Analysis of Redundancy(d) with Identical Replicas

Departure process of a single server queueing system with Markov renewal input and general service time distribution

The Stroop interference effect—slower responses to the color of an incongruent color-word stimulus (e.g., GREEN in red ink) compared to a neutral stimulus (e.g., XXXX in red ink)—is one of the most robust findings in cognitive psychology. This thesis investigated why the effect is so robust, focusing on the unique perceptual architecture of the Stroop stimulus. Specifically, in a Stroop stimulus, the visual word form and color are integrated into a visual object, and hence attending to the color inevitably results in the word receiving attention as well. Across four empirical chapters, I demonstrated that presenting the distractor word in advance of a Stroop stimulus (a manipulation termed <i>distractor priming</i>—e.g., presenting the word GREEN in white ink followed by the same word GREEN in red ink) significantly reduced—and in some cases eliminated—the Stroop interference effect. I argued that this occurred because the distractor prime broke the integration of the visual word form and color in the Stroop stimulus, allowing rapid disengagement of attention from the word. Crucially, I examined response time (RT) distributions. In standard Stroop stimuli, the interference effect increased with slower responses, yielding a positively sloped delta plot. When the integration was disrupted by distractor priming, the delta slope flattened—that is, the interference effect remained constant across the RT distribution. Chapter 2 introduced the distractor priming paradigm. Consistent with prior studies of distractor priming that used oral color-naming responses, distractor priming in a manual Stroop task sped up responses to the color. Importantly, it eliminated the Stroop interference effect across the entire RT distribution (i.e., it removed the positively sloped delta). Chapter 3 demonstrated that this reduction in Stroop interference and removal of the positively sloped delta occurred only with a prime visually identical to the distractor. Stroop interference remained large, and the positively sloped delta persisted when the prime was conceptually related but not visually identical to the distractor word (e.g., LEAF - GREENred) or an auditory word (e.g., hearing “green”). Chapter 4 explored the relative “automaticity” of the distractor priming benefit and demonstrated that even when the word precue was only 50% predictive of the upcoming distractor word (unlike 100% as in the previous studies), Stroop interference was reduced and the positively sloped delta removed. Additionally, this experiment showed that the delta slope pattern of the Stroop interference effect found in the distractor-primed (identity precue) condition differed from that in the response-precue condition (e.g., the word RED followed by the word GREEN in red ink). Chapter 5 tested the assumption underlying the distractor priming benefit—that the positively sloped delta was removed when the integration between the word and color was disrupted—using the primed Stroop task. This paradigm differed from the standard Stroop task in that the word and color were not integrated into a single visual object; instead, the distractor word was presented ahead of a row of colored Xs. Across two experiments, RT distributions in the primed Stroop task mirrored those in distractor priming, in that Stroop interference was small and the positively sloped delta was eliminated. Together, these findings indicate that the robustness of the Stroop interference effect stems from the perceptual integration of the distractor word and color into a single visual object. Chapter 6 provides a discussion of the implication of our interpretation of Stroop interference on how the Stroop task is used as a tool to study selective attention.

The ISDN has been developed to provide a variety of communication services. As a next step, the broadband ISDN with the asynchronous transfer mode is expected to contribute to the social infrastructure in the 21st century. For the optimum traffic and network designs of the ISDN and the BISDN, it is necessary to evaluate the GI/G/s general traffic (queuing) systems. The notation GI/G/s means s homogeneous parallel switching servers with the general independently distributed interarrival times and the general service time distribution. The exact analytical solution for the GI/G/s model has not yet been obtained. Until now, in order to evaluate the performance indexes of the GI/G/s traffic systems, we have presented the artificial neural method. The method uses a simple three layer feedfoward neural network trained with the backpropagation training algorithm. As a result of the training, the connective weight of the neural network accumulate a lot of information and knowledge relative to the behavior of the GI/G/s system. In this paper, the connective weight is analyzed and interpreted to find important traffic rules of the GI/G/s system. The discovery method is expected to contribute to the designs of 21st networks.

