Convex Optimization in Local Single-Threaded Parallel Mobile Computing Rashid Khogali Olivia Das Kaamran Raahemifar

Introduction “Single-threading Multi-buffer Scheduling & Processing Algorithm”(SMSP) ◦ Dictates which of the processors should process a given task based on classifying a set of minimized aggregate cost functions.  Each cost function is associated with a processing stream. ◦ Explicitly determines the processor’s optimum processing rate of executing the tasks. ◦ Multidimensional convex optimization problem.

Scenario Each processor has a memory queue that accommodates an arbitrary maximum number of tasks. Tasks and processors are heterogeneous.

Goal Find the optimized decision algorithm.  dictates which task goes to which processing stream.  “optimize” means to minimize both time and energy consumption. Determine the optimized processing rate of executing each task.

Assumptions Heterogeneous processors and tasks Online Constrained processing rates Energy cost affected by remaining energy level User determines unit cost of energy and time Stochastic availability Multiple energy sources

Definitions TaskT k = (m k, p μ,k, B k )  m k :memory requirement in bits.  p μ,k :minimum recommended execution rate of the task.  B k : number of base instructions. User ProfileU k = ( α ε,k, α t,k )  α ε,k : energy cost sensitivity factor($/Joule)  α t,k : time cost sensitivity factor($/Second)  α ε,k is treated with more objectivity than α t,k. Stream ProcessorP s,j  P s,j : operating frequency (base instructions/second)  p μ,k ≦ P s,j ≦ P Max,j

Definitions(cont.) Task’s Energy and Power Consumption ε k = λ j (p k ) 3 t k t k = B k / p k  ε k : expected energy consumption(Joules)  p k : actual execution rate  t k : actual execution time  B k : task’s number of base instructions  λ j : processor energy inefficiency coefficient ε k = λ j B k (p k ) 2

Constraints  M m : available memory  (E m,j – E θ,j ): usable battery energy of j th processing stream

Steps Assume the potential aggregate cost of introducing the task to each of the processing streams. Minimize the aggregate cost function by re-adjusting the processing rates of all tasks in the queue. Choose the stream with the lowest potential aggregate cost.

Cost Function C j : cost of the j th stream i j : # of task in queue ε %,j : remaining power A l,j : availability of executing T l in the j th stream t θ,r,j : overhead access time of a task T r to be accessed by P j

Cost Function(Cont.) Rearrange the cost function Assume A k,j = A j, ∀ k ∈ {1,2, …, i j } otherwise

Minimizing Cost Function “ i ” dimensional optimization problem for each stream. Adjustable parameter: p l Optimize C j

Minimizing Cost Function(Cont.)

Confirm Minima Use Hessian matrix [1] to confirm minima. [1] 海森矩陣：

Confirm Minima(Cont.)

Minimizing Constrained Cost Function Don’t forget “p μ,k ≦ P s,j ≦ P Max,j ”

Single-threading Multi-buffer Scheduling & Processing Algorithm User specifies α ε,k and α t,k for each T k ∈ T. For an arriving task T k ∈ T, evaluate and compare the minimum potential processing cost C min. T k is assigned to stream j* and to be processed at an adjusted optimum processing rate.

Single-threading Multi-buffer Scheduling & Processing Algorithm(Cont.) Execute T 1,j* at rate Update processing rate whenever a task is either introduced or deleted to Q s,j*.

Analytical Demonstration

Conclusion The authors propose a real-time multiprocessor scheduling algorithm(SMSP). The algorithm explicitly finds a globally optimum solution for each aggregate cost function. ◦ Minimizes the sum of both energy and execution time of tasks.

Assume ε %,j does not significant vary or is more or less a constant function of p k. ◦ The assumption is valid as long as the condition: ε k << E cap,j, is satisfied.