Continuous-Time Modeling of Zipfian Workload Locality

Traditional workload analysis uses discrete times measured by data accesses, including the subarea of workloads with stochastic and independent accesses. A precise analysis in this flavor is the classic independent reference model (IRM) (King, 1971) where data are treated as independent random variables with stationary access probabilities. Despite the exponential blow-up of the exact solution to IRM (King, 1971), effective approximations have been developed, notably by Dan and Towsley (1990) and by Denning and Schwartz (1972). While these techniques were unrelated in the past, they are of the same species using different discrete steps. This paper presents Cell, a time-continuous model of locality for workloads with stochastic access patterns adhering to IRM. To draw connections between theory and applications, we evaluate our model on the challenging Zipfian workloads, which are ubiquitously found in web requests and caching systems. With trace simulation, we compare the modeling results for the Cell model, as well as for those presented by the formerly mentioned related work. Additionally, we present a partitioning strategy to effectively approximate the locality of a large Zipfian workload with a small number of partitions using the proposed Cell model, further reducing the time of locality analysis for the practical task.

Economists model time as continuous or discrete. For long, either alternative has brought about relevant economic issues, from the implementation of the basic Solow and Ramsey models of growth and the business cycle, towards the issue of equilibrium indeterminacy and endogenous cycles. In this paper, we introduce to some of those relevant issues in economic dynamics. First, we describe a baseline continuous vs discrete time modelling setting relevant for questions in growth and business cycle theory. Then we turn to the issue of local indeterminacy in a canonical model of economic growth with a pollution externality whose size is related to the model period. Finally, we propose a growth model with delays to show that a discrete time representation implicitly imposes a particular form of time–to–build to the continuous time representation. Our approach suggests that the recent literature on continuous time models with delays should help to bridge the gap between continuous and discrete time representations in economic dynamics.

An analysis of 15 years of hourly observations of cloud cover at an airport location near Copenhagen, Denmark shows that the main part of the variations can be described by first-order homogeneous Markov models. Models in both discrete and continuous time are considered. Special emphasis is laid upon the representation of the variation by a Markov model in continuous time. The physical restrictions for transitions of cloud cover are investigated and it is shown that the natural restrictions imply a very simple structure for the matrix of transition rates corresponding to the embedded Markov process in continuous time. The maximum likelihood estimate of the matrix of transition rates is found by numerical methods and an indication of the estimation error under the model is given.

In this paper we investigate the return and volatility spillovers among equity and bond markets in the UK, USA, Germany and Japan, using continuous time models and discrete time multivariate GARCH modelling methods. Using weekly data over the period 2001 to 2011, empirical evidence of uni- and/or bi-directional return and volatility spillovers is provided. The continuous time analysis finds evidence of feedback effects in some cases. The discrete time results provide weak evidence of return spillovers, while volatility transmission among the majority of equity and fixed income markets is verified. Evidence shows that some of these relationships change in the post-crisis period.

We consider a model of deterministic one‐time parameter change in a continuous time autoregressive model around a deterministic trend function. The exact discrete time analogue model is detailed and compared to corresponding parameter change models adopted in the discrete time literature. The relationships between the parameters in the continuous time model and the discrete time analogue model are also explored. Our results show that the discrete time models used in the literature can be justified by the corresponding continuous time model, with a only a minor modification needed for the (most likely) case where the changepoint does not coincide with one of the discrete time observation points. The implications of our results for a number of extant discrete time models and testing procedures are discussed.

We study the time path of inflation and unemployment using the Blanchard treatment of the relationship between the two and taking the monetary policy condition into account. We solve the model both in continuous and discrete time and compare the results. The economic dynamics of inflation and unemployment shows that they fluctuate around their intertemporal equilibria, inflation around the growth rate of nominal money supply, respectively, and unemployment around the natural rate of unemployment. However, while the continuous-time case shows uniform and smooth fluctuation for both economic variables, in discrete time their time path is explosive and nonoscillatory. The hysteresis case shows dynamic stability and convergence for inflation and unemployment to their intertemporal equilibria both in discrete and continuous time. When inflation affects unemployment adversely the time paths of the two, both in discrete and continuous time, are dynamically unstable.

As mentioned, the transition of a continuous time model into a discrete time model is not an easy issue. We discuss here various discretization procedures to turn continuous time into discrete time models. There are many methods to convert continuous time models into discrete time variants. The main discretization methods are the Euler method, the Milstein method and a new local linearization method. All those will be illustrated here to obtain discrete-time approximate models.

