scholarly journals MAD Dispersion Measure Makes Extremal Queue Analysis Simple

2022 ◽  
Author(s):  
Wouter van Eekelen ◽  
Dick den Hertog ◽  
Johan S.H. van Leeuwaarden
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Queueing Theory ◽  
Upper Bounds ◽  
Dispersion Measure ◽  
Mean Absolute Deviation ◽  
Absolute Deviation ◽  
Worst Case ◽  
Expected Waiting Time ◽  
Computational Intractability

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival- and service-time distributions. We address this extremal queue problem by measuring dispersion in terms of mean absolute deviation (MAD) instead of the more conventional variance, making available methods for distribution-free analysis. Combined with random walk theory, we obtain explicit expressions for the extremal interarrival- and service-time distributions and, hence, the best possible upper bounds for all moments of the waiting time. We also obtain tight lower bounds that, together with the upper bounds, provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp also when the mean and MAD are not known precisely but are estimated based on available data instead. Summary of Contribution: Queueing theory is a classic OR topic with a central role for the GI/G/1 queue. Although this queueing system is conceptually simple, it is notoriously hard to determine the worst-case expected waiting time when only knowing the first two moments of the interarrival- and service-time distributions. In this setting, the exact form of the extremal distribution can only be determined numerically as the solution to a nonconvex nonlinear optimization problem. Our paper demonstrates that using mean absolute deviation (MAD) instead of variance alleviates the computational intractability of the extremal GI/G/1 queue problem, enabling us to state the worst-case distributions explicitly.

Simple approximations to the expected waiting time for a cluster of any given size, for point processes

1983 ◽  
Vol 15 (01) ◽  
pp. 21-38 ◽  
Author(s):  
Ester Samuel-Cahn
Keyword(s):  
Waiting Time ◽  
Point Processes ◽  
Upper Bounds ◽  
Poisson Processes ◽  
Lower And Upper Bounds ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
Interarrival Times ◽  
Expected Waiting Time ◽  
Simple Points

For point processes, such that the interarrival times of points are independently and identically distributed, let T(L, m) denote the time until at least points cluster within an interval of length at most L. Let τ (L, m) + 1 be the total number of points observed until the above happens. Simple approximations of Eτ (L, m) and ET(L, m) are derived, as well as lower and upper bounds for their value. Approximations to the variances are also given. In particular the Poisson, Bernoulli and compound Poisson processes are discussed in detail. Some numerical tables are included.

scholarly journals PROPOSTA DI UNA FUNZIONE UTILITÀ DIPENDENTE DALL’UTILITÀ ATTESA E DALLA DISPERSIONE DELLE UTILITÀ

2014 ◽  
Author(s):  
Vittorio B. Frosini
Keyword(s):  
Standard Deviation ◽  
Expected Utility ◽  
Dispersion Measure ◽  
Risk Averse ◽  
Mean Absolute Deviation ◽  
Summary Statistic ◽  
Absolute Deviation ◽  
The Mean ◽  
Condition C ◽  
Substitution Axiom

The author develops the properties and implications of a proposal, concerning a summary statistic of the random prospect of utilities. Following a suggestion of Maurice Allais, such a statistic is increasing with expected utility, and decreasing – for most people, who are risk averse – with the mean absolute deviation of utilities; a parameter multiplying this dispersion measure allows for risk averse or risk prone behaviour, according to its sign, and also for more or less departure from a certain prospect. It is demonstrated that this statistic (a) satisfies the first stochastic dominance, (b) satisfies the independence condition, (c) satisfies the so called “problem of probabilistic insurance”, (d) resolves the paradoxes of Allais, Ellsberg and Kahneman-Tversky (paradox of the substitution axiom), (e) the mean absolute deviation from the mean cannot be replaced by the standard deviation.