In this paper, the waiting time and response time processes in continuous-time two parallel queueing systems are analyzed as well as the system size process in discrete-time two parallel queueing systems. Upon arrival, each job splits into two tasks, denoted by task 1 and task 2. Task k, $k = 1,2$, is assigned to the kth queueing system to be served on a FCFS (first-come-first-served) basis. Each queueing system consists of a single server and an infinite-capacity queue. For the continuous-time queueing system, we assume that jobs arrive in a state-independent Poisson fashion with rate $\lambda $; the servers have exponential service times with rates $\mu _1 $ and $\mu _2 $, respectively. A closed-form expression for the Laplace transform of the ergodic bivariate waiting time distribution is derived. The distribution of the sum of the two waiting times is then obtained. The distribution of the response time when $\mu _1 = \mu _2 $ is derived analytically. For the discrete-time queueing systems, it is assu...

In this paper, we present a bounding methodology that allows to compute a tight bound on the throughput of fork-join queueing networks with blocking and with general service time distributions. No exact models exist for queueing networks with general service time distributions and, consequently, bounds are the only certain information available. The methodology relies on two ideas. First, probability mass fitting (PMF) discretizes the service time distributions so that the evolution of the modified system can be modelled by a discrete Markov chain. Second, we show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a bound on the original throughput. The tightness of the bound is shown on computational experiments (error on the order of one percent). Finally, we discuss the extension to split-and-merge networks and the approximate estimations of the throughput.

A diffusion approximation for batch queues of type [formula omitted

In this paper we consider a multi-server queue with a near general arrival process (represented as an arbitrary state-dependent Coxian distribution), a near general state-dependent Coxian service time distribution and a possibly finite queueing room. In addition to the dependence on the current number of customers in the system, the rate of arrivals and the progress of the service may depend on each other. We propose a semi-numerical method based on the use of conditional probabilities to compute the steady-state queue length distribution in such a queueing system. Our approach is conceptually simple, easy to implement and can be applied to both infinite and finite C m /C k /c-like queues. The proposed method uses a simple fixed-point iteration. In the case of infinite queues, it avoids the need for arbitrary truncation through the use of asymptotic conditional probabilities.This preliminary study examines the computational behavior of the proposed method with a Cox-2 service distribution. Our results indicate that it is robust and performs well even when the number of servers and the coefficient of variation of the service times are relatively high. The number of iterations to attain convergence varies from low tens to several thousand. For example, we are able to solve queues with 1024 servers and the coefficients of variation of the service time and of the time between arrivals set to 4 within 1100 iterations.KeywordsMulti-server queuegeneral arrivalsgeneral service timessteady-state queue length distributionsimple efficient semi-numerical solution

Task replication has recently been advocated as a practical solution to reduce latencies in parallel systems. In addition to several convincing empirical studies, analytical results have been provided, yet under some strong assumptions such as independent service times of the replicas, which may lend themselves to some contrasting and perhaps contriving behavior. For instance, under the independence assumption, an overloaded system can be stabilized by a replication factor, but can be sent back in overload through further replication. Motivated by the need to dispense with such common and restricting assumptions, which may cause unexpected behavior, we develop a unified and general theoretical framework to compute tight bounds on the distribution of response times in general replication systems. These results immediately lend themselves to the optimal number of replicas minimizing response time quantiles, depending on the parameters of the system (e.g., the degree of correlation amongst replicas).

In this paper we consider an open queueing network having multiple classes, priorities, and general service time distributions. In the case where there is a single bottleneck station we conjecture that normalized queue length and sojourn time processes converge, in the heavy traffic limit, to one-dimensional reflected Brownian motion, and present expressions for its drift and variance. The conjecture is motivated by known heavy traffic limit theorems for some special cases of the general model, and some conjectured “Heavy Traffic Principles” derived from them. Using the known stationary distribution of one-dimensional reflected Brownian motion, we present expressions for the heavy traffic limit of stationary queue length and sojourn time distributions and moments. For systems with Markov routing we are able to explicitly calculate the limits.

