Stochastic Event Graphs Research Articles

Performance safety enforcement in stochastic event graphs against boost and slow attacks

This paper studies a performance safety enforcing problem in stochastic event graphs, a subclass of stochastic Petri net models. We assume that an intruder can attack part of the transitions to increase/decrease their firing rate such that the performance of the system violates a given safety interval. The difficulty in solving this problem is that the capability of the intruder, i.e., the number of transitions that can be simultaneously attacked, is limited. The control aim is to find a protecting policy such that the performance of the protected plant is guaranteed to be in a given safety interval. We show that this problem can be formulated as a two-player game between the intruder and the operator of the plant. By using mixed integer linear programming technique, we develop a heuristic method to compute a protecting policy that is locally optimal.

Tail Asymptotics for Discrete Event Systems

In the context of communication networks, the framework of stochastic event graphs allows a modeling of control mechanisms induced by the communication protocol and an analysis of its performances. We concentrate on the logarithmic tail asymptotics of the stationary response time for a class of networks that admit a representation as (max,plus)-linear systems in a random medium. We are able to derive analytic results when the distribution of the holding times are light-tailed. We show that the lack of independence may lead in dimension bigger than one to non-trivial effects in the asymptotics of the sojourn time. We also study in detail a simple queueing network with multipath routing.

Tails for (max, plus) recursions under subexponentiality

We study the stationary solution of a (max, plus)-linear recursion. Under subexponentiality assumptions on the input to the recursion, we obtain the tail asymptotics of certain (max, plus)-linear functionals of this solution. (Max, plus)-linear recursions arise from FIFO queueing networks; more specifically, from stochastic event graphs. In the event graph setting, two special cases of our results are of particular interest and have already been investigated in the literature. First, the functional may correspond to the end-to-end sojourn time of a customer. Second, for two queues in tandem, the functional may correspond to the sojourn time in the second queue. Our results allow for more general networks, which we illustrate by studying the tail asymptotics of the resequencing delay due to multi-path routing.

An Algebra for Queueing Networks with Time-Varying Service and Its Application to the Analysis of Integrated Service Networks

We introduce a network model that allows us to capture the time-varying service delivered to a traffic stream due to the presence of random perturbations (e.g., cross-traffic in a communication network). We first present the model for a single network element and then describe how such elements may be interconnected using the operations of fork and join. Such networks may be seen as a generalization of stochastic event graphs and of the class of fork-join networks. The departure processes in such a network satisfy a system of equations along with the exogenous arrival processes and the service processes of the various network elements. This system can be seen as a general time-varying linear system in the (min, plus) semi-field. We obtain an explicit representation of the departure process in terms of the exogenous arrival processes and the service processes. Sufficient conditions are derived for this system to have a unique solution. We also study liveness and absence of explosion in this class of networks. Under appropriate stationarity and ergodicity assumptions, we establish stability theorems for such networks. To this end we first obtain a rate result for the departure process in terms of the rates of the exogenous arrival process and the throughput of a saturated system. We then use this result to show that in a network with a single exogenous arrival, all queue lengths are finite with probability one if the arrival rate is less than the throughput of the saturated system, and to give a representation of the queue-length process. This class of networks allows for a detailed description of controlled sessions in integrated service networks. We also show that it contains several earlier discrete event models of the literature pertaining to stochastic Petri nets, service curves, and fork-join networks and show how the present model unifies them in a single algebraic structure. This structure is that of a semi-ring of functions of two real variables, where the addition is the pointwise minimum and the multiplication a generalization of inf-convolution.

Asymptotics of Subexponential Max Plus Networks: the Stochastic Event Graph Case

We calculate the exact tail asymptotics of stationary response times for open stochastic event graphs, in the irreducible and reducible cases. These networks admit a representation as (max, plus)-linear systems in a random medium. We study the case of renewal input and i.i.d. service times with subexponential distributions. We show that the stationary response times have tail asymptotics of the same order as the integrated tail of service times. The mutiplicative constants only involve the intensity of the arrival process and the (max, plus)-Lyapunov exponents of the sequence of (max, plus)-matrices.

Regular Ordering and Applications in Control Policies

In this paper we introduce the notion of regular ordering for periodic sequences based on the gaps between the entries. We define the notion of regular preserving functions using Schur convexity. This is used to extend some optimization results in queuing control problems. In particular, we show that the maximal traveling time in a stochastic event graph as well as the transmission times in a channel with redundancy, decrease (in the stochastic sense) when the input sequence becomes more regular.

A Differential Calculus for Random Matrices with Applications to (max, +)-Linear Stochastic Systems

We introduce the concept of weak differentiability for random matrices and thereby obtain closed-form analytical expressions for derivatives of functions of random matrices. More specifically, we develop a calculus of weak differentiation for random matrices that resembles the standard calculus of differentiation. Our formalism enables us to (algebraically) calculate derivatives of finite-horizon performance measures of stochastic event graphs. More precisely, we develop a theory of weak differentiation for (max, +)-linear systems. The resulting derivatives provide unbiased estimators for gradients of finite-horizon performance measures. For various types of (max, +)-linear systems, we compute these estimators explicitly and state the corresponding gradient estimation algorithm.

