Approximate performability and dependability analysis using generalized stochastic Petri Nets

Since current day fault-tolerant and distributed computer and communication systems tend to be large and complex, their corresponding performability models will suffer from the same characteristics. Therefore, calculating performability measures from these models is a difficult and time-consuming task.\ud \ud To alleviate the largeness and complexity problem to some extent we use generalized stochastic Petri nets to describe to models and to automatically generate the underlying Markov reward models. Still however, many models cannot be solved with the current numerical techniques, although they are conveniently and often compactly described.\ud \ud In this paper we discuss two heuristic state space truncation techniques that allow us to obtain very good approximations for the steady-state performability while only assessing a few percent of the states of the untruncated model. For a class of reversible models we derive explicit lower and upper bounds on the exact steady-state performability. For a much wider class of models a truncation theorem exists that allows one to obtain bounds for the error made in the truncation. We discuss this theorem in the context of approximate performability models and comment on its applicability. For all the proposed truncation techniques we present examples showing their usefulness

Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results

Our model is a constrained homogeneous random walk in a nonnegative orthant Z_+^d. The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed by Meyn and Tweedie. In this paper we show that, for this type of random walks, computing the stationary probability exactly is an undecidable problem: no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this model is also an undecidable problem. We extend these results to a certain type of queueing systems. The implication of these results is that no useful formulas for computing stationary probabilities and large deviations rates can exist in these systems

Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansions

In this paper, we develop a numerical approach based on Chaos expansions to analyze the sensitivity and the propagation of epistemic uncertainty through a queueing systems with breakdowns. Here, the quantity of interest is the stationary distribution of the model, which is a function of uncertain parameters. Polynomial chaos provide an efficient alternative to more traditional Monte Carlo simulations for modelling the propagation of uncertainty arising from those parameters. Furthermore, Polynomial chaos expansion affords a natural framework for computing Sobol' indices. Such indices give reliable information on the relative importance of each uncertain entry parameters. Numerical results show the benefit of using Polynomial Chaos over standard Monte-Carlo simulations, when considering statistical moments and Sobol' indices as output quantities

Analysis of a class of distributed queues with application

Recently we have developed a class of media access control algorithms for different types of Local Area Networks. A common feature of these LAN algorithms is that they represent various strategies by which the processors in the LAN can simulate the availability of a centralized packet transport facility, but whose service incorporates a particular type of change over time known as 'moving sever' overhead. First we describe the operation of moving server systems in general, for both First-Come - First-Served and Head-of-the-Line orders of service, together with an approach for their delay analysis in which we transform the moving server queueing system into a conventional queueing system having proportional waiting times. Then we describe how the various LAN algorithms may be obtained from the ideal moving server system, and how a significant component of their performance characteristics is determined by the performance characteristics of that ideal system. Finally, we evaluate the compatibility of such LAN algorithms with separable queueing network models of distributed systems by computing the interdeparture time distribution for M/M/1 in the presence of moving server overhead. Although it is not exponential, except in the limits of low server utilization or low overhead, the interdeparture time distribution is a weighted sum of exponential terms with a coefficient of variation not much smaller than unity. Thus, we conjecture that a service centre with moving server overhead could be used to represent one of these LAN algorithms in a product form queueing network model of a distributed system without introducing significant approximation errors

The MVA Priority Approximation

A Mean Value Analysis (MVA) approximation is presented for computing the average performance measures of closed-, open-, and mixed-type multiclass queuing networks containing Preemptive Resume (PR) and nonpreemptive Head-Of-Line (HOL) priority service centers. The approximation has essentially the same storage and computational requirements as MVA, thus allowing computationally efficient solutions of large priority queuing networks. The accuracy of the MVA approximation is systematically investigated and presented. It is shown that the approximation can compute the average performance measures of priority networks to within an accuracy of 5 percent for a large range of network parameter values. Accuracy of the method is shown to be superior to that of Sevcik's shadow approximation

Generalized gap acceptance models for unsignalized intersections

This paper contributes to the modeling and analysis of unsignalized intersections. In classical gap acceptance models vehicles on the minor road accept any gap greater than the CRITICAL gap, and reject gaps below this threshold, where the gap is the time between two subsequent vehicles on the major road. The main contribution of this paper is to develop a series of generalizations of existing models, thus increasing the model's practical applicability significantly. First, we incorporate {driver impatience behavior} while allowing for a realistic merging behavior; we do so by distinguishing between the critical gap and the merging time, thus allowing MULTIPLE vehicles to use a sufficiently large gap. Incorporating this feature is particularly challenging in models with driver impatience. Secondly, we allow for multiple classes of gap acceptance behavior, enabling us to distinguish between different driver types and/or different vehicle types. Thirdly, we use the novel M$^X$/SM2/1 queueing model, which has batch arrivals, dependent service times, and a different service-time distribution for vehicles arriving in an empty queue on the minor road (where `service time' refers to the time required to find a sufficiently large gap). This setup facilitates the analysis of the service-time distribution of an arbitrary vehicle on the minor road and of the queue length on the minor road. In particular, we can compute the MEAN service time, thus enabling the evaluation of the capacity for the minor road vehicles

A Large Closed Queueing Network Containing Two Types of Node and Multiple Customer Classes: One Bottleneck Station

The paper studies a closed queueing network containing two types of node. The first type (server station) is an infinite server queueing system, and the second type (client station) is a single server queueing system with autonomous service, i.e. every client station serves customers (units) only at random instants generated by strictly stationary and ergodic sequence of random variables. It is assumed that there are $r$ server stations. At the initial time moment all units are distributed in the server stations, and the $i$th server station contains $N_i$ units, $i=1,2,...,r$, where all the values $N_i$ are large numbers of the same order. The total number of client stations is equal to $k$. The expected times between departures in the client stations are small values of the order $O(N^{-1})$ ~ $(N=N_1+N_2+...+N_r)$. After service completion in the $i$th server station a unit is transmitted to the $j$th client station with probability $p_{i,j}$ ~ ($j=1,2,...,k$), and being served in the $j$th client station the unit returns to the $i$th server station. Under the assumption that only one of the client stations is a bottleneck node, i.e. the expected number of arrivals per time unit to the node is greater than the expected number of departures from that node, the paper derives the representation for non-stationary queue-length distributions in non-bottleneck client stations.Comment: 39 pages, 5 figure

A Stochastic Resource-Sharing Network for Electric Vehicle Charging

We consider a distribution grid used to charge electric vehicles such that voltage drops stay bounded. We model this as a class of resource-sharing networks, known as bandwidth-sharing networks in the communication network literature. We focus on resource-sharing networks that are driven by a class of greedy control rules that can be implemented in a decentralized fashion. For a large number of such control rules, we can characterize the performance of the system by a fluid approximation. This leads to a set of dynamic equations that take into account the stochastic behavior of EVs. We show that the invariant point of these equations is unique and can be computed by solving a specific ACOPF problem, which admits an exact convex relaxation. We illustrate our findings with a case study using the SCE 47-bus network and several special cases that allow for explicit computations.Comment: 13 pages, 8 figure