the system, the server begins a vacation and the vacation duration V follows an exponential distribution with the parameter ฮธ. During a vacation period, arriving customers can be served at a mean rate of ๐œ‡

๐‘ฃ

. Upon the completion of a service in the vacation, if there are also customers in the queue, the server ends the vacation and comes back to the normal working level. When a vacation ends, the server begins a closed-down period. During a closed-down period, an arriving customer cannot be served immediately and experiences a period a set-up time, set-up duration S follows an exponential distribution with parameter ๐œ‚ and a regular busy period starts after a set-up period.

In this service discipline, the server may come back from the vacation without completing the vacation, and the server can only go on vacations if there is no customer left in the system upon the completion of a service. Meanwhile, the vacation service rate can be only applied to the first customer arrived during a vacation period. we assume that the inter-arrival times, the service times,and the working vacation times and set-up times are mutually independent. In addition, the service discipline is first in first out (FIFO).

Let Q(t) be the number of customers in the system at time t and let.

J(t)=

t at time period busy a in is regular a

in is system the

2,

t, at time period up - set a in is system the

1, ,

t, at time period vacation working

a in is system the

0,

Then {(๐‘„(๐‘ก), ๐ฝ(๐‘ก)); ๐‘ก โ‰ฅ 0} is a quasi birth and death process (QBD) with the state space.

ฮฉ = (๐‘˜, ๐‘—), ๐‘˜ โ‰ฅ 1, ๐‘— = {0,1,2} โˆช {(0,0), (0,1)}

where state (0,1) denotes that the system is in a set-up state, state (๐พ, 0), ๐พ โ‰ฅ 0 indicates that

the system is in vacation and there are K customers in the system are in set-up state and there

are K customers in the queue, state (๐พ, 1), ๐พ โ‰ฅ 1 indicates that the system is in set-up state

and there are K customers in the queue, state (๐พ, 2), ๐พ โ‰ฅ 1 indicates that the system is in

regular busy period state and there are K customers in the queue.

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Using the lexicographical sequence for the states the infinitesimal generator can be written as

Q=

(

๐ต

00

๐ด

00

โ‹ฏ

๐ถ

00

๐ต ๐ด โ‹ฏ

๐ถ ๐ต ๐ด โ‹ฏ

๐ถ ๐ต ๐ด โ‹ฏ

โ‹ฎ โ‹ฎ โ‹ฎ

) Where

๐ต

00

= ( 0

)

( ),๐ด

00

=

0 0

0 0

,, ,

C

00

=

0 0 0

0

b v

,

B=

) (

0 0

) ( 0

0 )

+ + (

-

v

b

,A=

0 0

0 0

0 0

,C=

b v

0 0

0 0 0

0 0

.

To analyze this QBD process, it is necessary to solve for the minimal non-negative solution of the matrix quadratic equation ๐‘…

2

๐ถ + ๐‘…๐ต + ๐ด = 0โ€ฆ.(1). This solution is called the rate matrix and denoted by R. Obviously, we have

Lemma 1:If ๐œŒ = ๐œ†(๐œ‡

๐‘

)

โˆ’1

< 1, the matrix equation (1) has the minimal non-negative solution

๐‘… =

0 0

) 1 0 (

) 1 (

) 0 (

r r r

b b

v

(2)where๐œ‚ =

๐œ†+ฮธ+๐œ‡๐œ†

๐‘ฃ

, ๐‘Ÿ =

๐œ†+๐›ฝ๐œ†

.

Proof. Since the matrices A,B and C are all Upper triangular, therefore R is upper triangular

๐‘… =

(

33

23 22

13 12 11

0 0 0

r r r

r r r

)

substituting ๐‘…

2

and R into (1), we get the following set of equations.

