Heavy Traffic Limit Theorems for
Real-Time Computer Systems
Presented by:
John Lehoczky
Carnegie Mellon
Co-authors: B.Doytchinov, J.Hansen,
L.Kruk, R. Rajkumar, C.Yeung, and H.Zhu
Presented at WORMS04
April 19, 2004
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Background: 1
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Real-time systems refer to computer and
communication systems in which the
applications/tasks/jobs/packets have explicit
timing requirements (deadlines).
These arise in (e.g.):
– voice and video transmission (e.g. videoconferencing)
– control systems (e.g. automotive)
– avionics systems
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Background: 2
We often distinguish different types of real-time
systems or tasks:
•Hard real-time: any failure to meet a deadline is
regarded as a system failure. (e.g. avionics or
control systems)
•Soft real-time: deadline misses or packet loss is
acceptable as long as it doesn’t reduce the QoS
below requirements (e.g. multi-media
applications).
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Goals
•
For a given workload model we want:
– to predict the fraction of the workload that will
miss its deadlines (end-to-end deadlines in the
network case),
– to design workload scheduling and control
policies that will ensure QoS guarantees (e.g. a
suitably small fraction miss their deadlines),
– to investigate network design issues, e.g.:
• Number of priority bits needed
• Cost/benefit from flow tables
• Cost/benefit from keeping lead-time
information
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Formulation
In the hard real-time formulation where no
deadlines misses are permitted, one must adopt a
worst case formulation:
• task arrivals occur as soon as possible,
• task services take on their maximum values,
• task deadlines are as short as possible.
One must bound the worst case utilization.
But it average case utilization is substantially less
than worst case utilization, the system will, on
average, be highly underutilized.
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Model
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Multiple streams in a multi-node acyclic network.
Independent streams of jobs.
Jobs in a stream form a renewal process and
have independent computational requirements at
each node
For a given stream, each job has an i.i.d. deadline
(different for different streams)
Node processing is EDF (Q-EDF), FIFO, PS, HOLPS, Fixed Priority.
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Analysis: 1
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In addition to tracking the workload at each node,
we need to track the lead-time (= time until
deadline elapses) for each task.
The dimensionality becomes unbounded, and
exact analysis is impossible.
We resort to a heavy traffic analysis. This is
appropriate for real-time problems. If we can
analyze and control under heavy traffic, moderate
traffic will be better.
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Analysis: 2
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Heavy traffic analysis (traffic intensity on each
node converges to 1)
One node – workload converges to Brownian
motion. Multiple nodes, workload may converge
to RBM (depending upon scheduling policy).
Conditional on the workload, lead-time profile
converges to a deterministic form depending
upon
– flow deadline distributions,
– scheduling policy
– traffic intensity
Combining the lead-time profile with the
equilibrium distribution of the workload process,
we can determine the lateness fraction for each
flow.
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Processor Sharing – Exp. Deadlines
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Processor Sharing – Exp. Deadlines
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Processor Sharing – Exp. Deadlines
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Processor Sharing – Exp. Deadlines
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Processor Sharing–Const. Deadlines
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Processor Sharing-Const. Deadlines
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Processor Sharing-Const. Deadlines
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EDF Miss Rate Prediction
=0.95
EDF scheduling
Uniform(10,x) deadlines
Internet
Exponential
Uniform
EDF Deadline
Miss Rate:
_
EDF  e D
: computed from
the first two
moments of task
inter-arrival times
and service times.
_
D : Mean Deadline
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Motivation/Payoff
CPU 1
CPU 2
Server 1
Stream 1
Server 1.1
FIFO
Stream 2
Server 4
FIFO
FIFO
Server 2
Server 5
FIFO
FIFO
Server 3
Stream 3
Server 6
Stream 4
FIFO
FIFO
Stream
Arrivals
Service Time
Bounds
Lower
Upper
Inter-arrival Time
Bounds
Lower
Upper
Traffic
Intensity
Avg.
Constant
Deadline
Worst
Case
Stream 1
Sporadic
1
5
7.5
12.5
0.300
0.667
100
Stream 2
Sporadic
2
8
15
25
0.250
0.533
200
Stream 3
Sporadic
20
23
45
55
0.430
0.511
500
.980
1.711
Worst case is not schedulable (util. exceeds 100%)
Miss rate is 10-14
Miss rate is only 10-7 even if deadlines are halved.
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