Queueing Petri nets with product form solution

Predictive performance models are used increasingly throughout the phases of the software engineering lifecycle of distributed systems. However, as systems grow in size and complexity, building models that accurately capture the different aspects of their behavior becomes a more and more challenging task. The challenge stems from the limited model expressiveness on the one hand and the limited scalability of model analysis techniques on the other. This chapter presents a novel methodology for modeling and performance analysis of distributed systems [Kounev, 2006]. The methodology is based on queueing Petri nets (QPNs) which provide greater modeling power and expressiveness than conventional modeling paradigms such as queueing networks and generalized stochastic Petri nets. Using QPNs, one can integrate both hardware and software aspects of system behavior into the same model. In addition to hardware contention and scheduling strategies, QPNs make it easy to model software contention, simultaneous resource possession, synchronization, blocking and asynchronous processing. These aspects have significant impact on the performance of modern distributed systems. To avoid the problem of state space explosion, our methodology uses discrete event simulation for model analysis. We propose an efficient and reliable method for simulation of QPNs [Kounev & Buchmann, 2006]. As a validation of our approach, we present a case study of a real-world distributed system, showing how our methodology is applied in a step-by-step fashion to evaluate the system performance and scalability. The system studied is a deployment of the industry-standard SPECjAppServer2004 benchmark. A detailed model of the system and its workload is built and used to predict the system performance for several deployment configurations and workload scenarios of interest. Taking advantage of the expressive power of QPNs, our approach makes it possible to model systems at a higher degree of accuracy providing a number of important benefits. The rest of this chapter is organized as follows. In Section 2, we give a brief introduction to QPNs. Following this, in Section 3, we present a method for quantitative analysis of QPNs based on discrete event simulation. The latter enables us to analyze QPN models of realistic size and complexity. In Section 4, we present our performance modeling methodology for distributed systems. The methodology is introduced in a step-by-step

The seminal paper of Jackson began a chain of research on queueing networks with product-form stationary distributions which continues strongly to this day. Hard on the heels of the early results on queueing networks followed a series of papers which discussed the relationship between product-form stationary distributions and the quasireversibility of network nodes. More recently, the definition of quasireversibility and the coupling mechanism between nodes have been extended so that they apply to some of the later product-form queueing networks incorporating negative customers, signals, and batch movements. In parallel with this research, it has been shown that some special queueing networks can have arrival and service parameters which depend upon the network state, rather than just the node state, and still retain a generalised product-form stationary distribution. In this paper we begin by offering an alternative proof of a product-form result of Chao and Miyazawa and then build on this proof by postulating a state-dependent coupling mechanism for a quasireversible network. Our main theorem is that the resultant network has a generalised product form stationary distribution. We conclude the paper with some examples.

Product form equilibrium distributions and a convolution algorithm for stochastic Petri nets

In this paper, we present an approach for model transformation from Queueing Network Models (QNMs) into Queueing Petri Nets (QPNs). The performance of QPNs can be analyzed using a powerful simulation engine, SimQPN, designed to exploit the knowledge and behavior of QPNs to improve the efficiency of simulation. When QNMs are transformed into QPNs, their performance can be analyzed efficiently using SimQPN. To validate our approach, we apply it to analyze the performance of several queueing network models including a model of a database system. The evaluation results show that the performance analysis of the transformed QNMs has high accuracy and low overhead. In this context, model transformation enables the performance analysis of queueing networks using different ways that can be more efficient.

Stochastic Petri Nets (SPNs) are used in engineering and systems analysis to model communication systems, computer systems and software, flexible manufacturing systems, and other applications. The objective of this thesis is to improve the solution techniques for SPN, in particular, to solve or alleviate the state space explosion problem--the bottleneck of SPN analysis. Several solution techniques have been examined, ranging from separable SPN with product form solution, Norton's aggregation, a new approximate convolution algorithm for solving large, colored SPNs, to the core of this thesis--the iterative SPN decomposition technique. This technique uses the concept of divide and conquer. It iteratively solves several size-reduced auxiliary SPN models for performance measures of the original SPN. The structure of the auxiliary SPNs is obtained by aggregating some subnets in the original SPN according to some formal PN reduction rules while preserving the external black-box equivalence of the subnets such that the structural and behavioral properties of the unaggregated part are not affected. The rate parameters of the auxiliary models are obtained iteratively. This approximate technique has good accuracy and large computation reduction, and has been applied to a large application example with success. The major contribution of this thesis is the proposed iterative decomposition technique, which is applicable to general classes of SPNs, and to large models. For the technique, this thesis defines the black-box equivalence criterion for aggregating subnets in PN so that the desired structural and behavioral properties can be preserved, and several formal reduction rules that are shown to preserve the black-box equivalence. The iterative parameter tuning process can be regarded as a generalization of Jacobson and Lazowska's surrogate delay method to SPN domain by allowing multi-arc connections between subnets, several subnets aggregated separately in one auxiliary model, several auxiliary models to control the degree of the structural reduction, and local marking dependent or constant rate for an aggregated transition with degrees of accuracies and complexities.

Several modeling and simulation approaches can be used for predicting network performance. However, each approach has its own formalism and accuracy rate, and their use requires different expertise and manual steps. In this paper, we present a network modeling language, named Network Modeling Language (NML), and a model-to-model transformation approach to overcome this challenge. NML offers a single formalism capable of obtaining prediction results based on multiple performance models. We demonstrate how to use NML to create Software-Defined Networking (SDN) models and transform them into Queueing Petri Nets (QPN) and Stochastic Petri Nets (SPN). In our analysis, we show that our automatically generated models could represent the network performance with a prediction error rate of 3%.