{

๐œ† โˆ’ ๐‘Ÿ

11

(๐œ† + ๐œ‡

๐‘ฃ

+ ฮธ) = 0

โˆ’๐‘Ÿ

12

= 0

๐œ‡

๐‘ฃ

๐‘Ÿ

112

+ ๐œ‡

๐‘

(๐‘Ÿ

11

๐‘Ÿ

13

+ ๐‘Ÿ

12

๐‘Ÿ

23

+ ๐‘Ÿ

13

๐‘Ÿ

33

) + ฮธ๐‘Ÿ

11

+ ๐›ฝ๐‘Ÿ

12

โˆ’ ๐‘Ÿ

13

(๐œ† + ๐œ‡

๐‘

) = 0 ๐œ† โˆ’ ๐‘Ÿ

22

(๐œ† + ๐›ฝ) = 0

๐œ‡

๐‘

(๐‘Ÿ

23

๐‘Ÿ

22

+ ๐‘Ÿ

23

๐‘Ÿ

33

) + ๐›ฝ๐‘Ÿ

22

โˆ’ ๐‘Ÿ

23

(๐œ† + ๐œ‡

๐‘

) = 0 ๐œ‡

๐‘

๐‘Ÿ

332

โˆ’ ๐‘Ÿ

33

(๐œ† + ๐œ‡) + ๐œ† = 0

(3)

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To obtain the minimal non-negative solution of (1),by using the equation๐‘Ÿ

22

=

๐œ†

๐œ†+๐›ฝ

= r.From the first equation we get ๐›ฝ =

๐œ†+๐œ‡๐œ†

๐‘ฃ+ฮธ

= ๐‘Ÿ

11

, ๐‘Ÿ

12

= 0 in the second equation of (3), ๐‘Ÿ

23

=

๐›ฝ๐‘Ÿ

๐œ‡๐‘(1โˆ’๐‘Ÿ)

= ๐œŒ in the fifth equation of (3), ๐‘Ÿ

33

= ๐œŒ in the last equation of (3), ๐‘Ÿ

13

=

๐œ‚(๐œ‚๐œ‡๐œ‡ ๐‘ฃ+ฮธ)

๐‘(1โˆ’๐œ‚)

in the third equation of(3).

Theorem 1. The QBD process {(๐‘„(๐‘ก), ๐ฝ(๐‘ก)); ๐‘ก โ‰ฅ 0} is positive recurrent if and only if ๐œŒ < 1.

Proof. In terms of the matrix geometric method of Neuts(1981), ๐ถ[๐‘…] =

B RC C

A B

00

00 00

๐ถ[๐‘…] = (

โˆ’(๐œ† + ) ๐œ† 0 0

0 โˆ’๐œ† 0 ๐œ† 0

๐œ‡

๐‘ฃ

0 โˆ’(๐œ† + ๐œ‡

๐‘ฃ

+ ) 0 (๐œ† + ๐œƒ)

0 0 0 โˆ’(๐œ† + ๐›ฝ) (๐œ† + ๐›ฝ)

๐œ‡

๐‘

0 0 0 โˆ’๐œ‡

๐‘

)

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๐ถ[๐‘…] is an irreducible and aperiodic generator with finite state. Therefore (๐‘ฅ

0

, ๐‘ฅ

1

, ๐‘ฅ

2

, ๐‘ฅ

3

, ๐‘ฅ

4

, ๐‘ฅ

5

, )๐ถ(๐‘…) = 0 has positive solution. Thus,process {(๐‘„(๐‘ก), ๐ฝ(๐‘ก)); ๐‘ก โ‰ฅ 0} is positive recurrent if and only if ๐‘†๐‘ƒ(๐‘…) = ๐‘š๐‘Ž๐‘ฅ(๐›ฝ, ๐‘Ÿ, ๐œŒ) < 1. note that 0 < ๐‘Ÿ < 1, the above relation means that ๐œŒ < 1.

