Research Article Fluid Queue Driven by an //1 Queue Subject to …downloads.hindawi.com/journals/aor/2016/2673017.pdf · 2019. 7. 30. · Fluid queues driven by an //1 queueing model

Research ArticleFluid Queue Driven by an𝑀/𝑀/1 QueueSubject to Bernoulli-Schedule-ControlledVacation and Vacation Interruption

Kolinjivadi Viswanathan Vijayashree and Atlimuthu Anjuka

Department of Mathematics, Anna University, Chennai 600025, India

Correspondence should be addressed to Kolinjivadi Viswanathan Vijayashree; [email protected]

Received 30 November 2015; Revised 12 March 2016; Accepted 16 March 2016

Academic Editor: Yi-Kuei Lin

Copyright © 2016 K. V. Vijayashree and A. Anjuka. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper deals with the stationary analysis of a fluid queue driven by an𝑀/𝑀/1 queueing model subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption. The model under consideration can be viewed as a quasi-birth and death process.The governing system of differential difference equations is solved using matrix-geometric method in the Laplacian domain. Theresulting solutions are then inverted to obtain an explicit expression for the joint steady state probabilities of the content of the bufferand the state of the background queueingmodel. Numerical illustrations are added to depict the convergence of the stationary buffercontent distribution to one subject to suitable stability conditions.

1. Introduction

In many real time situations, the server in the backgroundqueueing model may become unavailable for a randomperiod of time to perform a secondary task, when there are nocustomers in thewaiting line at the service completion epoch.Such period of server absence is termed as server vacation.Queueing models subject to various vacation policies areof interest to researchers in recent times owing to theirwidespread applicability.

There are different types of vacation queueing systems.In the single vacation scheme, the server takes a vacation ofsome random duration when the queue is empty. At the endof the vacation, the server returns to the queue. The serverresumes service if there is at least one customer waiting uponhis return from vacation. However, if the queue is emptyon the server’s return, the server waits to complete a busyperiod. In the multiple vacation scheme, if the server returnsfrom a vacation and finds the queue empty, he immediatelycommences another vacation. If there is at least one waitingcustomer, then he will commence the service. Queueingmodels subject to single or multiple exponential vacation

are apt to model many practical scenarios [1–3]. However,a better modeling assumption would be to assume that theserver works at a slower rate during vacation periods incomparison to that of a regular working period. Such modelsare classified as queues subject to working vacations [4–6]. Inaddition, the server can stop the vacation once some indicesof the system, such as the number of customers, achieve acertain value in the vacation period. Certainly, it is possiblefor the server to take an interrupted vacation, so we callthis policy vacation interruption. Li and Tian [7] studiedthe single server queueing model with working vacationand vacation interruption. The modulating queueing modelconsidered in this paper is an 𝑀/𝑀/1 queue wherein theserver is subject to regular vacation with probability 𝑝 orworking vacation with probability 1−𝑝. Further, the vacationduration of the server during working vacation epochmay beinterrupted due to vacation interruption.

Fluid queues have become a fascinating area of research inrecent years due to their widespread applicability in computerand communication systems [8, 9], manufacturing systems[10], and so forth. A stochastic fluid flow system is an input-output model where the input is modeled as a continuous

Hindawi Publishing CorporationAdvances in Operations ResearchVolume 2016, Article ID 2673017, 11 pageshttp://dx.doi.org/10.1155/2016/2673017

2 Advances in Operations Research

fluid that enters and leaves the storage device called a buffer,according to randomly varying rates. They are appropriatein a situation wherein the arrival is comprised of a discreteunit, but the interarrival time between successive arrivals isnegligible. Therefore, the arrivals can be approximated by acontinuous flow of fluid as individuals units have less impacton the performance of the system. In these models, a fluidbuffer is either filled or depleted or both at rates determinedby the current state of the background queueing model.Markov modulated fluid queues are a particular class of fluidmodels useful for modeling many physical phenomena andthey often allow tractable analysis. In addition, fluid modelsare quite useful as approximate models for certain queueingand inventory systems where the flow consists of discreteentities, but the behavior of the individual is not importantto identify the performance analysis. Certain interesting realworld applications of Markov Modulated Fluid Flow modelscan be found in [11–14]. Besides, fluid queues also havesuccessful applications in the field of congestion control[15] and risk processes [16]. More recently, Bosman andNunez-Queija [17] considered a tandem fluid queue modelto evaluate the performance of streaming media over anunreliable network.

For example, consider a production inventory modeloperating in a stochastic environment. The inventory levelincreases when the production rate exceeds the demand rateand decreases otherwise. The inventory level under contin-uous review can be viewed as a fluid process that fluctuatesaccording to the evolution of the underlying backgroundenvironment. For example, consider a machine shop with asingle server. When the server is busy, items are producedcontinuously at a rate 𝑟 and if he is idle, there is no production.However, for all practical reasons, the servermight either takea vacation of some random duration with probability 𝑝 ordecide to provide service at a reduced rate with probability1−𝑝. Further, by offering service at a reduced rate, the servermay continue to do so with probability 𝑞 or due to certainunforeseen reasons, like a sudden increase in the demand,interrupt the vacation with probability 1 − 𝑞, and continuethe busy period. The demands are assumed to vary fromtime to time at the rate 𝑑

𝑡independent of the state of the

server. The level of inventory thus oscillates between 𝑟 − 𝑑𝑡

and −𝑑𝑡depending on the busy or idle state of the server.

Such scenario can be modeled as a fluid queue driven byan𝑀/𝑀/1 queue subject to Bernoulli-Schedule-ControlledVacation and Vacation Interruption.

The stationary analysis of fluid queueing models in astochastic environment has been discussed by many authors.Fluid queues driven by an 𝑀/𝑀/1 queueing model areextensively studied in the literature. Various techniques havebeen employed by researchers to obtain the stationary buffercontent distribution. To mention a few, Adan and Resing[18] analyze the buffer content distribution by viewing thearrival process as an alternating renewal process andVirtamoand Norros [19] provide the buffer content distribution byfinding the spectrum of the eigenvalue equation and explicitexpressions for the corresponding eigenvectors in terms ofChebyshev polynomials of the second kind. Sericola andTuffin [20] express the stationary distribution of the buffer

occupancy in terms of a sequence of recursively definedpolynomials. Parthasarathy et al. [21] present an explicitexpression for the buffer content distribution in terms ofmodified Bessel function of the first kind using continuedfraction methodology.

