Generalized semi-Markov Process Research Articles

Insensitivity of Generalized Semi-Markov Processes Evolving in a Random Environment

Insensitivity of Generalized Semi-Markov Processes Evolving in a Random Environment

Algebraic structure of some stochastic discrete event systems, with applications

Generalized semi-Markov processes (GSMPs) and stochastic Petri nets (SPNs) are generally regarded as performance models (as opposed to logical models) of discrete event systems. Here we take the view that GSMPs and SPNS are essentially automata (generators) driven by input sequences that determine the timing of events. This view combines the deterministic, logical aspects and the stochastic, timed aspects of the two models. We focus on two conditions, (M) and (CX) (which we previously developed to study monotonicity and convexity properties of GSMPs), and the antimatroid and lattice structure they imply for the language generated by a GSMP or SPN. We illustrate applications of these structural properties in the areas of derivative estimation, simulation variance reduction, parallel simulation, and optimal control.

Further results on ASTA for general stationary processes and related problems

We consider the equivalence of state probabilities of a general stationary process at an arbitrary time and at embedded epochs of a given point process, which is called ASTA (Arrivals See Time Averages). By using an event-conditonal intensity, we give necessary and sufficient conditions for ASTA for a large class of state sets, which determines a state distribution. We do not need any additional assumptions except that the general process has left-hand limits at all points of time. Especially, for a stationary pure-jump process with a point process, ASTA is obtained for all state sets. As an application of those results, Anti-PASTA is obtained for a pure-jump Markov process and a certain class of GSMP (Generalized Semi-Markov Processes), where Anti-PASTA means that ASTA implies that the arrival process is Poisson.

Further results on ASTA for general stationary processes and related problems

We consider the equivalence of state probabilities of a general stationary process at an arbitrary time and at embedded epochs of a given point process, which is called ASTA (Arrivals See Time Averages). By using an event-conditonal intensity, we give necessary and sufficient conditions for ASTA for a large class of state sets, which determines a state distribution. We do not need any additional assumptions except that the general process has left-hand limits at all points of time. Especially, for a stationary pure-jump process with a point process, ASTA is obtained for all state sets. As an application of those results, Anti-PASTA is obtained for a pure-jump Markov process and a certain class of GSMP (Generalized Semi-Markov Processes), where Anti-PASTA means that ASTA implies that the arrival process is Poisson.

Sample path and performance homogeneity of discrete event dynamic systems

In this paper, a criterion for the applicability of the Infinitesimal Perturbation Analysis (IPA) is given. The concepts of sample path and performance homogeneity are discussed in terms of the model of the Generalized Semi-Markov process.

Sensitivity bounds on a GI/M/n/n queueing system

AbstractA method for determining the upper and lower bounds for performance measures for certain types of Generalised Semi-Markov Processes has been described in Taylor and Coyle [8]. A brief description of this method and its use in finding an upper bound for the time congestion of a GI/M/n/n queueing system will be given. This bound turns out to have a simple form which is quickly calculated and easy to use in practice.

A Quasi-Reversibility Approach to the Insensitivity of Generalized Semi-Markov Processes

This paper is concerned with a certain property of the stationary distribution of a generalized semi-Markov process (GSMP) known as insensitivity. It is well-known that the so-called Matthes' conditions form a necessary and sufficient algebraic criterion for insensitivity. Most proofs of these conditions are basically algebraic. By interpreting a GSMP as a simple queueing network, we are able to show that Matthes' conditions are equivalent to the quasi-reversibility of the network, thus obtaining another simple proof of the sufficiency of these conditions. Furthermore, we apply our method to find a simple criterion for the insensitivity of GSMP's with generalized routing (in a sense that is introduced in the paper).

Stochastic Petri net representation of discrete event simulations

In the context of discrete event simulation, the marking of a stochastic Petri net (SPN) corresponds to the state of the underlying stochastic process of the simulation and the firing of a transition corresponds to the occurrence of an event. A study is made of the modeling power of SPNs with timed and immediate transitions, showing that such Petri nets provide a general framework for simulation. The principle result is that for any (finite or) countable state GSMP (generalized semi-Markov process) there exists an SPN having a marking process that mimics the GSMP in the sense that the two processes (and their underlying general state-space Markov chains) have the same finite dimensional distributions. >

A GSMP formalism for discrete event systems

A precise mathematical framework for the study of discrete event systems is described. The idea is to define a particular type of stochastic process, called a generalized semi-Markov process (GSMP), which captures the essential dynamical structure of a discrete event system. An attempt is also made to give the flavor of the qualitative theory and numerical algorithms that can be obtained as a result of viewing discrete event systems as GSMPs. Likelihood ratio concepts for importance sampling are briefly described.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Modeling Power of Stochastic Petri Nets for Simulation

Generalized semi-Markov processes and stochastic Petri nets have been proposed as general frameworks for a discrete event simulation on a countable state space. The two formal systems differ, however, with respect to the clock setting (event scheduling) mechanism, the state transition mechanism, and the form of the state space. We obtain conditions under which the marking process of a stochastic Petri net “mimics” a generalized semi-Markov process in the sense that the two processes (and their underlying general state-space Markov chains) have the same finite dimensional distributions. The results imply that stochastic Petri nets have at least the modeling power of generalized semiMarkov processes for discrete event simulation.

