Simultaneous Resource Possession Research Articles

LN: A Flexible Algorithmic Framework for Layered Queueing Network Analysis

Layered queueing networks (LQNs) are an extension of ordinary queueing networks useful to model simultaneous resource possession and stochastic call graphs in distributed systems. Existing computational algorithms for LQNs have primarily focused on mean-value analysis. However, other solution paradigms, such as normalizing constant analysis and mean-field approximation, can improve the computation of LQN mean and transient performance metrics, state probabilities, and response time distributions. Motivated by this observation, we propose the first LQN meta-solver, called LN, that allows for the dynamic selection of the performance analysis paradigm to be iteratively applied to the submodels arising from layer decomposition. We report experiments where this added flexibility helps us to reduce the LQN solution errors. We also demonstrate that the meta-solver approach eases the integration of LQNs with other formalisms, such as caching models, enabling the analysis of more general classes of layered stochastic networks. Additionally, to support the accurate evaluation of the LQN submodels, we develop novel algorithms for homogeneous queueing networks consisting of an infinite server node and a set of identical queueing stations. In particular, we propose an exact method of moment algorithms, integration techniques for normalizing constants, and a fast non-iterative mean-value analysis technique.

Improved approximations for the Erlang loss model

Stochastic loss networks are often very effective models for studying the random dynamics of systems requiring simultaneous resource possession. Given a stochastic network and a multi-class customer workload, the classical Erlang model renders the stationary probability that a customer will be lost due to insufficient capacity for at least one required resource type. Recently a novel family of slice methods has been proposed by Jung et al. (Proceedings of ACM SIGMETRICS conference on measurement and modeling of computer systems, pp. 407---418, 2008) to approximate the stationary loss probabilities in the Erlang model, and has been shown to provide better performance than the classical Erlang fixed point approximation in many regimes of interest. In this paper, we propose some new methods for loss probability calculation. We propose a refinement of the 3-point slice method of Jung et al. (Proceedings of ACM SIGMETRICS conference on measurement and modeling of computer systems, pp. 407---418, 2008) which exhibits improved accuracy, especially when heavily loaded networks are considered, at comparable computational cost. Next we exploit the structure of the stationary distribution to propose randomized algorithms to approximate both the stationary distribution and the loss probabilities. Whereas our refined slice method is exact in a certain scaling regime and is therefore ideally suited to the asymptotic analysis of large networks, the latter algorithms borrow from volume computation methods for convex polytopes to provide approximations for the unscaled network with error bounds as a function of the computational costs.

Burst lost probabilities in a queuing network with simultaneous resource possession: a single-node decomposition approach

An efficient analytical method is presented for the calculation of blocking probabilities in a tandem queuing network with simultaneous resource possession. This queuing network model is motivated from the need to model optical burst switching networks, where the size of the data bursts varies and the link distance between two adjacent network elements also varies depending on the network's topology. A fast single-node decomposition algorithm is developed to compute the blocking probabilities in the network. The algorithm extends the popular link-decomposition method from teletraffic theory by allowing dynamic simultaneous link possession. Simulation is used to validate the accuracy of the algorithm.

Call blocking probabilities in a traffic-groomed tandem optical network

In this paper, we consider an optical network consisting of N nodes arranged in tandem. We assume traffic grooming, which permits multiple sub-rate traffic streams to be carried on the same wavelength. This optical network is modeled by a tandem queueing network of multi-rate Erlang loss nodes with simultaneous resource possession, with a view to calculating call blocking probabilities. The queueing network is analyzed by decomposition using a modified version of Courtois' algorithm. Numerical comparisons of the decomposition algorithm against simulation show that the algorithm has good accuracy. Also, it is shown that the proposed algorithm is an improvement over the well-known single-node decomposition algorithm based on the link-independence assumption.

Call blocking probabilities in a traffic-groomed tandem optical network

In this paper, we consider an optical network consisting of N nodes arranged in tandem. We assume traffic grooming, which permits multiple sub-rate traffic streams to be carried on the same wavelength. This optical network is modeled by a tandem queueing network of multi-rate Erlang loss nodes with simultaneous resource possession, with a view to calculating call blocking probabilities. The queueing network is analyzed by decomposition using a modified version of Courtois' algorithm. Numerical comparisons of the decomposition algorithm against simulation show that the algorithm has good accuracy. Also, it is shown that the proposed algorithm is an improvement over the well-known single-node decomposition algorithm based on the link-independence assumption.