Furthermore, fluid models driven by an 𝑀/𝑀/1 queuesubject to various vacation strategies were analyzed in steadystate byMao et al. [22] andWang et al. [23].Theworkwas fur-ther extended to the stationary analysis of fluid queues drivenby an𝑀/𝑀/1 queue with multiple exponential vacation and𝑁 policy [24]. Fluid model driven by an𝑀/𝑀/1 queue withworking vacations and vacation interruption was studied byXu et al. [25]. However, in most of the literature relating tofluid queues driven by vacation queueing models, the buffercontent distribution is expressed in the Laplace domain.More recently, Vijayashree and Anjuka [26, 27] presentedan explicit expression for the buffer content distribution ofa fluid queueing model modulated by an 𝑀/𝑀/1 queuesubject to catastrophes and subsequent repair and a fluidqueue driven by an 𝑀/𝐸

2/1 queueing model, respectively.

Also, Ammar [28] derives an explicit expression for the fluidqueue driven by an𝑀/𝑀/1 queue with multiple exponentialvacation using generating function methodology.

This paper presents an analytical solution for the fluidqueue driven by an 𝑀/𝑀/1 queue subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption instationary regime. When the background queueing modelis empty, the server will take either an ordinary vacationwith probability 𝑝 or a working vacation with probability1−𝑝. If the system is in a working vacation, upon completionof service, the server either ends the vacation and enters aregular busy period with probability 1 − 𝑞 or continues thevacation with probability 𝑞. It is assumed that the fluid in thebuffer content increases at a constant rate, 𝜎, when there isone or more customers in the background queueing modeland decreases at a constant rate, 𝜎

0, when the queue is empty.

The system of equations governing the process is modeledin terms of quasi-birth and death process and solved usingmatrix-geometric method. The stationary distribution of thebuffer content is thereby obtained in the Laplace domainand hence inverted to obtain explicit expressions for thejoint steady state probabilities of the state of the backgroundqueueing model and content of the buffer. Closed formexpressions help to gain a deeper insight into the model. As aspecial case, when 𝑝 = 0, 𝑞 = 0, and 𝑝 = 1 the theoreticalresults, so obtained, are seen to coincide with the existingresults of Xu et al. [25] and Wang et al. [23], respectively.

The rest of the paper is organized as follows: Sec-tion 2 gives a brief description of the background queueingmodel under consideration. Section 3 presents the sys-tem of differential difference equations that governs thefluid queueing model under steady state subject to suit-able stability conditions. Section 4 gives detailed deriva-tions of the closed form expressions for the joint steadystate probabilities of the state of the background queueingmodel and also the buffer content distributions. Section 5presents the numerical illustration of the stationary buffercontent distribution for suitable choice of the parametervalues.

Advances in Operations Research 3

2. Model Description

Consider an𝑀/𝑀/1 queueing model with infinite capacity.Let the customers arrive according to a Poisson process withparameter 𝜆. The server provides service to the arrivingcustomers according to an exponential distribution withparameter 𝜇. When the system becomes empty, the serverbegins a vacation of random length and takes an ordinaryvacation with probability 𝑝 or a working vacation with prob-ability 1 − 𝑝, where 0 ≤ 𝑝 ≤ 1. In an ordinary vacation, theserver will stop working even if there are new arrivals duringthe vacation period. In a working vacation, customers areserved at a lower rate 𝜇V < 𝜇. Further, in a working vacation,it is assumed that, at the instants of service completion,either the vacation is interrupted and the server resumes toa regular busy period with probability 1 − 𝑞 or the servercontinues the vacation with probability 𝑞. When the vacationperiod ends and the system is nonempty, a new busy periodstarts. The ordinary vacation times and the working vacationtimes are also assumed to be exponentially distributed withparameters 𝜃 and 𝜃V, respectively. Let𝑁(𝑡)denote the numberof customers at time 𝑡. Define

𝐽 (𝑡)

=

{{{{

{{{{

{

0, if the server is in a working vacation at time 𝑡,

1, if the server is in an ordinary vacation at time 𝑡,

2, if the server is in a regular busy period at time 𝑡.

(1)

It is well known that the process {(𝑁(𝑡), 𝐽(𝑡)), 𝑡 ≥ 0} is aquasi-birth and death (QBD) process with state space givenby

Ω

= {((0, 0) ∪ (0, 1)) ∪ (𝑘, 𝑗) , 𝑘 = 1, 2, . . . , 𝑗 = 0, 1, 2} .

(2)

The state transition diagram of the background queueingmodel is given in Figure 1. Let

𝜋𝑘,𝑗= lim𝑡→∞

𝑃 {𝑁 (𝑡) = 𝑘, 𝐽 (𝑡) = 𝑗} , (𝑘, 𝑗) ∈ Ω, (3)

represent the steady state probabilities of the backgroundqueueing model. Further, let 𝜋

0= (𝜋0,0

𝜋0,1) and 𝜋

𝑘=

(𝜋𝑘,0

𝜋𝑘,1

𝜋𝑘,2) for 𝑘 ≥ 1. Then, the stationary probability

vector is denoted by

𝜋 = (𝜋0, 𝜋1, 𝜋2, . . .) . (4)

It is readily seen that the system of equations governingthe background queueing model under steady state can bewritten in the form of matrix as

𝜋𝑄 = 0,

𝜋0𝑒1+

∞

∑

𝑘=1

𝜋𝑘𝑒2= 1,

(5)

3, 12, 10, 1 1, 1

3, 02, 00, 0 1, 0

3, 22, 21, 2

𝜆

𝜆 𝜆 𝜆

𝜆 𝜆 𝜆

𝜆

𝜆 𝜆 𝜆

𝜃 𝜃 𝜃

𝜇 𝜇 𝜇

p𝜇

𝜇�

𝜃� 𝜃� 𝜃�

q𝜇�

q𝜇�

q𝜇�

(1 − q)𝜇�

(1 − q)𝜇�

(1 − q)𝜇�

· · ·

· · ·

· · ·

(1 − p)𝜇

Figure 1: State transition diagram.

where 𝑒1= (1, 1)

𝑇, 𝑒2= (1, 1, 1)

𝑇, and 𝑄 = (𝐵0 𝐴0

𝐶0 𝐵 𝐴

𝐶 𝐵 𝐴

d d d).

Note that

𝐵0= (

−𝜆 0

0 −𝜆) ,

𝐴0= (

𝜆 0 0

0 𝜆 0) ,

𝐶0= (

𝜇V 0

0 0

(1 − 𝑝) 𝜇 𝑝𝜇

) ,

𝐴 = (

𝜆 0 0

0 𝜆 0

0 0 𝜆

) ,

𝐵 = (

− (𝜆 + 𝜇V + 𝜃V) 0 𝜃V

0 − (𝜆 + 𝜃) 𝜃

0 0 − (𝜆 + 𝜇)

) ,

𝐶 = (

𝑞𝜇V 0 (1 − 𝑞) 𝜇V

0 0 0

0 0 𝜇

) .

(6)

The steady state probabilities of the single server queueingmodels with Poisson arrival and exponentially distributedservice times subject to Bernoulli-Schedule-ControlledVaca-tion and Vacation Interruption were studied by Zhang andShi [29].

