Scheduling Two-Point Stochastic Jobs to Minimize the Makespan on Two Parallel Machines

1997 ◽  
Vol 11 (1) ◽  
pp. 95-105 ◽  
Author(s):  
Sem Borst ◽  
John Bruno ◽  
E. G. Coffman ◽  
Steven Phillips
Keyword(s):  
General Problem ◽  
Time Distribution ◽  
Hazard Rate ◽  
Parallel Machines ◽  
Discrete Distributions ◽  
Hazard Rates ◽  
Running Time ◽  
First Order ◽  
Optimal Policies ◽  
Expected Makespan

Simple optimal policies are known for the problem of scheduling jobs to minimize expected makespan on two parallel machines when the job running-time distribution has a monotone hazard rate. But no such policy appears to be known in general. We investigate the general problem by adopting two-point running-time distributions, the simplest discrete distributions not having monotone hazard rates. We derive a policy that gives an explicit, compact solution to this problem and prove its optimality. We also comment briefly on first-order extensions of the model, but each of these seems to be markedly more difficult to analyze.

Scheduling jobs by stochastic processing requirements on parallel machines to minimize makespan or flowtime

1982 ◽  
Vol 19 (1) ◽  
pp. 167-182 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Probability Distribution ◽  
Hazard Rate ◽  
Parallel Machines ◽  
Random Variables ◽  
Expected Value ◽  
Hazard Rates ◽  
Stochastic Processing ◽  
Identical Machines ◽  
Exponential Random Variables ◽  
Monotone Hazard Rate

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.

Scheduling jobs by stochastic processing requirements on parallel machines to minimize makespan or flowtime

1982 ◽  
Vol 19 (01) ◽  
pp. 167-182 ◽  
Author(s):  
Richard R. Weber
Keyword(s):  
Probability Distribution ◽  
Hazard Rate ◽  
Parallel Machines ◽  
Random Variables ◽  
Expected Value ◽  
Hazard Rates ◽  
Stochastic Processing ◽  
Identical Machines ◽  
Exponential Random Variables ◽  
Monotone Hazard Rate

A number of identical machines operating in parallel are to be used to complete the processing of a collection of jobs so as to minimize either the jobs' makespan or flowtime. The total processing required to complete each job has the same probability distribution, but some jobs may have received differing amounts of processing prior to the start. When the distribution has a monotone hazard rate the expected value of the makespan (flowtime) is minimized by a strategy which always processes those jobs with the least (greatest) hazard rates. When the distribution has a density whose logarithm is concave or convex these strategies minimize the makespan and flowtime in distribution. These results are also true when the processing requirements are distributed as exponential random variables with different parameters.

scholarly journals A micro-analysis of Irish firm deaths during the financial crisis (2006–2010)

2020 ◽  
Vol 39 (1) ◽  
pp. 1-16
Author(s):  
Bernadette Power ◽  
Geraldine Ryan ◽  
Justin Doran
Keyword(s):  
Financial Crisis ◽  
Firm Size ◽  
Hazard Rate ◽  
Labour Force ◽  
Considerable Effect ◽  
Hazard Rates ◽  
Economic Cycle ◽  
Established Firms ◽  
Micro Analysis ◽  
Industry Conditions

AbstractThis paper examines differences in the hazard rates of young, established and mature firms during the financial crisis, using microdata from more than 300,000 Irish firms. The findings confirm that firm size at the time of the crisis had the largest impact on the probability of exit. The liability of smallness was pronounced in mature cohorts. Industry conditions had a considerable effect on the hazard rate of young cohorts, as opposed to mature counterparts. Interestingly, agglomeration raised the hazard rates of younger cohorts only. By contrast, attributes of the labour force of the region largely influenced the hazard rates of more established firms. Firms founded before the crisis were significantly less likely to exit in the aftermath of the crisis, in comparison with firms founded just before or during the crisis, whereas more mature firms seem to be more sensitive to the economic cycle.

Scheduling jobs with exponential processing times on parallel machines

1988 ◽  
Vol 25 (04) ◽  
pp. 752-762 ◽  
Author(s):  
Tapani Lehtonen
Keyword(s):  
Binomial Distribution ◽  
Parallel Machines ◽  
Essential Feature ◽  
Stochastic Order ◽  
Partial Solution ◽  
Flow Time ◽  
Optimal Assignment ◽  
Processing Times ◽  
Expected Makespan

We consider a system where jobs are processed by parallel machines. The processing times are exponentially distributed. An essential feature is that the assignment of the jobs to the machines is decided before the system starts to work. We consider both the flow time and the makespan. In the case of the flow time we allow both the machines and the jobs to be non-homogeneous. The optimization is by minimizing the flow time in the sense of stochastic order and the optimal assignment is obtained for this case. The case of the makespan is harder. We consider the expected makespan and as a partial solution we prove an optimality result for the case where there are two non-homogeneous machines and the jobs are homogeneous. It turns out that the optimal assignment can be expressed by using a quantile of a binomial distribution.