3. QUEUE LENGTH DISTRIBUTION

If ๐œŒ < 1, let(๐‘„, ๐ฝ) be the stationary limit of the ๐‘„๐ต๐ท process {(๐‘„(๐‘ก), ๐ฝ(๐‘ก)); ๐‘ก โ‰ฅ 0}.Denote ๐œ‹

๐‘˜

= { (๐œ‹

00

, ๐œ‹

01

), ๐‘˜ = 0,

(๐œ‹

๐‘˜0

, ๐œ‹

๐‘˜1

, ๐œ‹

๐‘˜2

), ๐‘˜ โ‰ฅ 1,

๐œ‹

๐‘˜๐‘—

= ๐‘ƒ{๐‘„ = ๐‘˜, ๐ฝ = ๐‘—, ๐‘ก โ‰ฅ 0} = ๐‘ก โ†’

๐‘™๐‘ก

โˆž{๐‘ƒ๐‘„(๐‘ก) = ๐‘˜, ๐ฝ(๐‘ก) = ๐‘—, ๐‘ก โ‰ฅ 0}. (k,j) โˆˆ ฮฉUsing quasi birth and death process and the matrix-geometric solution method,it is easy to get the following theorem.

Theorem 2. If ๐œŒ < 1, the joint probability distribution of (๐‘„, ๐ฝ) is

. 1 ) ],

(

) ) (

1 ( ) ) (

) ( 1 (

) [ (

, 0 , 0 ,

1

1

0

1 1

j b

1 - k

0 j

1 1

j b

v 00

2

00, 1 k1

00 k0

k

r r r k

k

k b

k

j

k j k k

j k k

k k

(5)

(6)

Where,

00

= [

1โˆ’๐œ‚1

+

ฮธ๐›ฝ

+

ฮธ(ฮธ+๐œ‚๐œ‡๐œ‡ ๐‘ฃ)

๐‘(1โˆ’๐œ‚)2

+

(๐œ†+๐›ฝ)๐œ‡ ฮธ๐œ‚๐‘Ÿ

๐‘(1โˆ’๐‘Ÿ)2(1โˆ’๐œŒ)

+

(๐œ†+ฮธ)๐œ‚+ฮธ๐œ‡

๐‘(1โˆ’๐œŒ)

]

โˆ’1

. Proof.

With the matrix-geometric solution method by using (Neuts,1981), we have

๐œ‹

๐‘˜

= (๐œ‹

๐‘˜0

, ๐œ‹

๐‘˜1

, ๐œ‹

๐‘˜2

) = (๐œ‹

10

, ๐œ‹

11

, ๐œ‹

12

)๐‘…

๐‘˜โˆ’1

, ๐‘˜ โ‰ฅ 1. (6) and ๐œ‹

0

,๐œ‹

1

,๐œ‹

2

satisfy the set of equations(๐œ‹

00

, ๐œ‹

01

, ๐œ‹

11

, ๐œ‹

12

)๐ถ[๐‘…] = 0 substituting ๐ถ[๐‘…] in (5) into the above relation, we obtain

{

โˆ’(๐œ† + ฮธ)๐œ‹

00

+ ๐œ‡

๐‘ฃ

๐œ‹

10

+ ๐œ‡

๐‘

๐œ‹

12

= 0 ฮธ๐œ‹

00

โˆ’ ๐œ†๐œ‹

01

= 0

๐œ†๐œ‹

00

โˆ’ (๐œ† + ฮธ + ๐œ‡

๐‘ฃ

)๐œ‹

10

= 0 ๐œ†๐œ‹

01

โˆ’ (๐œ† + ๐›ฝ)๐œ‹

11

= 0

(๐œ† + ฮธ)๐œ‹

10

+ (๐œ† + ๐›ฝ)๐œ‹

11

โˆ’ ๐œ‡

๐‘

๐œ‹

12

= 0 Taking ๐œ‹

00

as a known constant, we get

(๐œ‹

00

, ๐œ‹

00

, ๐œ‹

01

, ๐œ‹

10

, ๐œ‹

11

, ๐œ‹

12

) = ๐œ‹

00

(1,

ฮธ๐œ†

, ๐›ฝ,

๐œ†+๐›ฝฮธ

,

(๐œ†+ฮธ)๐œ‚+ฮธ๐œ‡

๐‘

)

and ๐‘…

๐‘˜

= (

๐œ‚

๐‘˜

0

๐œ‚(ฮธ+๐œ‚๐œ‡๐‘ฃ)

