Efficient and Robust Allocation Algorithms in Clouds under Memory Constraints

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1 Efficient and Robust Allocation Algorithms in Clouds under Memory Constraints Olivier Beaumont,, Paul Renaud-Goud Inria & University of Bordeaux Bordeaux, France 9th Scheduling for Large Scale Systems Workshop

2 Provider-side allocation Typical Cloud Computing scenario A number of clients submit services to a provider Think of services as commercial websites Services have hardware requirements (CPU, memory, I/O,...) Clients express demands: for each service, enough CPU must be available (serve enough requests per second) Optimization Several services can be allocated to the same Physical Machine (PM) Optimize energy and resource usage: consolidation Allocation of services onto PMs: MultiDimensional Bin Packing Problem

3 Introducing machine failures Data centers are large Failures will happen Our approach: over-provisioning allocate additional capacity Clients express a reliability requirement expressed as a probability, or as a cost penalty Assumptions Static setting over a given time period (between two migration phases): Compute an allocation at the start of the period During the time period, some machines fail Services should still be running (enough instances) at the end Machines fail independently with probability f

4 Problem formulation Notations Identical machines, with capacity C and memory M ns services: demands d i, reliability requirement r i, memory m i Variables A i,j : CPU capacity for service i on machine j Alive cpu i = m j=1 is alive j A i,j, where is alive j = 0 with prob. f Formulation minimize m s.t. i, P (Alive cpu i < d i ) < r i j, A i,j >0 m i M j, ns i=1 A i,j C

5 Two-step algorithm 1 Load Balancing Relaxed formulation to compute resource usage (CPU, memory) for each service so as to satisfy reliability constraints 2 Packing Use column generation techniques to obtain a feasible packing

6 First step: focus on reliability Approximation by normal distribution Given an allocation A i,j, computing P (Alive cpu i < d i ) is #P-complete. (as hard as counting the # of solutions to a knapsack problem)

7 First step: focus on reliability Approximation by normal distribution Given an allocation A i,j, computing P (Alive cpu i < d i ) is #P-complete. P (Alive cpu i < d i ) < r i is approx. by m j=1 A i,j B i m j=1 A2 i,j K i

8 First step: focus on reliability Approximation by normal distribution Given an allocation A i,j, computing P (Alive cpu i < d i ) is #P-complete. P (Alive cpu i < d i ) < r i is approx. by m j=1 A i,j B i m j=1 A2 i,j K i Relaxed formulation (lower bound) minimize m s.t. i, m j=1 A m i,j B i j=1 A2 i,j K i j A i,j >0 m i mm ns j i=1 A i,j mc

9 First step: focus on reliability Approximation by normal distribution Given an allocation A i,j, computing P (Alive cpu i < d i ) is #P-complete. P (Alive cpu i < d i ) < r i is approx. by m j=1 A i,j B i m j=1 A2 i,j K i Relaxed formulation (lower bound) minimize m s.t. i, m j=1 A m i,j B i j=1 A2 i,j K i j A i,j >0 m i mm ns j i=1 A i,j mc Homogeneous allocations are dominant (A i,j = A i or 0) A i CPU #machines n i

10 First step: focus on reliability Approximation by normal distribution Given an allocation A i,j, computing P (Alive cpu i < d i ) is #P-complete. P (Alive cpu i < d i ) < r i is approx. by m j=1 A i,j B i m j=1 A2 i,j K i Relaxed formulation (lower bound) i, n i > B i i n im i mm mc i K i 1 B i ni Homogeneous allocations are dominant (A i,j = A i or 0) A i CPU Can be solved for fractional n i. n i #machines

11 Second step: packing Solve the homogeneous pb to get ideal allocations n i, A i Pack them onto individual machines Remember the approximate formulation i, m j=1 A m i,j B i j=1 A2 i,j K i minimize m s.t. j, A i,j >0 m i M j, ns i=1 A i,j C Splitting a service always improves the first inequality CPU A i #machines n i

12 Second step: packing Solve the homogeneous pb to get ideal allocations n i, A i Pack them onto individual machines Remember the approximate formulation i, m j=1 A m i,j B i j=1 A2 i,j K i minimize m s.t. j, A i,j >0 m i M j, ns i=1 A i,j C Splitting a service always improves the first inequality Splittable bin packing with memory constraints Allocate n i A i total capacity to service i Chunk of at most A i on each machine Each chunk uses m i memory At most M memory used and C capacity per machine

13 LP Formulation with configurations Valid configuration C c i, fraction x i,c of the maximum capacity A i devoted to service S i c is valid iff i x i,c C and x i,c >0 m i M Variable λ c : number of machines with this configuration Almost full configurations are enough: x i,c = 1 or 0 except for one service Configuration formulation minimize c F λ c st i, c F λ c x i,c n i

14 Column generation Primal: minimize c F λ c st i, c F λ c x i,c n i Dual: maximize i n i p i st c F, i x i,c p i 1 Principle: iteratively augment the set of configurations F Start with a small set of configurations Solve the primal, get a sub-optimal solution Find a violated constraint in the dual (splittable knapsack) Add this configuration to the set and loop

15 Splittable knapsack Splittable knapsack problem Given A i, m i, p i, M and C, Find J and x i such that i J m i M, and i J x ia i C So as to maximize i J x ip i Properties Almost full configurations are dominant. NP-hard (from 2-Part) O(MC) Dynamic Programming algorithm to solve optimally.

16 Summary Two-step heuristic Compute a homogeneous lower bound n i, A i (binary search) Solve the packing problem Start with a small set of configurations Solve the primal, get a sub-optimal solution Find a violated constraint in the dual (splittable knapsack) Add this configuration to the set and loop

17 Experimental evaluation Setting Memory and CPU usage from public Google trace 150 jobs account for 90% of resource usage r i = 10 X, with X U(2, 8) f = 0.01 NoSharing heuristic Computes homogeneous allocations with i, A i = C Exactly one service per machine

18 Experimental results: number of machines Ratio of machines used to lower bound Algorithm NoSharing ColumnGeneration Time interval Running time of ColumnGeneration: mean 14.6s, max 22.2s

19 Experimental results: execution time Time (seconds) Number of services Number of configurations generated Number of services

20 Conclusions Explored allocation problems with reliability constraints Binomial approximation allows a good lower bound Reformulations & column generation give good heuristics Further directions Extensions: multi-dimensional (CPU + IO), heterogeneous machines,... Migrations for dynamic setting Recycle the techniques for other problems

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