Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications

Many application require transient solutions of Markov chains. Several authors have considered efficient algorithms for the transient solution of acyclic Markov chains. In this paper, we investigate an approximate numerical technique for solving cyclic Markov chains. The basic approach is to break the Markov chain into two acyclic subchains, and iteratively combine their solution. An easily computable error bound allows the accuracy to be bounded before the computation begins.

A novel retrial multi-server queueing model with batch arrival of customers is considered. In contrast to the standard batch arrival when a whole batch arrives into the system at one epoch, we assume that the customers of a batch arrive one-by one in exponentially distributed times. Service time is exponentially distributed. Flow of batches is the stationary Poisson arrival process. Batch size distribution is geometric. The number of batches, which can be admitted into the system simultaneously, is subject of control. Analysis of the joint distribution of the number batches and customers in the system is implemented using the tool of the multi-dimensional asymptotically quasi-Toeplitz Markov chains.

The Neumann expansion of Bessel functions (of integer order) of a function \(g:\mathbb {C}\rightarrow \mathbb {C}\) corresponds to representing g as a linear combination of basis functions φ0, φ1, …, i.e., \(g(s)=\sum _{\ell = 0}^\infty w_\ell \varphi _\ell (s)\), where φi(s) = Ji(s), i = 0, …, are the Bessel functions. In this work, we study an expansion for a more general class of basis functions. More precisely, we assume that the basis functions satisfy an infinite dimensional linear ordinary differential equation associated with a Hessenberg matrix, motivated by the fact that these basis functions occur in certain iterative methods. A procedure to compute the basis functions as well as the coefficients is proposed. Theoretical properties of the expansion are studied. We illustrate that non-standard basis functions can give faster convergence than the Bessel functions.

Use of Lagrange Interpolating Polynomials in RLT. Systems in Series. Parallel Configurations. Spanning trees. Recycling. Split Cuts. Retrial Queues. Horse Racing. MACBETH (Measuring Attractiveness by a Categorical Based Evaluation Technique). Dual simplex. History of CP. Procurement Contracts. MRP. Just in Time. Push and Pull Production. MULTIVARIATE INPUT MODELING. Operational Research Society. The M/G/1 Queue. Drama Theory. Hyper-heuristics. Antithetic Variates. Music. Edgeworth market games. Operational Risk. Queueing Notation. Environmental impact. Why variance is not risk. Conjugate gradient methods. Birth-and-Death Processes. Surgery Planning and Scheduling. Large-scale linear models. Standby Systems. Billiards/Pool. AXIOMATIC MEASURES OF RISK AND RISK-VALUE MODELS. Age Replacement Policies. PASTA and Related Results. Collaborative Procurement. LP projection algorithms. R&D risk management. Bioterrorism. Vendor-Managed Inventory. Gomory Cuts. Travel demand modeling. System Availability. Infinite horizon problems. Trust. Bayesian Network Classifiers. Integer Programming Duality. Queueing Disciplines. Stakeholder participation. Models and Basic Properties. Parametric LP analysis. Operations Research and Golfing. Ice Hockey. Clustering. Finite-Population Models. Batch Arrivals and Service. Optimal Monitoring Strategies. Basic Polyhedral Theory. Operations Research and Art. Managing portfolios of risks. Measures of risk equity. Branch and Bound Algorithms. Analytics in Retail. Variational Inequalities. Cross-Entropy Method. TSP heuristics. Capacity allocation. Control Variates. Cover Inequalities. Direct Search Methods. NLP Software. Bilinear optimization. Discrete-Time Martingales. Branch and Price. Column generation. IP Preprocessing. Optimal Reliability Allocation. Simplex-based LP solvers. Instance Formats. Clique generalizations. Availability in Stochastic Models. Evolutionary Algorithms. Basic CP theory: Search. Multimethodology. Branch and Cut. Memetic Algorithms. Urban mass transit. Coherent Systems. Subgradient optimization. Klimov's Model. Credit risk assessment. Branch-width and Tangles. Stochastic Hazard Process. Deterministic Global Optimization. Multistage (stochastic) games. Basis Reduction Methods. ACCIDENT PRECURSORS AND WARNING SYSTEMS MANAGEMENT: A BAYESIAN APPROACH TO MATHEMATICAL MODELS. Genetic Algorithms. Newsvendor models. Parallel Systems. Opportunities in Tennis. Job-shop scheduling. Simplex method and complexity. Geometric programming. Evacuation Planning. The Performance? Allocation Games. Scenario Generation. Robustness Analysis. Dynamic auctions. Neuroeconomics and game theory. Common Random Numbers. Selective Support Vector Machines. Semi-Markov Processes. Risk Averse Models. Sampling Methods. Domination problems. Simulation of Rare Events. Introduction to Tabu Search. Basic Interdiction Models. Introductory Concepts. Remanufacturing. PAUSE procedure. Wardrop equilibria. Mass customization. Ellipsoidal algorithms. Benders decomposition. Dynamic vehicle routing. Warranty Modeling. Design for Network Resiliency. Passenger rail transportation. Vehicle Routing Problem. Supply Chain Outsourcing. Lovasz-Schriver Reformulation. Hazard Rate Function. Lot-sizing. Differential games. Regenerative Processes. Guided Local Search. Preferences for repeated gambles. Percolation Theory. Simulated Annealing. Scoring Rules. Mixing Sets. Cricket. Prospect Theory. Combining Forecasts. Graph search techniques. Quasi-Newton methods. Random search algorithms. Supply chain coordination. Service Outsourcing. Markov Renewal Processes. K-out-of-n Systems.