๐œ‡๐‘(1โˆ’๐œ‚)

โˆ‘

๐‘˜โˆ’1๐‘–=0

๐œ‚

๐‘—

๐œŒ

๐‘˜โˆ’1โˆ’๐‘—

0 ๐‘Ÿ

๐‘˜ ๐›ฝ๐‘Ÿ

๐œ‡๐‘(1โˆ’๐‘Ÿ)

โˆ‘

๐‘˜โˆ’1๐‘–=0

๐‘Ÿ

๐‘—

๐œŒ

๐‘˜โˆ’1โˆ’๐‘—

0 0 ๐œŒ

๐‘˜

)

,๐‘˜ โ‰ฅ 1

Substituting (๐œ‹

10

, ๐œ‹

11

, ๐œ‹

12

) and ๐‘…

๐‘˜โˆ’1

into (6), we obtain (5). Finally, ๐œ‹

00

can be determined by the normalization condition.with(5), the probabilities of the server in various state are as follows, respectively

P{J=0} = โˆ‘

โˆž๐‘˜=0

๐œ‹

๐‘˜0

= ๐œ‹

00 1

1โˆ’๐œ‚

P{the server is in a closed-down period} = ๐œ‹

01

= ๐‘˜

๐œ†

P{the server is in set-up period} = โˆ‘

โˆž๐‘˜=0

๐œ‹

๐‘˜1

= ๐‘˜

๐›ฝ

(7) P{the server is in regular service period} = โˆ‘

โˆž๐‘˜=0

๐œ‹

๐‘˜2

= ๐œ‹00[๐œ‚(๐œƒ + ๐œ‚๐œ‡๐‘ฃ) ๐œ‡๐‘(1 โˆ’ ๐›ฝ)

๐œ‚

(1 โˆ’ ๐œ‚)(1 โˆ’ ๐œŒ)+ ๐œƒ ๐œ† + ๐›ฝ

๐›ฝ ๐œ‡๐‘(1 โˆ’ ๐‘Ÿ)

๐‘Ÿ

(1 โˆ’ ๐‘Ÿ)(1 โˆ’ ๐œŒ)+(๐œ† + )๐œ‚ + ๐œ‡๐‘

1 1 โˆ’ ๐œŒ]

4. STOCHASTIC DECOMPOSITION RESULTS

The classical vacation queues have the stochastic decomposition property for the queue length and the sojourn time, denoted by Q and W, respectively.

Firstly, We discuss the distribution of the number of customers in the system.

Theorem 3. If ๐œŒ < 1 the stationary queue length Q can be decomposed into the sum of two

independent random variables: ๐‘„ = ๐‘„

0

+ ๐‘„

๐‘‘

, where ๐‘„

0

is the stationary queue length of a

classic ๐‘€/๐‘€/1 queue without vacations, and follows a geometric distribution with the

parameter 1 โˆ’ ๐œŒ; the additional queue length ๐‘„

๐‘‘

has a modified geometric distribution.

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๐‘ƒ{๐‘„

๐‘‘

= ๐‘˜} = {

๐พ๐›ฟ

1

, ๐‘˜ = 0, ๐พ๐›ฟ

2

, ๐‘˜ = 1,

๐พ๐›ฟ

3

(1 โˆ’ ๐œ‚)๐›ฝ

๐‘˜โˆ’1

+ ๐‘˜(1 โˆ’ ๐‘Ÿ)๐‘Ÿ

๐‘˜โˆ’1

, ๐‘˜ โ‰ฅ 2.

