Finite-buffer System Research Articles

A comprehensive study on the queue-size distribution in a finite-buffer system with a general independent input flow

A finite-buffer GI/M/1/N−type queueing model is considered. The explicit formula for the Laplace transform of the transient queue-size distribution, conditioned by the number of packets present in the system at the starting time, is derived. The shape of the formula allows for finding the stationary distribution by applying the key renewal theorem. Moreover, the convergence rate of the transient queue-size distribution to the stationary one is determined with the constant value given explicitly. Numerical example is attached as well.

Analysis of Finite Buffer Queue: Maximum Entropy Probability Distribution With Shifted Fractional Geometric and Arithmetic Means

A theoretical method based on maximum Shannon entropy framework (MSEF) in the presence of the geometric/or shifted fractional geometric mean of the queue size is applied to study the finite buffer system. Analytical expression of the loss probability for large buffer size is found to depict power law behavior. The maximum entropy framework is extended to incorporate additional shifted fractional arithmetic mean constraint to yield the expression of loss probability which is similar to the one derived by Kim and Shroff who have employed an entirely different approach based on maximum variance asymptotic (MVA) approximation. An important finding is the relationship between fractional order and the Hurst parameter. The advantage of MSEF is that it has enabled one to derive the analytical closed form generalized expression of the probability distribution of queue size in finite buffer system. Further, MSEF with shifted fractional geometric mean constraint gives the same distribution of queue size as the one obtained by maximizing Tsallis entropy subject to the fractional arithmetic mean of queue size.

Limiting Properties of Overloaded Multiuser Wireless Systems With Throughput-Optimal Scheduling

Throughput-optimal scheduling has been widely discussed due to its capability to stabilize single-hop multiuser wireless systems if possible. However, most of the previous discussions focused on the underloaded scenario, i.e., the arrival rate lies inside the achievable rate region. The behavior of throughput-optimal scheduling in overloaded multiuser wireless systems is the focus of this paper. We first show that, with the infinite buffer assumption, although all the queues are unstable, both the average throughput and a function of queue length converge as time evolves. In addition, the average throughput is the solution to a convex optimization problem whose objective is determined by the scheduling algorithm. By investigating the average throughput of two special throughput-optimal scheduling algorithms, i.e., the generalized MaxWeight and Log-Rule, we find that users can be prioritized by tuning the parameters associated with the scheduling algorithm, but the fairness is not likely to be guaranteed and some users may starve. Second, by studying the finite buffer system, we show that whether the buffer is dedicated to each queue or shared among queues has a great impact on the system performance, and the potential user starvation problem can be alleviated by a proper design.

Generalized sub-Gaussian fractional Brownian motion queueing model

It is well known that often the one-dimensional distribution of a queue content is not Gaussian but its tails behave like a Gaussian. We propose to consider a general class of processes, namely the class of \(\varphi \)-sub-Gaussian random processes, which is more general than the Gaussian one and includes non-Gaussian processes. The class of sub-Gaussian random processes contains Gaussian processes also and therefore is of special interest. In this paper we provide an estimate for the queue content distribution of a fluid queue fed by \(N\) independent strictly \(\varphi \)-sub-Gaussian generalized fractional Brownian motion input processes. We obtain an upper estimate of buffer overflow probability in a finite buffer system defined on any finite time interval \([a,b]\) or infinite interval \([0,\infty )\). The derived estimate captures more accurately the performance of the queueing system for a wider-range of input processes.