Two different techniques for bounding the steady-state solution of large Markov chains are presented. The first one is a verification technique based on monotone iterative methods. The second one is a decomposition technique based on the concepts of eigen-vector polyhedron. The computation of the availability of repairable fault-tolerant systems is used to illustrate the performances and the numerical aspects of both methods.

Markov chains with periodic graphs arise frequently in a wide range of modelling experiments. Application areas range from flexible manufacturing systems in which pallets are treated in a cyclic manner to computer communication networks. The primary goal of this paper is to show how advantage may be taken of this property to reduce the amount of computer memory and computation time needed to compute stationary probability vectors of periodic Markov chains. After reviewing some basic properties of Markov chains whose associated graph is periodic, we introduce a ‘reduced scheme’ in which only a subset of the probability vector need be computed. We consider the effect of the application of both direct and iterative methods to the original Markov chain (permuted to normal periodic form) as well as to the reduced system. We show how the periodicity of the Markov chain may be efficiently computed as the chain itself is being generated. Finally, some numerical experiments to illustrate the theory, are presented.

A number of algorithms involving Markov chains contain no subtractions. This property makes the analysis of rounding errors particularly simple. To show this, some principles for analyzing the propagation and generation of rounding errors in algorithms containing no subtraction are discussed first. These principles are then applied in the context of a simple recursive algorithm involving the transient solution of discrete-time Markov chains, Jensen's algorithm, and state reduction. Jensen's algorithm, also known as randomization or uniformization, is an algorithm for finding transient solutions of continuous-time Markov chains. State reduction is a method for finding equilibrium probabilities for discrete-time or continuous-time Markov chains.

Recently the Multi-Level algorithm was introduced as a general purpose solver for the solution of steady state Markov chains. In this paper we consider the performance of the Multi-Level algorithm for solving Nearly Completely Decomposable (NCD) Markov chains, for which special-purpose iterative aggregation/disaggregation algorithms such as the Koury-McAllister-Stewart (KMS) method have been developed that can exploit the decomposability of the the Markov chain. We present experimental results indicating that the general-purpose Multi-Level algorithm is competitive, and can be significantly faster than the special-purpose KMS algorithm when Gauss-Seidel and Gaussian Elimination are used for solving the individual blocks.KeywordsMarkov ChainFine LevelMarkov Chain ModelingGaussian EliminationCoarse LevelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A queueing model for performance analysis of goback- N (GBN) automatic repeat request (ARQ) protocols in wireless networks using dynamic radio link adaptation with fixed feedback was proposed. Firstly, the paper analyzes the major factors affecting residence time of the packets, aiming at the problem that packets in a batch arrival at relay. Secondly, the paper establishes Markov model of packet transmission and deduces the system state is a G/M/1 type Markov chains, and the delay statistics at relay node are figured out by using matrix geometric methods. At last, the paper analyzes the influence of the arrival rate, channel bit error rate and the doppler frequency shift on the system.

