Optimal Scheduling of Multiclass Stochastic Systems

1996 ◽  
Vol 10 (2) ◽  
pp. 229-241 ◽  
Author(s):  
Rhonda Righter
Keyword(s):  
Optimal Policy ◽  
General Class ◽  
Completion Time ◽  
Stochastic Systems ◽  
Random Variables ◽  
Random Variable ◽  
Optimal Scheduling ◽  
Parallel Server ◽  
Dependent Function ◽  
The Cost

We study the problem of scheduling multiclass parallel server queues where the cost for each job is a general class-dependent function of the job's completion time and a second random variable. We characterize the structure of the optimal policy under fairly general conditions on the random variables.

scholarly journals Nonrecurrence and Bell-like inequalities

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 762-768
Author(s):  
Douglas G. Danforth
Keyword(s):  
Infinite Number ◽  
General Class ◽  
Hidden Variables ◽  
Random Variables ◽  
Random Variable ◽  
Bell Inequalities ◽  
Usual Sense ◽  
Real Random Variable ◽  
Action At A Distance ◽  
The Continuum

AbstractThe general class,Λ, of Bell hidden variables is composed of two subclassesΛRandΛNsuch thatΛR⋃ΛN=ΛandΛR∩ΛN= {}. The classΛNis very large and contains random variables whose domain is the continuum, the reals. There are an uncountable infinite number of reals. Every instance of a real random variable is unique. The probability of two instances being equal is zero, exactly zero.ΛNinduces sample independence. All correlations are context dependent but not in the usual sense. There is no “spooky action at a distance”. Random variables, belonging toΛN, are independent from one experiment to the next. The existence of the classΛNmakes it impossible to derive any of the standard Bell inequalities used to define quantum entanglement.

Convex majorization with an application to the length of critical paths

1979 ◽  
Vol 16 (03) ◽  
pp. 671-677 ◽  
Author(s):  
Isaac Meilijson ◽  
Arthur Nádas
Keyword(s):  
Maximal Function ◽  
Completion Time ◽  
Random Variables ◽  
Random Variable ◽  
List Type ◽  
Type Number ◽  
Joint Distributions ◽  
Littlewood Maximal Function ◽  
Delay Times ◽  
Critical Paths

1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all . 2. Let H be the Hardy–Littlewood maximal function HY (x) = E(Y – X | Y > x). Then HY (Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y. 3. Let X 1 X 2 · ·· Xn be random variables with given marginal distributions, let I 1, I 2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij , and delay times Xi .) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi 's with specified marginals, and of the ‘bottleneck probability' of each path.

Convex majorization with an application to the length of critical paths

1979 ◽  
Vol 16 (3) ◽  
pp. 671-677 ◽  
Author(s):  
Isaac Meilijson ◽  
Arthur Nádas
Keyword(s):  
Maximal Function ◽  
Completion Time ◽  
Random Variables ◽  
Random Variable ◽  
List Type ◽  
Type Number ◽  
Joint Distributions ◽  
Littlewood Maximal Function ◽  
Delay Times ◽  
Critical Paths

1. (Y) for all non-negative, non-decreasing convex functions φ (X is convexly smaller than Y) if and only if, for all .2.Let H be the Hardy–Littlewood maximal function HY(x) = E(Y – X | Y > x). Then HY(Y) is the smallest random variable exceeding stochastically all random variables convexly smaller than Y.3.Let X1X2 · ·· Xn be random variables with given marginal distributions, let I1,I2, ···, Ik be arbitrary non-empty subsets of {1,2, ···, n} and let M = max (M is the completion time of a PERT network with paths Ij, and delay times Xi.) The paper introduces a computation of the convex supremum of M in the class of all joint distributions of the Xi's with specified marginals, and of the ‘bottleneck probability' of each path.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals Optimality of index policies for stochastic scheduling with switching penalties

1992 ◽  
Vol 29 (04) ◽  
pp. 957-966 ◽  
Author(s):  
Mark P. Van Oyen ◽  
Dimitrios G. Pandelis ◽  
Demosthenis Teneketzis
Keyword(s):  
Optimal Policy ◽  
Time Delays ◽  
Stochastic Scheduling ◽  
Optimal Scheduling ◽  
Scheduling Policies ◽  
Parallel Queues ◽  
Lump Sum ◽  
Index Policies ◽  
Service Policy ◽  
The Impact

We investigate the impact of switching penalties on the nature of optimal scheduling policies for systems of parallel queues without arrivals. We study two types of switching penalties incurred when switching between queues: lump sum costs and time delays. Under the assumption that the service periods of jobs in a given queue possess the same distribution, we derive an index rule that defines an optimal policy. For switching penalties that depend on the particular nodes involved in a switch, we show that although an index rule is not optimal in general, there is an exhaustive service policy that is optimal.

INFORMATION AND UNCERTAINTY IN A QUEUING SYSTEM

2007 ◽  
Vol 21 (3) ◽  
pp. 361-380 ◽  
Author(s):  
Refael Hassin
Keyword(s):  
Service Quality ◽  
Random Variables ◽  
Random Variable ◽  
Queuing System ◽  
Service Rate ◽  
Single Server ◽  
Profit Function

This article deals with the effect of information and uncertainty on profits in an unobservable single-server queuing system. We consider scenarios in which the service rate, the service quality, or the waiting conditions are random variables that are known to the server but not to the customers. We ask whether the server is motivated to reveal these parameters. We investigate the structure of the profit function and its sensitivity to the variance of the random variable. We consider and compare variations of the model according to whether the server can modify the service price after observing the realization of the random variable.

The central limit problem for trimmed sums

1987 ◽  
Vol 102 (2) ◽  
pp. 329-349 ◽  
Author(s):  
Philip S. Griffin ◽  
William E. Pruitt
Keyword(s):  
Distribution Function ◽  
Random Variables ◽  
Random Variable ◽  
Limit Problem ◽  
Absolute Value ◽  
Trimmed Sums ◽  
Common Distribution ◽  
Common Distribution Function ◽  
Central Limit Problem ◽  
Trimmed Sum

Let X, X1, X2,… be a sequence of non-degenerate i.i.d. random variables with common distribution function F. For 1 ≤ j ≤ n, let mn(j) be the number of Xi satisfying either |Xi| > |Xj|, 1 ≤ i ≤ n, or |Xi| = |Xj|, 1 ≤ i ≤ j, and let (r)Xn = Xj if mn(j) = r. Thus (r)Xn is the rth largest random variable in absolute value from amongst X1, …, Xn with ties being broken according to the order in which the random variables occur. Set (r)Sn = (r+1)Xn + … + (n)Xn and write Sn for (0)Sn. We will refer to (r)Sn as a trimmed sum.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

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