A Bayes Continuous Reliability Growth Model with Two Categories of Failures

1997 ◽  
Vol 04 (01) ◽  
pp. 1-15
Author(s):  
Daming Lin ◽  
W. K. Chiu
Keyword(s):  
Competing Risks ◽  
Growth Model ◽  
Prior Distribution ◽  
Hazard Rate ◽  
Failure Modes ◽  
Weighted Average ◽  
Hazard Rates ◽  
Development Phase ◽  
Reliability Growth ◽  
Time Invariant

A Bayesian continuous reliability growth model is presented. It is assumed that the development phase of a product consists of m stages. In each stage, the failure mechanism of the product follows a competing risks model with two specific failure modes: inherent and assignable-cause. The hazard rate for each mode is time-invariant within one stage. Under the assumption that modifications of the product improve its reliability, we assign a reasonable joint prior distribution for the hazard rates. Then Bayesian analysis is carried out using this prior distribution. It turns out that the posterior pdf of the hazard rates of interest is just a weighted average of pdf's which have the same form as the prior pdf. A numerical example is given for illustration.

A stochastic process for modeling failures of a system having a non-monotonic hazard rate function

2019 ◽  
Vol 233 (5) ◽  
pp. 731-746
Author(s):  
Lucianne Varn ◽  
Stefanka Chukova ◽  
Richard Arnold
Keyword(s):  
Stochastic Process ◽  
Hazard Rate ◽  
Repair Process ◽  
Rate Function ◽  
Hazard Rates ◽  
Failure Rates ◽  
Hazard Rate Function ◽  
Repairable Systems ◽  
System Failures ◽  
Quantities Of Interest

Reliability literature on modeling failures of repairable systems mostly deals with systems having monotonically increasing hazard/failure rates. When the hazard rate of a system is non-monotonic, models developed for monotonically increasing failure rates cannot be effectively applied without making assumptions on the types of repair performed following system failures. For instance, for systems having bathtub-shaped hazard rates, it is assumed that during the initial, decreasing hazard rate phase, all repairs are minimal. These assumptions on the type of general repair can be restrictive. In order to relax these assumptions, it has been suggested that general repairs in the initially decreasing phase can be modeled as “aging” the system. This approach however does not preserve the order of effectiveness of the types of general repair as defined in the literature. In this article, we develop a set of models to address these shortcomings. We propose a new stochastic process to model consecutive failures of repairable systems having non-monotonic, specifically bathtub-shaped, hazard rates, where the types of general repair are not restricted and the order of the effectiveness of the types of repair is preserved. The proposed models guarantee that a repaired system is at least as reliable as one that has not failed (or equivalently one that has been minimally repaired). To illustrate the models, we present multiple examples and simulate the failure-repair process and estimate the quantities of interest.

Geometric renewal convergence rates from hazard rates

2001 ◽  
Vol 38 (01) ◽  
pp. 180-194 ◽  
Author(s):  
Kenneth S. Berenhaut ◽  
Robert Lund
Keyword(s):  
Convergence Rate ◽  
Hazard Rate ◽  
Convergence Rates ◽  
Hazard Rates ◽  
Finite Support ◽  
Good Convergence ◽  
Renewal Sequence ◽  
Geometric Convergence ◽  
New Better Than Used ◽  
Better Than

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

Rigid homogeneous chains

1981 ◽  
Vol 89 (1) ◽  
pp. 07-17 ◽  
Author(s):  
A. M. W. Glass ◽  
Yuri Gurevich ◽  
W. Charles Holland ◽  
Saharon Shelah
Keyword(s):  
Automorphism Group ◽  
General Problem ◽  
Automorphism Groups ◽  
Ordered Sets ◽  
First Order ◽  
Image Position

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S(Ω), satisfiesthen Ω has cardinality at most ℵ0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S(Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o-primitive (i.e. it may have convex congruences). Fortunately, ‘o-primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o-primitive; the second is to determine how the o-primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

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