Towards Feasibility Region Calculus: An End-to-end

Schedulability Analysis of Real-Time Multistage Execution

William Hawkins and Tarek Abdelzaher

Presented By: Farhana Dewan

CSC 8260 2

Outline

IntroductionSystem ModelGeneralized Stage Delay TheoremProof of the TheoremUsage of Feasibility RegionSimulation ResultsConclusion

CSC 8260 3

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

CSC 8260 4

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

CSC 8260 5

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

CSC 8260 6

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

CSC 8260 7

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)

CSC 8260 8

Generalized Stage Delay Theorem

End to end schedulabiltiy condition: Σj Fj ≤1

CSC 8260 9

Proof

Stage j processing n concurrent tasks Instantaneous utilization at stage j for task Tm

To obtain lower bound, ignore lower priority tasks

CSC 8260 10

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,

CSC 8260 11

Proof (cont.)

Busy period

Rearranging and substituting, we obtain instantaneous utilization Uj

CSC 8260 12

Proof (cont.)

Worst case arrival sequence

T= A1j – Anj, Aij – Anj = T + Σh=1 Chj Qnj = T + Σi=1 Cij

CSC 8260 13

Proof (cont.)

CSC 8260 14

Proof (cont.)

Di is bounded by Dn/αj and Σ is minimized when Cij = C for i=1 to n-1

CSC 8260 15

Proof (cont.)

Delay:

To obtain fraction of deadline, divide by deadline

F to be worst case bound, last term must be maximized

CSC 8260 16

Generalized Stage Delay Theorem Corollary

CSC 8260 17

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

CSC 8260 18

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

CSC 8260 19

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 8260 20

CSC 8260 21

CSC 8260 22

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 8260 23

CSC 8260 24

CSC 8260 25

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 8260 26

CSC 8260 27

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 8260 28

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