Vidyadhar G. Kulkarni

Markov regenerative stochastic Petri nets

Abstract Stochastic Petri nets of various types (SPN, GSPN, ESPN, DSPN etc.) are recognized as useful modeling tools for analyzing the performance and reliability of systems. The analysis of such Petri nets proceeds by utilizing the underlying continuous-time stochastic processes — continuous-time Markov chains for SPN and GSPN, semi-Markov processes for a subset of ESPNs and Markov regenerative processes for DSPN. In this paper, we introduce a new class of stochastic Petri nets, called Markov Regenerative Stochastic Petri Nets (MRSPNs), that can be analyzed by means of Markov regenerative processes and constitutes a true generalization of all the above classes. The MRSPNs allow immediate transitions, exponentially distributed timed transitions and generally distributed timed transitions. With a restriction that at most one generally distributed timed transition be enabled in each marking, the transient and steady state analysis of MRSPNs can be carried out analytically-numerically rather than by simulation. Equations for the solution of MRSPNs are developed in this paper, and are applied to an example.

Dynamic Scheduling of Outpatient Appointments Under Patient No-Shows and Cancellations

This paper develops a framework and proposes heuristic dynamic policies for scheduling patient appointments, taking into account the fact that patients may cancel or not show up for their appointments. In a simulation study that considers a model clinic, which is created using data obtained from an actual clinic, we find that the heuristics proposed outperform all the other benchmark policies, particularly when the patient load is high compared with the regular capacity. Supporting earlier findings in the literature, we find that the open access policy, a recently proposed popular scheduling paradigm that calls for “meeting todays demand today,” can be a reasonable choice when the patient load is relatively low.

Markov and Markov-Regenerative pert Networks

This paper investigates pert networks with independent and exponentially distributed activity durations. We model such networks as finite-state, absorbing, continuous-time Markov chains with upper triangular generator matrices. The state space is related to the network structure. We present simple and computationally stable algorithms to evaluate the usual performance criteria: the distribution and moments of project completion time, the probability that a given path is critical, and other related performance measures. In addition, we algorithmically analyze conditional performance measures-for example, project completion time, given a critical path-and present computational results. We then study extensions both to resource-constrained pert networks and to a special class of nonexponential pert networks.

Fluid stochastic Petri nets : Theory, applications, and solution techniques

In this paper we introduce a new class of stochastic Petri nets in which one or more places can hold fluid rather than discrete tokens. We define a class of fluid stochastic Petri nets in such a way that the discrete and continuous portions may affect each other. Following this definition we provide equations for their transient and steady-state behavior. We present several examples showing the utility of the construct in communication network modeling and reliability analysis, and discuss important special cases. We then discuss numerical methods for computing the transient behavior of such nets. Finally, some numerical examples are presented.

FSPNs: Fluid Stochastic Petri Nets

In this paper we introduce a new class of stochastic Petri nets in which one or more places can hold fluid rather than discrete tokens. After defining the class of fluid stochastic Petri nets, we provide equations for their transient and steady-state behavior. We give two application examples. We hope that this paper will spur further research on this topic.

Retrial queues with server subject to breakdown and repairs

In this paper we consider a single server retrial queue where the server is subject to breakdowns and repairs. New customers arrive at the service station according to a Poisson process and demand i.i.d. service times. If the server is idle, the incoming customer starts getting served immediately. If the server is busy, the incoming customer conducts a retrial after an exponential amount of time. The retrial customers behave independently of each other. The server stays up for an exponential time and then fails. Repair times have a general distribution. The failure/repair behavior when the server is idle is different from when it is busy. Two different models are considered. In model I, the failed server cannot be occupied and the customer whose service is interrupted has to either leave the system or rejoin the retrial group. In model II, the customer whose service is interrupted by a failure stays at the server and restarts the service when repair is completed. Model II can be handled as a special case of model I. For model I, we derive the stability condition and study the limiting behavior of the system by using the tools of Markov regenerative processes.

A Classified Bibliography Of Research On Stochastic Pert Networks: 1966-1987

AbstractThe aim of this paper is to present a classified bibliography of research work on stochastic PERT networks organized into six sections — classical PERT problem, exact analysis, approximation and bounds, Monte Carlo sampling, miscellaneous papers, and books that deal with stochastic activity networks. Some of the above categories are further split into subcategodes when appropriate, and explanations of subject matter in each category and subcategory precede the actual list of references.

Transient Analysis of Deterministic and Stochastic Petri Nets

Deterministic and stochastic Petri nets (DSPNs) are recognized as a useful modeling technique because of their capability to represent constant delays which appear in many practical systems. If at most one deterministic transition is allowed to be enabled in each marking, the state probabilities of a DSPN can be obtained analytically rather than by simulation. We show that the continuous time stochastic process underlying the DSPN with this condition is a Markov regenerative process and develop a method for computing the transient (time dependent) behavior. We also provide a steady state solution method using Markov regenerative process theory and show that it is consistent with the method of Ajmone Marsan and Chiola.

On modelling the performance and reliability of multimode computer systems

We present an effective technique for the combined performance and reliability analysis of multimode computer systems. A reward rate (or a performance level) is associated with each mode of operation. The switching between different modes is characterized by a continuoustime Markov chain. Different types of service-interruption interactions (as a result of mode switching) are considered. We consider the execution time of a given job on such a system and derive the distribution of its completion time. A useful dual relationship, between the completion time of a given job and the accumulated reward up to a given time, is noted. We demonstrate the use of our technique by means of a simple example.

Shortest paths in networks with exponentially distributed arc lengths

This paper develops methods for the exact computation of the distribution of the length of the shortest path from a given source node s to a given sink node t in a directed network in which the arc lengths are independent and exponentially distributed random variables. A continuous time Markov chain with a single absorbing state is constructed from the original network such that the time until absorption into this absorbing state starting from the initial state is equal to the length of the shortest path in the original network. It is shown that the state space of this Markov chain is the set of all minimal (s, t) cuts in the network and that its generator matrix is upper triangular. Algorithms are described for computing the distribution and moments of the length of the shortest path based on this Markov chain representation. Algorithms are also developed for computing the probability that a given (s, t) path is the shortest path in the network and for computing the conditional distribution of the length of a path given that it is the shortest (s, t) path in the network. All algorithms are numerically stable and are illustrated by several numerical examples.