Buffer Content Distribution Research Articles

On the time-dependent solution of fluid models driven by an <i>M</i>/<i>M</i>/1 queue using a probabilistic approach

More study has been performed on the steady state analysis of fluid models. However, the transient analysis of fluid models has not been extensively carried due to complexity of the problem. This is because the system of conservation laws for which no closed form solution is available. The motivation behind this paper is to provide a new methodology to find transient distribution of buffer content for fluid models. In this paper, the time-dependent solution of a fluid model driven by an M/M/1 queue is derived using a simple probability approach. Finally, numerical results are illustration for the proposed approach.

Performance analysis of a fluid flow system modulated by a single server queue prone to catastrophic failures and repairs

<p style='text-indent:20px;'>This article deals with a new class of fluid flow queue regulated by an <inline-formula><tex-math id="M1">\begin{document}$ M/M/1 $\end{document}</tex-math></inline-formula> queueing system in the presence of active catastrophic failures and subsequent repairs of the server. The distribution of the stationary buffer-content is determined in terms of modified Bessel function of first kind. Besides, an explicit closed-form analytical expression for the probability density function of the fluid level in the stationary regime is derived in an elegant manner. Some important and essential performance measures are obtained. Finally, numerical results are provided to illustrate the behaviour of the buffer content distribution and the performance measures for the proposed fluid flow queueing model.</p>

Mean-Field Limits for Large-Scale Random-Access Networks

We establish mean-field limits for large-scale random-access networks with buffer dynamics and arbitrary interference graphs. Although saturated buffer scenarios have been widely investigated and yield useful throughput estimates for persistent sessions, they fail to capture the fluctuations in buffer contents over time and provide no insight in the delay performance of flows with intermittent packet arrivals. Motivated by that issue, we explore in the present paper random-access networks with buffer dynamics, where flows with empty buffers refrain from competition for the medium. The occurrence of empty buffers thus results in a complex dynamic interaction between activity states and buffer contents, which severely complicates the performance analysis. Hence, we focus on a many-sources regime where the total number of nodes grows large, which not only offers mathematical tractability but is also highly relevant with the densification of wireless networks as the Internet of Things emerges. We exploit timescale separation properties to prove that the properly scaled buffer occupancy process converges to the solution of a deterministic initial value problem and establish the existence and uniqueness of the associated fixed point. This approach simplifies the performance analysis of networks with huge numbers of nodes to a low-dimensional fixed-point calculation. For the case of a complete interference graph, we demonstrate asymptotic stability, provide a simple closed form expression for the fixed point, and prove interchange of the mean-field and steady-state limits. This yields asymptotically exact approximations for key performance metrics, in particular the stationary buffer content and packet delay distributions.

Stationary Solution of a Fluid Queue Driven by a Queue with Chain Sequence Rates and Controlled Input

A fluid queueing system in which the fluid flow in to the buffer is regulated by the state of the background queueing process is considered. In this model, the arrival and service rates follow chain sequence rates and are controlled by an exponential timer. The buffer content distribution along with averages are found using continued fraction methodology. Numerical results are illustrated to analyze the trend of the average buffer content for the model under consideration. It is interesting to note that the stationary solution of a fluid queue driven by a queue with chain sequence rates does not exist in the absence of exponential timer.

Exact stationary solution for a fluid queue driven by an <i>M/M/</i>1 queue with disaster and subsequent repair

This paper deals with the stationary analysis of a fluid queuing model driven by an M/M/1 queue subject to disaster and subsequent repair. Further, arrivals are allowed to join the background queuing model during the period of repair at a slower rate as compared to the arrivals during regular busy period of the server. Such a model was analysed earlier by Ammar (2014), however the model formulation and hence the main results are found to be incorrect. In this paper, the assumptions are suitably modified to ensure correctness, detailed mathematical analysis is carried out to find an explicit analytical expression for the buffer content distribution. The underlying system of differential difference equations that govern the process are solved using Laplace transform and generating function methodologies. The closed form expressions for the joint steady state probabilities of the state of the background queuing model and the content of the buffer are obtained in terms of modified Bessel function of the first kind.