ON CUSTOMERS ACTING AS SERVERS

We consider systems comprised of two interlacing M/M/ • /• type queues, where customers of each queue are the servers of the other queue. Such systems can be found for example in file sharing programs, SETI@home project, and other applications [Arazi, A, E Ben-Jacob and U Yechiali (2005). Controlling an oscillating Jackson-type network having state-dependant service rates. Mathematical Methods of Operations Research, 62, 453–466]. Denoting by Li the number of customers in queue i(Qi), i = 1, 2, we assume that Q1 is a multi-server finite-buffer system with an overall capacity of size N, where the customers there are served by the L2 customers present in Q2. Regarding Q2, we study two different scenarios described as follows: (i) All customers present in Q1 join hands together to form a single server for the customers in Q2, with service time exponentially distributed with an overall intensity μ2L1. That is, the service rate of the customers in Q2 changes dynamically, following the state of Q1. (ii) Each of the customers present in Q1individually acts as a server for the customers in Q2, with service time exponentially distributed with mean 1/μ2. In other words, the number of servers at Q2 changes according to the queue size fluctuations of Q1. We present a probabilistic analysis of such systems, applying both Matrix Geometric method and Probability Generating Functions (PGFs) approach, and derive the stability condition for each model, along with its two-dimensional stationary distribution function. We reveal a relationship between the roots of a given matrix, related to the PGFs, and the stability condition of the systems. In addition, we calculate the means of Li, i = 1, 2, along with their correlation coefficient, and obtain the probability of blocking at Q1. Finally, we present numerical examples and compare between the two models.

Slotted ALOHA performance for FU-FB in frequency selective fading environment

Slotted ALOHA is a simple and straightforward random multiple access technique, which has been used extensively in data and cellular networks as the protocol for random access. The complexity of state space-based analysis methods for finite user finite buffer systems increases exponentially with buffer size and number of users. The presence of multipath frequency selective fading channel further adds to the complexity, making the analysis practically intractable. This paper uses an approximate analysis technique called tagged user analysis TUA to analyze the performance parameters of slotted ALOHA over multipath and frequency selective fading channels for finite user finite buffer systems. In TUA, the steady state system performance is evaluated from the analysis of a single user. Moreover, the state flow graph of TUA has just four states, thus reducing the complexity of the analysis. Simulation results confirm the validity of the TUA analysis. Copyright © 2013 John Wiley & Sons, Ltd.

Network queue and loss analysis using histogram-based traffic models

This paper proposes a new performance analysis for obtaining the queue length distribution in a finite size buffer system. This analysis uses a simple traffic model based on histograms that captures the arrival rate distribution on a network model. In order to reflect the second-order statistics (long-range dependence and self-similarity) of network traffic this basic model has been extended using the Hurst parameter. The buffer occupancy distribution is obtained using these traffic models and defining a new stochastic process that works directly with histograms. The loss probability and network delay distribution can be obtained using this distribution. The estimation of the expected traffic loss probability is a key issue in provisioning Quality of Service in telecommunications networks. This paper shows that the proposed buffer occupancy distribution and loss probability calculations techniques are very precise.

Analysis of spatially distributed FU-FB S-ALOHA in fading channels using TUA

The tagged user analysis (TUA) is a generic approximate method developed to analyze random access protocols for finite-user finite-buffer systems that avoids complexities of Markovian analysis by using available queueing results. In this paper, TUA method is extended to the analysis of finite buffer S-ALOHA operating over flat fading radio channels when the user population is spatially distributed. The expressions for system performance indices are derived. It is shown that for moderate number of active users, the simulation and analytical results have a close fit.

An Approximate Analysis of Buffered S-ALOHA in Fading Channels Using Tagged User Analysis

Tagged user analysis (TUA) is a generic approximate method of analyzing random access protocols for finite-user finite-buffer systems. This technique decouples the channel contention behavior from the user queuing behavior and allows the use of classical queuing theory results to be directly applicable to the analysis of finite-user finite-buffer random access methods. In this paper, we extend TUA to analyze finite buffer S-ALOHA operating over flat fading radio channels and derive expressions for system performance indices like throughput, average packet delay, blocking probability and queue length. It is shown that for a moderate number of active users, the simulation and analytical results fit closely

Characterizing losses during busy periods in finite buffer systems

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ= 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and whenρ> 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, whenρ= 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

Characterizing losses during busy periods in finite buffer systems

For multiple-server finite-buffer systems with batch Poisson arrivals, we explore how the distribution of the number of losses during a busy period changes with the buffer size and the initial number of customers. We show that when the arrival rate equals the maximal service rate (ρ = 1), as the buffer size increases the number of losses in a busy period increases in the convex sense, and when ρ > 1, as the buffer size increases the number of busy period losses increases in the increasing convex sense. Also, the number of busy period losses is stochastically increasing in the initial number of customers. A consequence of our results is that, when ρ = 1, the mean number of busy period losses equals the mean batch size of arrivals regardless of the buffer size. We show that this invariance does not extend to general arrival processes.