In this paper, we present a tool (DIMON) for distributed modeling of nets such as queueing networks or Petri nets or combinations of both over an IP based computer network, together with a 3-D graphical editor (DIM3ON) for modeling hierarchical nets utilizing that tool. Both the tool and the editor are implemented in Java. In this implementation, the net concept is considered as a superclass and thus can be extended by inheritance to any more specialized concept, e.g., to a stochastic Petri net (note that the Petri net concept itself is a subclass of the net concept). The editor is designed to be extensible in a rather easy manner in order to consider yet unsupported net classes. The nets are specified using the Extensible Markup Language (XML) in order to facilitate storage and interchange across a computer network. Keywords—combined queueing network and Petri nets, modeling tool, distributed modeling, hierarchi-

We consider networks of queues in discrete time, where the steady state distribution can be computed explicitly in closed form (product form networks): (i) Closed cycles and open tandems of single server FCFS Bernoulli nodes with state dependent service probabilities, where customers flow linearly, (ii) networks of doubly stochastic and geometrical queues (which are discrete time analogues of Kelly’s symmetric, resp. general, servers), where customers of different types move through the network governed by a general routing mechanism and request for service according to general, resp. geometrical, distributions, (iii) networks with batch movements of customers and batch service, where the service and routing mechanism is defined via an abstract transition scheme.

Recently Miyazawa and Taylor (1997) proposed a new class of queueing networks with batch arrival batch service and assemble-transfer features. In such networks customers arrive and are served in batches, and may change size when a batch transfers from one node to another. With the assumption of an additional arrival process at each node when it is empty, they obtain a simple product-form steady-state probability distribution, which is a (stochastic) upper bound for the original network. This paper shows that this class of network possesses a set of non-standard partial balance equations, and it is demonstrated that the condition of the additional arrival process introduced by Miyazawa and Taylor is there precisely to satisfy the partial balance equations, i.e. it is necessary and sufficient not only for having a product form solution, but also for the partial balance equations to hold.

Stochastic Petri Net Package (SPNP) is a software package whose goal is to compute performance, availability or performability measures from Stochastic Petri Nets (SPN) and Fluid Stochastic Petri nets (FSPN). This software can use either analytic numeric methods, or simulation methods. Unfortunately, the standard discrete event simulation is inefficient to estimate the probabilities of rare events. For such rare event simulations, importance splitting technique is a good method to speedup the simulation. In the literature, two diffierent importance splitting techniques are known: RESTART and splitting. In this paper, we describe the application of these methods to (both fluid and discrete) Petri nets, their implementation in SPNP and we give some illustrations of the speed-up. The RESTART technique has already been applied in another Petri net package, TimeNet, but here we implement both RESTART and splitting, and we apply them to a more general class of Petri nets including the fluid ones.

short-paper Share on Tail asymptotics of the stationary distribution for a two-node generalized Jackson network Author: Masakiyo Miyazawa Tokyo University of Science, Noda, Chiba, Japan Tokyo University of Science, Noda, Chiba, JapanView Profile Authors Info & Claims ACM SIGMETRICS Performance Evaluation ReviewVolume 42Issue 2September 2014pp 70–72https://doi.org/10.1145/2667522.2667545Published:04 September 2014Publication History 1citation26DownloadsMetricsTotal Citations1Total Downloads26Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product formsolution for the network. This paper extends previous results to arbitrary service with finite mean (the nondifferentiable case included). This is done by introducing supplementary variables for the remaining service requirements at the various queues in the system. By using generalized function (in the sense of Schwartz) a new technique is presented for handling queueing networks in which service distributions may be general.

We examine product form equilibrium distributions for stochastic process algebra models, finding conditions that guarantee the existence of a solution for the traffic equations. These equations are a set of linear equations, which are the basis of exact analysis of product form models like queueing networks, stochastic Petri nets, as well as stochastic process algebras. This paper illustrates how the equilibrium distribution can be expressed as a product for a certain class of stochastic process algebra models that satisfy certain conditions. Although the product from criterion derived in this paper is developed in the context of Performance Evaluation Process Algebra (PEPA), the results can be easily generalised to any of the other stochastic process algebra.

Basic graph models of processes, such as Petri nets, have usually omitted the concept of time as a parameter. Time has been added to the Petri net model in two ways. The timed Petri net (TPN) uses a fixed number of discrete time intervals. The stochastic Petri net (SPN) uses an exponentially distributed random variable. In this paper, a discrete time stochastic Petri model is described. These discrete time SPN's fill the gap between TPN and normal SPN. However, the use of discrete time complicates the SPN model in that more than one transition may fire at a time step. Finally, an example of a live and bounded Petri net which has nonempty, disjoint, recurrent subsets of markings is given.

This paper presents a systematic survey of the existing database system performance evaluation models based on the queueing theory. The continuous evolution of the methodologies developed is classified according to the mathematical modeling language used. This survey covers formal models – from queueing systems and queueing networks to queueing Petri nets. Some fundamentals of the queueing system theory are presented and queueing system models are classified according to service time distribution. The paper introduces queueing networks and considers several classification criteria applicable to such models. This survey distinguishes methodologies, which evaluate database performance at the integrated system level. Finally, queueing Petri nets are introduced, which combine modeling power of queueing networks and Petri nets. Two performance models within this formalism are investigated. We find that an insufficient amount of research effort is directed into the area of NoSQL data stores. Vast majority of models developed focus on traditional relational models. These models should be adapted to evaluate performance of non-relational data stores.