(8) ๐‘Ÿ =

๐œ†+๐›ฝ๐œ†

,

๐›ฟ

1

=

(๐œ†+ฮธ)

๐œ†

(1 โˆ’ ๐‘Ÿ)(1 โˆ’ ๐œ‚), ๐›ฟ

2

= [(๐œ‚ โˆ’ ๐œŒ) +

ฮธ

๐œ†

(๐‘Ÿ โˆ’ ๐œŒ) +

(๐œ†+ฮธ)๐œ‚+ฮธ

๐œ‡๐‘

(1 โˆ’ ๐‘Ÿ)(1 โˆ’ ๐œ‚)], ๐›ฟ

3

= [(๐œ‚ โˆ’ ๐œŒ) +

๐œ‚(ฮธ+๐œ‚๐œ‡๐‘ฃ)

๐œ‡๐‘(1โˆ’๐œ‚)

](1 โˆ’ ๐‘Ÿ) ๐›ฟ

4

= [

ฮธ๐œ†

(๐‘Ÿ โˆ’ ๐œŒ) +

(๐œ†+๐›ฝ)๐œ‡ฮธ๐›ฝ

๐‘(1โˆ’๐‘Ÿ)

](1 โˆ’ ๐œ‚) ๐พ =

(1โˆ’๐‘Ÿ)(1โˆ’๐œŒ)(1โˆ’๐œ‚)๐œ‹00

Proof. Denote

๐œ†

๐œ†+๐›ฝ

= ๐‘Ÿ with (5), the probability generating function of Q can be written as

๐‘„(๐‘ง) = โˆ‘โˆž๐‘˜=0๐œ‹๐‘˜0๐‘ง๐‘˜+ โˆ‘โˆž๐‘˜=0๐œ‹๐‘˜1๐‘ง๐‘˜+ โˆ‘โˆž๐‘˜=0๐œ‹๐‘˜2๐‘ง๐‘˜

= ๐œ‹00[1โˆ’๐œ‚๐‘ง1 +ฮธ๐œ†1โˆ’๐‘Ÿ๐‘ง1 +๐œ‡ ๐œ‚2(ฮธ+๐œ‚๐œ‡๐‘ฃ)๐‘ง2

๐‘(1โˆ’๐œ‚)(1โˆ’๐œ‚๐‘ง)(1โˆ’๐œŒ๐‘ง)+(๐œ†+๐›ฝ)๐œ‡ ฮธ๐›ฝ๐‘Ÿ๐‘ง2

๐‘(1โˆ’๐‘Ÿ)(1โˆ’๐‘Ÿ๐‘ง)(1โˆ’๐œŒ๐‘ง)+((๐œ†+ฮธ)๐œ‚+ฮธ)๐‘ง(1โˆ’๐‘Ÿ)(1โˆ’๐œ‚)

๐œ‡๐‘ ]

๐‘„(๐‘ง) =

(1โˆ’๐œŒ)

1โˆ’๐œŒ๐‘ง

๐พ[

(1โˆ’๐‘Ÿ)(1โˆ’๐œ‚)(1โˆ’๐œ‚๐‘ง)

+

ฮธ

๐œ†

(1โˆ’๐‘Ÿ)(1โˆ’๐œ‚)(1โˆ’๐œŒ๐‘ง)

1โˆ’๐‘Ÿ๐‘ง

+

๐œ‚2(ฮธ+๐œ‚๐œ‡๐‘ฃ)(1โˆ’๐‘Ÿ)๐‘ง2

๐œ‡๐‘(1โˆ’๐œ‚๐‘ง)

+

(๐œ†+๐›ฝ)๐œ‡ฮธ๐›ฝ

๐‘

๐‘Ÿ(1โˆ’๐œ‚)๐‘ง2

(1โˆ’๐‘Ÿ๐‘ง)

+

(๐œ†+ฮธ)๐œ‚+ฮธ๐œ‡

๐‘

๐‘ง(1 โˆ’ ๐‘Ÿ)