Stationary analysis of a fluid queue driven by an M / M / 1 queue with working vacation

This paper studies a fluid queueing model driven by an M / M / 1 queue subject to working vacation. The underlying system of differential difference equations that governs the process is solved using continued fraction and generating function methodologies. Explicit expressions for the joint steady-state probabilities of the state of the background queueing model and the content of the buffer are obtained in terms of modified Bessel function of the first kind. Numerical illustrations are added to depict the variations of the state probabilities and the buffer content distributions against varying values of the parameters subject to the stability conditions.

A unified algorithm for finite and infinite buffer content distribution of Markov fluid models

This paper is interested in the asymptotic distribution of fluid models with finite or infinite buffer capacity. The probability distribution of the buffer content is controlled by a linear differential system with specific boundary conditions. All earlier methods are forced to test the singularity of the drift matrix before starting any computation. Also, the finite and infinite buffer capacity cases are treated differently. Our approach leads to a unified algorithm which is able to compute simultaneously the buffer workload distribution for finite and infinite buffer capacity without looking into the drift matrix singularity. The proposed method is not spectral and it is based on the uniformization technique. An example of a video traffic is firstly modeled by a Markovian fluid and secondly analyzed.

Fluid Queue Driven by anM/M/1Queue Subject to Bernoulli-Schedule-Controlled Vacation and Vacation Interruption

This paper deals with the stationary analysis of a fluid queue driven by anM/M/1queueing 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.

Fluid queue driven by an M/M/1 disasters queue

This paper considers a fluid queue driven by an M/M/1 disaster queue. We derive explicit expression for the stationary distribution function of the buffer content in terms of modified Bessel function of first kind by using generating function technique. Numerical results are presented to illustrate the behavior of the buffer content distribution.

Open problems in Gaussian fluid queueing theory

We present three challenging open problems that originate from the analysis of the asymptotic behavior of Gaussian fluid queueing models. In particular, we address the problem of characterizing the correlation structure of the stationary buffer content process, the speed of convergence to stationarity, and analysis of an asymptotic constant associated with the stationary buffer content distribution (the so-called Pickands constant).

Effective load for flow-level performance modelling of file transfers in wireless LANs

Today, a wide range of 802.11-based Wireless LANs (WLANs) have become dominant to provide wireless Internet access for file transfers. For engineering purposes, there is a need for very simple, explicit, yet accurate, models that predict the performance of WLANs under anticipated load conditions. In this context, several detailed packet-level models have been proposed, based on fixed-point equations. Despite the fact that these models generally lead to accurate performance predictions, they do not lead to simple explicit expressions for the performance of WLANs. Motivated by this, we propose a new analytic model that captures the highly complex combined dynamics and protocol overhead of the 802.11 MAC, IP, TCP and application-layer into an explicit expression for a single parameter which will be called the effective service time. Based on the effective service time, we define the effective load to describe the flow-level performance of file transfers over WLANs with an M/ G/1 Processor Sharing (PS) model. Using the M/ G/1 PS model properties we propose a simple analytic model to obtain WLAN AP buffer content distribution. Despite the fact that PS models are heavily used in modelling flow-level performance in communication networks, an extensive validation of such models has not been published in the field, or context, of WLAN. To this end, our model is validated extensively by comparing the model-based average response times against simulations. The results show that the model leads to highly accurate predictions over a wide range of parameter combinations, including light- and heavy-tailed file-size distributions and light- and heavy-load scenarios. The simplicity and accuracy of the model make the results of high practical relevance and useful for performance engineering purposes.

