Stochastic flowshop no-wait scheduling

1985 ◽  
Vol 22 (1) ◽  
pp. 240-246 ◽  
Author(s):  
E. Frostig ◽  
I. Adiri
Keyword(s):  
Independent Random Variable ◽  
Processing Time ◽  
Random Variables ◽  
Random Variable ◽  
Schedule Length ◽  
Special Cases ◽  
No Wait ◽  
Completion Times ◽  
Sum Of Completion Times

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti. It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti, (i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.

Stochastic flowshop no-wait scheduling

1985 ◽  
Vol 22 (01) ◽  
pp. 240-246
Author(s):  
E. Frostig ◽  
I. Adiri
Keyword(s):  
Independent Random Variable ◽  
Processing Time ◽  
Random Variables ◽  
Random Variable ◽  
Schedule Length ◽  
Special Cases ◽  
No Wait ◽  
Completion Times ◽  
Sum Of Completion Times

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti . It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti , ( i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.

scholarly journals On Exact Sampling of Nonnegative Infinitely Divisible Random Variables

2012 ◽  
Vol 44 (3) ◽  
pp. 842-873 ◽  
Author(s):  
Zhiyi Chi
Keyword(s):  
Basic Result ◽  
Random Variables ◽  
Random Variable ◽  
Fixed Number ◽  
Rejection Sampling ◽  
Infinitely Divisible ◽  
Exact Sampling ◽  
Special Cases ◽  
Wide Range ◽  
Lévy Density

Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variable X with its Lévy density explicitly specified, if its distribution conditional on X ≤ r can be sampled exactly, where r > 0 is any fixed number, then X can be sampled exactly using rejection sampling, without knowing the explicit expression of the density of X. We show that variations of the result can be used to sample various nonnegative i.d. random variables.

Subsequence principles for vector-valued random variables

1979 ◽  
Vol 86 (2) ◽  
pp. 301-312 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Principal Result ◽  
Moment Condition ◽  
Special Cases ◽  
Cesaro Averages ◽  
Image Position ◽  
Vector Valued ◽  
Subsequence Principle ◽  
Independent Identically Distributed

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such thatalmost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

scholarly journals The Law of the Iterated Logarithm for Linear Processes Generated by a Sequence of Stationary Independent Random Variables under the Sub-Linear Expectation

Entropy ◽  
2021 ◽  
Vol 23 (10) ◽  
pp. 1313
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Independent Random Variable ◽  
Random Variables ◽  
Random Variable ◽  
Second Order ◽  
Law Of Iterated Logarithm ◽  
Independent Random Variables ◽  
Linear Processes ◽  
Iterated Logarithm ◽  
The Law

In this paper, we obtain the law of iterated logarithm for linear processes in sub-linear expectation space. It is established for strictly stationary independent random variable sequences with finite second-order moments in the sense of non-additive capacity.

scholarly journals On Exact Sampling of Nonnegative Infinitely Divisible Random Variables

2012 ◽  
Vol 44 (03) ◽  
pp. 842-873 ◽  
Author(s):  
Zhiyi Chi
Keyword(s):  
Basic Result ◽  
Random Variables ◽  
Random Variable ◽  
Fixed Number ◽  
Rejection Sampling ◽  
Infinitely Divisible ◽  
Exact Sampling ◽  
Special Cases ◽  
Wide Range ◽  
Lévy Density

Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variableXwith its Lévy density explicitly specified, if its distributionconditionalonX≤rcan be sampled exactly, wherer&gt; 0 is any fixed number, thenXcan be sampled exactly using rejection sampling, without knowing the explicit expression of the density ofX. We show that variations of the result can be used to sample various nonnegative i.d. random variables.

scholarly journals ANALYTICAL EVALUATION OF PROCESSING TIME OF A QUERY IN AN INFORMATION SYSTEM