2

(1 โˆ’ ๐œ‚)

2

]

1โˆ’๐‘Ÿ

1โˆ’๐‘Ÿ๐‘ง(1 โˆ’ ๐‘ง๐œŒ) = (1 โˆ’ ๐‘Ÿ) + (๐‘Ÿ โˆ’ ๐œŒ)(1โˆ’๐‘Ÿ)๐‘ง1โˆ’๐‘Ÿ๐‘ง

1โˆ’๐œ‚

1โˆ’๐œ‚๐‘ง(1 โˆ’ ๐œŒ๐‘ง) = (1 โˆ’ ๐œ‚) + (๐œ‚ โˆ’ ๐œŒ)(1โˆ’๐œ‚)๐‘ง1โˆ’๐œ‚๐‘ง

Q(z) =1โˆ’๐œŒ๐‘ง1โˆ’๐œŒ ๐พ[(1 โˆ’ ๐‘Ÿ)(1 โˆ’ ๐œ‚) + (1 โˆ’ ๐‘Ÿ)(๐œ‚ โˆ’ ๐œŒ)(1โˆ’๐œ‚)๐‘ง1โˆ’๐œ‚๐‘ง +ฮธ๐œ†(1 โˆ’ ๐œ‚)(1 โˆ’ ๐‘Ÿ) +ฮธ๐œ†(1 โˆ’ ๐œ‚)(๐‘Ÿ โˆ’ ๐œŒ)(1โˆ’๐‘Ÿ)๐‘ง1โˆ’๐‘Ÿ๐‘ง +๐œ‚(ฮธ+๐œ‚๐œ‡๐œ‡ ๐‘ฃ)(1โˆ’๐‘Ÿ)

๐‘(1โˆ’๐œ‚)

(1โˆ’๐œ‚)๐œ‚ 1โˆ’๐œ‚๐‘ง ๐‘ง2 +(๐œ†+๐›ฝ)๐œ‡ฮธ๐›ฝ(1โˆ’๐›ฝ)

๐‘(1โˆ’๐‘Ÿ) (1โˆ’๐‘Ÿ)๐‘Ÿ

1โˆ’๐‘Ÿ๐‘ง ๐‘ง2+(๐œ†+ฮธ)๐œ‚+ฮธ๐œ‡

๐‘ ๐‘ง(1 โˆ’ ๐‘Ÿ)2(1 โˆ’ ๐›ฝ2)]

=1โˆ’๐œŒ๐‘ง1โˆ’๐œŒ ๐พ{๐›ฟ1+ ๐›ฟ2๐‘ง + ๐›ฟ3๐œ‚(1โˆ’๐œ‚)

1โˆ’๐œ‚๐‘ง ๐‘ง2+ ๐›ฟ4๐‘Ÿ(1โˆ’๐‘Ÿ) 1โˆ’๐‘Ÿ๐‘ง ๐‘ง2} =1โˆ’๐œŒ๐‘ง1โˆ’๐œŒ ๐‘„๐‘‘(๐‘ง)

๐ธ(๐‘„๐‘‘) = ๐พ[๐›ฟ2+2๐œ‚โˆ’๐œ‚1โˆ’๐œ‚2๐›ฟ3+2๐‘Ÿโˆ’๐‘Ÿ1โˆ’๐‘Ÿ2๐›ฟ4] ๐ธ(๐‘„) =1โˆ’๐œŒ๐œŒ + ๐ธ(๐‘„๐‘‘)

5. WAITING TIME DISTRIBUTION

Denote the waiting time of a customer in the system by W, we have the following

stochastic decomposition results.

