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.

STOCHASTIC ORDER OF SAMPLE RANGE FROM HETEROGENEOUS EXPONENTIAL RANDOM VARIABLES

2008 ◽  
Vol 23 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Peng Zhao ◽  
Xiaohu Li
Keyword(s):  
Hazard Rate ◽  
Stochastic Order ◽  
Random Variables ◽  
Hazard Rates ◽  
Minimum Order ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Usual Stochastic Order ◽  
Exponential Random Variables ◽  
Sample Range

Let X1, …, Xn be independent exponential random variables with their respective hazard rates λ1, …, λn, and let Y1, …, Yn be independent exponential random variables with common hazard rate λ. Denote by Xn:n, Yn:n and X1:n, Y1:n the corresponding maximum and minimum order statistics. Xn:n−X1:n is proved to be larger than Yn:n−Y1:n according to the usual stochastic order if and only if $\lambda \geq \left({\bar{\lambda}}^{-1}\prod\nolimits^{n}_{i=1}\lambda_{i}\right)^{{1}/{(n-1)}}$ with $\bar{\lambda}=\sum\nolimits^{n}_{i=1}\lambda_{i}/n$. Further, this usual stochastic order is strengthened to the hazard rate order for n=2. However, a counterexample reveals that this can be strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo and Khaledi and Kochar.

COMMENTS ON THE SURVEY BY BALAKRISHNAN AND ZHAO

2013 ◽  
Vol 27 (4) ◽  
pp. 445-449 ◽  
Author(s):  
Moshe Shaked
Keyword(s):  
Order Statistics ◽  
Negative Binomial ◽  
Random Variables ◽  
Hazard Rates ◽  
Proportional Hazard ◽  
Stochastic Comparisons ◽  
Exponential Random Variables ◽  
Binomial Random Variables

N. Balakrishnan and Peng Zhao have prepared an outstanding survey of recent results that stochastically compare various order statistics and some ranges based on two collections of independent heterogeneous random variables. Their survey focuses on results for heterogeneous exponential random variables and their extensions to random variables with proportional hazard rates. In addition, some results that stochastically compare order statistics based on heterogeneous gamma, Weibull, geometric, and negative binomial random variables are also given. In particular, the authors of have listed some stochastic comparisons that are based on one heterogeneous collection of random variables, and one homogeneous collection of random variables. Personally, I find these types of comparisons to be quite fascinating. Balakrishnan and Zhao have done a thorough job of listing all the known results of this kind.

On the optimality of static priority policies in stochastic scheduling on parallel machines

1987 ◽  
Vol 24 (02) ◽  
pp. 430-448 ◽  
Author(s):  
Thomas Kämpke
Keyword(s):  
Optimal Policy ◽  
Parallel Machines ◽  
Random Variables ◽  
Stochastic Scheduling ◽  
Sufficient Condition ◽  
Exponential Random Variables ◽  
Holding Cost ◽  
Special Case ◽  
Weighted Flowtime

n jobs are to be preemptively scheduled for processing on n machines. The machines may have differing speeds and the jobs have processing requirements which are distributed as independent exponential random variables with different means. Holding cost g(U) is incurred per unit time that the set of uncompleted jobs is U and it is desired to minimize the total expected holding cost which is incurred until all jobs are complete. We show that if g satisfies certain simple conditions then the optimal policy is one which takes the jobs in the order 1, 2, ···, n and assigns each uncompleted job in turn to the fastest available machine. In the special case in which the objective is to minimize the expected weighted flowtime, where there is a holding cost of wi while job i is incomplete, the sufficient condition is simply w1 ≧ … ≧ wn and λ1 w1 ≧ … ≧ λn wn .

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.

On the optimality of static priority policies in stochastic scheduling on parallel machines

1987 ◽  
Vol 24 (2) ◽  
pp. 430-448 ◽  
Author(s):  
Thomas Kämpke
Keyword(s):  
Optimal Policy ◽  
Parallel Machines ◽  
Random Variables ◽  
Stochastic Scheduling ◽  
Sufficient Condition ◽  
Exponential Random Variables ◽  
Holding Cost ◽  
Special Case ◽  
Weighted Flowtime

n jobs are to be preemptively scheduled for processing on n machines. The machines may have differing speeds and the jobs have processing requirements which are distributed as independent exponential random variables with different means. Holding cost g(U) is incurred per unit time that the set of uncompleted jobs is U and it is desired to minimize the total expected holding cost which is incurred until all jobs are complete. We show that if g satisfies certain simple conditions then the optimal policy is one which takes the jobs in the order 1, 2, ···, n and assigns each uncompleted job in turn to the fastest available machine. In the special case in which the objective is to minimize the expected weighted flowtime, where there is a holding cost of wi while job i is incomplete, the sufficient condition is simply w1 ≧ … ≧ wn and λ1 w1 ≧ … ≧ λn wn.

