Heavy-tailed asymptotics for a fluid model driven by an $M/G/1$ queue

Abstract

In this paper, an infinite-buffer fluid queue driven by an $M/G/1$ queue is discussed. The Laplace transform of the distribution of the stationary buffer content is expressed through the minimal positive solution to a crucial equation, similar to the fundamental equation satisfied by the busy period of an $M/G/1$ queue. Furthermore, the distribution of the stationary buffer content is shown to be regularly varying with index $-\alpha +1$ if the distribution of the service times is regularly varying with index $-\alpha <-1$. Meanwhile, the first $\lceil \alpha \rceil -2$ moments of the stationary buffer content are given, where $\lceil x\rceil$ is the ceiling function of the real number $x$.

Introduction

In the past two decades, fluid queues have attracted considerable interest. A fluid queue is an input–output system, where a continuous fluid enters and leaves a storage device, called a buffer, according to a randomly varying rate regulated by an external stochastic environment. Such fluid queues have been well accepted as a useful mathematical tool for modeling, for example, packet voice and video systems with or without background data, computer networks including call admission control, traffic shaping and modeling of TCP, and production and inventory systems. Readers may refer to Anick, Mitra and Sondhi [1], Mitra [2] and Kulkarni [3] for more details.

There have been many studies on fluid queues derived by a continuous-time Markov chain with finite states. Examples include Anick, Mitra and Sondhi [1], van Doorn [4], Mitra [2], Stern Elwalid [5], Kontovasilis and Mitrou [6], Asmussen [7], Karandikar and Kulkarni [8], Blaabjerg, Andersson and Andersson [9], Kulkarni [3] and Lenin [10]. In these works, the spectral-analytic method is often used with the computations of the eigenvalues and eigenvectors of a coefficient matrix.

For fluid queues driven by a Markov chain with countably-infinite states, the available results and methods are few. Virtamo and Norros [11] analyzed a fluid queue driven by an $M/M/1$ queue and proposed a spectral-decomposition method. The key of the method is to express the generalized eigenvalues explicitly using the Chebyshew Polynomials of the second kind. Subsequent papers have been published on this theme, including Adan and Resing [12], presenting an embedded points method, and Parthasarathy, Vijayashree and Lenin [13], providing a continued fraction method. For more general fluid queues, Konovalov [14] discussed the stability issue of a fluid queue driven by a $\mathrm{GI}/G/1$ queue. Van Doorn [15] studied a fluid queue driven by an irreducible birth–death process, and used the orthogonal polynomials to derive the distribution of the stationary buffer content. Sericola and Tuffin [16] provided an iteratively stable algorithm to study the distribution of the stationary buffer content in a fluid queue driven by a Markovian queue such as a $\mathrm{PH}/\mathrm{PH}/N/L$ queue for both $L<+\infty$ and $L=+\infty$. Barbot and Sericola [17] gave a new analytic expression for a stationary solution to the fluid queue driven by an $M/M/1$ queue. Li and Zhao [18] considered a fluid queue driven by a continuous-time level-dependent quasi-birth-and-death (QBD) process with either finitely-many levels or infinitely-many levels. They found that the Laplace transform of the distribution of the stationary buffer content is in a simple matrix-structured form.

Since the publication of Leland, Taqqu, Willinger and Wilson [19] on Ethernet data network in 1993, it has been well known that queues fed by long range dependent traffic may produce heavy-tailed performance measures such as busy periods, waiting times and queue lengths, while queues with heavy-tailed distributions can result in self-similar or long range dependent traffic. Readers may refer to Norros [20], Heath, Resnick and Samorodnitsky [21] and some survey papers in the two books edited by Adler, Feldman and Taqqu [22] and Park and Willinger [23]. In particular, examples on waiting times include Erramill, Narayan and Willinger [24], Boxma and Cohen [25], Whitt [26] and Takine [27]; examples on queue lengths include Resnick [28], Roughan, Veitch and Rumsewicz [29], Jelenković [30], Asmussen, Klüppelberg and Sigman [31], and Shang, Liu and Li [32]; and works on the more general heavy-tailed asymptotics of stationary probability vectors of Markov chains of $\mathrm{GI}/G/1$ type were presented in Li and Zhao [33] and Takine [34]. For a regularly varying tail of the busy period, Meyer and Teugels [35] and Zwart [36] studied the $M/G/1$ queue and the $\mathrm{GI}/G/1$ queue, respectively. Boxma and Dumas [37] provided an excellent overview on fluid queues fed by $N$ independent sources that alternate between silent and active periods, where the distribution of the active period is long-tailed. They considered the effect of this long-tailed behavior on the stationary buffer content distribution and the busy period distribution.

The main contribution of this paper is twofold. First, we derive a simple expression for the Laplace transform of the stationary buffer content distribution based on the minimal positive solution to a crucial equation, which is similar to the fundamental equation satisfied by the busy period of an $M/G/1$ queue. Based on the minimal positive solution, we then obtain some useful results for the fluid queue (e.g., Lemma 1 and Theorem 3) and derive regularly varying asymptotics of the stationary buffer content in Theorem 4. Second, we provide a novel approach for calculating regularly varying asymptotics of the stationary buffer content, from which we can obtain the moments of the stationary buffer content if the distribution of the service times is regularly varying. Roughan, Veitch and Rumsewicz [29] analyzed an $M/G/1$ queue with regularly varying index of the service times restricted to the range $-\alpha \in (-2,-1)$. With the new approach, we are able to generalize the range of the regularly varying index to $-\alpha <-1$. This approach can also be applied to discussing regularly varying asymptotics of other stochastic models.

This paper consists of 4 sections. Section 2 describes an infinite-buffer fluid queue driven by an $M/G/1$ queue and sets up a system of differential equations satisfied by the distribution of the stationary buffer content. Using the solution to the system of differential equations, we provide a simple expression for the Laplace transform of the distribution of the stationary buffer content. In Section 3, we provide the regularly varying asymptotic analysis for the distribution of the stationary buffer content and then derive the moments of the stationary buffer content. Section 4 concludes the paper.