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On the Minimization of Weighted On the Minimization of Weighted Waiting Time VarianceWaiting Time Variance

Xueping Li

Dept. of Industrial & Information Engineering

University of Tennessee, Knoxville

INFORMSSan Francisco 2005

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Outline

Introduction & Motivation Formulation of Weighted Waiting Time

Variance minimization problem Optimal sequences analysis Development of WVS & WSS

algorithms Results & discussions Q/A

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5One Waiting Time Variance One Waiting Time Variance (WTV) Minimization Example(WTV) Minimization Example

Jobs 1 2 3 4 5

Sequence 1 5 4 3 2 1

Waiting Time 0 5 9 12 14

Mean WT = 8

WTV = 31.5

Sequence 2 4 2 1 3 5

Waiting Time 0 4 6 7 10

Mean WT = 5.4

WTV = 13.8

… Which one of the 5! sequences is optimal? WTV is NP-hard [4].

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Introduction to WWTV

Jobs with higher priority need more consistent services

Mission critical applications & High priority jobs Classification of the weight

User type (VIP, normal, etc.), status IP headers, Protocols (HTTP, RTP…)

Closely related to Earliness/Tardiness Penalty and Weighted Common Due date problems in manufacturing systems

Later than d: Loss of user satisfaction, reputation, etc. Early than d: Inventory cost, insurance, etc.

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5Quality of Service (QoS) for Information Infrastructure

Attributes of QoS [1,2] Timeliness Precision

Accuracy

Great needs for QoS We rely more and more on the Internet for e-business, entertainment, education, etc. High priority requests need guaranteed service

Current Internet can’t provide reliable, dependable service

High volume traffics while limited web resources Cyber attacks & “Best Effort” service model

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Pursue for Stability

Stability is a key measurement of QoS timeliness which enables predictable, dependable services

Service stability of an individual resource through minimizing job waiting time variance such that each resource becomes a “standard part”

Earliness/Tardiness Penalty & Just-in-time (JIT) philosophy

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Assumptions

Single Machine WTV Problem: 1 || WWTV

Assumptions n jobs to be processed on a single machine

which can process only one job at a time All jobs are available at time zero The processing time of each job is fixed and

known Setup time is zero

No preemption is allowed

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5WWTV FormulationWWTV Formulation

Objective function:

Subject to:

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Lit. Review

Key Papers WTV is antithetical to Completion Time Variance (CTV) [3] The longest jobs must be the last one to process [5] There exists one optimal sequence like 2nd….,3rd, 1st [6] Optimal sequences are V-Shaped which means that the jobs before the shortest job are descending sorted while the jobs after are ascending sorted [7] E & C scheduling methods (E & C 1.1, 1.2) [8] Agreeably Weighted WTV problem [9]

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5Weight Settings

Weight Settings of the jobs Positively correlated weight (PW):: smaller

processing time, lower weight Negatively correlated weight (NW):: smaller

processing time, higher weight* Random weight:: randomly weighted (RW)

*: Agreeably weighted in [9]

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Optimal WWTV Sequences Analysis

Compared to equal weighted WTV problems, the optimal sequences of WWTV problems

Neither the largest job nor largest weighted job has to be the last job in the sequence

“Dual” property doesn’t hold true V-shape property still hold in NW scenarios, but not in PW

and RW scenarios with respect to weighted processing time

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Strong V-Shape Tendency

PW NW

RW

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5Development of WWTV Scheduling Methods

Weighted Verified Spiral (WVS) Step 1: Sort the jobs with respect to the weighed

processing times P’i = Pi/Vi. Fix the positions of two largest jobs & the smallest job as P’n-1, P’1, P’n.

Step 2: Insert the remaining jobs with larger weighted processing time one by one either to the exactly left or right of P’1 depending on which sequence produces smaller WWTV till all jobs are inserted.

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5Development of WWTV Scheduling Methods

Weighted Simplified Spiral (WSS) Step 1: Sort the jobs with respect to the weighed processing

times P’i = Pi/Vi.

Step 2: Place P’n to the last position of the sequence, P’n-1 at the first position, P’n-2 to the last but one, P’n-3 to the first but one … and so on in a spiral fashion till all jobs are placed.

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WWTV Testing Scenarios

Scheduling Methods FIFO, WSPT WSS WVS

Small-size problems WWTV Difference WMWT Difference

Large-size problems Normal, Exponential, Uniform, Pareto 1000 problems in each scenario Pair comparison and mean WWTV comparison

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5Testing Results:Small-size problems

WWTVD and WMWTD

PW NW

RW

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5Testing Results:Large-size problems

Pair Comparison & Mean WWTV Comparison

NormalPareto

PW NW RW

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5WWTV Computational Time Comparison

Computational Time Comparison of the Scheduling Methods (in Milliseconds)

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Conclusion

Weighted Waiting Time Variance Minimization Problems

WVS & WSS are able to reduce WWTV compared to the existing scheduling methods

WSS outperforms WVS in NW scenario Future Study

Processing times follow other probability distributions Class-based WWTV Stochastic WWTV

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5References

1. N. Ye, 2002. QoS-Centric stateful resource management in information systems. Information Systems Frontiers, Vol. 4, No. 2, pp. 149-160

2. Lawrence, T. F., 1997. The quality of service model and high assurance. In Proceedings of the IEEE High Assurance Systems Engineering Workshop.

3. A.G. Merten and M.E. Muller, 1972. Variance minimization in single machine sequencing problems, Management Science 18, pp. 518-528.

4. W. Kubiak, 1993. Completion time variance minimization on a single machine is difficult. Operations Research Letters 14, pp. 49-59.

5. L. Schrage, 1975. Minimizing the time-in-system variance for a finite jobset. Management Science 21, pp.540-543.

6. Hall, N.G., Kubiak, W. Proof of a conjecture of Schrage about the completion time variance problem. Operations Research Letters. Vol.10, Issue 8, 1991. pp. 467-472.

7. V. Vani and M. Raghavachari, 1987, Deterministic and random single machine sequencing with variance minimization, Oper. Res. 35 (1987), pp. 111-120.

8. Eilon S., Chowdhury I.G., 1977. Minimizing Waiting Time Variance in the Single Machine Problem, Management Science, Vol. 23, pp. 567-574.

9. X. Cai, 1995. Minimization of agreeably weighted variance in single machine systems. European Journal of Operational Research, pp. 576-592.

10. W. Kubiak, Cheng J. and Kovalyov M.Y., 2002. Fast fully polynomial approximation schemes for minimizing completion time variance. European Journal of Operational Research, Vol. 137, Issue 2, pp. 303-309

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Q / A

Thanks for listening!