3. Analysis of Fluid Queue

This section deals with the stationary analysis of a fluidqueue modulated by an𝑀/𝑀/1 queueing model subject toBernoulli-Schedule-Controlled Vacation and Vacation Inter-ruption. Let 𝐶(𝑡) be the content of the buffer at time 𝑡. Fur-thermore, it is assumed that the content of the buffer increasesat the rate 𝜎, when there are customers in the background


Variations of the content of the buffer

(0, 0)(1, 0)(2, 0)

(1, 2)(2, 2)

(0, 1)(1, 1)(2, 1)

t

𝜎𝜎𝜎

𝜎

𝜎0

𝜎0 𝜎0

𝜎0 if N(t) = 0

𝜎 if N(t) > 0

· · ·

· · ·

· · ·

States of the background queueingmodel

Figure 2: Interaction between fluid model and the background queueing model.

queueingmodel, while the buffer content decreases at the rate𝜎0, when the system is empty. The rate at which the content

of the buffer varies with time is given by

𝑑𝐶 (𝑡)

𝑑𝑡

=

{{{{{{{

{{{{{{{

{

0, (𝑁 (𝑡) , 𝐽 (𝑡)) = (0, 0) , 𝐶 (𝑡) = 0

𝜎0, (𝑁 (𝑡) , 𝐽 (𝑡)) = (0, 0) , 𝐶 (𝑡) > 0

𝜎0, (𝑁 (𝑡) , 𝐽 (𝑡)) = (0, 1) , 𝐶 (𝑡) > 0

𝜎, (𝑁 (𝑡) , 𝐽 (𝑡)) = (𝑘, 𝑗) , 𝑘 ≥ 1, 𝑗 = 0, 1, 2,

(7)

where 𝜎0< 0 and 𝜎 > 0. Figure 2 depicts the interaction

between the buffer content process and the backgroundqueueing model. It is seen that the content of the infinitecapacity buffer decreases at the rate 𝜎

0< 0 when the back-

ground queueing model is empty with no waiting customersand it increases at the rate 𝜎 when 𝑁(𝑡) ̸= 0 irrespective ofthe states of 𝐽(𝑡).

Clearly the 3-dimensional process {(𝑁(𝑡), 𝐽(𝑡), 𝐶(𝑡)), 𝑡 ≥0} represents a fluid queue driven by an 𝑀/𝑀/1 queuewith Bernoulli-Schedule-Controlled Vacation and VacationInterruption. As the content of the buffer varies dynamically,it is necessary that the net effective rate of the fluid remainsnegative to ensure the stability of the process in a long run.Hence, the stability condition is given by

𝑑 = 𝜎0(𝜋0,0+ 𝜋0,1) + 𝜎

∞

∑

𝑘=1

2

∑

𝑗=0

𝜋𝑘,𝑗< 0. (8)

Define the joint probability distribution functions of theMarkov process {(𝑁(𝑡), 𝐽(𝑡), 𝐶(𝑡)), 𝑡 ≥ 0} at time 𝑡 as

𝐹𝑘,𝑗(𝑡, 𝑥) = Pr {𝑁 (𝑡) = 𝑘, 𝐽 (𝑡) = 𝑗, 𝐶 (𝑡) ≤ 𝑥} ,

(𝑘, 𝑗) ∈ Ω, 𝑥 ≥ 0.

(9)

When the process {(𝑁(𝑡), 𝐽(𝑡), 𝐶(𝑡)), 𝑡 ≥ 0} is stable, itsstationary random vector is denoted by (𝑁, 𝐽, 𝐶). Understeady state conditions, let

𝐹𝑘,𝑗(𝑥) = lim

𝑡→+∞

Pr {𝑁 (𝑡) = 𝑘, 𝐽 (𝑡) = 𝑗, 𝐶 (𝑡) ≤ 𝑥} ,

𝑥 > 0, (𝑘, 𝑗) ∈ Ω.

(10)

Using standard methods, the system of differential differenceequations that governs the process {(𝑁(𝑡), 𝐽(𝑡), 𝐶(𝑡)), 𝑡 ≥ 0}is given by

𝜎0

𝑑𝐹0,0(𝑥)

𝑑𝑥= −𝜆𝐹

0,0(𝑥) + 𝜇V𝐹1,0 (𝑥)

+ (1 − 𝑝) 𝜇𝐹1,2(𝑥) ,

𝜎0

𝑑𝐹0,1(𝑥)

𝑑𝑥= −𝜆𝐹

0,1(𝑥) + 𝑝𝜇𝐹

1,2(𝑥) ,

𝜎𝑑𝐹𝑘,0(𝑥)

𝑑𝑥= − (𝜆 + 𝜇V + 𝜃V) 𝐹𝑘,0 (𝑥) + 𝜆𝐹𝑘−1,0 (𝑥)

+ 𝑞𝜇V𝐹𝑘+1,0 (𝑥) 𝑘 ≥ 1,


𝑑𝑥= − (𝜆 + 𝜃) 𝐹

𝑘,1(𝑥) + 𝜆𝐹

𝑘−1,1(𝑥) 𝑘 ≥ 1,

𝜎𝑑𝐹1,2(𝑥)

𝑑𝑥= − (𝜆 + 𝜇) 𝐹

1,2(𝑥) + 𝜃𝐹

1,1(𝑥)

+ 𝜃V𝐹1,0 (𝑥) + (1 − 𝑞) 𝜇V𝐹2,0 (𝑥)

+ 𝜇𝐹2,2(𝑥) ,


𝑑𝑥= − (𝜆 + 𝜇) 𝐹

𝑘,2(𝑥) + 𝜃𝐹

𝑘,1(𝑥)

+ 𝜃V𝐹𝑘,0 (𝑥) + (1 − 𝑞) 𝜇V𝐹𝑘+1,0 (𝑥)

+ 𝜆𝐹𝑘−1,2

(𝑥) + 𝜇𝐹𝑘+1,2

(𝑥) 𝑘 ≥ 2,

(11)

with the boundary conditions

𝐹0,0(0) = 𝑎

1,

𝐹0,1(0) = 𝑎

2,

𝐹𝑘,𝑗(0) = 0, (𝑘, 𝑗) ∈ Ω \ {(0, 0) ∪ (0, 1)} .

(12)

The constants 𝑎1and 𝑎2are such that 0 < 𝑎

1< 1 and 0 < 𝑎

2<

1. Since wemake an assumption that the content of the bufferincreases at the rate 𝜎 when there is one or more customersin the background queueing model, it is impossible to havethe buffer empty when the modulating process is in any of


the states other than (0, 0) and (0, 1). However, when thereare no customers in the background queueing model, thebuffer content depletes at rate 𝜎

0< 0 and hence with some

positive probability, it is possible that the content of the bufferis empty.Therefore the boundary conditions given by (12) arevalid.