Clustering of events in a stochastic process

1981 ◽  
Vol 18 (1) ◽  
pp. 268-275 ◽  
Author(s):  
Joseph Glaz
Keyword(s):  
Stochastic Process ◽  
Waiting Time ◽  
Service Time ◽  
Exact Distribution ◽  
Fixed Interval ◽  
Server Queue ◽  
Clustering Of Events ◽  
Poisson Arrivals ◽  
Expected Waiting Time ◽  
Expected Duration

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.

scholarly journals SISTEM PENDUKUNG KEPUTUSAN PREDIKSI JUMLAH PENUMPANG UNTUK EVALUASI KAPASITAS HALTE BUS TRANS JOGJA DENGAN METODE EXPONENTIAL SMOOTHING DAN LEAST SQUARE

Compiler ◽  
2013 ◽  
Vol 2 (1) ◽  
Author(s):  
Dwi Prasetiyo ◽  
Anton Setiawan Honggowibowo ◽  
Yuliani Indrianingsih
Keyword(s):  
Time Series ◽  
Waiting Time ◽  
Exponential Smoothing ◽  
Small Error ◽  
Least Square ◽  
Mean Absolute Deviation ◽  
Absolute Deviation ◽  
Bus Stop ◽  
Trend Forecasting ◽  
Bus Stops

The increasing number o f passengers Trans Jogja bus stops can result in the existing capacity can not accommodate the number of passengers comfortably. Problems that often arise include delays resulting bus passenger waiting time is longer and there is a buildup of the number of passengers at stops. As a result of these problems, the capacity o f passenger stops can be full so that prospective passengers waiting outside the bus stop. Forecasting is one very important element in the decision. In this study using stationary and trend forecasting the data because the data are not significant changes between time and swell in certain periods and a normal in periods others. Time series methods for forecasting the number o f passengers on the Trans Jogja stop using exponential smoothing calculation and least square. From these calculations the value sought MAD (Mean Absolute Deviation) or least square error is exponential smoothing and forecasting results with small error. Forecasting will be better if it contains fewer possible errors.

Clustering of events in a stochastic process

1981 ◽  
Vol 18 (01) ◽  
pp. 268-275 ◽  
Author(s):  
Joseph Glaz
Keyword(s):  
Stochastic Process ◽  
Waiting Time ◽  
Service Time ◽  
Exact Distribution ◽  
Fixed Interval ◽  
Server Queue ◽  
Clustering Of Events ◽  
Poisson Arrivals ◽  
Expected Waiting Time ◽  
Expected Duration

In this paper we derive bounds for the expected waiting time of clustering of at least n events of a stochastic process within a fixed interval of length p. Using this approach of clustering, we derive bounds for the expected duration of the period of time that at least n servers are busy in an ∞-server queue with constant service time. For the case of Poisson arrivals we derive the exact distribution of the duration of that period.

Simple approximations to the expected waiting time for a cluster of any given size, for point processes

1983 ◽  
Vol 15 (1) ◽  
pp. 21-38 ◽  
Author(s):  
Ester Samuel-Cahn
Keyword(s):  
Waiting Time ◽  
Point Processes ◽  
Upper Bounds ◽  
Poisson Processes ◽  
Lower And Upper Bounds ◽  
Compound Poisson ◽  
Compound Poisson Processes ◽  
Interarrival Times ◽  
Expected Waiting Time

For point processes, such that the interarrival times of points are independently and identically distributed, let T(L, m) denote the time until at least points cluster within an interval of length at most L. Let τ (L, m) + 1 be the total number of points observed until the above happens. Simple approximations of Eτ (L, m) and ET(L, m) are derived, as well as lower and upper bounds for their value. Approximations to the variances are also given. In particular the Poisson, Bernoulli and compound Poisson processes are discussed in detail. Some numerical tables are included.

