State-dependent Limited Processor Sharing - Approximations and Optimal Control

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1 State-dependent Limited Processor Sharing - Approximations and Optimal Control Varun Gupta University of Chicago Joint work with: Jiheng Zhang (HKUST)

2 State-dependent Limited Processor Sharing Processor Sharing μ n jobs at server service rate μ/n per job 2

3 State-dependent Limited Processor Sharing State-dependent Processor Sharing n jobs at server service rate /n per job

4 State-dependent Limited Processor Sharing FCFS buffer Sharing Limit Control (L) State-dependent Processor Sharing Motivation Congestion based friction e.g., server thread-pool management Service systems with human agents GOAL: Design of good control policies 4

5 State-dependent Limited Processor Sharing FCFS buffer Sharing Limit Control (L) State-dependent Processor Sharing Straw-man: Set to maximum efficiency point (L*) Depends on arrival rate and variance of job size distribution Low L FCFS dominates good for low variance High L PS dominates good for high variance 5

6 State-dependent Limited Processor Sharing FCFS PS GOAL : Design good control policies Two classes 1. Static control policy Sub-goal: Approximation for a given control L 2. Dynamic control policy Sub-goal: Numerical algorithm to solve a dynamic control problem Setting: Analysis under diffusion scaling Sharing Limit Control (L) 6

7 1. Diffusion Approximation for Static policies 7

8 Approximation for static sharing limit FCFS PS General i.i.d. interarrivals λ = arrival rate c a = coefficient of variation General i.i.d. service requirements c s = coefficient of variation L = static sharing limit 1.3 Sharing Limit Control (L) 1.1 Q: A meaningful asymptotic regime? λ That is, construct a sequence of LPS systems that faithfully approximates the original system L 8

9 Q: A meaningful asymptotic regime? Proposal #1: Conventional heavy traffic λ (r) μ L λ Does not capture original system! L 9

10 Q: A meaningful asymptotic regime? Proposal #2: Scale sharing limit L (r) = Lr λ Stretch service rate curve μ r (rx) μ x where μ x is a continuous extension of μ n L Also not faithful enough! In this example, system gets stuck at 0. 10

11 An axiomatic approach for state-dependent systems IDEA: Define what we mean by faithful And reverse engineer the asymptotic scaling Proposal Feed the original system M/M/ input 0.6 Let H(n) 0.4 H n = Prob N n 0.2 and H a continuous differentiable interpolation L 11

12 An axiomatic approach for state-dependent systems IDEA: Define what we mean by faithful And reverse engineer the asymptotic scaling Proposal Feed the original system M/M/ input Let H(n) H n = Prob N n and H a continuous differentiable interpolation SCALING: In the r th system L (r) = Lr Under M/M/ input: H r (rx) H x L H x 12

13 SCALING: In the r th system L (r) = Lr Under M/M/ input: H r (rx) H x We guarantee a limit that depends on the entire μ n Reverse engineered service rates: H(n) L H x lim r λ μ r rx = λ r d log( h x ) dx θ(x) drift function 1.3 Entire μ r curve collapses to λ at rate 1/r λ

14 Final Approximation for mean number in system: E N n=0 (n L) π(n) n=0 π(n) c 2 s +1 c 2 s +c2 a c 2 s +1 c 2 s +c2 a + c s n=0 (n L) + π(n) n=0 π(n) c 2 s +1 c 2 s +c2 a c 2 s +1 c 2 s +c2 a where π(n) is probability mass function for the original system under M/M/ input E[N] Sharing Limit (L) Weibull Pareto (1.1) LogNormal Approx. 14

15 2. Diffusion control problem for dynamic policies 15

16 Dynamic policies Static policies too sensitive to λ IDEA: Dynamically adjust L based on congestion EXAMPLE: Drift function Policy (c s 2 = c a 2 = 10) Θ(x) L(n) L* L* x = number at server n = number in system 16

17 Dynamic policies Static policies too sensitive to λ IDEA: Dynamically adjust L based on congestion EXAMPLE: Drift function Policy (c s 2 = c a 2 = 0.3) Θ(x) L(n) L* x = number at server n = number in system 17

18 Dynamic policies CRUX: an average cost diffusion control problem Avg. cost of opt policy cost function w: v = min k A(w) c w, k θ k G w + σ2 2 G (w) workload (state space) Action space Gradient of relative value function Arbitrary θ(k) Must resort to numerical methods State of the art: discretize into a locally consistent MDP (see Kushner, Dupuis) 18

19 Average cost diffusion control problem w: v = min k A(w) c w, k θ k G w + σ2 2 G (w) Would be done if knew G(0) and v FACT: G(0) = 0 since the diffusion reflects at w = 0 What about v? AVG. COST ITERATION ALGORITHM: 1. Guess v 2. Check if v < v, or v > v 3. Refine guess and iterate until ε-optimal 19

20 w: v = min k A(w) c w, k θ k G w + σ2 2 G (w) AVG. COST ITERATION ALGORITHM: 1. Guess v 2. Check if v < v, or v > v 3. Refine guess and iterate until ε-optimal v < v THM: Value fn. is non-decreasing G() is non-negative v > v Continuation envelope! G(w) w G(w) w! 20

21 w: v = min k A(w) c w, k θ k G w + σ2 2 G (w) ALGORITHM 1: 1. Guess v 2. Check if v < v, or v > v - Test events occur for w = O log 1 v v 3. Refine guess and iterate until ε-optimal - O log 1 ε iterations using binary search ALGORITHM 2: More sophisticated; based on Newton- Raphson root finding method needs O log log 1 iterations ε 21

22 SUMMARY Two general concepts for control of systems with statedependent parameters 1. An axiomatic approach to asymptotic scaling Fix the limit under a tractable arrival process, and reverse engineer the sequence to guarantee the limit 2. A numerical tool Average cost iteration algorithm for 1-dimensional diffusions with one known reflection 22

Designing load balancing and admission control policies: lessons from NDS regime

Designing load balancing and admission control policies: lessons from NDS regime VARUN GUPTA University of Chicago Based on works with : Neil Walton, Jiheng Zhang ρ K θ is a useful regime to study the