Sensitivity Analysis of Joint Characteristics of (Max,+)-Linear Queueing Networks

We introduce the concept of D-differentiability of matrices over the (max,+) algebra. Specifically, we view the stochastic (max,+)-linear system x(k + 1) = A(k)⊗ x(k), for k ≥ 0 with x(0) = x0, as a Markov chain the transition dynamic of which is given through the matrices A(k). Elaborating on the product rule of D-differentiability for Markov kernels, we obtain results on differentiability of (max,+)-linear systems and unbiased gradient estimators as well. Moreover, we establish sufficient conditions for deducing the differentiability of a (max,+)-linear system from that of the firing time distributions of the corresponding stochastic event graph. The results hold uniformly on a predefined class of performance functions. We illustrate our approach with an analysis of joint characteristics of waiting times in a (max,+)-linear queueing network.

Balanced sequences and optimal routing

The objective pursued in this paper is two-fold. The first part addresses the following combinatorial problem: is it possible to construct an infinite sequence over n letters where each letter is distributed as “evenly” as possible and appears with a given rate? The second objective of the paper is to use this construction in the framework of optimal routing in queuing networks. We show under rather general assumptions that the optimal deterministic routing in stochastic event graphs is such a sequence.

Manufacturing flow control using fluid event graphs

Flow control of manufacturing systems subject random events is considered in this paper using a fluid stochastic event graph model that is a decision-free Petri net In this model, tokens are considered as continuous flows. A transition can be in operating state or in failure state. A transition in operating state can fire at its maximal speed and a transition in failure state cannot fire. Transitions between failure and operating states are independent of the firing conditions and the sojourn time in each state is a random variable of general distribution. For the purpose of performance evaluation, a set of evolution equations that determines continuous state variables at epochs of discrete events is established. Gradient estimators of the cumulative firings with respect to the system parameters are derived. Finally, an optimisation problem of the initial marking that maximises a concave function of throughput rate and the initial marking is addressed.

Admission control in stochastic event graphs

We first show that the expectation of convex increasing functions of the workload (or waiting time) in (max, +) linear systems, under a single input sequence, is multi-modular. This is done using a coupling argument and a vectorial version of Lindley's equation. Next, we use this result and the optimization theory based on multi-modular costs to construct the optimal open-loop admission control in general (max, +) linear systems under admission rate constraints. This optimization result only requires stationarity assumptions on the arrival process and on the service times of the servers in the system.

Stochastic cyclic flow lines: non-blocking, Markovian models

In this paper we consider stochastic cyclic flow lines where identical sets of jobs are repeatedly produced in the same loading and processing sequence. Each machine has an input buffer with enough capacity. Processing times are stochastic. We model the shop as a stochastic event graph, a class of Petri nets. We characterise the ergodicity condition and the cycle time. For the case where processing times are exponentially distributed, we present a way of computing queue length distributions. For two-machine cases, by the matrix geometric method, we compute the exact queue length distributions. For general cases, we present two methods for approximately decomposing the line model into two-machine submodels, one based on starvation propagation and the other based on transition enabling probability propagation. We experiment our approximate methods for various stochastic cyclic flow lines and discuss performance characteristics as well as accuracy of the approximate methods. Finally, we discuss the effects of job processing sequences of stochastic cyclic flow lines.

Expansions for steady-state characteristics of (max, +)-linear systems

An expansion formula is given for the expected value of a general class of functions of steady state variables in open, stochastic (max, +)-linear systems with Poisson input. Expansions for Laplace transforms, moments, distribution functions and tail probabilities of these steady state variables are considered as specific instances of our general expansion formula. Such (max, +)-linear systems are known to allow one to represent a class of discrete event networks called stochastic event graphs. Examples of such event graphs pertaining to queueing theory and Petri net theory are given in the paper in order to illustrate the proposed expansion method. We also include some comments on numerical analysis

Products of Irreducible Random Matrices in the (Max, +) Algebra

We consider the recursive equation x(n + 1)= A(n)⊗x(n), where x(n + 1) and x(n) are ℝ k -valued vectors and A(n) is an irreducible random matrix of size k × k. The matrix-vector multiplication in the (max, +) algebra is defined by (A(n)⊗x(n))= maxj (Aij (n) + xj (n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices such that and the matrices have a unique periodic regime.

Products of Irreducible Random Matrices in the (Max, +) Algebra

We consider the recursive equation x(n + 1)= A(n)⊗x(n), where x(n + 1) and x(n) are ℝk-valued vectors and A(n) is an irreducible random matrix of size k × k. The matrix-vector multiplication in the (max, +) algebra is defined by (A(n)⊗x(n))= maxj (Aij (n) + xj(n)). This type of equation can be used to represent the evolution of stochastic event graphs which include cyclic Jackson networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ ℕ} is i.i.d. or, more generally, stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices such that and the matrices have a unique periodic regime.

Cycle time of stochastic event graphs: evaluation and marking optimization

Addresses the performance evaluation and optimization of a strongly connected event graph with random firing times. The authors proposed an upper bound and a lower bound for the average cycle time of event graphs knowing the initial marking. The authors then prove that, under some weak conditions on the firing time distributions, it is possible to reach, on average, a cycle time smaller than any given value C* with a finite marking, assuming that C* is greater than the greatest firing time. Finally, the authors propose an heuristic algorithm to reach such an average cycle time while minimizing a p-invariant criterion.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Ergodic Theory of Stochastic Petri Networks

Stochastic Petri nets are a general formalism for describing the dynamics of discrete event systems. The present paper focuses on a subclass of stochastic Petri nets called stochastic event graphs. Under the assumption that the variables used for the timing of an event graph form stationary and ergodic sequences of random variables, we make use of an associated stochastic recursive equation in order to construct its stationary and ergodic regime. In particular, we determine the conditions under which the existence of this regime is guaranteed.