Regenerative generalized semi-markov processes

Haas, Peter J.; Shedler, Gerald S.

doi:10.1080/15326348708807064

Cited by 32 publications

(30 citation statements)

References 14 publications

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“…In this Suppose (throughout this section) that the GSMP has a single-state g ~ S, that is, a state g in which only one event is active so that E(g) = {~} for some ~ 6 E. Also suppose that the single-state g is recurrent in the sense that the chain {(Sn, Cn): n > 0} hits the set of states {(s, c) E ~: s = g} infinitely often with probability 1, so that the GSMP hits g infinitely often with probability 1; cf (Haas and Shedler 1987b, 1993a. Next, suppose that the joint distribution of the initial state So and the vector of clock readings Co are such that the GSMP behaves as i~f at time 0 the state is ~ and event ~ occurs.…”

Section: One-dependent Cycles and Estimation Methodsmentioning

confidence: 99%

Estimation methods for passage times using one-dependent cycles

Haas

Shedler

1996

Discrete Event Dyn Syst

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Abetraet. The lengths of certain passage-time intervals (random time intervals) in discrete-event stochastic systems correspond to delays in computer, communication, manufacturing, and transportation systems. Simulation is often the only available means for analyzing a sequence of such lengths. It is sometimes possible to obtain meaningful estimates for the limiting average delay indirectly, that is, without measuring lengths of individual passage-time intervals. For general time-average limits of a sequence of delays, however, it is necessary to measure individual lengths and combine them to form point and interval estimates. We consider sequences of delays determined by state transitions of a generalized semi-Markov process and introduce a recursively-generated sequence of realvalued random vectors, called start vectors, to provide the link between the starts and terminations of passage-time intervals. This method of start vectors for measuring delays avoids the need to "tag" entities in the system. We show that if the generalized semi-Markov process has a recurrent single-state, then the sample paths of any sequence of delays can be decomposed into one-dependent, identically distributed cycles. We then show that an extension of the regenerative method for analysis of simulation output can be used to obtain meaningful point estimates and confidence intervals for time-average limits. This estimation procedure is valid not only when there are no ongoing passage times at any regeneration point but, unlike previous methods, also when the sequence of delays does not inherit regenerative structure. Application of these methods to a manufacturing cell with robots is discussed.Keywords: discrete-event stochastic systems, stochastic simulation, regenerative processes, regenerative simulation, generalized semi-Markov processes, passage times, Little's Law, networks of queues 1, IntroductionAnalysis of delay phenomena is often necessary for assessment of the performance of computer, communication, manufacturing, and transportation systems. Examples of delays in such systems include , the time to produce an item in a flexible manufacturing system; ® the time to compute the answer to a query in a database management system; and the time to transmit a message from one node to another in a communication network.This paper deals with simulation methods that are applicable to delays when the system under study is modeled as a discrete-event stochastic system. We suppose that the delays of interest correspond to lengths of certain passage-time intervals (random time intervals) determined by state transitions of the system. Simulation is often the only available means for analyzing a sequence of such lengths. A discrete-event stochastic system makes state transitions when events associated with the occupied state occur; events occur only at an increasing sequence of random times. The underlying stochastic process of a discrete-event stochastic system records the state of the

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Section: One-dependent Cycles and Estimation Methodsmentioning

confidence: 99%

Estimation methods for passage times using one-dependent cycles

Haas

Shedler

1996

Discrete Event Dyn Syst

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“…Verification of Condition (iii) in Theorem 4.20 can sometimes be simplified when the reachability set is finite, each clock-setting distribution has finite second moment, and there exists an increasing sequence { v(n): n > 0 ) of a.s. finite random indices along with a positive number a such that (4.7)holds. It then suffices to show that supk,, -Ep [ ( s ( k ) -v(k -I))'] < OLI for some r > 2; cf[14, Prop. 4.111.…”

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confidence: 99%

Passage times in colored stochastic petri nets

Haas

Shedler

1993

Communications in Statistics. Stochastic Models

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Passage times in colored stochastic Petri nets correspond to delays in discrete-event stochastic systems. Formal definition of a sequence of passage times in a colored stochastic Petri net is in terms of the underlying general state space Markov chain of the marking process. Using symmetry of the net with respect to color, we provide conditions under which a sequence of passage times is a regenerative process in discrete time with finite cycle-length moments. The regenerative property implies time-average convergence, convergence in distribution, and a central limit theorem for sequences of passage times. It follows that strongly consistent point estimates and asymptotic confidence intervals for general characteristics of passage times can be obtained by simulating a finite portion of a single sample path of the underlying general state space Markov chain. Using regenerative structure of the marking process and a version of Little's Law given by Glynn and Whitt, we also obtain conditions under which a sequence of passage times converges in a time-average sense and the limit can be expressed as a ratio of expected values. The resulting estimation procedure for the limiting average passage time requires no measurement of individual passage times and is valid even if there are no regeneration points for the sequence of passage times.In practice, the most difficult part of the argument required to establish the applicability of regenerative methods for simulation of passage times in colored stochastic Petri nets is to show that certain fixed sets of markings are recurrent. We show that representations for certain conditional distributions of clock readings can be combined with a "geometric trials" criterion to establish recurrence.

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“…Probabilistic semantics of timed automata was proposed in [14,15], and a more general model of stochastic games over timed automata was considered in [16]. In this paper we build mainly on the previous work about GSMPs [1,2,3] and RTPs [4,17] and interpret timed automata as a model-independent specification language which can express important properties of timed systems. This view is adopted also in [18] where continuous-time Markov chains are checked against timed-automata specifications.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we study stochastic real-time games (SRTGs) which are obtained as a natural game-theoretic extension of generalized semi-Markov processes (GSMP) [1,2,3] or real-time probabilistic processes (RTP) [4]. Intuitively, all of these formalisms model systems which react to certain events, such as message receipts, subsystem failures, timeouts, etc.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Real-Time Games with Qualitative Timed Automata Objectives

Brázdil

Krčál

Křetínský

et al. 2010

CONCUR 2010 - Concurrency Theory

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We consider two-player stochastic games over real-time probabilistic processes where the winning objective is specified by a timed automaton. The goal of player is to play in such a way that the play (a timed word) is accepted by the timed automaton with probability one. Player aims at the opposite. We prove that whenever player has a winning strategy, then she also has a strategy that can be specified by a timed automaton. The strategy automaton reads the history of a play, and the decisions taken by the strategy depend only on the region of the resulting configuration. We also give an exponential-time algorithm which computes a winning timed automaton strategy if it exists.

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Regenerative generalized semi-markov processes

Cited by 32 publications

References 14 publications

Estimation methods for passage times using one-dependent cycles

Estimation methods for passage times using one-dependent cycles

Passage times in colored stochastic petri nets

Stochastic Real-Time Games with Qualitative Timed Automata Objectives

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