On the asymptotic relationship between the overflow probability and the loss ratio

In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log P{Q > x} and logP L (x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them.

On the asymptotic relationship between the overflow probability and the loss ratio

In this paper we study the asymptotic relationship between the loss ratio in a finite buffer system and the overflow probability (the tail of the queue length distribution) in the corresponding infinite buffer system. We model the system by a fluid queue which consists of a server with constant rate c and a fluid input. We provide asymptotic upper and lower bounds on the difference between log P{Q > x} and logPL(x) under different conditions. The conditions for the upper bound are simple and are satisfied by a very large class of input processes. The conditions on the lower bound are more complex but we show that various classes of processes such as Markov modulated and ARMA type Gaussian input processes satisfy them.

Loss probability calculations and asymptotic analysis for finite buffer multiplexers

We propose an approximation for the loss probability, P/sub L/(x), in a finite buffer system with buffer size x. Our study is motivated by the case of a high-speed network where a large number of sources are expected to be multiplexed. Hence, by appealing to central limit theorem type of arguments, we model the input process as a general Gaussian process. Our result is obtained by making a simple mapping from the tail probability in an infinite buffer system to the loss probability in a finite buffer system. We also provide a strong asymptotic relationship between our approximation and the actual loss probability for a fairly large class of Gaussian input processes. We derive some interesting asymptotic properties of our approximation and illustrate its effectiveness via a detailed numerical investigation.

The significance of finite buffer edge effects in histogram measurements of queues

The empirically oberved "fractal" or "self-similar" nature of packet traffic implies heavy tailed queue processes for such traffic. However, based on our simulation analysis using real network data as well as standard models, we have found that the actual losses sustained are remarkably lower than those suggested by the heavy tail distribution. This can be explained by an effect observed in the tail of the histogram of a finite buffer queue process, which we call "tail-raising", which contains information pertinent to performance estimation. This effect is also responsible for a significant reduction in packet losses for finite buffer systems, than would be otherwise predicted by the buffer overflow probability for heavy-tailed queues. We define a new parameter X B on the histogram of a queue process for a finite buffer system, to calculate the tail of the queue process based on the information available in the histogram on the finite buffer. We propose an estimator that approximates X B , namely, X min?, which is measurable because of the tail-raising effect and has a robust measurement method. The proposed estimator shows promise as a good predictor for performance metrics of queueing systems. We propose an innovative packet loss ratio estimation technique which uses histogram measurements combined with a virtual buffer scheme to find and extrapolate the objective packet loss rate using a binning strategy for histogram measurement, namely, Symmetric Logarithmic Binning (SLB).

SMAQ: a measurement-based tool for traffic modeling and queuing analysis. I. Design methodologies and software architecture

SMAQ is a measurement-based tool for integration of traffic modeling and queuing analysis. There are three basic components in SMAQ. In the design of the first component, statistic measurement, the most critical issues are to identify the important traffic statistics for queuing analysis in a finite buffer system and then to build a measurement structure to collect them. Our study indicates that both first- and second-order traffic statistics, measured within a given frequency-window, have a very significant impact on the queue length and loss rate performance. In the design of the second component, matched modeling, the focal point is to construct a stochastic model that can match a wide range of important statistics collected in various applications. New methodologies and fast algorithms are developed for such construction on the basis of a circulant modulated Poisson process (CMPP). For the third component, queuing solutions, the basic requirement is to provide numerical solutions of the queue length and loss rate for transport of given traffic in a finite buffer system. A fast and stable computation method, called a Folding algorithm, is applied to provide both steady-state and transient solutions of various kinds, including congestion control performance where arriving traffic are selectively discarded based on queue thresholds. We provide both design methodologies and software architectures of these three components, with discussion of practical engineering issues for the use of the SMAQ tool.

