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Queueing Systems - Vol. 1: Theory
01 Jan 2013-
About: The article was published on 2013-01-01 and is currently open access. It has received 540 citations till now. The article focuses on the topics: Layered queueing network & Bulk queue.
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IBM1
TL;DR: A procedure based on Schruben's Brownian bridge model for the detection of nonstationarity and a spectral method for estimating the variance of the sample mean are explored for estimation of the steady state mean of an output sequence from a discrete event simulation.
Abstract: This paper studies the estimation of the steady state mean of an output sequence from a discrete event simulation. It considers the problem of the automatic generation of a confidence interval of prespecified width when there is an initial transient present. It explores a procedure based on Schruben's Brownian bridge model for the detection of nonstationarity and a spectral method for estimating the variance of the sample mean. The procedure is evaluated empirically for a variety of output sequences. The performance measures considered are bias, confidence interval coverage, mean confidence interval width, mean run length, and mean amount of deleted data. If the output sequence contains a strong transient, then inclusion of a test for stationarity in the run length control procedure results in point estimates with lower bias, narrower confidence intervals, and shorter run lengths than when no check for stationarity is performed. If the output sequence contains no initial transient, then the performance measures of the procedure with a stationarity test are only slightly degraded from those of the procedure without such a test. If the run length is short relative to the extent of the initial transient, the stationarity tests may not be powerful enough to detect the transient, resulting in a procedure with unreliable point and interval estimates.
1,237 citations
TL;DR: This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions and describes two methods for inverting Laplace transform based on the Post-Widder inversion formula.
Abstract: This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this purpose, we also describe two methods for inverting Laplace transforms based on the Post-Widder inversion formula. The overall procedure is illustrated by several queueing examples.
726 citations
TL;DR: The model is motivated by applications in which the objective is to minimize the wait for service in a stochastic and dynamically changing environment, a departure from classical vehicle routing problems where one seeks to minimize total travel time in a static, deterministic environment.
Abstract: We propose and analyze a generic mathematical model for dynamic, stochastic vehicle routing problems, the dynamic traveling repairman problem (DTRP). The model is motivated by applications in which the objective is to minimize the wait for service in a stochastic and dynamically changing environment. This is a departure from classical vehicle routing problems where one seeks to minimize total travel time in a static, deterministic environment. Potential areas of application include repair, inventory, emergency service and scheduling problems. The DTRP is defined as follows: Demands for service arrive in time according to a Poisson process, are independent and uniformly distributed in a Euclidean service region, and require an independent and identically distributed amount of on-site service by a vehicle. The problem is to find a policy for routing the service vehicle that minimizes the average time demands spent in the system. We propose and analyze several policies for the DTRP. We find a provably optima...
447 citations
TL;DR: It is shown that a first come, first served (FCFS) policy that schedules the video with the longest outstanding request can perform better than the maximum queue length (MQL) policy, and multicasting is better exploited by scheduling playback of the most popular videos at predetermined, regular intervals (hence, termed FCFS-n).
Abstract: In a video-on-demand environment, continuous delivery of video streams to the clients is guaranteed by sufficient reserved network and server resources. This leads to a hard limit on the number of streams that a video server can deliver. Multiple client requests for the same video can be served with a single disk I/O stream by sending (multicasting) the same data blocks to multiple clients (with the multicast facility, if present in the system). This is achieved by batching (grouping) requests for the same video that arrive within a short time. We explore the role of customerwaiting time and reneging behavior in selecting the video to be multicast. We show that a first come, first served (FCFS) policy that schedules the video with the longest outstanding request can perform better than the maximum queue length (MQL) policy that chooses the video with the maximum number of outstanding requests. Additionally, multicasting is better exploited by scheduling playback of the n most popular videos at predetermined, regular intervals (hence, termed FCFS-n). If user reneging can be reduced by guaranteeing that a maximum waiting time will not be exceeded, then performance of FCFS-n is further improved by selecting the regular playback intervals as this maximum waiting time. For an empirical workload, we demonstrate a substantial reduction (of the order of 60%) in the required server capacity by batching.
409 citations
01 May 2007
TL;DR: A simple stochastic fluid model is developed that accounts for many of the essential features of a P2P streaming system, including the peers' realtime demand for content, peer churn, peers with heterogeneous upload capacity, limited infrastructure capacity, and peer buffering and playback delay.
Abstract: We develop a simple stochastic fluid model that seeks to expose the fundamental characteristics and limitations of P2P streaming systems. This model accounts for many of the essential features of a P2P streaming system, including the peers' realtime demand for content, peer churn (peers joining and leaving), peers with heterogeneous upload capacity, limited infrastructure capacity, and peer buffering and playback delay. The model is tractable, providing closed-form expressions which can be used to shed insight on the fundamental behavior of P2P streaming systems. The model shows that performance is largely determined by a critical value. When the system is of moderate-to-large size, if a certain ratio of traffic loads exceeds the critical value, the system performs well; otherwise, the system performs poorly. Furthermore, large systems have better performance than small systems since they are more resilient to bandwidth fluctuations caused by peer churn. Finally, buffering can dramatically improve performance in the critical region, for both small and large systems. In particular, buffering can bring more improvement than can additional infrastructure bandwidth.
365 citations
Cites background from "Queueing Systems - Vol. 1: Theory"
...Let Pi(t) be the number of class-i peers in the system at time t. Clearly, P1(t) and P2(t) are two independent M/G/∞ processes [ 10 ]....
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