Soohan Ahn

Fluid Flow Models and Queues—A Connection by Stochastic Coupling

We establish in a direct manner that the steady state distribution of Markovian fluid flow models can be obtained from a quasi birth and death queue. This is accomplished through the construction of the processes on a common probability space and the demonstration of a distributional coupling relation between them. The results here provide an interpretation for the quasi-birth-and-death processes in the matrix-geometric approach of Ramaswami and subsequent results based on them obtained by Soares and Latouche.

Transient Analysis of Fluid Flow Models via Stochastic Coupling to a Queue

Abstract Markovian fluid flow models are used extensively in performance analysis of communication networks. They are also instances of Markov reward models that find applications in several areas like storage theory, insurance risk and financial models, and inventory control. This paper deals with the transient (time dependent) analysis of such models. Given a Markovian fluid flow, we construct on the same probability space a sequence of queues that are stochastically coupled to the fluid flow in the sense that at certain selected random epochs, the distribution of the fluid level and the phase (the state of the modulating Markov chain) is identical to that of the work in the queue and the phase. The fluid flow is realized as a stochastic process limit of the processes of work in the system for the queues, and the latter are analyzed using the matrix-geometric method. These in turn provide the needed characterization of transient results for the fluid model.

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 (nλ). The n-th approximating queue’s system size is bounded at a value qn such that the corresponding expected amount of work qn/(nλ) → 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.

Transient Analysis of Fluid Models via Elementary Level-Crossing Arguments

An analysis of the time-dependent evolution of the canonical Markov modulated fluid flow model is presented using elementary level-crossing arguments.

Bilateral phase type distributions

Abstract A new class of probability distributions called “bilateral phase type distributions (BPH)” on (−∞, ∞) is defined as a generalization of the versatile class of phase type (PH) distributions on [0, ∞) introduced by Marcel F. Neuts. We derive the basic descriptors of such distributions in an algorithmically tractable manner and show that this class has many interesting closure properties and is dense in the class of all distributions on the real line. Based on the established versatility and tractability of phase type distributions, we believe that this class has high potential for general use in statistics, particularly to cover non-normal distributions, and also that its inherent connection to Markov chains may make it suitable for inference based on hidden Markov chain methods and MCMC type approaches.

GOP ARIMA: Modeling the nonstationarity of VBR processes

In this work, we develop a stochastic model, GOP ARIMA (autoregressive integrated moving average for a group of pictures) for VBR processes with a regular GOP pattern. It explicitly incorporates the deterministic time-dependent behavior of frame-level VBR traffic. The GOP ARIMA model elaborately represents the inter- and intra-GOP sample autocorrelation structures and provides a physical explanation of observed stochastic characteristics of the empirical VBR process. We explain stochastic characteristics of the empirical VBR process, e.g., slowly decaying sample autocorrelations and strong correlations at the lags, based on the aspect of nonstationarity of the underlying process. The GOP ARIMA model generates synthetic traffic, which has the same multiplicative periodic sample autocorrelation structure as well as slowly decaying autocorrelations of the empirical VBR process. The simulation results show that the GOP ARIMA process very well captures the behavior of the empirical process in various respects: packet loss, packet delay, and frame corruption. Our work makes a contribution not only toward providing a theoretical explanation of the observed characteristics of the empirical VBR process but also toward the development of an efficient method for generating a more realistic synthetic sequence for various engineering purposes and for predicting future bandwidth requirements.

A Markov‐modulated fluid flow queueing model under D‐policy

We consider a Markov-modulated fluid flow queueing model under D-policy. As soon as the fluid level reaches zero, the server becomes idle. During the idle period, fluid arrives from outside according to an underlying continuous time Markov chain (UMC) and the idle server does not process the fluid. We consider two increase patterns of fluid during the idle period: vertical increase (Type-V) and linear increase (Type-L). The idle server is reactivated only when the cumulative fluid level in the system exceeds a predetermined threshold value D. We derive the distributions of fluid level and mean performance measures for both types. We also present cost optimization model to minimize average operating cost per unit time. Copyright

Analysis of the M/D/1-type queue based on an integer-valued first-order autoregressive process

In this paper, we propose a queueing model based on an integer-valued first-order autoregressive(INAR(1)) process. We derive the queue length distribution and its asymptotic decay rate of the proposed model. Also, our numerical study shows that the new model can be considered as an alternative approach to the well-known MMPP/D/1 queue in terms of performance and amount of computational work.

A factorization property for BMAP/G/1 vacation queues under variable service speed

Abstract This paper proposes a simple factorization principle that can be used efficiently and effectively to derive the vector generating function of the queue length at an arbitrary time of the BMAP/G/1/ queueing systems under variable service speed. We first prove the factorization property. Then we provide moments formula. Lastly we present some applications of the factorization principle.

A workload factorization for BMAP/G/1 vacation queues under variable service speed

Abstract This paper proposes a simple factorization property for the workload distribution of the BMAP/G/1/ vacation queues under variable service speed. The server provides service at different service speeds depending on the phases of the underlying Markov chain. Using the factorization principle, the workload distribution at any arbitrary time point can be easily derived only by obtaining the distribution during the idle period. We prove the factorization property and the moments formula. Lastly, we provide some applications of our factorization principle.