Fast simulation of Markov fluid models

In this paper we study continuous flow finite buffer systems with input rates modulated by Markov chains. Discrete event simulations are applied for estimating loss probabilities. The simulations are executed under a twisted version of the original probability measure (importance sampling). We present a simple rule for determining a new measure, then show that the new measure matches the ‘most likely' empirical measure that we expect from large deviations arguments, and finally prove optimality of the new measure.

Fast simulation of Markov fluid models

In this paper we study continuous flow finite buffer systems with input rates modulated by Markov chains. Discrete event simulations are applied for estimating loss probabilities. The simulations are executed under a twisted version of the original probability measure (importance sampling). We present a simple rule for determining a new measure, then show that the new measure matches the ‘most likely' empirical measure that we expect from large deviations arguments, and finally prove optimality of the new measure.

Link capacity allocation by input power spectrum

Consider a multimedia input traffic stream, described by a stationary random process. Our previous study indicates that second-order input statistics, which are described by a power spectrum in the frequency domain, play a dominant role in queueing performance as compared with higher-order input statistics. In this paper, we show the significant effect of an input power spectrum on link capacity allocation in a finite-buffer system to achieve a desired performance criterion. The study also explores a fundamental limit of buffer sizing to improve link capacity when traffic contains significant low-frequency power.

Performance impacts of self-similarity in traffic

Recent measurement studies in Bellcore and elsewhere have convincingly established the presence of statistical self similarity in high-speed network traffic. What is less clear --- and as such the subject of intense current research --- is the impact of the self-similarity on network performance. Given that traditional queueing models of network performance do not model self-similarity, the validity of traditional models to predict network performance would be supported if it is shown that self-similarity does not have measurable impacts on performance. On the other hand, if the converse of this assertion were true, it would have significant impacts on the way networks are designed and analyzed, as well as open up new areas of research in mathematical modeling, queueing analysis, network design and control. The issues addressed in this session are therefore of fundamental importance in high-speed network research.Given that queueing behavior is dominated by traffic characteristics over the time scales of busy periods, it has been argued that phenomena that span many time scales, such as self-similarity, should not be relevant for queueing performance. However, the paper by Narayan, Erramilli and Willinger presents evidence that for data traffic, the long range dependence (which is related to the self-similarity in traffic) can dominate queueing behavior under a variety of conditions. Specifically, it is shown based on a series of carefully designed simulation experiments with actual traffic traces, that the queueing behavior with actual traces is considerably heavier than that predicted by traditional theory, and that these differences are attributable to long range dependence. The paper by Heyman and Lakshman investigates modeling of video traffic to predict cell loss performance with finite buffer systems, and they conclude that long-range dependence is not a crucial property in determining the finite buffer behavior of video conferences. In particular, a Markov chain model that does not model long-range dependence is nevertheless able to reproduce various operating characteristics over a wide range of loadings obtained with the actual video trace. Mukherjee, Adas, Klivansky and Song investigate the performance impacts of short-range and long-range correlation components using simulations with a fractional ARIMA model. They also discuss a strategy to provide quality of service guarantees with long range dependent traffic, as well as recent results on NSFNET traffic. Finally, the paper by Li describes a frequency-domain based analytical tool that matches a special class of Markov chains with traces exhibiting a variety of characteristics, including long-range dependence. Good agreement is reported between analytical queueing solutions of the matched Markov chains, and simulation results obtained video and data traffic traces.This session therefore brings together a wide range of viewpoints on this issue. Resolution of such seemingly conflicting conclusions lies in the fact that in performance analysis, answers sensitively depend on the specific details of a problem. Thus the proper question to ask is not whether or not self-similarity matters in queueing; but under what conditions it matters. Likewise, the question to ask is not whether a class of models is invalid; but to identify the conditions under which traditional Markov or self-similar traffic models are expected to be valid. Finally, given an understanding of statistical features that are relevant to a given problem, the challenge is to model these accurately and parsimoniously so that the model is useful in practical performance analysis. The work outlined in the abstracts below adds significantly to our understanding of these issues.