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

Preface to the Second Edition. Preface to the First Edition. 1. Introduction. 1.1 Motivation. 1.2 Methodological Background. 1.3 Basics of Probability and Statistics. 2. Markov Chains. 2.1 Markov Processes. 2.2 Performance Measures. 2.3 Generation Methods. 3. Steady-State Solutions of Markov Chains. 3.1 Solution for a Birth Death Process. 3.2 Matrix-Geometric Method: Quasi-Birth-Death Process. 3.3 Hessenberg Matrix: Non-Markovian Queues. 3.4 Numerical Solution: Direct Methods. 3.5 Numerical Solution: Iterative Methods. 3.6 Comparison of Numerical Solution Methods. 4. Steady-State Aggregation/Disaggregation Methods. 4.1 Courtois' Approximate Method. 4.2 Takahashi's Iterative Method. 5. Transient Solution of Markov Chains. 5.1 Transient Analysis Using Exact Methods. 5.2 Aggregation of Stiff Markov Chains. 6. Single Station Queueing Systems. 6.1 Notation. 6.2 Markovian Queues. 6.3 Non-Markovian Queues. 6.4 Priority Queues. 6.5 Asymmetric Queues. 6.6 Queues with Batch Service and Batch Arrivals. 6.7 Retrial Queues. 6.8 Special Classes of Point Arrival Processes. 7. Queueing Networks. 7.1 Definitions and Notation. 7.2 Performance Measures. 7.3 Product-Form Queueing Networks. 8. Algorithms for Product-Form Networks. 8.1 The Convolution Algorithm. 8.2 The Mean Value Analysis. 8.3 Flow Equivalent Server Method. 8.4 Summary. 9. Approximation Algorithms for Product-Form Networks. 9.1 Approximations Based on the MVA. 9.2 Summation Method. 9.3 Bottapprox Method. 9.4 Bounds Analysis. 9.5 Summary. 10. Algorithms for Non-Product-Form Networks. 10.1 Nonexponential Distributions. 10.2 Different Service Times at FCFS Nodes. 10.3 Priority Networks. 10.4 Simultaneous Resource Possession. 10.5 Prograrns with Internal Concurrency. 10.6 Parallel Processing. 10.7 Networks with Asymmetric Nodes. 10.8 Networks with Blocking. 10.9 Networks with Batch Service. 11. Discrete-Event Simulation. 11.1 Introduction to Simulation. 11.2 Simulative or Analytic Solution? 11.3 Classification of Simulation Models. 11.4 Classification of Tools in DES. 11.5 The Role of Probability and Statistics in Simulation. 11.6 Applications. 12. Performance Analysis Tools. 12.1 PEPSY. 12.2 SPNP. 12. 3 MOSEL-2. 12.4 SHARPE. 12.5 Characteristics of Some Tools. 13. Applications. 13.1 Case Studies of Queueing Networks. 13.2 Case Studies of Markov Chains. 13.3 Case Studies of Hierarchical Models. Glossary. Bibliography. Index.

Approximation methods for two-layer queueing models

We discuss methods to calculate performance measures of queueing models with simultaneous resource possession in which customers who possess different types of passive resources compete with other customers for the same active resources. In this paper, we propose a two-layer queueing model which can adequately model such systems and two extended flow-equivalent methods for approximating the performance measures of two-layer models. We give an example of the model, and we describe the results of a study of the accuracy of the approximation methods.

