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.

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 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.

THE TOTAL TIME ON TEST STATISTICS AND l1-ISOTROPY

2000 ◽  
Vol 07 (02) ◽  
pp. 143-151 ◽  
Author(s):  
YU HAYAKAWA
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Test Statistics ◽  
Exponential Random Variables ◽  
Characterization Property ◽  
Mutually Independent ◽  
Special Case

In the literature on the total time on test statistics, it is often assumed that the random variables are mutually independent. It is well known that the scaled total time on test statistics of i.i.d. exponential random variables are the order statistics of independent uniform random variables on (0, 1). We show that this is in fact a characterization property of the l1-isotropic sequence of random variables, which includes the sequence of i.i.d. exponential random variables as a special case.

A limit theorem with applications in order statistics

1974 ◽  
Vol 11 (1) ◽  
pp. 219-222 ◽  
Author(s):  
János Galambos
Keyword(s):  
Limit Theorem ◽  
Order Statistics ◽  
Limit Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Analytic Method ◽  
Sufficient Condition ◽  
Dependent Random Variables ◽  
Special Case ◽  
Infinite Sequences

Let A1, A2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n), as n→ + ∞, in terms of limits of the Sk(n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X1, X2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

A limit theorem with applications in order statistics

1974 ◽  
Vol 11 (01) ◽  
pp. 219-222
Author(s):  
János Galambos
Keyword(s):  
Limit Theorem ◽  
Order Statistics ◽  
Limit Distribution ◽  
Probability Space ◽  
Random Variables ◽  
Analytic Method ◽  
Sufficient Condition ◽  
Dependent Random Variables ◽  
Special Case ◽  
Infinite Sequences

Let A 1, A 2, ···, An be events on a given probability space and let Br, n be the event that exactly r of the A's occur. Let further Sk (n) be the kth binomial moment of the number of the A's which occur. A sufficient condition is given for the existence of lim P (Br,n ), as n→ + ∞, in terms of limits of the Sk (n)'s and a formula is given for the limit above. This formula for the limit is similar to the sieve theorem of Takács (1967) for infinite sequences of events and in the proof we make use of Takács's analytic method. The result is immediately applicable to the limit distribution of the maximum of (dependent) random variables X 1, X 2, ···, Xn by choosing Aj = {Xj ≧ x}. Our main theorem is reformulated for this special case and an example is given for illustration.

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 Admission controls for Erlang's loss system with service times distributed as a finite sum of exponential random variables

1998 ◽  
Vol 2 (2) ◽  
pp. 119-132 ◽  
Author(s):  
Brian Moretta ◽  
Ilze Ziedins
Keyword(s):  
Optimal Policy ◽  
Random Variables ◽  
Loss System ◽  
Standard Assumption ◽  
Threshold Policy ◽  
Exponential Random Variables ◽  
Reservation Policy ◽  
Service Times ◽  
Trunk Reservation ◽  
Finite Sum

It is known that a threshold policy (or trunk reservation policy) is optimal for Erlang’s loss system under certain assumptions. In this paper we examine the robustness of this policy under departures from the standard assumption of exponential service times (call holding times) and give examples where the optimal policy has a generalized trunk reservation form.

A Note on Stochastic Scheduling on a Single Machine Subject to Breakdown–The Preemptive Repeat Model

1991 ◽  
Vol 5 (3) ◽  
pp. 349-354 ◽  
Author(s):  
Esther Frostig
Keyword(s):  
Single Machine ◽  
Random Variables ◽  
Stochastic Scheduling ◽  
Independent Random Variables ◽  
Processing Times ◽  
Optimal Policies ◽  
Number Of Late Jobs ◽  
One Machine ◽  
Late Jobs ◽  
Weighted Flowtime

This paper considers scheduling n jobs on one machine to minimize the expected weighted flowtime and the number of late jobs. The processing times of the jobs are independent random variables. The machine is subject to failure and repair where the uptimes are exponentially distributed. We find the optimal policies for the preemptive repeat model.

A Classification and Closure Properties of Languages for Describing Concurrent System Behaviours

1981 ◽  
Vol 4 (3) ◽  
pp. 531-549 ◽  
Author(s):  
Miklós Szijártó
Keyword(s):  
Formal Languages ◽  
Parallel Program ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Concurrent System ◽  
Closure Properties ◽  
Sequential Program ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Independence Relations

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

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