The basic equations for a supplemented GSMP and its applications to queues

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

The basic equations for a supplemented GSMP and its applications to queues

A supplemented GSMP (generalized semi-Markov process) is a useful stochastic process for discussing fairly general queues including queueing networks. Although much work has been done on its insensitivity property, there are only a few papers on its general properties. This paper considers a supplemented GSMP in a general setting. Our main concern is with a system of Laplace–Stieltjes transforms of the steady state equations called the basic equations. The basic equations are derived directly under the stationary condition. It is shown that these basic equations with some other conditions characterize the stationary distribution. We mention how to get a solution to the basic equations when the solution is partially known or inferred. Their applications to queues are discussed.

Regenerative generalized semi-markov processes

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 simulation methods for local area computer networks

Local area computer network simulations are inherently non-Markovian in that the underlying stochastic process cannot be modeled as a Markov chain with countable state space. We restrict attention to local network simulations whose underlying stochastic process can be represented as a generalized semi-Markov process (GSMP). Using new better than used distributional assumptions and sample path properties of the GSMP, we provide a geometric trials criterion for recurrence in this setting. We also provide conditions which ensure that a GSMP is a regenerative process and that the expected time between regeneration points is finite. Steady-state estimation procedures for ring and bus network simulations follow from these results.

Semi-Markov processes on a general state space: α-theory and quasi-stationarity

AbstractBy amalgamating the approaches of Tweedie (1974) and Nummelin (1977), an α-theory is developed for general semi-Markov processes. It is shown that α-transient, α-recurrent and α-positive recurrent processes can be defined, with properties analogous to those for transient, recurrent and positive recurrent processes. Limit theorems for α-positive recurrent processes follow by transforming to the probabilistic case, as in the above references: these then give results on the existence and form of quasistationary distributions, extending those of Tweedie (1975) and Nummelin (1976).

Continuity of Generalized Semi-Markov Processes

It is shown that sequences of generalized semi-Markov processes converge in the sense of weak convergence of random functions if associated sequences of defining elements (initial distributions, transition functions and clock time distributions) converge. This continuity or stability is used to obtain information about invariant probability measures. It is shown that there exists an invariant probability measure for any finite-state generalized semi-Markov process in which each clock time distribution has a continuous c.d.f. and a finite mean. For generalized semi-Markov processes with unique invariant probability measures, sequences of invariant probability measures converge when associated sequences of defining elements converge. Hence, properties of invariant measures can be deduced from convenient approximations. For example, insensitivity properties established for special classes of generalized semi-Markov processes by Schassberger (Schassberger, R. 1977. Insensitivity of steady-state distributions of generalized semi-Markov processes, I. Ann. Probab. 5 87–99.), (Schassberger, R. 1978. Insensitivity of steady-state distributions of generalized semi-Markov processes, II. Ann. Probab. 6 85–93.), Konig and Jansen (König, D., U. Jansen. 1976. Eine Invarianzeigenschaft Zufälliger Bedienungs-Prozesse mit Positiven Geschwindigkeiten. Math Nachr. 70 321–364.), and Burman (Burman, D. Y. 1981. Insensitivity in queueing systems. Adv. Appl. Probab. To appear.) extend to a larger class of generalized semi-Markov processes.

Insensitivity of Steady-state Distributions of Generalized Semi-Markov Processes. Part II

In a number of well-known applied probability models certain steady-state probabilities display an insensitivity property: they depend only on the means of certain lifetime distributions entering the definition of the model, not on their exact shapes. This phenomenon has been studied by Matthes and co-workers in a general framework. New and simple proofs are given for their essential results.

Hardware simulation of semi-Markov and related processes: Part I: a versatile generator

After a brief review and classification of the known generators, this part I of 2 parts paper investigates carefully a new programmable generator of general ordinary renewal processes in discrete time. The generator is based on the concept of failure rate and its implementation is completely digital. The effects of time and amplitude quantization are analyzed in detail. Statistical tests, checking distribution and independence, are partly developed and then applied to numerous examples; results are satisfactory. The distributions of the generated random time intervals are not truncated. Thereafter the generator of renewal processes is extended to allow the hardware simulation of general semi-Markov processes in discrete time. (The generation of related processes will be considered in part II, together with applications.)

Insensitivity of Steady-State Distributions of Generalized Semi-Markov Processes. Part I

New proofs are given for results by Matthes concerning a certain insensitivity phenomenon displayed, for instance, by many well-known stochastic models from the areas of reliability and telephone engineering.

On the Equilibrium Distribution of a Class of Finite-State Generalized Semi-Markov Processes

An algorithm is presented for computing the stationary distribution of a certain class of stochastic processes. The class comprises many known or potentially useful processes in the fields of operations research and applied probability.