Generalized semi-Markov Process Research Articles

On simulation output analysis for generalized semi-markov processes

The usual model for the underlying process of a discrete-event stochastic system is the generalized semi-Markov process (GSMP). A GSMP is defined in terms of a general state space Markov chain that describes the process at successive state-transition times. We provide conditions on the clock-setting distributions and state-transition probabilities of a finite state GSMP under which this underlying chain is φ-irreducible and satisfies a drift criterion for stability due to Meyn and Tweedie. If the GSMP also has a “single state” in which exactly one event is scheduled to occur, then this state is hit infinitely often with probability I and the time between successive hits has finite second moment. It follows that the standard regenerative method for analysis of simulation output can be used to obtain strongly consistent point estimates and asymptotic confidence intervals for time-average limits of the process. We also show that, under our conditions, point estimates and confidence intervals for time-average limits can be obtained using methods based on standardized time series. In particular, the method of batch means is applicable. Our results rest on a new functional central limit theorem for GSMP's together with results of Glynn and Iglehart. The standardized-time-series methods apply even when the GSMP does not have a single state or indeed any type of classical regenerative structure

Likelihood Ratio Derivative Estimation for Finite-Time Performance Measures in Generalized Semi-Markov Processes

This paper investigates the likelihood ratio method for estimating derivatives of finite-time performance measures in generalized semi-Markov processes (GSMPs). We develop readily verifiable conditions for the applicability of this method. Our conditions mainly place restrictions on the basic building blocks (i.e., the transition probabilities, the distribution and density functions of the event lifetimes, and the initial distribution) of the GSMP, which is in contrast to the structural conditions needed for infinitesimal perturbation analysis. We explicitly show that our conditions hold in many practical settings, and in particular, for large classes of queueing and reliability models. One intermediate result we obtain in this study, which is of independent value, is to formally show that the random variable representing the number of occurring events in a GSMP in a finite time horizon, has finite exponential moments in a neighborhood of zero.

Perturbation analysis of stochastic Petri nets

This paper addresses the sensitivity analysis of stochastic Petri nets (SPNs) using simulations. The goal is to evaluate the derivatives of performance measures with respect to timing parameters. To characterize the underlying stochastic processes of SPNs, we use a generalized semi-Markov process (GSMP) representation and propose a new representation, called GSMP*, which differs from GSMP in the routing mechanism. By using existing results on perturbation analysis of GSMPs and by extending them to GSMP*, unbiased sensitivity estimators are obtained for SPNs simulated under a GSMP or GSMP* framework. Most importantly, we prove that only one simulation run is needed for evaluating both the performance measures and their derivatives for a class of free-choice nets simulated under a GSMP framework and for any free-choice net simulated under a GSMP* framework.

STOCHASTIC PROPERTIES OF FORK/JOIN MULTI-STAGE PRODUCTION SYSTEMS WITH GENERAL BLOCKING

For tandem queues and forkljoin queueing networks with communication blocking, the stochas- tic comparison, reversibility and other equivalence properties have been studied. In this paper, we consider a forkljoin multi-stage production system with general blocking which includes the above models as special cases. Under the weak conditions of the initial numbers of items, we formulate this system into a generalized semi-Markov process (GSMP). Then we show the convex property of the GSMP, the stochastic comparison with respect to stochastic and convex ordering, the reversibility and the structural equivalence.

Event coupling and performance sensitivity analysis of generalized semi-Markov processes

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.

Event coupling and performance sensitivity analysis of generalized semi-Markov processes

We study a fundamental feature of the generalized semi-Markov processes (GSMPs), called event coupling. The event coupling reflects the logical behavior of a GSMP that specifies which events can be affected by any given event. Based on the event-coupling property, GSMPs can be classified into three classes: the strongly coupled, the hierarchically coupled, and the decomposable GSMPs. The event-coupling property on a sample path of a GSMP can be represented by the event-coupling trees. With the event-coupling tree, we can quantify the effect of a single perturbation on a performance measure by using realization factors. A set of equations that specifies the realization factors is derived. We show that the sensitivity of steady-state performance with respect to a parameter of an event lifetime distribution can be obtained by a simple formula based on realization factors and that the sample-path performance sensitivity converges to the sensitivity of the steady-state performance with probability one as the length of the sample path goes to infinity. This generalizes the existing results of perturbation analysis of queueing networks to GSMPs.