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Theorem4. If ๐œŒ < 1 and ๐œ‡

๐‘

> ๐œ‡

๐‘ฃ

the stationary waiting time W can be decomposed into the sum of two independent random variables: ๐‘Š = ๐‘Š

0

+ ๐‘Š

๐‘‘

, where ๐‘Š

0

is the virtual time of a customer in a corresponding classic M/M/1 queue and has an exponential distribution with parameter ๐œ‡

๐‘

(1 โˆ’ ๐œŒ); the additional delay ๐‘Š

๐‘‘

has the LST

๐‘Š

๐‘‘โˆ—

(๐‘ ) = ๐‘˜[๐›ผ

1

+ ๐›ผ

2 ๐œ‡๐‘ฃ+ฮธ

๐œ‡๐‘ฃ+ฮธ+๐‘ 

+ ๐›ผ

3 ๐›ฝ

๐›ฝ+๐‘ 

]. (9) ๐›ผ

1

= ๐›ฟ

1

+ ๐›ฟ

2

โˆ’

(1โˆ’๐œ‚)(2๐œ†+๐œ‡๐‘ฃ+ฮธ)

๐œ†

๐›ฟ

3

โˆ’

(1โˆ’๐‘Ÿ)(2๐œ†+๐›ฝ)

๐œ†

๐›ฟ

4

,๐›ผ

2

=

1

๐œ‚

๐›ฟ

3

, ๐›ผ

3

=

1

๐‘Ÿ

๐›ฟ

4

.

Proof.The M/M/1 model, the queue length and the waiting time of a customer have the relationship below as in the other vacation queues: ๐‘„(๐‘ง) = ๐‘Š

โˆ—

(๐œ†(1 โˆ’ ๐‘ง))From the above Theorem 4, the PGF of the number of customers Q Can be written as

๐‘„(๐‘) =

1โˆ’๐œŒ

1โˆ’๐œŒ๐‘ง

๐พ {๐›ฟ

1

+ ๐›ฟ

2

๐‘ง + ๐›ฟ

3๐œ‚(1โˆ’๐œ‚)

1โˆ’๐œ‚๐‘ง

๐‘ง

2

+ ๐›ฟ

4๐‘Ÿ(1โˆ’๐‘Ÿ)

1โˆ’๐‘Ÿ๐‘ง

๐‘ง

2

} (10) Taking ๐‘ง = 1 โˆ’

๐‘ 

๐œ†

in (8), note that

๐‘ง2

(๐œ† + ๐›ฝ) โˆ’ ๐œ†๐‘ง|๐‘ง=1โˆ’๐‘ 

๐œ†=(1 โˆ’๐‘ ๐œ†)2 ๐›ฝ + ๐‘  = 1

๐œ†2[(๐œ† + ๐›ฝ)2

๐›ฝ + ๐‘  โˆ’ (2๐œ† + ๐›ฝ) + ๐‘ ],

๐‘ง2 1โˆ’๐œ‚๐‘ง|๐‘ง=1โˆ’๐‘ 

๐œ†= (1โˆ’

๐‘  ๐œ†)2

๐œ‡๐‘ฃ+ฮธ+๐‘ =๐œ†12[(๐œ†+๐œ‡๐œ‡ ๐‘ฃ+ฮธ)2

๐‘ฃ+ฮธ+๐‘  โˆ’ (2๐œ† + ๐œ‡๐‘ฃ+ ฮธ) + ๐‘ ], 1โˆ’๐œŒ๐‘ง1โˆ’๐œŒ|๐‘ง=1โˆ’s

๐œ†= ๐œ‡๐‘(1โˆ’๐œŒ)

๐œ‡๐‘(1โˆ’๐œŒ)+๐‘ 

Substituting the above results into (8), we have

๐‘Šโˆ—(๐‘ ) = ๐œ‡๐‘(1 โˆ’ ๐œŒ)

๐œ‡๐‘(1 โˆ’ ๐œŒ) + ๐‘ ๐พ{๐›ฟ1+ ๐›ฟ2(1 โˆ’๐‘ 

๐œ†) + ๐›ฟ3(1 โˆ’ ๐œ‚)