Many stochastic systems face a time-dependent demand. Especially in stochastic service systems, for example, in call centers, customers may leave the queue if their waiting time exceeds their personal patience. As discussed in the extant literature, it can be useful to use general distributions to model such customer patience. This paper analyzes the time-dependent performance of a multiserver queue with a nonhomogeneous Poisson arrival process with a time-dependent arrival rate, exponentially distributed processing times, and generally distributed time to abandon. Fast and accurate performance approximations are essential for decision support in such queueing systems, but the extant literature lacks appropriate methods for the setting we consider. To approximate time-dependent performance measures for small- and medium-sized systems, we develop a new stationary backlog-carryover (SBC) approach that allows for the analysis of underloaded and overloaded systems. Abandonments are considered in two steps of the algorithm: (i) in the approximation of the utilization as a reduced arrival stream and (ii) in the approximation of waiting-based performance measures with a stationary model for general abandonments. To improve the approximation quality, we discuss an adjustment to the interval lengths. We present a limit result that indicates convergence of the method for stationary parameters. The numerical study compares the approximation quality of different adjustments to the interval length. The new SBC approach is effective for instances with small numbers of time-dependent servers and gamma-distributed abandonment times with different coefficients of variation and for an empirical distribution of the abandonment times from real-world data obtained from a call center. A discrete-event simulation benchmark confirms that the SBC algorithm approximates the performance of the queueing system with abandonments very well for different parameter configurations. Summary of Contribution: The paper presents a fast and accurate numerical method to approximate the performance measures of a time‐dependent queueing system with generally distributed abandonments. The presented stationary backlog carryover approach with abandonment combines algorithmic ideas with stationary queueing models for generally distributed abandonment times. The reliability of the method is analyzed for transient systems and numerically studied with real‐world data.

Single server queueing models are important in the study of a wide variety of stochastic systems. A particularly important example is the study of task schedules for computer systems with real-time applications. In this paper, we present a class of approximations for the waiting time distribution in single server queueing systems with general independent (renewal) input and general (independent) service time distributions. These approximations allow the analyst to use as much (or as little) of the structure of the input and service processes as desired. Moreover, they also allow him to concentrate on specific quantities associated with the delay distribution that may represent the most relevant performance criteria. Examples include probability of delay, mean delay, and tails of the delay distribution. The methods given in this paper have been used to analyze the performance of task schedules in a variety of processor-based systems.

A new proof of the stability of closed Jackson-type queueing networks (with general service-time distributions) is given and sufficient conditions are given for obtaining Cesaro, weak and total variation convergence of the continuous-time joint queue length and residual service-time process to a limiting distribution. The result weakens the sufficient conditions (for stability) of Borovkov (1986) by allowing more general service-time distributions.

We introduce and analyze a deterministic fluid model that serves as an approximation for the Gt/GI/st + GI manyserver queueing model, which has a general time-varying arrival process (the Gt), a general service-time distribution (the first GI), a time-dependent number of servers (the st) and allows abandonment from queue according to a general abandonment-time distribution (the +GI). This fluid model approximates the associated queueing system when the arrival rate and number of servers are both large. We also show that the system dynamics greatly simplifies in two special cases: (i) when the service time distribution is exponential (M) and (ii) when the service time distribution is deterministic (D) and the model is stationary. We develop an efficient algorithm to compute all standard performance functions in both cases. In case (i), we establish an asymptotic loss of memory (ALOM) property, i.e., asymptotic independence from the initial conditions as time evolves. We show that the difference in the performance functions with different initial conditions dissipates over time exponentially fast, under regularity conditions. In contrast, in case (ii) we show that ALOM fails dramatically. Instead, although all model parameters are constants, we show that the performance rapidly approaches a periodic steady state (PSS) with a period equal to the service time, whenever the system does not start with the unique stationary distribution. Moreover, the form of the PSS depends on the initial condition. Simulation and a heavy-traffic limit confirm that this anomalous behavior also occurs in the large-scale queueing model.

We consider a single-server infinite-capacity queueing sysem with Poisson arrivals of customer groups of random size and a general service time distribution, the server of which applies a general exhaustive service vacation policy. We are concerned with the steady-state distribution of the actual waiting time of a customer arriving while the server is active.

A service facility with multiple customer classes is considered. Each class forms an independent Poisson arrival process with admission controlled by a threshold parameter and is characterized by a general service time distribution. Gradient estimators are derived for performance measures of the system (mean system time, throughput, blocking probabilities) using perturbation analysis techniques. The approach is based on exploiting alternative sample path representations of generalized semi-Markov process models of the system, such that the resulting sample functions are continuous with respect to parameters of interest. The unbiasedness of the estimators is proved. Simulation examples are included to illustrate their properties. >