Varsha Mainkar

Techniques and tools for reliability and performance evaluation: problems and perspectives

Modelling techniques and tools of the future must meet the challenges presented by todays highly demanding and schedule-oriented developing environment. With the emergence of high performance and reliability systems the problem of how to analyze such systems has become increasingly more difficult. Traditional assumptions of independent events, exponential distributions and other such “convenient” assumptions no longer model systems realistically. Nevertheless, the demand for answering performance and reliability related questions during the design process has increased. In this paper we discuss some of the issues involved in integrating modeling and design during a product development process. We present a broad range of existing techniques of systems analysis. We also describe a variety of tools that have been developed to make the analysis process simpler.

Reliability modeling of life-critical, real-time systems

We discuss the role of modeling in the design and validation of life-critical, real-time systems. The basics of Markov, Markov reward, and stochastic reward net models are covered. An example of a nuclear power plant cooling system is developed in detail. Multilevel models, model calibration, and model validation are also discussed. >

Numerical computation of response time distributions using stochastic reward nets

We consider the numerical computation of response time distributions for closed product form queueing networks using thetagged customer approach. We map this problem on to the computation of the time to absorption distribution of a finite-state continuous time Markov chain. The construction and solution of these Markov chains is carried out using a variation of stochastic Petri nets called stochastic reward nets (SRNs). We examine the effects of changing the service discipline and the service time distribution at a queueing center on the response time distribution. A multiserver queueing network example is also presented. While the tagged customer approach for computing the response time distribution is not new, this paper presents a new approach for computing the response time distributions using SRNs.

Approximate analysis of priority scheduling systems using stochastic reward nets

Presents a performance analysis of a heterogeneous multiprocessor system where tasks may arrive from Poisson sources as well as by spawning and probabilistic branching of other tasks. Non-preemptive priority scheduling is used between different tasks. Stochastic reward nets are used as the system model, and are solved analytically by generating the underlying continuous-time Markov chain. An approximation technique is used, that is based on fixed-point iteration to avoid the problem of a large underlying Markov chain. The iteration scheme works reasonably well, and the existence of a fixed point for the iterative scheme is guaranteed under certain conditions.<<ETX>>

Sensitivity analysis of Markov regenerative stochastic Petri nets

Sensitivity analysis, i.e., the analysis of the effect of small variations in system parameters on the output measures, can be studied by computing the derivatives of the output measures with respect to the parameter. An algorithm for parametric sensitivity analysis of Markov regenerative stochastic Petri nets (MRSPN) is presented. MRSPNs are a true generalization of stochastic Petri nets, in that they allow for transitions to have generally distributed firing times (under certain conditions). The expressions for the steady state probabilities of MRSPNs were developed by H. Choi et al. (1993). The authors extend the steady state analysis and present equations for sensitivity of the steady state probabilities with respect to an arbitrary system parameter. Sensitivity functions of the performance measures can accordingly be expressed in terms of the sensitivity functions of the steady state probabilities. The authors present an application of our algorithm by finding an optimizing parameter for a vacation queue.<<ETX>>

Transient Analysis of Real-Time Systems Using Deterministic and Stochastic Petri Nets

We present an analysis of a real-time system which has fixed-priority scheduling of aperiodic and periodic tasks. We assume that the cycle times of periodic tasks are multiples of each other, task execution times have phase-type distributions and aperiodic tasks arrive from Poisson sources. Since interarrival times of periodic tasks are constant, deterministic and stochastic Petri nets (DSPN) are an appropriate model. Using Markov regenerative theory to solve the DSPN model, the time-dependent behavior of the real-time system is studied. The steady-state behavior of such a system is shown to be periodic, i.e., there is no limiting probability distribution. Further, the response time distribution of an aperiodic task is also computed numerically. To our knowledge, this is the first time an analytical evaluation of a real-time system has been carried out at this level of detail.

Approximate Computation of Sojourn Time Distribution in Open Queueing Networks

The method of decomposition of queues has been widely used in solution of large and complex queueing networks for which exact solutions do not exist. We apply the basic paradigm of decomposition in computing approximations to the sojourn-time distribution in open queueing networks in which the service times and arrival processes are non-Markovian. For doing so we have made use of existing results on sojourn time distribution at a single queue. Using these, a queueing network is translated into a semi-Markov chain, whose absorption time distribution approximates the sojourn time distribution of the queueing network. However, the semi-Markov model does not represent the state of the queueing network (i.e., number of jobs at each queue). The state-space size of the semi-Markov models is thus linear in the number of queues in the network. This is achieved by having one state in the semi-Markov model corresponding to each queue in the queueing network, and one absorbing state to denote exit out of the network. The states are then connected together according to the topology of the network. The holding time distribution of a state is the sojourn time distribution at the corresponding queue. This sojourn time distribution must be computed by considering each queue in isolation. We approximate the arrival process to each queue to a phase-type arrival process, and then compute the sojourn time distribution assuming it is a PH/G/1 queue. Once we have the holding time distributions and the routing probability matrix, the absorption time distribution of the semi-Markov chain can be computed. The absorption time distribution approximates the sojourn time distribution of the queueing network.