4. Solution Methodology

The stationary distributions of the fluid process play a vitalrole as they give more information relating to quantitiesof interest for practical applications like tail probabilities,expected buffer content, traffic intensity, expected delay, andsojourn time. This section presents explicit expressions forthe joint steady state probabilities of the background queue-ing model and the content of the buffer in terms of modifiedBessel function of the first kind. The governing system ofequations in the Laplace domain is expressed as a systemof matrix equations. The minimal nonnegative solution ofthe matrix quadratic equation is determined. The stationaryjoint probability distributions are expressed in terms of thisminimal nonnegative solution and are shown to satisfy thegoverning system of matrix equations. This section presentsan explicit analytical solution to the governing system ofequations represented by (11). In this sequel, define

𝐹0(𝑥) = (𝐹

0,0(𝑥) , 𝐹

0,1(𝑥)) ,

𝐹𝑘(𝑥) = (𝐹

𝑘,0(𝑥) , 𝐹

𝑘,1(𝑥) , 𝐹

𝑘,2(𝑥)) 𝑘 = 1, 2, . . . .

(13)

Let 𝐹(𝑥) = (𝐹0(𝑥), 𝐹1(𝑥), 𝐹2(𝑥), . . .).The system of equations

represented by (11) can be written in matrix form as

𝐹(𝑥) Λ = 𝐹 (𝑥)𝑄, (14)

where

Λ =(

Σ

Σ

Σ

d

),

Σ= (

𝜎00

0 𝜎0

) , Σ = (

𝜎 0 0

0 𝜎 0

0 0 𝜎

) .

(15)

Let �̂�(𝑠) = (�̂�0(𝑠), �̂�1(𝑠), �̂�2(𝑠), . . .) represent the Laplace tran-

sform of 𝐹(𝑥). Then, the Laplace transform of (14) yields

�̂� (𝑠) (𝑄 − 𝑠Λ) = (−𝑎 0 0 ⋅ ⋅ ⋅ 0) , (16)

where 𝑎 = (𝜎0𝑎1𝜎0𝑎2) .Thegoverning systemof differential

difference equations in the Laplace domain is then given by

�̂�0(𝑠) (𝐵

0− 𝑠Σ) + �̂�1(𝑠) 𝐶0= −𝑎, (17)

�̂�0(𝑠) 𝐴0+ �̂�1(𝑠) (𝐵 − 𝑠Σ) + �̂�

2(𝑠) 𝐶 = 0, (18)

�̂�𝑘−1(𝑠) 𝐴 + �̂�

𝑘(𝑠) (𝐵 − 𝑠Σ) + �̂�

𝑘+1(𝑠) 𝐶 = 0

for 𝑘 = 2, 3, . . . .(19)

Our objective is to solve the above system ofmatrix differenceequations to obtain explicit expressions for the stationaryprobability distribution and hence determine the stationarybuffer content distribution. Towards that end, we present alemma below followed by a theorem.

Lemma 1. The matrix quadratic equation

𝐴 + 𝑅 (𝑠) (𝐵 − 𝑠Σ) + 𝑅2(𝑠) 𝐶 = 0 (20)

has the minimal nonnegative solution given by

𝑅 (𝑠) = (

𝑟 (𝑠) 0 𝛽 (𝑠)

0 𝜓 (𝑠) 𝛾 (𝑠)

0 0 𝑧 (𝑠)

) , (21)

where

𝑟 (𝑠)

=

(𝜆 + 𝜇V + 𝜃V + 𝑠𝜎) − √(𝜆 + 𝜇V + 𝜃V + 𝑠𝜎)2

− 4𝜆𝑞𝜇V

2𝑞𝜇V,

𝛽 (𝑠) =𝑟 (𝑠) 𝑧 (𝑠) [(1 − 𝑞) 𝜇V𝑟 (𝑠) + 𝜃V]

𝜆 − 𝜇𝑟 (𝑠) 𝑧 (𝑠),

𝜓 (𝑠) =𝜆

𝜆 + 𝜃 + 𝑠𝜎,

𝛾 (𝑠) =𝜃𝑧 (𝑠)

𝜆 + 𝜃 + 𝑠𝜎 − 𝜇𝑧 (𝑠),

𝑧 (𝑠) =

(𝜆 + 𝜇 + 𝑠𝜎) − √(𝜆 + 𝜇 + 𝑠𝜎)2

− 4𝜆𝜇

2𝜇.

(22)

Proof. Since 𝐴, (𝐵 − 𝑠Σ), and 𝐶 are all upper triangularmatrices, we can assume that the solution 𝑅(𝑠) has the samestructure as

𝑅 (𝑠) = (

𝑟11(𝑠) 𝑟12(𝑠) 𝑟13(𝑠)

0 𝑟22(𝑠) 𝑟23(𝑠)

0 0 𝑟33(𝑠)

) . (23)

Note that the first row and second column elements of all thematrices of 𝐴, (𝐵 − 𝑠Σ), and 𝐶 are zero, so the element 𝑟

12(𝑠)

in 𝑅(𝑠) is also zero. Therefore 𝑟12(𝑠) = 0. Substituting for 𝑅(𝑠)

into (20) leads to

𝜆 − (𝜆 + 𝜇V + 𝜃V + 𝑠𝜎) 𝑟11 (𝑠) + 𝑞𝜇V (𝑟11 (𝑠))2

= 0, (24)

𝜆 − (𝜆 + 𝜇 + 𝑠𝜎) 𝑟33(𝑠) + 𝜇 (𝑟

33(𝑠))2

= 0, (25)

𝜆 − (𝜆 + 𝜃 + 𝑠𝜎) 𝑟22(𝑠) = 0, (26)

(1 − 𝑞) 𝜇V (𝑟11 (𝑠))2

+ 𝜇𝑟13(𝑠) 𝑟11(𝑠) + 𝜇𝑟

13(𝑠) 𝑟33(𝑠)

+ 𝜃V𝑟11 (𝑠) − (𝜆 + 𝜇 + 𝑠𝜎) 𝑟13 (𝑠) = 0,(27)

𝜃𝑟22(𝑠) − (𝜆 + 𝜇 + 𝑠𝜎) 𝑟

23(𝑠)

+ 𝜇 [𝑟22(𝑠) 𝑟23(𝑠) + 𝑟

23(𝑠) 𝑟33(𝑠)] = 0.

(28)


The solutions of (24) and (25) are given by

𝑟11(𝑠)

=

(𝜆 + 𝜇V + 𝜃V + 𝑠𝜎) ± √(𝜆 + 𝜇V + 𝜃V + 𝑠𝜎)2

− 4𝜆𝑞𝜇V

2𝑞𝜇V,

𝑟33(𝑠) =

(𝜆 + 𝜇 + 𝑠𝜎) ± √(𝜆 + 𝜇 + 𝑠𝜎)2

− 4𝜆𝜇

2𝜇.

