Class Of Stochastic Petri Nets Research Articles

A modeling framework to implement preemption policies in non-Markovian SPNs

Petri nets represent a useful tool for performance, dependability, and performability analysis of complex systems. Their modeling power can be increased even more if nonexponentially distributed events are considered. However, the inclusion of nonexponential distributions destroys the memoryless property and requires to specify how the marking process is conditioned upon its past history. We consider, in particular, the class of stochastic Petri nets whose marking process can be mapped into a Markov regenerative process. An adequate mathematical framework is developed to deal with the considered class of Markov Regenerative Stochastic Petri Nets (MRSPN). A unified approach for the solution of MRSPNs where different preemption policies can be defined in the same model is presented. The solution is provided both in steady-state and in transient condition. An example concludes the paper.

Asymptotics of stochastic networks with subexponential service times

We analyse the tail behaviour of stationary response times in the class of open stochastic networks with renewal input admitting a representation as (\max,+)-linear systems. For a K-station tandem network of single server queues with infinite buffer capacity, which is one of the simplest models in this class, we first show that if the tail of the service time distribution of one server, say server i_0\in\{1,\ldots,K\}, is subexponential and heavier than those of the other servers, then the stationary distribution of the response time until the completion of service at server j\geq i_0 asymptotically behaves like the stationary response time distribution in an isolated single-server queue with server i_0. Similar asymptotics are given in the case when several service time distributions are subexponential and asymptotically tail-equivalent. This result is then extended to the asymptotics of general (\max,+)-linear systems associated with i.i.d. driving matrices having one (or more) dominant diagonal entry in the subexponential class. In the irreducible case, the asymptotics are surprisingly simple, in comparison with results of the same kind in the Cramer case: the asymptotics only involve the excess distribution of the dominant diagonal entry, the mean value of this entry, the intensity of the arrival process, and the Lyapunov exponent of the sequence of driving matrices. In the reducible case, asymptotics of the same kind, though somewhat more complex, are also obtained. As a direct application, we give the asymptotics of stationary response times in a class of stochastic Petri nets called event graphs. This is based on the assumption that the firing times are independent and that the tail of the firing times of one of the transitions is subexponential and heavier than those of the others. An extension of these results to nonrenewal input processes is discussed. Asymptotics of queue size processes are also considered.

Analysis of windowing mechanisms with infinite-state stochastic Petri nets

In this paper we present a performance evaluation of windowing mechanisms in world-wide web applications. Previously, such mechanisms have been studied by means of measurements only, however, given suitable tool support, we show that such evaluations can also be performed conveniently using infinite-state stochastic Petri nets. We briefly present this class of stochastic Petri nets as well as the approach for solving the underlying infinite-state Markov chain using matrix-geometric methods. We then present a model of the TCP slow-start congestion avoidance mechanism, subject to a (recently published) typical worldwide web workload. The model is parameterized using measurement data for a national connection and an overseas connection. Our study shows how the maximum congestion window size, the connection release timeout and the packet loss probability influence the expected number of buffered segments at the server, the connection setup rate and the connection time.

Fluid stochastic Petri nets: Theory, applications, and solution techniques

In this paper we introduce a new class of stochastic Petri nets in which one or more places can hold fluid rather than discrete tokens. We define a class of fluid stochastic Petri nets in such a way that the discrete and continuous portions may affect each other. Following this definition we provide equations for their transient and steady-state behavior. We present several examples showing the utility of the construct in communication network modeling and reliability analysis, and discuss important special cases. We then discuss numerical methods for computing the transient behavior of such nets. Finally, some numerical examples are presented and evidence of the accuracy of the fluid approximation is given.

An efficient two-stage iterative method for the steady-state analysis of Markov regenerative stochastic Petri net models

Abstract To enhance the modeling power of stochastic Petri nets (SPNs), new steady-state analysis methods have been proposed for nets that include non-exponential transitions. The underlying stochastic process is a Markov regenerative process (MRP) when at most one non-exponential transition is enabled in each marking. Time-efficient algorithms for constructing and solving the MRP have been developed. However, the space required to solve such models is often extremely large. This largeness is due to the large number of transitions in the MRP. Traditional analysis methods require that all these transitions be stored in primary memory for efficient computation. If the size of available memory is smaller than that needed to store these transitions, a time-efficient computation is impossible using these methods. To use this class of SPNs to model realistic systems, the space complexity of MRP analysis algorithms must be reduced. In this paper, we propose a new steady-state analysis method that is both time and space efficient. The new method takes advantage of the structure of the underlying process to reduce both computation time and required memory. The performance of the proposed method is compared to existing methods using several SPN examples.

