Markov-modulated Fluid Model Research Articles

Singularly Perturbed Markov Modulated Fluid Queues

We consider a Markov modulated fluid queue for which the environment is nearly completely decomposable. Under the basic assumption that both the nearly completely decomposable Markov modulated fluid model and the unperturbed fluid models are positive recurrent, we show that the stationary density of the level can be expanded as convergent power series of the aggregated stationary densities. We go further in the analysis by assuming that one or more of the unperturbed fluid queues is not necessarily positive recurrent. We provide numerical illustration.

RARE EVENT ANALYSIS AND EFFICIENT SIMULATION FOR A MULTI-DIMENSIONAL RUIN PROBLEM

This paper focuses on the evaluation of the probability that both components of a bivariate stochastic process ever simultaneously exceed some large level; a leading example is that of two Markov fluid queues driven by the same background process ever reaching the set (u, ∞)×(u, ∞), for u>0. Exact analysis being prohibitive, we resort to asymptotic techniques and efficient simulation, focusing on large values of u. The first contribution concerns various expressions for the decay rate of the probability of interest, which are valid under Gärtner–Ellis-type conditions. The second contribution is an importance-sampling-based rare-event simulation technique for the bivariate Markov modulated fluid model, which is capable of asymptotically efficiently estimating the probability of interest; the efficiency of this procedure is assessed in a series of numerical experiments.

Estimation of quality of service parameters for OFDM–TDMA networks using Markov chains and service curve

SummaryThis paper presents a model based on Queueing Theory for Orthogonal Frequency‐Division Multiplexing–Time‐Division Multiple Access (OFDM–TDMA) networks. This network model requires a Markovian model to the traffic processes. Thus, we present a Markov Modulated Fluid Model (MMFM) with autocorrelation function fitting for the network traffic, in order to evaluate the performance of the transmission link in OFDM–TDMA systems. That is, we deduce equations for estimating QoS parameters of OFDM–TDMA systems. The obtained results confirm that the proposed network model is efficient in describing the OFDM based link performance. A novel equation is also introduced for estimating the buffer overflow probability in OFDM–TDMA systems using a proposed network service curve. The computational effort is reduced compared to the Queueing Theory based method of overflow probability estimation. Besides, the results show that the estimates are very close to those obtained by simulations.

Performance Analysis and Modeling of OFDM-TDMA Systems Using Queueing Theory and a Markov Modulated Fluid Model

This paper presents a model based on Queuing Theory, Markov Chains and MMFM in order to evaluate the performance of the transmission link in OFDM-TDMA systems. To this end, a MMFM traffic model with autocorrelation function fitting is proposed. From the Queueing model, it is derived multiple QoS parameter estimators. The obtained results confirm that the proposed model is efficient in describing the link performance indicators. The use of MMFM to model the arrival process improves the QoS parameter estimates of the queueing model making their values very close to those of the simulation results.

Alternative fluid approximation approach for the steady-state distribution of the two-sided reflected Markov modulated Brownian motion and its computation

The steady-state distribution of the two-sided reflected Markov Modulated Brownian motion is derived through an alternative fluid approximation approach. In this paper, we have shown that the distribution can be obtained by taking limit on the steady-state distribution of a weakly convergent Markov modulated fluid model. In addition, we present how the duality theorem developed by Ahn and Ramaswami is applied to get the limit result. For computation, we introduce a quadratically convergent algorithm and its related theories which rely on the fluid approximation approach. Through numerical examples, we finally illustrate behaviors of the steady-state distribution and also some performances of the algorithm.

Response time distribution of flash memory accesses

Flash memory is becoming an increasingly important storage component among non-volatile storage devices. Its cost is decreasing dramatically and its performance continues to improve, which makes it a serious competitor for disks and a candidate for enterprise-tier storage devices of the future. Consequently, it is important to devise models and tools to analyse its behaviour and to evaluate its effects on a system’s performance. We propose a Markov modulated fluid model with priority classes to investigate the response time characteristics of Flash memory accesses. This model can represent well the Flash access operation types, respecting the erase/write/read relative priorities and autocorrelations. We apply the model to estimate response time densities at the chip for an OLTP-type of workload and indicate the magnitude of the penalty suffered by writes under priority scheduling of read operations. The model is validated against a customised hardware simulator that uses input-traces typical of our Markovian workload description.

Asymptotic analysis of a fluid model modulated by an M/M/1 queue

We analyze asymptotically a differential-difference equation that arises in a Markov-modulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.

Asymptotic analysis of a fluid model modulated by an M/M/1 queue

We analyze asymptotically a differential-difference equation that arises in a Markov-modulated fluid model. We use singular perturbation methods to analyze the problem with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.