An approximate solution for queuing network models of a burst-level traffic control scheme in ATM WAN

This paper proposes a closed queuing network model for a burst-level traffic control scheme in ATM WAN environments, and explores solution techniques for this model. An approximate solution technique is developed to solve the closed queuing networks with multiple chains, chain-dependent service times, MSFIFO (multiple servers first-in first-out) queues, and simultaneous resource possession. Results of this study show that, with respect to the subnetwork algorithm of Souza e Silva and Muntz [Approximate solution for a class of non-product form queuing network models, Performance Evaluation (7) (1987) 221–242], our algorithm is advantageous, because (1) it is applicable to queuing network models with nested first-allocated/first-released passive resources; (2) it greatly reduces the complexity of the subnetwork algorithm by not decomposing the network and representing the complements; (3) it achieves higher accuracy and provides more reliable conservative estimates; and (4) it is easy to understand due to its straightforward interpretation.

Order independent loss queues

This paper presents an analysis of generalized Order Independent (OI) loss queues serving customers belonging to different types (classes) where limits are placed on the number of customers of each type that may be present in the system. We prove that such queues satisfy partial balance and we present their stationary distribution. OI loss queues can be used to model blocking systems with simultaneous resource possession with the option of queueing blocked customers. The OI loss queue thus extends previous loss models where customers are rejected when processing resources are not available.

Mean-value analysis of multi-facility blocking models with state-dependent arrivals

A new mean-value type of algorithm is developed for analyzing multi-facility blocking models with state-dependent arrival rates. It can be applied to a broad class of blocking systems with simultaneous resource possession including, for example, circuit-switched networks. The underlying recursion is cast in terms of blocking probabilities and marginal state probabilities. The developments made here generalize previous results that were restricted to the case of constant arrival rates.

Modeling operator/workstation interference in asynchronous automatic assembly systems

In several instances ofdiscrete event dynamic systems (DEDS), jobs sometimes require service from two or more resources at the same time. When queueing network models are used to study DEDS, this feature ofsimultaneous resource possession is often ignored because it is difficult for the models to handle. In some DEDS, this feature of a job demanding several resources simultaneously can have a significant effect on system performance, especially if there is a limited amount of one or more of these resources. For example, in an asynchronous automatic assembly system, an assembly at a workstation needs an operator when it experiences a jam (a random phenomenon) in order to clear the jam. Due to the presence of a limited (small) number of operators, an assembly may have to wait for an operator. This waiting orinterference time has a significant effect on the system production rate. This paper develops an analytical approximation method that can be used to determine the steady-state performance of automatic assembly systems for a given assignment of operators. The analytical method involves the simultaneous solution of two “coupled” queueing models; one of the models calculates the waiting time for an operator resource, while the other computes the waiting time for a workstation resource. The solution technique developed can be adapted to study instances of simultaneous resource possession in other DEDS, such as flexible manufacturing systems and computer/communication networks.

Approximate analysis of a cyclic queuing network with applications to a simultaneous resource possession problem

In this paper we develop very accurate approximations for all mean performance measures of interest, for a class of cyclic queuing networks (CQNs). The number of nodes K, and the closed chain population N, are finite but arbitrary. Each node processes type t customers from the single closed chain, and type o customers from a dedicated open chain. The queuing discipline at each node is FIFO and the service time distribution for both type t and o customers is exponential, but with distinct service rates for each. Service rates may differ from node to node. Although the structure of this class of queuing networks is simple, very little is known in the literature except for the special case where the network reduces to a product form network. An analytic study of the approximation demonstrates that it mirrors the expected behavior of the CQN in many essential respects: monotonicity, bottleneck and asymptotic behavior. Moreover, in the case of balanced CQNs, the approximation yields simple and explicit expressions for all quantities of interest. This analytic study complements the numerical experiments presented and suggests that the approximation captures the essential structure of the CQN, insofar as mean performance quantities are concerned.

A conservation law based approximate analysis for a class of simultaneous resource possession problems

In this paper we analyze a single server queueing model in which there are two types of jobs, one of which must wait in an external queue until a token is available, and only then may join the service queue. The interarrival times and service requirements for both types of jobs are assumed to be independent and exponentially distributed. We derive the stability condition for such a model where the service queue discipline is either FCFS (First-Come-First-Serve) or PS (Processor-Sharing). We then propose analytic approximations for the mean waiting times for both types of jobs, relying heavily on the M/G/1 conservation law. Numerical results show that our approximations are very accurate (within a few percent of the simulated results) even when the system is heavily loaded. The approximations are also shown to be asymptotically exact as the number of tokens N → ∞.

