Advances in Operations Research

Volume 2016, Article ID 2673017, 11 pages

http://dx.doi.org/10.1155/2016/2673017

## Fluid Queue Driven by an Queue Subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption

Department of Mathematics, Anna University, Chennai 600025, India

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

Academic Editor: Yi-Kuei Lin

Copyright © 2016 Kolinjivadi Viswanathan Vijayashree and Atlimuthu Anjuka. 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 deals with the stationary analysis of a fluid queue driven by an 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. The resulting solutions are then inverted to obtain an explicit expression for the joint steady state probabilities of the content of the buffer and the state of the background queueing model. Numerical illustrations are added to depict the convergence of the stationary buffer content distribution to one subject to suitable stability conditions.

#### 1. Introduction

In many real time situations, the server in the background queueing model may become unavailable for a random period of time to perform a secondary task, when there are no customers in the waiting line at the service completion epoch. Such period of server absence is termed as server vacation. Queueing models subject to various vacation policies are of interest to researchers in recent times owing to their widespread applicability.

There are different types of vacation queueing systems. In the single vacation scheme, the server takes a vacation of some random duration when the queue is empty. At the end of the vacation, the server returns to the queue. The server resumes service if there is at least one customer waiting upon his return from vacation. However, if the queue is empty on the server’s return, the server waits to complete a busy period. In the multiple vacation scheme, if the server returns from a vacation and finds the queue empty, he immediately commences another vacation. If there is at least one waiting customer, then he will commence the service. Queueing models 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 the server works at a slower rate during vacation periods in comparison to that of a regular working period. Such models are classified as queues subject to working vacations [4–6]. In addition, the server can stop the vacation once 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. Li and Tian [7] studied the single server queueing model with working vacation and vacation interruption. The modulating queueing model considered in this paper is an queue wherein the server is subject to regular vacation with probability or working vacation with probability . Further, the vacation duration of the server during working vacation epoch may be interrupted due to vacation interruption.

Fluid queues have become a fascinating area of research in recent years due to their widespread applicability in computer and 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 fluid that enters and leaves the storage device called a buffer, according to randomly varying rates. They are appropriate in a situation wherein the arrival is comprised of a discrete unit, but the interarrival time between successive arrivals is negligible. Therefore, the arrivals can be approximated by a continuous flow of fluid as individuals units have less impact on the performance of the system. In these models, a fluid buffer is either filled or depleted or both at rates determined by the current state of the background queueing model. Markov modulated fluid queues are a particular class of fluid models useful for modeling many physical phenomena and they often allow tractable analysis. In addition, fluid models are quite useful as approximate models for certain queueing and inventory systems where the flow consists of discrete entities, but the behavior of the individual is not important to identify the performance analysis. Certain interesting real world applications of Markov Modulated Fluid Flow models can be found in [11–14]. Besides, fluid queues also have successful applications in the field of congestion control [15] and risk processes [16]. More recently, Bosman and Nunez-Queija [17] considered a tandem fluid queue model to evaluate the performance of streaming media over an unreliable network.

For example, consider a production inventory model operating in a stochastic environment. The inventory level increases when the production rate exceeds the demand rate and decreases otherwise. The inventory level under continuous review can be viewed as a fluid process that fluctuates according to the evolution of the underlying background environment. For example, consider a machine shop with a single server. When the server is busy, items are produced continuously at a rate and if he is idle, there is no production. However, for all practical reasons, the server might either take a vacation of some random duration with probability or decide to provide service at a reduced rate with probability . Further, by offering service at a reduced rate, the server may continue to do so with probability or due to certain unforeseen reasons, like a sudden increase in the demand, interrupt the vacation with probability , and continue the busy period. The demands are assumed to vary from time to time at the rate independent of the state of the server. The level of inventory thus oscillates between and depending on the busy or idle state of the server. Such scenario can be modeled as a fluid queue driven by an queue subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption.