Asymptotic behavior of the loss rate for Markov-modulated fluid queue with a finite buffer

We consider a Markov-modulated fluid queue with a finite buffer. It is assumed that the fluid flow is modulated by a background Markov chain which may have different transitions when the buffer content is empty or full. In Sakuma and Miyazawa (Asymptotic Behavior of Loss Rate for Feedback Finite Fluid Queue with Downward Jumps. Advances in Queueing Theory and Network Applications, pp. 195---211, Springer, Cambridge, 2009), we have studied asymptotic loss rate for this type of fluid queue when the mean drift of the fluid flow is negative. However, the null drift case is not studied. Our major interest is in asymptotic loss rate of the fluid queue with a finite buffer including the null drift case. We consider the density of the stationary buffer content distribution and derive it in matrix exponential forms from an occupation measure. This result is not only useful to get the asymptotic loss rate especially for the null drift case, but also it is interesting in its own light.

Fluid Model Driven by an M/M/1/N Queue with Single Exponential Vacation

This paper studies a fluid model driven by an M/M/1/N queue with single exponential vacation. With the standard spectral method, the exact expression of the steady-state distribution of the buffer content is acquired. Then the performance measure-mean buffer content is obtained. Finally, using the numerical example, we present the parameter effect on the stationary distribution of the buffer content.

ON A GENERIC CLASS OF LÉVY-DRIVEN VACATION MODELS

This article analyzes a generic class of queuing systems with server vacation. The special feature of the models considered is that the duration of the vacations explicitly depends on the buffer content evolution during the previous active period (i.e., the time elapsed since the previous vacation). During both active periods and vacations, the buffer content evolves as a Lévy process. For two specific classes of models, the Laplace–Stieltjes transform of the buffer content distribution at switching epochs between successive vacations and active periods, and in steady state, is derived.

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

We consider a Levy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper, we consider only light-tailed inputs. For the Brownian input case, we derive exact tail asymptotics for the marginal stationary distribution of the second buffer content, while weaker asymptotic results are obtained for the general Levy input case. The results generalize those of Lieshout and Mandjes from the recent papers (Lieshout and Mandjes in Math. Methods Oper. Res. 66:275---298, 2007 and Queueing Syst. 60:203---226, 2008) for the corresponding tandem queue without an intermediate input.

Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. We also study networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property.

Heavy-tailed asymptotics for a fluid model driven by an [formula omitted] queue

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 − α + 1 if the distribution of the service times is regularly varying with index − α < − 1 . Meanwhile, the first ⌈ α ⌉ − 2 moments of the stationary buffer content are given, where ⌈ x ⌉ is the ceiling function of the real number x .

A discrete-time queueing model of EMD policy in high-speed networks

In this paper, we consider a discrete-time queue for the Early Message Discarding (EMD) policy in high-speed networks under bursty arrival. This system is analyzed as a quasi-birth-and-death (QBD) process. We derive the steady-state probability distribution of buffer content by using generating-functions approach. Finally, we obtain the useful performance measure— goodput ratio and give some numerical results.

Ray Solution of a Singularly Perturbed Elliptic PDE with Applications to Communications Networks

We analyze a second order, linear, elliptic PDE with mixed boundary conditions. This problem arose as a limiting case of a Markov‐modulated queueing model for data handling switches in communications networks. We use singular perturbation methods to analyze the problem. In particular we use the ray method to solve the PDE in the limit where convection dominates diffusion. We show that there are both interior and boundary caustics, as well as a cusp point where two caustics meet, an internal layer, boundary layers, and a corner layer. Our analysis leads to approximate formulas for the queue length (or buffer content) distribution at the switch.

Time to stationarity for general Markov fluid models

AbstractIn this paper, we analyse the buffer content behaviour for general Markov fluid models. Departing from a uniformization‐based solution in transient regime, we deduce the steady‐state buffer content distribution and we give a simple expression measuring a time to stationarity. Moreover, for a fixed period of time, an upper bound for the absolute difference between the transient solution and the asymptotic one is proposed. As an application example, a heterogeneous fluid‐flow model is considered and analysed with our proposed method. Copyright © 2005 John Wiley &amp; Sons, Ltd.