2018 ◽  
pp. 69-73 ◽  
Author(s):  
M. M. Butaev ◽  
A. A. Tarasov
Keyword(s):  
Information Systems ◽  
Normal Distribution ◽  
Processing Time ◽  
Random Variables ◽  
Initial Approximation ◽  
Random Variable ◽  
Sequential Processing ◽  
Processing Times ◽  
Distribution Laws ◽  
Interval Distribution

The normal distribution of a random variable is usually used in studies of the probabilistic characteristics of information systems. However, the approximation by the normal distribution of distributions determined on a limited interval distorts the physical meaning of the model and the numerical results, and it can only be used as an initial approximation. The aim of the work is to improve the methods for calculating the probabilistic characteristics of information systems. The object of the study is an analytical method for calculating the processing time of the query in the system. The subject of the study are formulas for calculating the duration of sequential processing of the query by elements of the system with uniformly distributed random processing times. In deriving the formulas for calculating the probability characteristics of a sum of independent uniformly distributed random variables, the methods of the theory of probability and statistics are applied. It is proposed for random variables, determined only on the positive coordinate axis, to use finite-interval distribution laws, for example, beta distribution. Density formulas and probability functions for sums of two, three and four independent uniformly distributed random variables are derived.

Stochastically Minimizing Total Delay of Jobs Subject to Random Deadlines

1991 ◽  
Vol 5 (3) ◽  
pp. 333-348 ◽  
Author(s):  
Susan H. Xu
Keyword(s):  
Time Distribution ◽  
Processing Time ◽  
Fixed Number ◽  
Parallel Processors ◽  
Threshold Strategy ◽  
Total Delay ◽  
Scheduling System ◽  
Processing Times ◽  
Completion Times ◽  
Sum Of Completion Times

This paper analyzes a scheduling system where a fixed number of nonpreemptive jobs is to be processed on multiple parallel processors with different processing speeds. Each processor has an exponential processing time distribution and the processors are ordered in ascending order of their mean processing times. Each job has its own deadline that is exponentially distributed with rate ß1, independent of the deadlines of other jobs and also independent of job processing times. A job departs the system as soon as either its processing completes or its deadline occurs. We show that there exists a simple threshold strategy that slochastically minimizes the total delay of all jobs. The policy depends on distributions of processing times and deadlines, but is independent of the rate of deadlines. When the rate of the deadline distribution is 0 (no deadlines), the total delay reduces to the flowtime (the sum of completion times of all jobs). If each job has its own probability of being correctly processed, then an extension of this policy stochastically maximizes the total number of correctly processed, nontardy jobs. We discuss possible generalizations and limitations of this result.

scholarly journals Explorable Uncertainty in Scheduling with Non-uniform Testing Times

2021 ◽  
pp. 127-142
Author(s):  
Susanne Albers ◽  
Alexander Eckl
Keyword(s):  
Processing Time ◽  
Randomized Algorithm ◽  
General Setting ◽  
Unit Size ◽  
Deterministic Case ◽  
Upper Limit ◽  
Uncertainty Models ◽  
Competitive Algorithms ◽  
Completion Times ◽  
Sum Of Completion Times

AbstractThe problem of scheduling with testing in the framework of explorable uncertainty models environments where some preliminary action can influence the duration of a task. In the model, each job has an unknown processing time that can be revealed by running a test. Alternatively, jobs may be run untested for the duration of a given upper limit. Recently, Dürr et al. [4] have studied the setting where all testing times are of unit size and have given lower and upper bounds for the objectives of minimizing the sum of completion times and the makespan on a single machine. In this paper, we extend the problem to non-uniform testing times and present the first competitive algorithms. The general setting is motivated for example by online user surveys for market prediction or querying centralized databases in distributed computing. Introducing general testing times gives the problem a new flavor and requires updated methods with new techniques in the analysis. We present constant competitive ratios for the objective of minimizing the sum of completion times in the deterministic case, both in the non-preemptive and preemptive setting. For the preemptive setting, we additionally give a first lower bound. We also present a randomized algorithm with improved competitive ratio. Furthermore, we give tight competitive ratios for the objective of minimizing the makespan, both in the deterministic and the randomized setting.

Sign in / Sign up

Export Citation Format

Share Document