Scheduling Methods for Disk Requests

2011 ◽  
pp. 212-223
Author(s):  
Phillip K.C. Tse
Keyword(s):  
Response Time ◽  
Waiting Time ◽  
Queueing Theory ◽  
Access Time ◽  
Disk Scheduling ◽  
Scheduling Policy ◽  
Incoming Order ◽  
Scheduling Method ◽  
Expected Waiting Time ◽  
First In First Out

Disk scheduling changes the sequence order to serve the requests that are waiting in the queue. While data placement reduces the access time of a disk request, scheduling reduces the waiting time of a request. Thus, the response time is found as: Response time = Waiting time + Access time The longer the waiting queue, the more useful is the scheduling method. When there is no waiting queue, any scheduling methods perform the same. Expected waiting time and queue length can be found using queueing theory. The queueing theory is out of the scope of this book. In general, a disk scheduling policy changes the service order of waiting requests. A disk scheduling policy accepts the waiting requests and serves them in the new service sequence. Notice that the service sequence may or may not be the same as the incoming order of the waiting requests. In this chapter, we shall describe two common disk scheduling methods. First, we shall describe the simple first-in-first-out method. After that, we shall describe the efficient SCAN algorithm in the following sections.

Minimally Empirical Double Hybrid Functionals Trained Against the GMTKN55 Database: revDSD-PBEP86-D4, revDOD-PBE-D4, and DOD-SCAN-D4

2019 ◽  
Author(s):  
Golokesh Santra ◽  
Nitai Sylvetsky ◽  
Gershom Martin
Keyword(s):  
Substantial Improvement ◽  
Viable Alternative ◽  
Mean Absolute Deviation ◽  
Dispersion Correction ◽  
Training Set ◽  
Weighted Mean ◽  
Absolute Deviation ◽  
Hybrid Functionals ◽  
Scaling Algorithms

We present a family of minimally empirical double-hybrid DFT functionals parametrized against the very large and diverse GMTKN55 benchmark. The very recently proposed wB97M(2) empirical double hybrid (with 16 empirical parameters) has the lowest WTMAD2 (weighted mean absolute deviation over GMTKN55) ever reported at 2.19 kcal/mol. However, our xrevDSD-PBEP86-D4 functional reaches a statistically equivalent WTMAD2=2.22 kcal/mol, using just a handful of empirical parameters, and the xrevDOD-PBEP86-D4 functional reaches 2.25 kcal/mol with just opposite-spin MP2 correlation, making it amenable to reduced-scaling algorithms. In general, the D4 empirical dispersion correction is clearly superior to D3BJ. If one eschews dispersion corrections of any kind, noDispSD-SCAN offers a viable alternative. Parametrization over the entire GMTKN55 dataset yields substantial improvement over the small training set previously employed in the DSD papers.

Minimally Empirical Double Hybrid Functionals Trained Against the GMTKN55 Database: revDSD-PBEP86-D4, revDOD-PBE-D4, and DOD-SCAN-D4

2019 ◽  
Author(s):  
Golokesh Santra ◽  
Nitai Sylvetsky ◽  
Gershom Martin
Keyword(s):  
Substantial Improvement ◽  
Viable Alternative ◽  
Mean Absolute Deviation ◽  
Dispersion Correction ◽  
Training Set ◽  
Weighted Mean ◽  
Absolute Deviation ◽  
Hybrid Functionals ◽  
Scaling Algorithms

We present a family of minimally empirical double-hybrid DFT functionals parametrized against the very large and diverse GMTKN55 benchmark. The very recently proposed wB97M(2) empirical double hybrid (with 16 empirical parameters) has the lowest WTMAD2 (weighted mean absolute deviation over GMTKN55) ever reported at 2.19 kcal/mol. However, our xrevDSD-PBEP86-D4 functional reaches a statistically equivalent WTMAD2=2.22 kcal/mol, using just a handful of empirical parameters, and the xrevDOD-PBEP86-D4 functional reaches 2.25 kcal/mol with just opposite-spin MP2 correlation, making it amenable to reduced-scaling algorithms. In general, the D4 empirical dispersion correction is clearly superior to D3BJ. If one eschews dispersion corrections of any kind, noDispSD-SCAN offers a viable alternative. Parametrization over the entire GMTKN55 dataset yields substantial improvement over the small training set previously employed in the DSD papers.

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