On the Minimization of Weighted Waiting Time Variance

1. INFORMS San Francisco 2005 On the Minimization of Weighted Waiting Time Variance Xueping Li Dept. of Industrial & Information Engineering University of Tennessee, Knoxville

2. Outline • Introduction & Motivation • Formulation of Weighted Waiting Time Variance minimization problem • Optimal sequences analysis • Development of WVS & WSS algorithms • Results & discussions • Q/A

3. One Waiting Time Variance (WTV) Minimization Example 2 4 5 3 1 Jobs Sequence 1 5 4 2 1 3 Mean WT = 8 Waiting Time WTV = 31.5 0 5 9 12 14 Sequence 2 4 2 3 5 1 Mean WT = 5.4 Waiting Time WTV = 13.8 0 4 6 7 10 … Which one of the 5! sequences is optimal? WTV is NP-hard [4].

4. 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.

5. Quality 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

6. 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

7. 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

8. WWTV Formulation • Objective function: • Subject to:

9. 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]

10. Weight 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]

11. 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

12. Strong V-Shape Tendency PW NW RW

13. Development 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.

14. Development 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.

15. 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

16. Testing Results:Small-size problems • WWTVD and WMWTD PW NW RW

17. Testing Results:Large-size problems • Pair Comparison & Mean WWTV Comparison Pareto Normal PW NW RW

18. WWTV Computational Time Comparison • Computational Time Comparison of the Scheduling Methods (in Milliseconds)

19. 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

20. References • N. Ye, 2002. QoS-Centric stateful resource management in information systems. Information Systems Frontiers, Vol. 4, No. 2, pp. 149-160 • Lawrence, T. F., 1997. The quality of service model and high assurance. In Proceedings of the IEEE High Assurance Systems Engineering Workshop. • A.G. Merten and M.E. Muller, 1972. Variance minimization in single machine sequencing problems, Management Science 18, pp. 518-528. • W. Kubiak, 1993. Completion time variance minimization on a single machine is difficult. Operations Research Letters 14, pp. 49-59. • L. Schrage, 1975. Minimizing the time-in-system variance for a finite jobset. Management Science 21, pp.540-543. • 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. • V. Vani and M. Raghavachari, 1987, Deterministic and random single machine sequencing with variance minimization, Oper. Res. 35 (1987), pp. 111-120. • Eilon S., Chowdhury I.G., 1977. Minimizing Waiting Time Variance in the Single Machine Problem, Management Science, Vol. 23, pp. 567-574. • X. Cai, 1995. Minimization of agreeably weighted variance in single machine systems. European Journal of Operational Research, pp. 576-592. • 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

21. Q / A Thanks for listening!