On the structure of an insensitive generalized semi-Markov process with reallocation and point-process input

A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

On the structure of an insensitive generalized semi-Markov process with reallocation and point-process input

A generalized semi-Markov process with reallocation (RGSMP) was introduced to accommodate a large class of stochastic processes which cannot be analyzed by the well-known model of an ordinary generalized semi-Markov process (GSMP). For stationary RGSMP whose initial distribution has a product form, we show that, for a randomly chosen clock of a fixed insensitive type, if the lifetime of this clock is changed to infinity, then the background process is stationary under a certain time change. This implies that the expected time required for the tagged clock to consume a given amount x of resource, called the attained sojourn time, is a linear function of x. Such stationarity and linearity results are known for two special RGSMPs: ordinary GSMP and Kelly's symmetric queue. Our results not only extend them to a general RGSMP but also give more detailed formulas, which allow us to calculate for instance the expected attained sojourn time while the background process is in a given state. Furthermore, we remark that analogous results hold for GSMP with point-process input, in which the lifetimes of clocks of a fixed type form an arbitrary stationary sequence (of not necessarily independent random variables).

Tight bounds on the sensitivity of generalised semi-Markov processes with a single generally distributed lifetime

There are some generalised semi-Markov processes (GSMP) which are insensitive, that is the value of some performance measures for the system depend only on the mean value of lifetimes and not on their actual distribution. In most cases this is not true and a performance measure can take on a number of values depending on the lifetime distributions. In this paper we present a method for finding tight bounds on the sensitivity of performance measures for the class of GSMPs with a single generally distributed lifetime. Using this method we can find upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to find bounds on the average queue length of the Engset queue and the time congestion in the GI/M/n/n queueing system.

Tight bounds on the sensitivity of generalised semi-Markov processes with a single generally distributed lifetime

There are some generalised semi-Markov processes (GSMP) which are insensitive, that is the value of some performance measures for the system depend only on the mean value of lifetimes and not on their actual distribution. In most cases this is not true and a performance measure can take on a number of values depending on the lifetime distributions. In this paper we present a method for finding tight bounds on the sensitivity of performance measures for the class of GSMPs with a single generally distributed lifetime. Using this method we can find upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to find bounds on the average queue length of the Engset queue and the time congestion in theGI/M/n/nqueueing system.

Insensitivity in Discrete-Time Generalized Semi-Markov Processes Allowing Multiple Events and Probabilistic Service Scheduling

In this paper we treat a discrete-time generalized semi-Markov process in which simultaneous deaths of more than one lifetime and simultaneous arrivals are permitted and service scheduling is probabilistic. Necessary and sufficient conditions for insensitivity are derived and a simple algorithmic procedure provided whereby the equilibrium probabilities can be furnished when insensitivity holds. Some particular cases of special interest are examined in detail.

Monotone Optimal Control of Permutable GSMPs

We consider Markovian GSMPs (generalized semi-Markov processes) in which the rates of events are subject to control. A control is monotone if the rate of one event is increasing or decreasing in the number of occurrences of other events. We give general conditions for the existence of monotone optimal controls. The conditions are functional properties for the one-step cost functions and, more importantly, structural properties for the GSMP. The main conditions on costs are submodularity or supermodularity with respect to pairs of events. The key structural condition is strong permutability, requiring that the state at any time be determined by the number of events of each type that have occurred, regardless of their order. This permits a reformulation of the original control problem into one based only on event counting processes. This reformulation leads to a unified treatment of a broad class of models and to meaningful generality beyond existing results.