Non-Markovian Petri nets

Non-Markovian models allow us to capture a very wide range of circumstances in which it is necessary to model phenomena whose times to occurrence is not exponentially distributed. Events such as timeouts in a protocol, service times at a machine performing the same task on each part, and memory access or instruction execution in a low-level h/w or s/w model, have durations which are constant or with a very low variance. Phase-type distributions can be used to approximate a non-exponential, but they increase the size of the state space.The analysis of stochastic systems with non-exponential timing is of increasing interest in the literature and requires the development of suitable modeling tools. Recently, some effort has been devoted to generalize the concept of Stochastic Petri Nets (SPN), by allowing the firing times to be generally distributed.A particular case of non-Markovian SPN, is the class of Deterministic and SPN (DSPN) [1]. A DSPN is a non-Markovian SPN where, in each marking, at most one transition is allowed to have a deterministic firing time with enabling memory policy.A new class of stochastic Petri nets has recently been defined [2, 3] by generalizing the deterministic firing times of the DSPN to generally distributed firing times. The underlying stochastic process for these classes of Petri nets is a Markov Regenerative Process (MRGP). This observation has opened a very fertile line of research aimed at the definition of solvable classes of models whose underlying marking process is an MRGP, and therefore referred to as Markov Regenerative Stochastic Petri Nets (MRSPN). Some of the results in this filed will be described in the session. In particular, Ciardo investigates stochastic confusion by defining the selection probability for transitions attempting to fire at the same time. German introduces the "method of supplementary variables" for the derivation of state equations describing the transient behavior of the marking process. Puliafito describes how, under some constraints, concurrent enabling of several generally distributed timed transitions is allowed. Bobbio and Telek discuss how age memory policy can be included to capture preemptive mechanisms of the resume (prs) type.

Arrival theorems for product-form stochastic Petri nets

We consider a particular class of Stochastic Petri Nets whose stationary probabilities at arbitrary instants exhibit a product form. We study these nets at specific instants in the steady state that occur directly after the firing of a transition. We focus our attention on the instant after tokens are removed from the places specified by a transition's input bag and just before tokens are entered into the places specified by the same transition's output bag. We show that the stationary probabilities at “arrival instants” are related to corresponding stationary probabilities at arbitrary instants in net(s) with lower load. We then show how one of the “arrival” theorems can be applied to the derivation of a formula for the mean sojourn time of a token in a place at steady state. This is the basis for the development of a Mean Value Analysis algorithm for the computation of performance indices for Product-Form Stochastic Petri Nets.

Markov regenerative stochastic Petri nets

Stochastic Petri nets of various types (SPN, GSPN, ESPN, DSPN etc.) are recognized as useful modeling tools for analyzing the performance and reliability of systems. The analysis of such Petri nets proceeds by utilizing the underlying continuous-time stochastic processes — continuous-time Markov chains for SPN and GSPN, semi-Markov processes for a subset of ESPNs and Markov regenerative processes for DSPN. In this paper, we introduce a new class of stochastic Petri nets, called Markov Regenerative Stochastic Petri Nets (MRSPNs), that can be analyzed by means of Markov regenerative processes and constitutes a true generalization of all the above classes. The MRSPNs allow immediate transitions, exponentially distributed timed transitions and generally distributed timed transitions. With a restriction that at most one generally distributed timed transition be enabled in each marking, the transient and steady state analysis of MRSPNs can be carried out analytically-numerically rather than by simulation. Equations for the solution of MRSPNs are developed in this paper, and are applied to an example.

Performance evaluation of client-server systems

A client-server system is a distributed system where a server station receives requests from its client stations, processes the requests and returns replies to the requesting stations. The authors consider client-server systems in which a set of workstations access a file server over a local area network. The systems are modelled by a class of stochastic Petri nets. The mean response time, the throughput and the parametric sensitivities are evaluated for a client-server system based on token ring network and a system based on CSMA/CD network. These models are different from the prevalent performance models of token ring or CSMA/CD network systems because of the message interdependencies introduced by the clients-server structure. An approximate analytic-numeric method rather than simulation is used to solve the models. The solution method and the accuracy of approximation are also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stochastic high-level Petri nets and applications

A class of stochastic Petri nets called stochastic high-level Petri nets (SHLPNs) is proposed. SHLPNs are high-level Petri nets augmented with exponentially distributed firing times. SHLPNs generally lead to models with a smaller state space. A computer marking concept is introduced that allows a considerable reduction of the number of states and induces a correct grouping of states in the Markov-domain SHLPN models of multiprocessor systems. The main advantage of modeling homogeneous systems using SHLPNs is that the resulting models are simpler and more intuitive and have a smaller number of states, as shown by examples. >