Analysis of a polling system for telephony traffic with application to wireless LANs

Recently, polling has been included as a resource sharing mechanism in the medium access control (MAC) protocol of several communication systems, such as the IEEE 802.11 wireless local area network, primarily to support real-time traffic. Furthermore, to allow these communication systems to support multimedia traffic, the polling scheme often coexists with other MAC schemes such as random access. Motivated by these systems, we develop a model for a polling system with vacations, where the vacations represent the time periods in which the resource sharing mechanism used is a non-polling mode. The real-time traffic served by the polling mode in our study is telephony. We use an on-off Markov modulated fluid (MMF) model to characterize telephony sources. Our analytical study and a counterpart validating simulation study show the following. Since voice codec rates are much smaller than link transmission rates, the queueing delay that arises from waiting for a poll dominates the total delay experienced by a voice packet. To keep delays low, the number of telephone calls that can be admitted must be chosen carefully according to delay tolerance, loss tolerance, codec rates, protocol overheads and the amount of bandwidth allocated to the polling mode. The effect of statistical multiplexing gain obtained by exploiting the on-off characteristics of telephony traffic is more noticeable when the impact of polling overhead is small

Steady State Analysis of Finite Fluid Flow Models Using Finite QBDs

The Markov modulated fluid model with finite buffer of size β is analyzed using a stochastic discretization yielding a sequence of finite waiting room queueing models with iid amounts of work distributed as exp (n?). The n-th approximating queue's system size is bounded at a value qn such that the corresponding expected amount of work qn/(n?) ? β as n ? ?. We demonstrate that as n ? ?, we obtain the exact performance results for the finite buffer fluid model from the processes of work in the system for these queues. The necessary (strong) limit theorems are proven for both transient and steady state results. Algorithms for steady state results are developed fully and illustrated with numerical examples.

Geometrical Optics Approach to Markov-Modulated Fluid Models

We analyze asymptotically a differential-difference equation, that arises in a Markov-modulated fluid model. Here, there are N identical sources that turn on and off, and when on they generate fluid at unit rate into a buffer, which processes the fluid at a rate c < N. In the steady-state limit, the joint probability distribution of the buffer content and the number of active sources satisfies a system of N + 1 ODEs, that can also be viewed as a differential-difference equation analogous to a backward/forward parabolic PDE. We use singular perturbation methods to analyze the problem for N → oo, with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.

On the Tail of the Waiting Time in a Markov-Modulated M/G/1 Queue

We show that the “exponential decay parameter” of the waiting time in a Markov-modulated M/G/1 queue is no larger than that of the corresponding M/G/1 queue with “averaged” parameters, and we give a necessary and sufficient condition for equality. We also explore the effect of speeding up the modulation process. A key tool is a Markov-modulated fluid model.

Burst-level congestion control using hindsight optimization

We consider the burst-level congestion-control problem in a communication network with multiple traffic sources, each modeled as a fully controllable stream of fluid traffic. The controlled traffic shares a common bottleneck node with high-priority cross traffic described by a Markov-modulated fluid (MMF). Each controlled source is assumed to have a unique round-trip delay. The goal is to maximize a linear combination of the throughput, delay, traffic-loss rate, and a fairness metric at the bottleneck node. We introduce a simulation-based congestion-control scheme capable of performing effectively under rapidly varying cross traffic by making use of the provided MMF model of that variation. The control problem is posed as a finite-horizon Markov decision process, and is solved heuristically using a technique called hindsight optimization. We provide a detailed derivation of our congestion-control algorithm based on this technique. Our empirical study shows that the control scheme performs significantly better than the conventional proportional-derivative congestion-control method.

Multiplexing of video and data sources in an ATM access network

ATM-over-DSL is a promising access technology for real-time MPEG video. In this paper, we first describe a configuration of an ATM-over-VDSL access network that can handle VBR video and ABR data calls. Then, we analyze a priority multiplexing strategy for video and data calls at the ATM-based DSL access multiplexor. Using a Markov-modulated fluid model for MPEG video calls we present a computationally efficient fluid analysis of cell loss for video and data traffic at the multiplexor. We derive the output fluid process of the video buffer and approximate it with a 2-state model. This 2-state model is used to analyze cell loss in the low-priority data buffer. We also obtain data cell loss by a first-order approximation method that is commonly used in priority queues. Our results indicate that our 2-state approximation is fairly accurate while computationally efficient compared to the full model. In addition, the results show that the commonly used first-order approximation can be excessively inaccurate.

Analysis of a Markov-Modulated Fluid Model for Multimedia Traffic with Loss and Delay Priorities

This paper presents a complete fluid-flow analysis of a traffic control mechanism in an ATM multiplexer, known as Nested Threshold Cell Discarding with Multiple Buffers. NTCD/MB is a buffer management scheme that implements loss and delay priority queueing strategies in order to satisfy a wide range of quality-of-service requirements. We consider a heterogeneous mix of traffic which consists of real-time traffic with a stringent cell delay requirement, and nonreal-time traffic with a stringent cell loss requirement. Moreover, a proportion of real-time cells has an additional stringent loss requirement. Cell arrivals are mode led using a fluid-flow modeling approach. The system we analyze consists of two buffers interacting via a common server. The steady-state probability distribution for the buffers' content as well as other performance statistics are obtained. Our approach is based on computing the conditional probability distributions of the buffers and using this information to find the desired unconditional stationary distributions. This approach can be used, in general, to analyze multiple dependent systems of buffers. Numerical examples based on the analysis are used along with simulations to study the performance of NTCD/MB. It is shown that NTCD/MB can be used effectively to provide performance guarantees on different cell loss and delay requirements of the traffic.