On using continuous flow lines to model discrete production lines

We extend previous research on the use of models ofcontinuous tandem (CT) lines for performance analysis ofdiscrete tandem (DT) production lines. We formalize the translation of input parameters from the DT line to the CT model, as well as the translation of performance measures (PMs) obtained from the CT model back to the DT line. We show that although the CT model conceptually represents a line with continuous fluid, it can be represented as a generalized semi-Markov process (GSMP). This representation leads to a simple and concise simulation algorithm for a CT model. We investigate the accuracy of the CT model for prediction of PMs in the DT line, and show that, with proper translation of parameters and PMs, the CT model provides reasonable estimates for the DT line PMs. We provide preliminary results on gradient estimation for CT models via infinitestimal perturbation analysis. The aim of the paper is to provide a firm foundation for the future exploration of CT models as a means to parameter optimization for DT lines.

A transient solution method for semi-Markov systems

We study semi-Markov processes with finite state spaces to obtain a transient solution. In the first place, we obtain Kolmogorov's equation from which, in the case of a less general semi-Markov process, we obtain an explicit formula for transient state probabilities. From this point we give a method to obtain a solution of general semi-Markov processes.

Analysis of a tandem queue with state-dependent general blocking: A GSMP perspective

In this paper, we study a tandem queue where there is a finite number of buffer positions at each stage. The blocking scheme is general in the sense that it can model a number of classical blocking schemes, including communication, manufacturing and kanban blockings as special cases. The system considered here differs from the conventional system in two aspects: (1) departure of jobs from the system is determined by an external arrival process of another queue lying parallel to the tandem queue; (2) the control parameters of the blocking scheme are state-dependent in that they may change values depending on the state of the system. We model this system as a generalized semi-Markov process (GSMP), we study the structural properties of its scheme, and establish monotonicity and convexity properties of event times with respect to both the clock times and the integer blocking parameters. In particular, we demonstrate that the state-dependent scheme is “better”, in certain sense, than the corresponding static scheme. Our results also recover the structural properties previously established for the classical blockings.

Generalized Semi-Markov Processes: Antimatroid Structure and Second-Order Properties

A generalized semi-Markov scheme models the structure of a discrete event system, such as a network of queues. By studying combinatorial and geometric representations of schemes we find conditions for second-order properties—convexity/concavity, sub/supermodularity—of their event epochs and event counting processes. A scheme generates a language of feasible strings of events. We show that monotonicity of the event epochs is equivalent to this language forming an antimatroid with repetition. This connection gives rise to a rich combinatorial structure, and serves as a starting point for other properties. For example, by strengthening the antimatroid condition we give several equivalent characterizations of the convexity of event epochs within a scheme. All of these correspond, in slightly different ways, to making a certain score space a lattice, to closing an ordinary antimatroid under intersections. We also establish second-order properties across schemes tied together through a synchronization mechanism. A geometric view based on the score space facilitates verification of these properties in certain queueing systems.

Monotonicity in Generalized Semi-Markov Processes

We establish stochastic monotonicity of the event epoch sequences of generalized semi-Markov processes through the structure of the generalized semi-Markov schemes on which they are based. Our main condition states, roughly, that the occurrence of more events in the short run never leads to the activation of less events in the long run. We consider monotonicity with respect to two types of inputs: clock times (which translate to, e.g., service and interarrival times in queueing systems), and structural parameters (which translate to, e.g., buffer size, number of servers, and job population). For the second type of comparison, we replace a structural change with an equivalent change in clock times to reduce the comparison to one of the first type. When applied to queueing systems, our results yield new comparisons and also unify several existing results previously established using special properties of individual systems.

Bounds on the Sensitivity of Generalised Semi-Markov Processes with a Single Generally Distributed Lifetime

In this paper we present a method for finding bounds on the sensitivity of Generalised Semi-Markov Processes with a single generally distributed lifetime. This method gives upper and lower bounds for the value of a function of the stationary distribution as the distribution of the general lifetime ranges over a set of distributions with fixed mean. The method is applied to some simple queueing systems.

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

The theory of insensitivity within generalized semi-Markov processes is extended to cover the case where such a process evolves in a random environment; that is, when the decay rates and transition probabilities are functions of the state of an extraneous environmental process.

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

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