A note on the normalised moments of distributions with non-monotonic hazard rate

1981 ◽  
Vol 18 (2) ◽  
pp. 530-535
Author(s):  
H. L. MacGillivray
Keyword(s):  
Hazard Rate ◽  
Random Variables ◽  
Hazard Rates ◽  
Regular Property

For distributions of non-negative random variables with a monotonic hazard rate, it is well known that the normalised moments have the same sign-regular property as the distribution. This note extends this correspondence in properties for some common distributions with non-monotonic hazard rates, linking a change in sign-regular properties.

Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (4) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X1,…,Xn be independent exponential random variables with Xi having hazard rate . Let Y1,…,Yn be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1nλi)1/n, the geometric mean of the λis. Let Xn:n = max{X1,…,Xn}. It is shown that Xn:n is greater than Yn:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of Xn:n and an upper bound on the hazard rate function of Xn:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference65, 203–211), which are in terms of the arithmetic mean of the λis. Furthermore, let X1*,…,Xn∗ be another set of independent exponential random variables with Xi∗ having hazard rate λi∗, i = 1,…,n. It is proved that if (logλ1,…,logλn) weakly majorizes (logλ1∗,…,logλn∗, then Xn:n is stochastically greater than Xn:n∗.

ORDERING CONVOLUTIONS OF HETEROGENEOUS EXPONENTIAL AND GEOMETRIC DISTRIBUTIONS REVISITED

2010 ◽  
Vol 24 (3) ◽  
pp. 329-348 ◽  
Author(s):  
Tiantian Mao ◽  
Taizhong Hu ◽  
Peng Zhao
Keyword(s):  
Hazard Rate ◽  
Residual Life ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Mean Residual Life ◽  
Reversed Hazard Rate ◽  
Exponential Random Variables ◽  
The Mean ◽  
Geometric Variables ◽  
Fail Safe

Let Sn(a1, …, an) be the sum of n independent exponential random variables with respective hazard rates a1, …, an or the sum of n independent geometric random variables with respective parameters a1, …, an. In this article, we investigate sufficient conditions on parameter vectors (a1, …, an) and $(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ under which Sn(a1, …, an) and $S_{n}(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ are ordered in terms of the increasing convex and the reversed hazard rate orders for both exponential and geometric random variables and in terms of the mean residual life order for geometric variables. For the bivariate case, all of these sufficient conditions are also necessary. These characterizations are used to compare fail-safe systems with heterogeneous exponential components in the sense of the increasing convex and the reversed hazard rate orders. The main results complement several known ones in the literature.

Some new results on stochastic comparisons of parallel systems

2000 ◽  
Vol 37 (04) ◽  
pp. 1123-1128 ◽  
Author(s):  
Baha-Eldin Khaledi ◽  
Subhash Kochar
Keyword(s):  
Lower Bound ◽  
Hazard Rate ◽  
Geometric Mean ◽  
Random Variables ◽  
Parallel Systems ◽  
Rate Function ◽  
Arithmetic Mean ◽  
Hazard Rate Function ◽  
Stochastic Comparisons ◽  
Exponential Random Variables

Let X 1,…,X n be independent exponential random variables with X i having hazard rate . Let Y 1,…,Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i=1 n λ i )1/n , the geometric mean of the λis. Let X n:n = max{X 1,…,X n }. It is shown that X n:n is greater than Y n:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X n:n and an upper bound on the hazard rate function of X n:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference 65, 203–211), which are in terms of the arithmetic mean of the λ i s. Furthermore, let X 1 *,…,X n ∗ be another set of independent exponential random variables with X i ∗ having hazard rate λ i ∗, i = 1,…,n. It is proved that if (logλ1,…,logλ n ) weakly majorizes (logλ1 ∗,…,logλ n ∗, then X n:n is stochastically greater than X n:n ∗.

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