(29)

Let 𝑟(𝑠) (𝑟1(𝑠)) and 𝑧(𝑠) (𝑧

1(𝑠)) denote the negative (positive)

roots of 𝑟11(𝑠) and 𝑟

33(𝑠), respectively. Considering the root

that lies inside the unit circle, 𝑟(𝑠) and 𝑧(𝑠) are taken forfurther analysis. Further 𝑧(𝑠) satisfies the following relations:

𝑠𝜎 + 𝜆 + 𝜇 (1 − 𝑧 (𝑠)) =𝜆

𝑧 (𝑠)= 𝜇 +

𝑠𝜎

1 − 𝑧 (𝑠). (30)

From (26), we get

𝑟22(𝑠) = 𝜓 (𝑠) =

𝜆

𝜆 + 𝜃 + 𝑠𝜎. (31)

Substituting for 𝑟11(𝑠) and 𝑟

33(𝑠) into (27) yields

𝑟13(𝑠) = 𝛽 (𝑠) =

𝑟 (𝑠) [(1 − 𝑞) 𝜇V𝑟 (𝑠) + 𝜃V]

𝜆 + 𝜇 + 𝑠𝜎 − 𝜇 (𝑟 (𝑠) + 𝑧 (𝑠)). (32)

Using the relation given by (30) in the above leads to

𝑟13(𝑠) = 𝛽 (𝑠) =

𝑟 (𝑠) 𝑧 (𝑠) [(1 − 𝑞) 𝜇V𝑟 (𝑠) + 𝜃V]

𝜆 − 𝜇𝑟 (𝑠) 𝑧 (𝑠). (33)

Again substituting for 𝑟22(𝑠) and 𝑟

33(𝑠) into (28) leads to

𝑟23(𝑠) = 𝛾 (𝑠)

=𝜆𝜃

(𝑠𝜎 + 𝜆 + 𝜇) (𝑠𝜎 + 𝜆 + 𝜃) − 𝜆𝜇 − 𝜇𝑧 (𝑠) (𝑠𝜎 + 𝜃 + 𝜆).

(34)

Using the relation given by (30) in the above equation yields

𝑟23(𝑠) = 𝛾 (𝑠) =

𝜃𝑧 (𝑠)

𝜆 + 𝜃 + 𝑠𝜎 − 𝜇𝑧 (𝑠). (35)

This completes the proof. Note that 𝑅𝑘(𝑠) for 𝑘 = 1, 2, 3, . . .can be simplified as

𝑅𝑘(𝑠)

=(

(

𝑟𝑘(𝑠) 0 𝛽 (𝑠)

𝑘

∑

𝑖=1

𝑟𝑘−𝑖(𝑠) 𝑧𝑖−1(𝑠)

0 𝜓𝑘(𝑠) 𝛾 (𝑠)

𝑘

∑

𝑗=1

𝜓𝑘−𝑗(𝑠) 𝑧𝑗−1(𝑠)

0 0 𝑧𝑘(𝑠)

)

)

.

(36)

Also 𝑅(0) = 𝑅, where 𝑅 is given by

𝑅 = (

𝑟 0𝜆 − 𝜇V𝑟

𝜇

0𝜆

𝜆 + 𝜃𝜌

0 0 𝜌

) , (37)

with𝜌 = 𝜆/𝜇 and 𝑟 = ((𝜆+𝜇V+𝜃V)−√(𝜆 + 𝜇V + 𝜃V)2− 4𝜆𝑞𝜇V)/

2𝑞𝜇V.

Theorem 2. The stationary joint probability distributions ofthe content of the buffer and the state of the backgroundqueueing model in Laplace domain are given by

�̂�𝑘(𝑠) = �̂�

0(𝑠) 𝑒𝑅

𝑘(𝑠) 𝑓𝑜𝑟 𝑘 = 1, 2, 3, . . . , (38)

�̂�0(𝑠) =

𝑎

𝑠Σ − 𝐵0− 𝑒𝑅 (𝑠) 𝐶

0

, (39)

where 𝑒 = ( 1 0 00 1 0

) .

Proof. Assume that

�̂�𝑘(𝑠) = �̂�

𝑘−1(𝑠) 𝑅 (𝑠) for 𝑘 = 1, 2, 3, . . . . (40)

Then, it can be recursively written as

�̂�𝑘(𝑠) = �̂�

0(𝑠) 𝑒𝑅

𝑘(𝑠) for 𝑘 = 1, 2, 3, . . . . (41)

Below, we verify that (40) satisfies (18) and (19). Substituting(41) into (18) leads to

�̂�0(𝑠) 𝐴0+ �̂�1(𝑠) (𝐵 − 𝑠Σ) + �̂�

2(𝑠) 𝐶

= �̂�0(𝑠) 𝑒𝐴 + �̂�

0(𝑠) 𝑒𝑅 (𝑠) (𝐵 − 𝑠Σ)

+ �̂�0(𝑠) 𝑒𝑅

2(𝑠) 𝐶

= �̂�0(𝑠) 𝑒 [𝐴 + 𝑅 (𝑠) (𝐵 − 𝑠Σ) + 𝑅

2(𝑠) 𝐶]

= 0 (by Lemma 1) .

(42)

Similarly, substituting (40) into (19) leads to

�̂�𝑘−1(𝑠) 𝐴 + �̂�

𝑘(𝑠) (𝐵 − 𝑠Σ) + �̂�

𝑘+1(𝑠) 𝐶

= �̂�𝑘−1(𝑠) 𝐴 + �̂�

𝑘−1(𝑠) 𝑅 (𝑠) (𝐵 − 𝑠Σ)

+ �̂�𝑘−1(𝑠) 𝑅2(𝑠) 𝐶

= �̂�𝑘−1(𝑠) [𝐴 + 𝑅 (𝑠) (𝐵 − 𝑠Σ) + 𝑅

2(𝑠) 𝐶]

= 0 (by Lemma 1) .

(43)

From (17), we get

�̂�0(𝑠) (𝐵

0− 𝑠Σ) + �̂�0(𝑠) 𝑒𝑅 (𝑠) 𝐶

0= −𝑎. (44)

Therefore

�̂�0(𝑠) =

𝑎

𝑠Σ − 𝐵0− 𝑒𝑅 (𝑠) 𝐶

0

, (45)

which upon simplification leads to

�̂�0,0(𝑠) =

∞

∑

𝑘=0

(𝛿 (𝑠))𝑘[

𝑎1𝜎0

𝑠𝜎0+ 𝜆 − 𝜇V𝑟 (𝑠)

+ 𝑎2𝜎0(1 − 𝑝) 𝜇𝛾 (𝑠)𝐷 (𝑠)] ,

(46)

�̂�0,1(𝑠) =

∞

∑

𝑘=0

(𝛿 (𝑠))𝑘

⋅ [(𝑎1𝜎0𝑝𝜇 − 𝑎

2𝜎 (1 − 𝑝) 𝜇) 𝛽 (𝑠)𝐷 (𝑠)

+𝑎2𝜎0

𝑠𝜎0+ 𝜆 − 𝑝𝜇𝛾 (𝑠)

] ,

(47)


where

𝛿 (𝑠) =𝛽 (𝑠) (1 − 𝑝) 𝜇 (𝑠𝜎

0+ 𝜆)

(𝑠𝜎0+ 𝜆 − 𝑝𝜇𝛾 (𝑠)) (𝑠𝜎

0+ 𝜆 − 𝜇V𝑟 (𝑠))

,

𝐷 (𝑠) =1

(𝑠𝜎0+ 𝜆 − 𝑝𝜇𝛾 (𝑠)) (𝑠𝜎

0+ 𝜆 − 𝜇V𝑟 (𝑠))

.

