Makespan scheduling problems are in the mainstream of operations research, industrial engineering, and computer science. A basic multiprocessor version requires that n tasks be scheduled on m identical processors so as to minimize the makespan, i.e., the latest task finishing time. In the standard probability model considered here, the task durations are i.i.d. random variables with a distribution F, and the objective is to estimate the distribution of the makespan as a function of m, n, and F. This paper surveys probabilistic results for the multiprocessor scheduling problem and an important variant known as the permutation flow-shop problem. Several of the results are new; the others have appeared in the last few years. Because of the difficulty of exact analysis, the results take the form of limits as n! 1 or as both m! 1 and n! 1 with m! n. Some highlights of the survey are: a new asymptotic analysis of the on-line greedy scheduling policy, the resolution of a longstanding open problem in the analysis of off-line policies, new applications of central limit theorems to makespan scheduling, and limit theorems giving the asymptotic behavior under the greedy and optimal policies for the flow-shop problem. Open problems and modeling issues are also discussed.
|
7715
|
Computers and Intractability: A Guide to the Theory of NP-Completeness
– Garey, Johnson
- 1979
|
|
1329
|
Convergence of Probability Measures
– Billingsley
- 1968
|
|
979
|
An introduction to probability theory and its application, volume I
– Feller
- 1967
|
|
436
|
Optimization by Simulated Annealing: An Experimental Evaluation
– Johnson, Aragon, et al.
- 1991
|
|
335
|
Applied Probability and Queues
– Asmussen
- 1987
|
|
257
|
Interacting Particle Systems
– Liggett
- 1985
|
|
239
|
Bounds for certain multiprocessing anomalies
– Graham
- 1966
|
|
235
|
Sequencing and scheduling: algorithms and complexity
– Lawler, Lenstra, et al.
- 1993
|
|
58
|
The complexity of flowshop and jobshop scheduling
– Garey, Johnson, et al.
- 1976
|
|
46
|
Multiple channel queues in heavy traffic
– Iglehart, Whitt
- 1970
|
|
43
|
The Differencing Method of Set Partitioning
– Karmarkar, Karp
- 1982
|
|
37
|
On-line load balancing and network flow
– Phillips, Westbrook
- 1993
|
|
28
|
Point processes, regular variation, and weak convergence
– Resnick
- 1986
|
|
27
|
Departure from Many Queues in Series
– Glynn, Whitt
- 1991
|
|
26
|
Approximate distributions of order statistics
– Reiss
- 1989
|
|
23
|
Probabilistic analysis of optimum partitioning
– Karmarkar, Karp, et al.
- 1986
|
|
17
|
Asymptotic enumeration of partial orders on a finite set
– Kleitman, Rothschild
- 1975
|
|
12
|
Optimal Two and Three Stage Production Schedules with Set-up Times Included
– Johnson
- 1954
|
|
10
|
Queues in series via interacting particle systems
– Srinivasan
- 1993
|
|
7
|
Using distributed-event parallel simulation to study departures from many queues in series
– Greenberg, Schlunk, et al.
- 1993
|
|
6
|
Tight bounds and probabilistic analysis of two heuristics for parallel processor scheduling
– Loulou
- 1984
|
|
6
|
Random Orders
– Winkler
- 1985
|
|
5
|
A note on expected makespans for largest-first sequences of independent tasks on two processors
– Jr, Frederickson, et al.
- 1984
|
|
5
|
A probabilistic analysis of 2-machine flowshops
– Ramudhin, Bartholdi, et al.
- 1996
|
|
4
|
Expected makespans for largest-first multiprocessor scheduling
– Jr, Flatto, et al.
- 1984
|
|
4
|
On the Expected Relative Performance of List Scheduling
– Gilbert
- 1985
|
|
4
|
A note on the average-case behavior of a simple differencing method for partitioning
– Lueker
- 1987
|
|
3
|
Palm Probabilities and Stationary Queueing Systems
– Baccelli, Br'emaud
- 1987
|
|
3
|
Personal communication
– Bentley
- 1983
|
|
3
|
Probabilistic Bounds on the Performance of List Scheduling
– Bruno, Downey
- 1986
|
|
3
|
The asymptotic optimality of the LPT rule
– Frenk, Kan, et al.
- 1987
|
|
2
|
A Probabilistic Analysis of the LPT Scheduling Rule
– Boxma
- 1984
|
|
2
|
A Probabilistic Analysis of Multiprocessor List Scheduling: the Erlang Case, Stochastic Models
– Boxma
- 1985
|
|
2
|
Probabilistic Analysis of Packing and Related Partitioning Problems
– Lueker
- 1991
|
|
2
|
On the Asymptotic Optimality of Multiprocessor Scheduling Heuristic Algorithms
– Han, Hong, et al.
- 1992
|
|
2
|
On approximations for queues I: extremal distributions
– Whitt
- 1984
|
|
1
|
Superposition of Point Processes
– inlar, E
- 1972
|
|
1
|
Limit Laws for Multiprocessor Scheduling," in preparation
– Coffman, Flatto, et al.
- 1993
|
|
1
|
Load Balancing on Two Identical Facilities
– Wright
- 1992
|
|
1
|
Polynomial Instances and Approximation Results for the Scheduling Problem on Three Dedicated Processors
– Dell'Olmo, Speranza, et al.
- 1993
|
|
1
|
On the Rate of Convergence to Optimality of the LPT Rule," Discrete Applied Mathematics
– Kan, G, et al.
- 1986
|
|
