Starts with a brief review of the history and the application areas considered in the literature. The author then proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs, the concurrent system model most amenable to analysis. Introductory discussions on stochastic nets with their application to performance modeling, and on high-level nets with their application to logic programming, are provided. Also included are recent results on reachability criteria. Suggestions are provided for further reading on many subject areas of Petri nets. >

/pdf/petri-nets-properties-analysis-and-applications-43jwyjdxr0.pdf

Petri nets: Properties, analysis and applications

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Non-Uniform Random Variate Generation.

This is a survey of the main methods in non-uniform random variate generation, and highlights recent research on the subject. Classical paradigms such as inversion, rejection, guide tables, and transformations are reviewed. We provide information on the expected time complexity of various algorithms, before addressing modern topics such as indirectly specified distributions, random processes, and Markov chain methods. Authors’ address: School of Computer Science, McGill University, 3480 University Street, Montreal, Canada H3A 2K6. The authors’ research was sponsored by NSERC Grant A3456 and FCAR Grant 90-ER-0291. 1. The main paradigms The purpose of this chapter is to review the main methods for generating random variables, vectors and processes. Classical workhorses such as the inversion method, the rejection method and table methods are reviewed in section 1. In section 2, we discuss the expected time complexity of various algorithms, and give a few examples of the design of generators that are uniformly fast over entire families of distributions. In section 3, we develop a few universal generators, such as generators for all log concave distributions on the real line. Section 4 deals with random variate generation when distributions are indirectly specified, e.g, via Fourier coefficients, characteristic functions, the moments, the moment generating function, distributional identities, infinite series or Kolmogorov measures. Random processes are briefly touched upon in section 5. Finally, the latest developments in Markov chain methods are discussed in section 6. Some of this work grew from Devroye (1986a), and we are carefully documenting work that was done since 1986. More recent references can be found in the book by Hörmann, Leydold and Derflinger (2004). Non-uniform random variate generation is concerned with the generation of random variables with certain distributions. Such random variables are often discrete, taking values in a countable set, or absolutely continuous, and thus described by a density. The methods used for generating them depend upon the computational model one is working with, and upon the demands on the part of the output. For example, in a ram (random access memory) model, one accepts that real numbers can be stored and operated upon (compared, added, multiplied, and so forth) in one time unit. Furthermore, this model assumes that a source capable of producing an i.i.d. (independent identically distributed) sequence of uniform [0, 1] random variables is available. This model is of course unrealistic, but designing random variate generators based on it has several advantages: first of all, it allows one to disconnect the theory of non-uniform random variate generation from that of uniform random variate generation, and secondly, it permits one to plan for the future, as more powerful computers will be developed that permit ever better approximations of the model. Algorithms designed under finite approximation limitations will have to be redesigned when the next generation of computers arrives. For the generation of discrete or integer-valued random variables, which includes the vast area of the generation of random combinatorial structures, one can adhere to a clean model, the pure bit model, in which each bit operation takes one time unit, and storage can be reported in terms of bits. Typically, one now assumes that an i.i.d. sequence of independent perfect bits is available. In this model, an elegant information-theoretic theory can be derived. For example, Knuth and Yao (1976) showed that to generate a random integer X described by the probability distribution {X = n} = pn, n ≥ 1, any method must use an expected number of bits greater than the binary entropy of the distribution, ∑

Non-uniform random variate generation

design issues. Specific aspects of cache memories tha t are investigated include: the cache fetch algorithm (demand versus prefetch), the placement and replacement algorithms, line size, store-through versus copy-back updating of main memory, cold-start versus warm-start miss ratios, mulhcache consistency, the effect of input /output through the cache, the behavior of split data/instruction caches, and cache size. Our discussion includes other aspects of memory system architecture, including translation lookaside buffers. Throughout the paper, we use as examples the implementation of the cache in the Amdahl 470V/6 and 470V/7, the IBM 3081, 3033, and 370/168, and the DEC VAX 11/780. An extensive bibliography is provided.

Cache Memories

Based on a statistical analysis of actual computer program address traces, some results are presented of a study aimed at deriving empirically valid stochastic models for program reference patterns in a computer system operating under demand paging. For the address traces examined, a semi-Markov model for the (univariate) point process of page exceptions is formulated and found to be an adequate characterization of the data.

Empirically derived micromodels for sequences of page exceptions

A new method for simulating a nonhomogeneous Poisson process with rate function λ(t) = exp{α0 + α1t + a2t2} in a fixed time interval (0, t0] is given. The method is based on a decomposition of the process, it employs a rejection technique, in conjunction with a method for simulating the nonhomogeneous Poisson process with rate function exp {γ0 + γ1t} by generation of gap statistics from a random number of exponential random variables with suitably chosen parameters. The proposed method for the degree-two exponential polynomial model is more efficient than time-scale transformation of a homogeneous Poisson process, and should be applicable to other rate function models.

Simulation of Nonhomogeneous Poisson Processes with Degree-Two Exponential Polynomial Rate Function

This paper presents results of an approximation study of cyclic queueing phenomena that occur in multiprogrammed computer systems. Based on Wald’s identity and using ideas of diffusion, the objective is to develop convenient and nearly explicit formulas relating processor utilization in such systems to simple program parameters and the level of multiprogramming. Some numerical results to indicate the quality of the proposed approximation are given.

Approximate models for processor utilization in multiprogrammed computer systems

A generalized semi-Markov process (GSMP) is the usual model for a discrete event simulation. The standard formulation assumes that the current state of the process uniquely determines the set of events that are scheduled to occur and that the state transition probabilities depend only on the current state, the next state, and the trigger event. We provide a formulation of a GSMP that avoids these restrictions, thus permitting formal specification of many non-Markovian simulation models. Using a monotone replacement property of “new better than used” distributions and sample path properties of the GSMP, we provide a criterion for recurrence in this setting and conditions under which a GSMP is a regenerative process and the time between regeneration points has finite moments. Steady state estimation methods for non-Markovian stochastic network simulations follow.

Regenerative generalized semi-markov processes

This paper considers access control for local area computer communication networks. We propose two distributed access control schemes for a bus network. The schemes are simple and asynchronous, and provide for collision-free communication among ports. In addition, one of the schemes provides a bounded, guaranteed time to transmisidon for each port. We also show that this scheme is efficient in the use of the bus bandwidth, in the sense that there is only a small fraction of time during which the bus is idle when there is at least one packet available for transmission.

Gerald S. Shedler

Papers

Empirically derived micromodels for sequences of page exceptions

Simulation of Nonhomogeneous Poisson Processes with Degree-Two Exponential Polynomial Rate Function

Approximate models for processor utilization in multiprogrammed computer systems

Regenerative generalized semi-markov processes

Collision-Free Access Control for Computer Communication Bus Networks