(48)

Hence all the joint stationary probabilities, �̂�𝑘(𝑠), for 𝑘 =

1, 2, 3, . . . are in terms of �̂�0(𝑠), where �̂�

0(𝑠) is given by (45).

This completes the proof.

Having determined all the joint steady state probabilitiesin the Laplace domain, we now present the explicit analyticalsolution by inverting using transform techniques. From (41),we get

[�̂�𝑘,0(𝑠) �̂�

𝑘,1(𝑠) �̂�

𝑘,2(𝑠)] = (�̂�

0,0(𝑠) �̂�

0,1(𝑠)) (

1 0 0

0 1 0)(

(

𝑟𝑘(𝑠) 0 𝛽 (𝑠)

𝑘

∑

𝑖=1

𝑟𝑘−𝑖(𝑠) 𝑧𝑖−1(𝑠)

0 𝜓𝑘(𝑠) 𝛾 (𝑠)

𝑘

∑

𝑗=1

𝜓𝑘−𝑗(𝑠) 𝑧𝑗−1(𝑠)

0 0 𝑧𝑘(𝑠)

)

)

, (49)

which can be written as

�̂�𝑘,0(𝑠) = 𝑟 (𝑠)

𝑘�̂�0,0(𝑠) ,

�̂�𝑘,1(𝑠) = 𝜓 (𝑠)

𝑘�̂�0,1(𝑠) ,

�̂�𝑘,2(𝑠) = 𝛽 (𝑠)

𝑘

∑

𝑖=1

𝑟𝑘−𝑖(𝑠) 𝑧𝑖−1(𝑠) �̂�0,0(𝑠)

+ 𝛾 (𝑠)

𝑘

∑

𝑗=1

𝜓𝑘−𝑗(𝑠) 𝑧𝑗−1(𝑠) �̂�0,1(𝑠) .

(50)

With𝛼 = 2√𝜆𝜇/𝜎 and𝛽 = 2√𝜆𝑞𝜇V/𝜎, inversion of (46), (47),and (50) yields

𝐹0,0(𝑥) =

∞

∑

𝑘=0

𝛿 (𝑥)∗𝑘∗ [𝑎1

∞

∑

𝑙=0

(𝜇V

𝜎0

)

𝑙

𝑒−(𝜆/𝜎0)𝑥

𝑥𝑙

𝑙!

∗𝑙𝐼𝑙(𝛽𝑥) 𝛽

𝑙

𝑥(𝜎

2𝑞𝜇V)

𝑙

𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥 + 𝑎

2𝜎0(1

− 𝑝) 𝜇𝛾 (𝑥) ∗ 𝐷 (𝑥)] ,

(51)

𝐹0,1(𝑥) =

∞

∑

𝑘=0

𝛿 (𝑥)∗𝑘∗ [(𝑎

1𝜎0𝑝𝜇 − 𝑎

2𝜎 (1 − 𝑝) 𝜇)

⋅ 𝛽 (𝑥) ∗ 𝐷 (𝑥) + 𝑎2

∞

∑

𝑘=0

(𝑝𝜇

𝜎0

)

𝑘


𝑥𝑘

𝑘!

∗ 𝛾 (𝑥)∗𝑘] ,

(52)

𝐹𝑘,0(𝑥) =

𝑘𝐼𝑘(𝛽𝑥) 𝛽

𝑘

𝑥(𝜎

2𝑞𝜇V)

𝑘

𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥

∗ 𝐹0,0(𝑥) 𝑘 = 1, 2, 3, . . . ,

(53)

𝐹𝑘,1(𝑥) = (

𝜆

𝜎)

𝑘

𝑒−((𝜆+𝜃)/𝜎)𝑥 𝑥

𝑘−1

(𝑘 − 1)!∗ 𝐹0,1(𝑥)

𝑘 = 1, 2, 3, . . . ,

(54)

𝐹𝑘,2(𝑥) = (𝛽 (𝑥) ∗

𝑘

∑

𝑖=1

(𝜎

2𝑞𝜇V)

𝑘−𝑖

⋅(𝑘 − 𝑖) 𝐼

𝑘−𝑖(𝛽𝑥) 𝛽

𝑘−𝑖

𝑥𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥 ∗ (

𝜎

2𝜇)

𝑖−1

⋅(𝑖 − 1) 𝐼

𝑖−1(𝛼𝑥) 𝛼

𝑖−1

𝑥𝑒−((𝜆+𝜇)/𝜎)𝑥

) + (𝛾 (𝑥)

∗

𝑘

∑

𝑗=1

(𝜆

𝜎)

𝑘−𝑗

𝑒−((𝜆+𝜃)/𝜎)𝑥 𝑥

𝑘−𝑗−1

(𝑘 − 𝑗 − 1)!∗ (

𝜎

2𝜇)

𝑗−1

⋅

(𝑗 − 1) 𝐼𝑗−1(𝛼𝑥) 𝛼

𝑗−1


∗ 𝐹0,1(𝑥))

𝑘 ≥ 1,

(55)

where

𝛿 (𝑥) = (1 − 𝑝) 𝜇𝛽 (𝑥) ∗

∞

∑

𝑖=0

∞

∑

𝑗=0

(𝑝𝜇)𝑖

𝜇𝑗

V

𝜎𝑖+𝑗+1

0


⋅𝑥𝑖+𝑗

(𝑖 + 𝑗)!∗ 𝛾 (𝑥)

∗𝑖∗

𝑗𝐼𝑗(𝛽𝑥) 𝛽

𝑗

𝑥(𝜎

2𝑞𝜇V)

𝑗

⋅ 𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥

𝐷 (𝑥) =

∞

∑

𝑖=0

∞

∑

𝑗=0

(𝑝𝜇)𝑖

𝜇𝑗

V

𝜎𝑖+𝑗+2

0


𝑥𝑖+𝑗+1

(𝑖 + 𝑗 + 1)!∗ 𝛾 (𝑥)

∗𝑖

∗

𝑗𝐼𝑗(𝛽𝑥) 𝛽

𝑗

𝑥(𝜎

2𝑞𝜇V)

𝑗

𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥,

𝛽 (𝑥) = (1 − 𝑞)