The stationary analysis of fluid queueing models in a stochastic environment has been discussed by many authors. Fluid queues driven by an queueing model are extensively studied in the literature. Various techniques have been employed by researchers to obtain the stationary buffer content distribution. To mention a few, Adan and Resing [18] analyze the buffer content distribution by viewing the arrival process as an alternating renewal process and Virtamo and Norros [19] provide the buffer content distribution by finding the spectrum of the eigenvalue equation and explicit expressions for the corresponding eigenvectors in terms of Chebyshev polynomials of the second kind. Sericola and Tuffin [20] express the stationary distribution of the buffer occupancy in terms of a sequence of recursively defined polynomials. Parthasarathy et al. [21] present an explicit expression for the buffer content distribution in terms of modified Bessel function of the first kind using continued fraction methodology.

Furthermore, fluid models driven by an queue subject to various vacation strategies were analyzed in steady state by Mao et al. [22] and Wang et al. [23]. The work was further extended to the stationary analysis of fluid queues driven by an queue with multiple exponential vacation and policy [24]. Fluid model driven by an queue with working vacations and vacation interruption was studied by Xu et al. [25]. However, in most of the literature relating to fluid queues driven by vacation queueing models, the buffer content distribution is expressed in the Laplace domain. More recently, Vijayashree and Anjuka [26, 27] presented an explicit expression for the buffer content distribution of a fluid queueing model modulated by an queue subject to catastrophes and subsequent repair and a fluid queue driven by an queueing model, respectively. Also, Ammar [28] derives an explicit expression for the fluid queue driven by an queue with multiple exponential vacation using generating function methodology.

This paper presents an analytical solution for the fluid queue driven by an queue subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption in stationary regime. When the background queueing model is empty, the server will take either an ordinary vacation with probability or a working vacation with probability . If the system is in a working vacation, upon completion of service, the server either ends the vacation and enters a regular busy period with probability or continues the vacation with probability . It is assumed that the fluid in the buffer content increases at a constant rate, , when there is one or more customers in the background queueing model and decreases at a constant rate, , when the queue is empty. The system of equations governing the process is modeled in terms of quasi-birth and death process and solved using matrix-geometric method. The stationary distribution of the buffer content is thereby obtained in the Laplace domain and hence inverted to obtain explicit expressions for the joint steady state probabilities of the state of the background queueing model and content of the buffer. Closed form expressions help to gain a deeper insight into the model. As a special case, when , , and the theoretical results, so obtained, are seen to coincide with the existing results of Xu et al. [25] and Wang et al. [23], respectively.

The rest of the paper is organized as follows: Section 2 gives a brief description of the background queueing model under consideration. Section 3 presents the system of differential difference equations that governs the fluid queueing model under steady state subject to suitable stability conditions. Section 4 gives detailed derivations of the closed form expressions for the joint steady state probabilities of the state of the background queueing model and also the buffer content distributions. Section 5 presents the numerical illustration of the stationary buffer content distribution for suitable choice of the parameter values.

#### 2. Model Description

Consider an queueing model with infinite capacity. Let the customers arrive according to a Poisson process with parameter . The server provides service to the arriving customers according to an exponential distribution with parameter . When the system becomes empty, the server begins a vacation of random length and takes an ordinary vacation with probability or a working vacation with probability , where . In an ordinary vacation, the server will stop working even if there are new arrivals during the vacation period. In a working vacation, customers are served at a lower rate . Further, in a working vacation, it is assumed that, at the instants of service completion, either the vacation is interrupted and the server resumes to a regular busy period with probability or the server continues the vacation with probability . When the vacation period ends and the system is nonempty, a new busy period starts. The ordinary vacation times and the working vacation times are also assumed to be exponentially distributed with parameters and , respectively. Let denote the number of customers at time . Define It is well known that the process is a quasi-birth and death (QBD) process with state space given by The state transition diagram of the background queueing model is given in Figure 1. Let represent the steady state probabilities of the background queueing model. Further, let and for . Then, the stationary probability vector is denoted by