Ruth J. Williams University of California, San Diego math.ucsd/~williams

1. DYNAMIC SCHEDULING OF A PARALLEL SERVER SYSTEM IN HEAVY TRAFFIC UNDER A COMPLETE RESOURCE POOLING CONDITION 1 I 1 K Ruth J. Williams University of California, San Diego http://math.ucsd.edu/~williams Collaborators: Steven Bell, Maury Bramson

2. HEAVY TRAFFIC APPROACH TO DYNAMIC SCHEDULING 1. Formulate stochastic network model 2. Define heavy traffic (flexible servers) 3. Formal diffusion approximation: BCP (Brownian control problem) 4. Reduce to EWF (equivalent workload formulation) 5. Solve the BCP (or EWF) 6. Interpret the solution of the BCP 7. Analyze the performance of this policy • OVERALL APPROACH: Harrison (‘88), Laws (‘92), Kelly-Laws (‘93), Harrison-Van Mieghem (‘97), Harrison (‘00) • SOME RELATED WORK ON OPTIMAL CONTROL OF PARALLEL SERVER MODELS: Harrison (‘98), Harrison-Lopez (‘99), W (‘00), Squillante, Xia, Yao, Zhang (‘00), Bell-W (‘01). • Convex holding cost: Stolyar (‘04), Mandelbaum-Stolyar (‘04). • Kushner-Chen (‘00), Ata-Kumar (‘04).

3. PARALLEL SERVER SYSTEM 1 I 1 K total time allocated to activity j up to t

4. Model Primitives Stochastic Processes • E I -diml renewal arrival process, rate • S J-diml renewal potential service process, rate Assume certain independence and finite second moments of interarrival and service times Structure Assume jobs in buffers queued in FIFO order (HL service)

5. Heavy traffic (Harrison, ’00) Linear Program (LP) Heavy traffic (HT) There is a unique solution of the LP and Proposition(Bramson-W, ‘03) The heavy traffic condition holds iff there is a unique vector (of allocation rates) such that & in addition, (full server utilization)

6. Two-server example: Harrison ‘98 1 2 1 2 Holding cost per unit of queuelength per unit time:

7. Greedy Scheduling Policy • Simulation of this policy goes here

8. Diffusion approximation • Assume HT henceforth • For a suitable cost function, is the only reasonable “fluid” allocation • How should one achieve using a policy in the original system? • Diffusion scaling: • Basic activities: Non-basic activities: J • Cost function:

9. (Formal) Brownian Control Problem (Harrison, ‘00) where is a Brownian motion and

10. “Canonical” non-negative workload(Harrison-Van Mieghem ‘97, Harrison ’00) Dual program (DP) Extreme point solutions: Max. lin. indept. subset of : Theorem: and are all non-negative. If we let have rows given by , have rows given by , then there is a matrix :

11. COMPLETE RESOURCE POOLING 1 I 1 K Theorem (Harrison-Lopez ‘99) The following are equivalent. (i) the workload is one-dimensional, (ii)B=I+K-1, (iii) there is a unique solution of (DP), (iv) all servers communicate via basic activities. In fact, under any of these conditions, the server-buffer graph with basic activities as edges is a tree. (W, ’00; Squillante-Xia-Yao-Zhang, ’00)

12. Solution of the BCP under complete resource pooling (Harrison-Lopez ’99) • Unique soln of (DP): • One-dimensional workload process: • Holding cost: • Minimum workload: • Optimal queuelength and idletime:

13. How can one interpret the solution of the BCP? • Seek a policy that (a) keeps the bulk of the work in a buffer i* with smallest ratio of holding cost to workload contribution, (b) incurs idleness only when the system is nearly empty, (c) incurs the bulk of the idleness at a server k* that serves i* via a basic activity.

14. Two-server example Threshold policy(Bell-W, ‘01) Asymptotically optimal if (and finite exponential moments) 1 2 1 2

15. Parallel server exampleThreshold policy: W ‘00 3 4 1 2 1 2 3 3 T 3 4 2 2 T 1 1 Suppose i*= 4 Then k*=3 Threshold policy: Server 3 is the root. Buffer 4 has lowest priority. Place thresholds on transition activities. Servers give priority to transition activities below them in the tree, except suspend such an activity when the associated buffer is below threshold. Next priority goes to non-transition activities below server. Lowest priority to activities aboveserver.

16. Asymptotic optimality of tree based threshold policy • Theorem (Bell-W, ’04) Assume finite exponential moments. • If denotes the threshold control in the rth system, then for any other sequence of control policies we have where

18. Complete Resource Pooling (CRP) CRP: workload is one-dimensional and non-negative. Solution of the Brownian control problem “keep all of the work in the buffer i* with the smallest ratio of holding cost to workload contribution , and only allow server idling when the whole system is empty. How should one interpret this solution? Parallel server system:P=0 • Harrison-Lopez ‘99: • CRP iff all servers communicate via basic activities • proposed discrete review policy (proof of asymptotic optimality – special two server case: Harrison ’98) • W. ‘00 (see also Squillante, Xia, Yao, Zhang ‘00): • CRP iff server-buffer graph with edges given by basic activities is a tree • proposed threshold policy (continuous review) • Bell-W. ’01, ‘04 • Proof that threshold policy of W. ‘00 is asymptotically optimal (two server (AAP) & multiserver (in prep.)) General network with CRP: • S. Kumar‘99: proposed discrete review policy (example) • Ata-Kumar ’04: discrete review policy and proof of asymptotic optimality • Kang-W. (in prep.): proposed threshold policy (continuous review)