⋅ (𝜇V

∞

∑

𝑘=0

𝜇𝑘

𝜆𝑘+1

(𝑘 + 2) 𝐼𝑘+2(𝛽𝑥) 𝛽

𝑘+2

𝑥(𝜎

2𝑞𝜇V)

𝑘+2


⋅ 𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥 ∗ (

𝜎

2𝜇)

𝑘+1

⋅(𝑘 + 1) 𝐼

𝑘+1(𝛼𝑥) 𝛼

𝑘+1


)

+ 𝜃V (

∞

∑

𝑘=0

𝜇𝑘

𝜆𝑘+1

(𝑘 + 1) 𝐼𝑘+1(𝛽𝑥) 𝛽

𝑘+1

𝑥(𝜎

2𝑞𝜇V)

𝑘+1

⋅ 𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥 ∗ (

𝜎

2𝜇)

𝑘+1

⋅(𝑘 + 1) 𝐼


𝑘+1


) ,

𝛾 (𝑥) = 𝜃

∞

∑

𝑘=0

𝜇𝑘

𝜎𝑘+1𝑒−((𝜆+𝜃)/𝜎)𝑥 𝑥

𝑘

𝑘!∗ (

𝜎

2𝜇)

𝑘+1

⋅(𝑘 + 1) 𝐼


𝑘+1


.

(56)

Thus all the joint steady state probabilities of the state of thesystem and the content of the buffer are explicitly obtained interms of modified Bessel function of the first kind.

Remark 3. When 𝑝 = 0 and 𝑞 = 0, the model under consid-eration reduces to a fluid queue driven by an𝑀/𝑀/1 queuewith working vacation and vacation interruption discussedby Xu et al. [25]. With 𝑎

2= 0, the expression for �̂�

0,0(𝑠) from

(39) yields

�̂�0,0(𝑠) =

𝑎1𝜎0

𝜆 + 𝑠𝜎0− 𝜇V𝑟 (𝑠) − 𝜇𝛽 (𝑠)

, (57)

where

𝑟 (𝑠) =𝜆

𝜆 + 𝜇V + 𝜃V + 𝑠𝜎,

𝛽 (𝑠) =𝑟 (𝑠) [𝜇V𝑟 (𝑠) + 𝜃V]

𝜇 [𝑧1(𝑠) − 𝑟 (𝑠)]

,

𝑧1(𝑠) =

(𝜆 + 𝜇 + 𝑠𝜎) + √(𝜆 + 𝜇 + 𝑠𝜎)2

− 4𝜆𝜇

2𝜇.

(58)

Observe that (57) is seen to coincide with the first expressionin (6) of [25].

Remark 4. When 𝑝 = 1, the model under considerationreduces to a fluid queuemodulated by an𝑀/𝑀/1 queue withmultiple exponential vacation discussed by Wang et al. [23].

Let 𝑎1= 0 and 𝑧(𝑠) = (𝜆/𝜇)𝑧

0(𝑠), where 𝑧

0(𝑠) = ((𝜆 +

𝜇 + 𝑠𝑟) − √(𝜆 + 𝜇 + 𝑠𝑟)2− 4𝜆𝜇)/2𝜆. Then the expression for

�̂�0,1(𝑠) from (39) becomes

�̂�0,1(𝑠) =

𝑎2𝜎0

𝜆 + 𝑠𝜎0− 𝜇𝛾 (𝑠)

, (59)

where 𝛾(𝑠) = 𝜆𝜃𝑧0(𝑠)/𝜇(𝜆 + 𝜃 + 𝑠𝜎 − 𝜆𝑧

0(𝑠)). Substituting

𝛾(𝑠) in the above equation and after certain simplification, weobtain�̂�0,1(𝑠)

=𝑎2𝜎0[𝑠𝜎 + 𝜆 (1 − 𝑧

0(𝑠)) + 𝜃]

(𝑠𝜎0+ 𝜆) (𝑠𝜎 + 𝜆 (1 − 𝑧

0(𝑠)) + 𝜃) − 𝜆𝜃𝑧

0(𝑠).

(60)

Observe that (60) is seen to coincide with (16) of [23].

Buffer Content Distribution. The stationary buffer contentdistribution of the fluid model under consideration is givenby

𝐹 (𝑥) = 𝑃 {𝐶 ≤ 𝑥}

=

∞

∑

𝑘=0

𝐹𝑘,0(𝑥) +

∞

∑

𝑘=0

𝐹𝑘,1(𝑥) +

∞

∑

𝑘=1

𝐹𝑘,2(𝑥) .

(61)

Taking Laplace transform of the above equation yields

�̂� (𝑠) =

∞

∑

𝑘=0

�̂�𝑘,0(𝑠) +

∞

∑

𝑘=0

�̂�𝑘,1(𝑠) +

∞

∑

𝑘=1

�̂�𝑘,2(𝑠) . (62)

In matrix notation, the above equation can be rewritten as

�̂� (𝑠) = �̂�0(𝑠) 𝑒1+

∞

∑

𝑘=1

�̂�𝑘(𝑠) 𝑒2. (63)

Then,

�̂� (𝑠) = �̂�0(𝑠) 𝑒1+ �̂�0(𝑠) 𝑒

∞

∑

𝑘=1

(𝑅 (𝑠))𝑘𝑒2

(f rom (41))

= �̂�0(𝑠) 𝑒1+ �̂�0(𝑠) 𝑒 [𝐼 − 𝑅 (𝑠)]

−1𝑒2,

(64)

where[𝐼 − 𝑅 (𝑠)]

−1

=(

(

1

1 − 𝑟 (𝑠)0

𝛽 (𝑠)

(1 − 𝑟 (𝑠)) (1 − 𝑧 (𝑠))

0𝑠𝜎 + 𝜆 + 𝜃

𝑠𝜎 + 𝜃

𝛾 (𝑠) (𝑠𝜎 + 𝜆 + 𝜃)

(1 − 𝑧 (𝑠)) (𝑠𝜎 + 𝜃)

0 01

1 − 𝑧 (𝑠)

)

)

.