Solving a workhorse incomplete markets model in continuous time is many times faster compared to its discrete time counterpart. This paper dissects the computational discrepancies and identifies the key bottlenecks. The implicit finite difference method – a commonly used tool in continuous time – accounts for a large share of the difference. This method is shown to be a special case of Howard’s improvement algorithm, efficiently implemented by relying on sparse matrix operations. By representing the policy function with a transition matrix it is possible to formulate a similar procedure in discrete time, which effectively eliminates the differences in run-times entirely.

American option pricing with discrete and continuous time models: An empirical comparison

This paper considers discrete time GARCH and continuous time SV models and uses these for American option pricing. We perform a Monte Carlo study to examine their differences in terms of option pricing, and we study the convergence of the discrete time option prices to their implied continuous time values. Finally, a large scale empirical analysis using individual stock options and options on an index is performed comparing the estimated prices from discrete time models to the corresponding continuous time model prices. The results indicate that, while the differences in performance are small overall, for in the money options the continuous time SV models do generally perform better than the discrete time GARCH specifications.

Many important factors affect the spread of childhood infectious disease. To better understand the fundamental drivers of infectious disease spread, several researchers have estimated seasonal transmission coefficients in discrete-time models. In this paper, we build upon this previous work and also develop a framework for efficient estimation using continuous differential equation models. We introduce nonlinear programming formulations to efficiently estimate model parameters and seasonal transmission profiles from existing case count data for the childhood disease, measles. We compare results from discrete time and continuous time models and address several shortcomings of the discrete-time method, including removing the need for the data reporting interval to match the time between successive cases in the chain of transmission or serial interval of the disease. Using a simultaneous approach for optimization of differential equation systems, we demonstrate that seasonal transmission parameters can be effectively estimated using continuous time models instead of discrete-time models.

Verification methodologies for real-time systems can be classified according to whether they are based on a continuous time model or a discrete time model. Continuous time often provides a more accurate model of physical reality, while discrete time can be more efficient to implement in an automatic verifier based on state exploration techniques. Choosing a model appears to require a compromise between efficiency and accuracy. We avoid this compromise by constructing discrete time models that are conservative approximations of appropriate continuous time models. Thus, if a system is verified to be correct in discrete time, then it is guaranteed to also be correct in continuous time. We also show that models with explicit simultaneity can be conservatively approximated by models with interleaving semantics. Proving these results requires constructing several different domains of agent models. We have devised a new method for simplifying this task, based on abstract algebras we call trace algebra and trace structure algebra. A trace algebra has a set of traces as its carrier, along with operations of projection and renaming on traces. A trace can be any mathematical object that satisfies certain simple axioms, so the theory is quite general. A trace structure consists, in part, of a subset of the set of traces from some trace algebra. In a trace structures algebra, operations of parallel composition, projection and renaming are defined on trace structures, in terms of the operations on traces. General methods for constructing conservative approximations are described and are applied to several specific real-time models. We believe that trace algebra is a powerful tool for unifying many models of concurrency and abstraction beyond the particular ones described in this thesis. We also describe an automatic verifier based on the theory, and give examples of using it to verify speed-dependent asynchronous circuits. We analyze how several different delay models, including a new model called chaos delay, affect the verification results. The circuits and their specifications are represented in discrete time, but because of our conservative approximations, circuits that are verified correct are also correct in continuous time.

We present new finite dimensional filters for estimating the state of Markov jump linear systems, given noisy measurements of the Markov chain. Discrete time as well as continuous time models are considered. A robust version of the continuous time filters is used to derive a discretization which links the continuous and discrete time results. Simulations compare the robust discretization with direct numerical solutions of the filtering equations. The new filters have applications in the passive tracking of maneuvering targets and speech coding.

Quadratic term structure models in discrete time

We demonstrate the important implications of the assumptions of discrete time in many sticky price models of the macroeconomy. For a given level of menu costs, discrete time models imply longer average contract length but smaller real effects of both trend inflation and monetary shocks than continuous time models. It is also feasible for a firm to enjoy full price flexibility in discrete time, while this would require paying infinite menu costs in continuous time, a distinction that is most important at high levels of trend inflation. Copyright © 2007 John Wiley & Sons, Ltd.