Mark Shifrin. YEQT 2015 Eurandom, Eindhoven 12th November 2015

Transcription

1 Asymptotically optimal policies for the multi-class finite queue (partially joint with Rami Atar) Technion Israel Institute of Technology BGU Ben Gurion University Mark Shifrin YEQT 2015 Eurandom, Eindhoven 12th November 2015

2 Lets start with motivational use-case: Hybrid Cloud: Private + Public Private No outsourcing cost Low time response vs Public Unlimited capacity Pay per usage Private Only: Have to supply additional processing resources for the peak load

3 Cloudbursting: exploit both of the worlds The demand should be met => excessive demand goes to the Public capacity limit 3

4 Differentiated task types problem The pricing (if cloudburst), service rates (if locally processed) are different. Reserve space for more expensive tasks? Objective: Find the optimal rule how to cloudburst. (M. Shifrin, R. Atar, I. Cidon. "Optimal scheduling in the hybrid-cloud")

5 Outline Hybrid Cloud motivational example Model definition (2 configurations) Diffusion scaling (Reduced) Brownian control problem (RBCP) Limit solution and numerical study AO policies and theorem Optimality gap

6 Model: The multi-class queue with finite buffers The admission/scheduling problem: How to optimally decide on admission and service allocation 6

7 Type 1 Type 2 task arrivals Separate buffers Type K DECISION $ $ Cloud Computing Resourses $ Fraction 1 Fraction 2 Fraction K SERVER 7

8 Type 1 Type 2 task arrivals Type K Shared buffer DECISION $ $ Cloud Computing Resourses $ Fraction 1 Fraction 2 Fraction K SERVER 8

9 Processes, HT condition 9

10 Diffusion model

11 Why diffusion limit? Reason of Math: Under heavy traffic condition we have limit problem which the can solve. Reason of Practice: The limit means increase service and the arrival complies with modern computational scenarios. Especially: cloud computing.

12 The Brownian Control Problem (BCP) The solution to this is well understood, via 2 steps (1) Workload reduction Equivalence of BCP with Reduced BCP (RBCP) where RBCP is 1-Dimensional problem, V BCP =V RBCP Atar Rami, Sh.M, 2014 (2) The Harrison-Taksar free boundary value problem (1983) Provides the solution for the processes related to the workload

13 Reduced BCP (RBCP) and its solution Uniquness: Atar,Budhiraja,Willams, 2007

14 Queue-length vs. Workload rectangular case

15 The triangular case Cost minimization problem problem is written Which is translated into LP With canonical form P. R. Thie and G. E. Keough. An introduction to linear programming and game theory.

16 Solution to the LP At most two tasks are present (simplex method) Recursive formula for the priorities: Given find by

17 Queue-length vs. Workload- Triangular case (schematically)

18 Numerical solution of the HJB Rectangular: 3 task classes

19 ODE rectangular case

20 V and V numerical solutions

21 The optimal curve

22 Numerical solution of the HJB Triangular: 3 task types

23 ODE triangular case

24 V and V numerical solutions

25 The optimal curve, order of accumulation

26 An AO policy rectangular case Difficulty: being close to Solution: work with while avoiding forced rejections

27 AO policy examples (rectangle)

28 An AO policy triangular case Same difficulty:, avoid forced rejections For each define appropriate and approximate Rejection: similar to rectangular case Service: serve high priority. Low priority only for 2 classes which fill the buffer, j and j-1 (recall, ) :

29 AO policy examples (triangle)

30 Asymptotic optimality full theorem General lower bound (LB) Consider any admissible control U n : Theorem: Proof see Atar R. & S. M., 2014 Upper bound (specific for each case) Define AO policy U n (Є) control Theorem: Proof (rectangular) - Atar R. & S. M., 2014

31 Upper Bound proof: two major steps Step 1 The workload process converges to RBM => workload complies with the solution according to H-T in the asymptotical sense Step 2 (state space collapse) is close to the minimizing curve => multi-d process is closed to the processes which correspond to the solution of BCP

32 Step 1 (conclusion and summary) Work conserving: Boundary on a*: The definition of the Skhorohod problem is valid. is continuous so the has cont. paths => convergence to the RBM which is solution to H-T free boundary problem.

33 Step 2 We show Reminder the minimizing curve:

34 Step 2 (technique & conclusion) Use C-tightness of and work as in Step 1 We have convergence of the workload By providing additional technical results the theorem follows:

35 Simulation study: When diffusion approximation becomes practical? R1: Observe how optimal policy solved by MDP becomes closed to the solution in the limit 3 cases: B=15, 50, 125 square root of n R2: implement the proposed policy increase n, compare to the optimal solution See the optimality gap

36 B=15 ( )

37 B=50 ( )

38 B=125 ( )

39 The proposed policy

40 The proposed policy #2

41 The proposed policy #3

42 Optimality gap Ratio between the (simulated) cost under the proposed policy and the (computed) optimal cost as a function of sq. root(n). The graph shows values for sq. root(n) = 3, 5, 10 and 20. The corresponding buffer sizes are given by 5*(sq. root(n)), namely 15 15, 25 25, and ,respectively.

43 Conclusions & future work We presented treatment of multi-class finite queue by heavy traffic analyses: Two specific cases (triangular and rectangular) AO policy guidelines for the proof Relevance for the large networks, especially Cloud Computing Many more settings are relevant: Tasks take different number of slots (is it solvable?) Other limits (number of serving machines)

44 Thanks! 44