Tandem queues are good mathematical models for description of information transmission in various communication systems and networks. These queues play also an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. So, their investigation is interesting for theory and applications. In this paper, we consider tandem queue suitable for modeling the systems and networks where information flows are correlated and bursty what is typical for many modern telecommunication networks. Possible correlation of customers inter-arrival times and batch arrivals are taken into account via of consideration of the Batch Markovian Arrival Process (BMAP) as input stream to the system. The system consists of two stations. The service time at the station 1 is assumed to be generally distributed. There is no buffer at this station, and customers who meet the busy server repeat attempts to enter the system in random time intervals. The service process at the station 2 is assumed to be described by the continuous time Markov chain with a finite state space. This assumption holds good, e.g., if the station 2 has a finite buffer, consists of a finite number of identical or heterogeneous servers where the service time distribution is assumed to be of PH (PHase) type. Markov chain embedded at service completion epochs at the station 1 and the process of system states at arbitrary time are under study. Ergodicity condition and algorithms for computing the steady state probabilities are presented.

In this paper we investigate the efficiency of two solution approaches to infinite stochastic Petri nets: the matrix-geometric method and the spectral expansion method. We first informally present infinite stochastic Petri nets, after which we describe, using uniform notation, the matrix-geometric and the spectral expansion method. We put special emphasis on the numerical aspect of the solution procedures. Then, we investigate the suitability of these approaches to account for batch-movements of tokens. We then compare the two solution approaches when applied to a larger modelling study of a fault-tolerant computer system. It turns out that the spectral expansion method is favorable in all cases, especially when more heavily loaded systems are studied and when batch arrivals are incorporated in the model. To the best of our knowledge, this paper is the first to compare the spectral expansion method, as advocated by Mitrani and Chakka, with the Latouche-Ramaswami algorithm for the matrix-geometric case. Furthermore, our comparisons go well beyond the usual textbook cases, since we are able to generate, with our tool SPN2MGM, models that are much larger than those that have been assessed in the past.

An established class of product-form queueing networks with batch arrivals and batch services is extended to closed networks of discrete-time nodes with finite capacity and shared buffers. A direct application is in modelling ATM networks.

Service systems with queueing often have both batch arrivals and batch services. This paper focuses on the number of busy servers and waiting customers in theGIX/MR/∞ system. For the number of busy servers, we obtain a recursive relation for the partial binomial moments in terms of matrices and explicit expressions for the marginal binomial moments. Special cases are also discussed to provide a more heuristic understanding of the model.

Inspired by the need for performability models for HSDPA user equipment, a Markovian queue with varying number of servers is conceived. The arrival and the service processes, the number of allocated or active servers of the queue are inherently, and independently (or jointly) Markov modulated. Batch arrivals, batch services, autocorrelation of inter-arrival times, and autocorrelation of batch sizes can be accommodated in the queue, by a suitable use of Markov modulation and generalized exponential distribution. The queue has a provision for negative customers too. Transformations of the balance equations into a computable form are proposed in order to obtain the steady state probabilities with the Spectral Expansion method. This queue is used to model the High Speed Downlink Packet Access (HSDPA) wireless networks. The model is an integrated one with respect to HSDPA, capable of accommodating many of the intricate aspects of HSDPA such as, channel allocation policy, loss of packets due to channel fading, bursty and correlated traffic. Good agreement is observed between the numerical results of the proposed analytical model and those of an independent simulator of real HSDPA and radio channel behaviors. The comparison of the terminal categories specified by the 3rd Generation Partnership Project (3GPP) is also presented.

Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering, and underlies the solution of time-homogeneous linear ordinary differential equations, simulation of discrete-time Markov chains, or discovering the spectral properties of matrices with iterative methods. In this paper, we investigate the possibility of speeding up matrix powering of sparse stable Hermitian matrices on a quantum computer. We present two quantum algorithms that can achieve speedup over the classical matrix powering algorithms -- (i) an adaption of quantum-walk based fast forwarding algorithm (ii) an algorithm based on Hamiltonian simulation. Furthermore, by mapping the N-bit parity determination problem to a matrix powering problem, we provide no-go theorems that limit the quantum speedups achievable in powering non-Hermitian matrices.