(65)

Upon simplification, we get

�̂� (𝑠) =1

1 − 𝑟 (𝑠)�̂�0,0(𝑠)

+𝛽 (𝑠)

(1 − 𝑟 (𝑠)) (1 − 𝑧 (𝑠))�̂�0,0(𝑠)

+(𝑠𝜎 + 𝜆 + 𝜃)

(𝑠𝜎 + 𝜃)�̂�0,1(𝑠)

−𝛾 (𝑠) (𝑠𝜎 + 𝜆 + 𝜃)

(1 − 𝑧 (𝑠)) (𝑠𝜎 + 𝜃)�̂�0,1(𝑠)

=

∞

∑

𝑘=0

(𝑟 (𝑠))𝑘�̂�0,0(𝑠)


+ 𝛽 (𝑠)

∞

∑

𝑘=0

(𝑟 (𝑠))𝑘

∞

∑

𝑗=0

(𝑧 (𝑠))𝑗�̂�0,0(𝑠) + �̂�

0,1(𝑠)

+𝜆

𝑠𝜎 + 𝜃�̂�0,1(𝑠) + 𝛾 (𝑠)

∞

∑

𝑘=0

(𝑧 (𝑠))𝑘�̂�0,1(𝑠)

+ 𝛾 (𝑠)

∞

∑

𝑘=0

(𝑧 (𝑠))𝑘 𝜆

𝑠𝜎 + 𝜃�̂�0,1(𝑠)

(66)

which on inversion leads to

𝐹 (𝑥) =

∞

∑

𝑘=0

(𝜎

2𝑞𝜇V)

𝑘𝑘𝐼𝑘(𝛽𝑥) 𝛽

𝑘

𝑥𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥

∗ 𝐹0,0(𝑥) + 𝛽 (𝑥)

∗

∞

∑

𝑘=0

(𝜎

2𝑞𝜇V)

𝑘𝑘𝐼𝑘(𝛽𝑥) 𝛽

𝑘

𝑥𝑒−((𝜆+𝜇V+𝜃V)/𝜎)𝑥

∗

∞

∑

𝑗=0

(𝜎

2𝜇)

𝑗 𝑗𝐼𝑗(𝛼𝑥) 𝛼

𝑗


∗ 𝐹0,0(𝑥)

+ 𝐹0,1(𝑥) +

𝜆

𝜎𝑒−(𝜃/𝜎)𝑥

∗ 𝐹0,1(𝑥) + 𝛾 (𝑥)

∗

∞

∑

𝑘=0

(𝜎

2𝜇)

𝑘𝑘𝐼𝑘(𝛼𝑥) 𝛼

𝑘


∗ 𝐹0,1(𝑥)

+ 𝛾 (𝑥) ∗

∞

∑

𝑘=0

(𝜎

2𝜇)

𝑘𝑘𝐼𝑘(𝛼𝑥) 𝛼

𝑘


∗𝜆

𝜎𝑒−(𝜃/𝜎)𝑥

∗ 𝐹0,1(𝑥) ,

(67)

where 𝐹0,0(𝑥) and 𝐹

0,1(𝑥) are given by (51) and (52), respec-

tively. Thus all the joint steady state probabilities and thebuffer content distribution of the fluid queue driven byan𝑀/𝑀/1 queue subject to Bernoulli-Schedule-ControlledVacation and Vacation Interruption are explicitly obtainedunder steady state. Explicit analytical expressions help thepractitioner to better understand the behavior of any quantityof interest, like the mean buffer content, for varying values ofthe parameters involved in the model.

5. Numerical Illustrations

This section illustrates the variation of the buffer contentdistribution against the content of the buffer for 𝜆 = 1, 𝜇 = 2,𝜇V = 1.1, 𝜃 = 0.9, 𝜃V = 0.6, 𝑝 = 0.5, 𝑞 = 0.5, 𝜎 = 1 and varyingvalues of 𝜎

0. The choice of 𝜎

0is relatively high as compared

to 𝜎 because of our assumptions that 𝜎 happens when thebackground queueing model is nonempty and 𝜎

0happens

otherwise. To compensate for the rarity in the occurrence of𝜎0, it is assumed to be larger.Figure 3 depicts the behavior of the buffer content

distribution, 𝐹(𝑥), against 𝑥 for the above choice of theparameter values with 𝜎

0= −450.9. For this choice of the

parameters, it is seen that 𝑑 = −140.86 < 0. Therefore, thestability condition is satisfied. It is seen that 𝐹(𝑥) increases

Table 1: Convergence of stationary buffer content distribution forvarying values of 𝜎

0.

𝑥𝐹(𝑥)

𝜎0= −140.9 𝜎

0= −250.9 𝜎

0= 350.9 𝜎

0= −450.9

0 0.3091 0.3112 0.3120 0.31240.5 0.4408 0.4434 0.4444 0.44491.0 0.5373 0.5403 0.5414 0.54211.5 0.6116 0.6148 0.616 0.61672.0 0.6706 0.674 0.6752 0.67592.5 0.7185 0.7219 0.7232 0.72383.0 0.7580 0.7614 0.7626 0.76333.5 0.7909 0.7942 0.7955 0.79614.0 0.8185 0.8218 0.823 0.82374.5 0.8419 0.8452 0.8463 0.8475.0 0.8618 0.865 0.8662 0.86686.0 0.8936 0.8966 0.8977 0.898311.0 0.9670 0.9694 0.9703 0.970814.0 0.9819 0.9842 0.9850 0.985517.0 0.9892 0.9914 0.9922 0.992620.0 0.9928 0.9950 0.9958 0.996225.0 0.9954 0.9976 0.9983 0.998730.0 0.9962 0.9984 0.9992 0.999634.0 0.9964 0.9987 0.9994 0.999836.0 0.9964 0.9988 0.9995 0.9999

0 5 10 15 20 25 30

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x)=P(C<x)

Buffer content, x

Figure 3: Variations of buffer content distribution against 𝑥.

with increase in the value of 𝑥 and converges to 1 as 𝑥 tendsto infinity. Observe that

lim𝑥→∞

𝑃 (𝐶 < 𝑥)

= lim𝑥→∞

(𝐹0,0(𝑥) + 𝐹

0,1(𝑥) +

∞

∑

𝑘=1

2

∑

𝑗=0

𝐹𝑘,𝑗(𝑥))

= 𝜋0,0+ 𝜋0,1+

∞

∑

𝑘=1

2

∑

𝑗=0

𝜋𝑘,𝑗= 1.

(68)

Furthermore, as the value of 𝜎0greatly affects buffer content

distribution, its variation against 𝑥 for a different value of 𝜎0

is presented in Table 1.


6. Conclusion

Markov Modulated Fluid Flows (MMFF) are a class offluid models wherein the rates at which the content of thefluid buffer varies are modulated by the Markov processevolving in the background. This paper studies a fluid modeldriven by an 𝑀/𝑀/1 queue subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption. The studyof such models provides greater flexibility to the designand control of input and output rates of fluid flow therebyadapting the fluid models to wider application background.The governing system of infinite differential difference equa-tions is explicitly solved using Laplace transform andmatrix-geometric methodology. Most of the existing results in theliterature pertaining to MMFF have presented the solutionto the buffer content distribution in the Laplace domain.However, closed form analytical solutions help to gain adeeper insight into the model and other related performancemeasures. The current findings can be thought of as oneof the key contributions to the theoretical development ofMMFF rather than the practical context. The theoreticalresults so obtained are verified with the existing results in theliterature as a special case. The variations of the stationarybuffer content distribution against the content of the bufferfor varying values of 𝜎

0are numerically illustrated.

Competing Interests

The authors declare that they have no competing interests.

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