Towards Feasibility Region Calculus: An End-to-end Schedulability Analysis of Real- Time Multistage Execution William Hawkins and Tarek Abdelzaher Presented By: Farhana Dewan

Outline Introduction System Model Generalized Stage Delay Theorem Proof of the Theorem Usage of Feasibility Region Simulation Results Conclusion 2CSC 8260

Introduction Aperiodic distributed system- system is large complex, workload irregular Less attention than periodic counter part This paper presents ◦Analytic framework for computing end-to-end feasibility ◦Fixed-priority scheduling Based on generalized stage delay theorem ◦Maximum fraction of end-to-end deadline a task can spend at a resource as a function of utilization of that resource ◦Sum of such fractions are less than 1 Feasibility region is considered as a volume in multi- dimensional space where each dimension is utilization of one resource Extends uni-dimensional schedulable region to multi- dimensional representation for distributed systems Generalizes concurrent infinitesimal tasks to arbitrary set of finite tasks 3CSC 8260

Introduction Distributed real-time systems ◦Performance sensitive server farms ◦Radar data processing back ends ◦Sensor networks Different classes of traffic traverse several stages of distribute processing Task must exit the system within specified per- class end 2 end latency constraints Utilization bound of resource for centralized system ◦U ≤ U bound For distributed systems resource stage i has utilization U i ◦f(U 1 …,U n ) ≤ C bound ◦C bound systems capacity to meet deadlines 4CSC 8260

Introduction Goal: ◦Simple schedulability analysis technique for distributed rts to satisfy e2e timing constraints ◦Conditions are sufficient ◦Fast dynamic admission control Acyclic resource system ◦No feedback cycle in overall task flow graph ◦Synthetic utilization Non-acyclic resource system ◦Instantaneous utilization 5CSC 8260

System Model Distributed system, task T i arrive, require execution to N (subset of) resources A ij arrival time of T i at stage j, 1≤j ≤N A i arrival time of task to the system, A i1 D i e2e deadline for T i C ij computation time of T i at stage j Set of current tasks V(t)={T i |A i ≤t<A i +D i } Instantaneous utilization U j Synthetic utilization U j 6CSC 8260

Definitions Urgency inversion factor α j for stage j ◦ Less urgent task assigned greater priority ◦ α j = min (D lo /D hi ) over all tasks executing at stage j such that priority(T hi )>priority(T lo ) Blocking factor β ij ◦ Maximum amount of time task i can be blocked at stage j due to lower priority task holding critical resource Maximum normalized blocking factor ◦ γ j = max (β ij /D i ) CSC 82607

Generalized Stage Delay Theorem End to end schedulabiltiy condition: Σ j F j ≤1 8CSC 8260

Proof Stage j processing n concurrent tasks Instantaneous utilization at stage j for task T m To obtain lower bound, ignore lower priority tasks CSC 82609

Proof (cont.) Consider task T n at stage j Worst case delay at stage j, Q nj B is the end of last processor gap t f time at which T n departs stage j L = A n –B offset of arrival of T n on j For worst case arrival scenario, L=0 Max amount of time critical task is preempted by tasks with absolute deadline prior to t f, CSC

Proof (cont.) Busy period Rearranging and substituting, we obtain instantaneous utilization U j CSC

Proof (cont.) Worst case arrival sequence T= A 1j – A nj, A ij – A nj = T + Σ h=1 C hj Q nj = T + Σ i=1 C ij CSC

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Proof (cont.) D i is bounded by D n /α j and Σ is minimized when C ij = C for i=1 to n-1 CSC

Proof (cont.) Delay: To obtain fraction of deadline, divide by deadline F to be worst case bound, last term must be maximized CSC

Generalized Stage Delay Theorem Corollary 16CSC 8260

Usage of Feasibility Region Stages of computation: resources can be CPU, communication links, disks ◦Scheduling policy at each resource ◦α j and γ j must be pre computed Admission controller: based on generalized stage delay theorem or corollary Feasibility region calculus: to build admission controller ◦Each task arriving the system, utilization is added to U j of each stage to be traversed by the task ◦Check fractional delay, if greater than 1, don’t admit, reverse the utilization modification ◦AC checks that the system operates in feasibility region Complexity: linear in terms of number of stages, fractional stage delay in constant time 17CSC 8260

Simulation Results Simulator: distributed real-time system with arbitrary tasks Admission Controller: for either of the cases For each arriving task, its utilization is tentatively added to every stage j it will traverse during computation. The generalized stage delay theorem, or its corollary if applicable, is used to check whether Σ j F j ≤ 1 over the stages to be traversed. If so, the task is admitted. If not, the task is rejected and its utilization is removed from further consideration. Task granularity: ratio of total computation time and deadline Load: sum of computation time of all tasks divided by simulation time 18CSC 8260

Acyclic Task System Resources has increasing ids, a task leaving stage x never requires resource from stage i, 0 ≤i≤x Pipeline, 1 to 5 stages, each task must be executed by each stage from 1 to 5 in order Deadlines are drawn from uniform distribution Task granularity is 1/100 Load is varied from 60% to 200% Corollary of generalized stage delay theorem is used No task misses deadline Each point in the plot is average of 100 simulation runs Utilization is high for all offered loads, independent of no of stages, AC is not pessimistic CSC

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Non-Acyclic Task System Task may receive computation from same resource more than once (Ex- resource is database) Each task in the experiment traverses more than 1 stage in the system Task granularity 1/100, computation time approximately equal at each stages Load is varied between 60% and 200% Utilization of system with 1 stage is higher than that of 2,3 or 4 stages Lower priority task suffer from delay, whether delay is from higher priority task in same stage or other Stage delay corollary can be used as heuristic in admission controller to improve utilization CSC

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Comparing Stage Delay Theorem with Generalized Stage Delay Theorem Stage delay theorem ◦System with very large number of concurrent task ◦Calculates feasibility region based on utilization in the stages Generalized Stage delay theorem ◦Calculates feasibility region based on utilization and concurrent tasks in the stages Two stage pipeline distributed rts 6 task classes, arrival time and deadline from uniform distribution For moderate number of tasks and very small granularity gsdt performs better CSC

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Conclusion Presented: Analytical framework for computing the end-to-end feasibility regions of distributed aperiodic task systems under independent fixed-priority scheduling Extended: the previous derivations of uni- dimensional schedulability regions for single processors Generalized: the results for infinite number of concurrent liquid tasks to arbitrary sets of finite tasks Applicable to more realistic acyclic and non- acyclic workloads CSC

Future Work The results can be extended to: ◦Other categories of scheduling policies such as EDF ◦Systems that accept some percentage of deadline misses (soft real-time systems), relaxed schedulability conditions can be derived ◦System where tasks need multiple resources simultaneously CSC