๐œ† [(๐œ† + ๐œ‡๐‘ฃ+ ๐œƒ)2

๐œ‡๐‘ฃ+ ๐œƒ๐‘ฃ+ ๐‘  โˆ’ (2๐œ† + ๐œ‡๐‘ฃ+ ฮธ) + ๐‘ ]

+๐›ฟ4[(๐œ† + ๐›ฝ)2

๐›ฝ + ๐‘  โˆ’ (2๐œ† + ๐›ฝ) + ๐‘ ]}

๐‘Š

โˆ—

(๐‘†) =

๐œ‡๐‘(1โˆ’๐œŒ)

๐œ‡๐‘(1โˆ’๐œŒ)+๐‘†

๐‘Š

๐‘‘โˆ—

(๐‘ ), where Note,

๐ธ(๐‘Š

๐‘‘

) = ๐พ{๐›ผ

2

1

๐œ‡

๐‘ฃ

+ ๐œƒ + ๐›ผ

3

1 ๐›ฝ }

๐ธ(๐‘Š) = 1

๐œ‡

๐‘

(1 โˆ’ ๐œŒ) + ๐ธ(๐‘Š

๐‘‘

) 6. NUMERICAL RESULTS

In section 4 and 5, we obtain the mean system length and the mean waiting time of an arbitrary customer. In this section, we illustrate the results obtained above numerically and discuss the effect of system parameters on system performance indices.

We assume that the service rate ๐œ‡

๐‘

in a regular busy period equals 0.9, arrival rate ๐œ† equals 0.2, at the same time, we assume that set-up time is an exponential distributed random variable with mean ๐œ‚ = 0.6. We plot the values of mean queue length ๐ธ(๐‘„) and mean waiting time ๐ธ(๐‘Š) by changing the service rate ๐œ‡

๐‘ฃ

in a vacation period: Meanwhile, to investigate the influence of the mean length

1

๐œƒ

of a vacation.

(9)

FIGURES

Figure 1 The changing curve of E(Q) Figure 2 The changing curve of E(W)

7. CONCLUSIONS

In this paper, we propose a model with single working vacation interruptions and set- up times. Using quasi birth and death process and matrix-geometric solution method, the distributions for the stationary queue length and waiting time of a customer in the system are derived. Stochastic decomposition structures of stationary indices are obtained; meanwhile, the distributions of the additional queue length and additional delay are gained. Finally, numerical examples demonstrate the results.

REFERENCES

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Journal of systems science and systems Engineering, 1(16), 121-127 (2007).

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8. Li,J., Tian,N., Zhang,Z.G., and Lu,H.P. Analysis of the M/G/1 queue with exponential working vacations-A matrix Analytic Approch,'' Queueing Systems, (61),1139-166 (2009).

9. Liu,W.,Xu,X., and Tian,N. Some results on the M/M/1 queue with working vacations, Operations Research Letters, 1(1), 595-600 (2007).

10. Shakir Majid., and Manoharan. P. Analysis of the M/M/1 queue with single working vacation and vacation interruption, International Journal of Mathematics Trends and Technogy, 1(47) (2017).

11. Servi,L.D., and Finn,S.G. M/M/1 queues with working vacations(M/M/1/WV), Performance Evaluation, 1(50), 41-52 (2002).

12. Sreenivasan,C., Chakravarthy,S.R., and Krishnamoorthy, A. MAP/PH/1 queue with working vacations, vacation interupptions and N Policy, Applied Mathematical Modelling, Applications, 6(37), 3879-3893 (2013).

13. Takagi, H. Queuing Analysis, A Foundation of performance Analysis, vol.1 of vacation and priority,'' Elsevier science publishers, B.V., Amsterdam.The Netherlands (1991).

14. Takagi,H. Priority queues with set-up times, Operations Research,4(38),667-677 (1990).

15. Tian, N., and Zhang, G. Vacation queueing Models, Theory and Applications, Springer- Verlay, New york (2006).

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vacation interruption, Journal of Computational and Applied Mathematics,10 (234), 2977-

2985 (2010).

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