Approximate solutions for a class of non-product form queueing network models

We investigate approximate solution techniques for solving a class of non-product form closed queueing networks with simultaneous resource possession. We propose a simple and accurate solution technique which can handle simultaneously held resources dedicated to a single type of user as well as resources shared among different types of users. The proposed algorithm can also handle the case of nested passive resources. An approximate solution for non-product form networks with first-come-first-serve multiple server service centers with different service times for different types of users is also developed, since this solution is needed for our main algorithm. Comparison with simulation and other approximation techniques demonstrates the good accuracy and performance of our solution method.

Blocking When Service Is Required From Several Facilities Simultaneously

This paper analyzes a mathematical model of a blocking system with simultaneous resource possession. There are several multiserver service facilities without extra waiting space at which several classes of customers arrive in independent Poisson processes. Each customer requests service from one server in each facility in a subset of the service facilities, with the subset depending on the customer class. If service can be provided immediately upon arrival at all required facilities, then service begins and all servers assigned to the customer start and finish together. Otherwise, the attempt is blocked (lost without generating retrials). The problem is to determine the blocking probability for each customer class. An exact expression is available, but it is complicated. Hence, this paper investigates approximation schemes.

On the applicability of using multiprogramming level distributions

A computer system's workload is represented by its multiprogramming level, which is defined as the number of tasks (jobs, customers) which actively compete for resources within the system. In a product-form queuing network model of the system, the workload is modeled by assuming that the multiprogramming level is either fixed (i.e., closed model) or that the multiprogramming level depends upon an outside arrival process (i.e., open model). However, in many actual systems, closed and open models are both inappropriate since the multiprogramming level is neither fixed nor governed by an outside arrival process. In an actual system., the multiprogramming level varies due to features such as task spawning, killing, blocking, parallel processing, and/or simultaneous resource possession. The multiprogramming level is a random variable with an associated distribution. This paper demonstrates that improved models can result from using this multiprogramming level distribution information. Several examples relative to open versus closed models, subsystem models, actual system models, and blocking models are given which demonstrate the applicability of using multiprogramming level distributions. This applicability, shown via the examples, is the main contribution of the paper. The examples also motivate interesting theoretical results relating to open models, closed models, and subsystem models.

A decomposition approach to modelling high service time variability

The predictive ability of queueing network models can be greatly enhanced if these models include the effects of system characteristics such as high service time variability and simultaneous resource possession, which violate the assumptions required for their efficient exact solution. In this paper we present a new approximate solution technique for queueing networks that include Coxian servers to represent resources at which customers have high service time variability. Our approach is unique in several respects: it is based directly on the theory of near-complete decomposability, it is non-iterative (performance measures for the queueing network of interest are expressed as linear combinations of the performance measures of a set of separable queueing networks), and it is conceptually and computationally simple.

A decomposition approach to modelling high service time variability

Abstract The predictive ability of queueing network models can be greatly enhanced if these models include the effects of system characteristics such as high service time variability and simultaneous resource possession, which violate the assumptions required for their efficient exact solution. In this paper we present a new approximate solution technique for queueing networks that include Coxian servers to represent resources at which customers have high service time variability. Our approach is unique in several respects: it is based directly on the theory of near-complete decomposability, it is non-iterative (performance measures for the queueing network of interest are expressed as linear combinations of the performance measures of a set of separable queueing networks), and it is conceptually and computationally simple.

Analyzing queueing networks with simultaneous resource possession

Analyzing queueing networks with simultaneous resource possession

Approximate Solution of Queueing Networks with Simultaneous Resource Possession

Queueing networks are important as Performance models of computer and communication systems because the performance of these systems is usually principally affected by contention for resources. Exact numerical solution of a queueing network is usually only feasible if the network has a product form solution in the sense of Jackson. An important network characteristic which apparently precludes a product form solution is simultaneous resource possession, e.g., a job holds memory and processor simultaneously. This paper extends previous methods for approximate numerical solution of queueing networks with homogeneous jobs and simultaneous resource possession to networks with heterogeneous jobs and simultaneous resource possession.