Towards Feasibility Region Calculus: An End-to-end
Schedulability Analysis of Real-Time Multistage Execution
William Hawkins and Tarek Abdelzaher
Presented By: Farhana Dewan
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Outline
IntroductionSystem ModelGeneralized Stage Delay TheoremProof of the TheoremUsage of Feasibility RegionSimulation ResultsConclusion
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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
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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 ≤ Ubound
For distributed systems resource stage i has utilization Ui ◦ f(U1 …,Un) ≤ Cbound
◦ Cbound systems capacity to meet deadlines
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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
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System Model
Distributed system, task Ti arrive, require execution to N (subset of) resources
Aij arrival time of Ti at stage j, 1≤j ≤N Ai arrival time of task to the system, Ai1
Di e2e deadline for Ti Cij computation time of Ti at stage j Set of current tasks V(t)={Ti|Ai≤t<Ai+Di} Instantaneous utilization Uj
Synthetic utilization Uj
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Definitions
Urgency inversion factor αj for stage j◦Less urgent task assigned greater priority◦αj = min (Dlo /Dhi ) over all tasks executing at stage j such that priority(Thi)>priority(Tlo)
Blocking factor βij ◦Maximum amount of time task i can be blocked at stage j due to lower priority task holding critical resource
Maximum normalized blocking factor◦γj = max (βij /Di)
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Generalized Stage Delay Theorem
End to end schedulabiltiy condition: Σj Fj ≤1
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Proof
Stage j processing n concurrent tasks Instantaneous utilization at stage j for task Tm
To obtain lower bound, ignore lower priority tasks
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Proof (cont.)
Consider task Tn at stage j Worst case delay at stage j, Qnj
B is the end of last processor gap tf time at which Tn departs stage j L = An –B offset of arrival of Tn on j
For worst case arrival scenario, L=0 Max amount of time critical task is preempted by tasks with
absolute deadline prior to tf,
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Proof (cont.)
Busy period
Rearranging and substituting, we obtain instantaneous utilization Uj
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Proof (cont.)
Worst case arrival sequence
T= A1j – Anj, Aij – Anj = T + Σh=1 Chj Qnj = T + Σi=1 Cij
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Proof (cont.)
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Proof (cont.)
Di is bounded by Dn/αj and Σ is minimized when Cij = C for i=1 to n-1
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Proof (cont.)
Delay:
To obtain fraction of deadline, divide by deadline
F to be worst case bound, last term must be maximized
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Generalized Stage Delay Theorem Corollary
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Usage of Feasibility Region
Stages of computation: resources can be CPU, communication links, disks◦ Scheduling policy at each resource◦ αj and γj 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 Uj 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
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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 Σj Fj≤ 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
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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
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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
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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
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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
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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