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Paper Presentation Amo Guangmo Tong University of Taxes at Dallas gxt140030@utdallas.edu January 24, 2014 Amo Guangmo Tong (UTD) January 24, 2014 1 / 30 Overview Tardiness Bounds under Global EDF Scheduling on a Multiprocessor 1 An O(m)
Amo Guangmo Tong
University of Taxes at Dallas gxt140030@utdallas.edu
January 24, 2014
Amo Guangmo Tong (UTD) January 24, 2014 1 / 30
1
Tardiness Bounds under Global EDF Scheduling on a Multiprocessor
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An O(m) Analysis Technique for Supporting Real-Time Self-Suspending Task Systems
Amo Guangmo Tong (UTD) January 24, 2014 2 / 30
Tardiness Bounds under Global EDF Scheduling on a Multiprocessor UmaMaheswari C. Devi and James H. Anderson
Amo Guangmo Tong (UTD) January 24, 2014 3 / 30
Abstact
The major topic in this paper is the scheduling of soft real-time sporadic task systems on a multiprocessor. This paper derives the tardiness bound under preemptive or non-preemptive global earliest-deadline-first (EDF) schedule when the total utilization is less than or even equal to the number of processors.
Amo Guangmo Tong (UTD) January 24, 2014 4 / 30
Soft real-time system T n sporadic tasks Tn m identical processors ,m > 1 minimum inter-arrival time or period, pi > 0 execution cost ei < pi relative deadline Di = pi utilization ui = ei/pi Usum = ui ≤ m Umax(k) denotes the subset of k tasks with highest utilizations in T Emax(k) denotes the subset of k tasks with highest executions in T
Amo Guangmo Tong (UTD) January 24, 2014 5 / 30
Processor Sharing(PS) schedule In a PS schedule, each job of Ti is allocated a fraction ui of a processor at each instant. Every job will complete executing exactly at its deadline. As long as Usum = ui ≤ m , a valid PS schedule exists. Earliest-Deadline-First(EDF) schedule In a EDF schedule, the job with earlier deadline has higher priority. Non-preemptive Earliest-Deadline-First(NP-EDF) schedule In a NP-EDF schedule, the job with earlier deadline has higher priority and a job cannot be paused until its completion.
Amo Guangmo Tong (UTD) January 24, 2014 6 / 30
For any given soft real-time system T A(PS, Ti, t) denotes the allocation of resource to task Ti within the interval [0, t] under PS schedule; A(EDF, Ti, t) denotes the allocation
lag(Ti, t) = A(PS, Ti, t) − A(EDF, Ti, t) LAG(T, t) =
i≤n lag(Ti, t)
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Step 1 Let td = di,j denotes the deadline of Ti,j. Notice that jobs with lower priority have no influence on the schedule of jobs with higher priority. We ignore the jobs with priority lower than Ti,j. In this case, LAG(T, td) represents the workload remaining to be done after td in order to finish all the execution. Intuitively, the tardiness bound
Amo Guangmo Tong (UTD) January 24, 2014 8 / 30
Step 2 Lemma 1 Let the tardiness of Tk with a deadline less than td be at most x + ek, where x ≥ 0, for all 1 ≤ i ≤ n under the EDF schedule for system T on m processors. If LAG(T, td) ≤ mx + ei . Then, the tardiness of Ti,j is at most x + ei. We denote mx + ei as Lower Bound.
Amo Guangmo Tong (UTD) January 24, 2014 9 / 30
Step 3 Lemma 2 Let the tardiness of Tk with a deadline less than td be at most x + ek, where x ≥ 0, for all 1 ≤ i ≤ n under the EDF schedule for system T on m processors. And Usum ≤ m. Then LAG(T, td) ≥ x ·
Ti∈Umax(m−1) ui + Ti∈Emax(m−1) ei
We denote x ·
Ti∈Umax(m−1) ui + Ti∈Emax(m−1) ei as Upper Bound.
Amo Guangmo Tong (UTD) January 24, 2014 10 / 30
Step 4 Let the UpperBound not more than LowerBound (x ·
Ti∈Umax(m−1) ui + Ti∈Emax(m−1) ei) ≤ (mx + ei)
Solve this inequation. Then, x ≥ [ (
Ti∈Emax(m−1) ei) − emin
m −
Ti∈Umax(m−1) ui
], in which case the condition in Lemma 1 can be satisfied. According to the Lemma 1, the tardiness of job Ti,j in task Ti is at most [ (
Ti∈Emax(m−1) ei) − emin
m −
Ti∈Umax(m−1) ui
] + ei. By induction, the tardiness bound of all of the jobs in task Ti is [ (
Ti∈Emax(m−1) ei) − emin
m −
Ti∈Umax(m−1) ui
] + ei.
Amo Guangmo Tong (UTD) January 24, 2014 11 / 30
The tardiness bounds under global EDF for task Ti in a sporadic real-time systems on multiprocessors is [ (
Ti∈Emax(m−1) ei) − emin
m −
Ti∈Umax(m−1) ui
] + ei, when the total utilization of a task system is not more than the number of processors, m, which means the resource can be fully utilized in the long run. The tardiness bound under non-preemptive global EDF can be derived by the similar framework with slight difference in the prove
Amo Guangmo Tong (UTD) January 24, 2014 12 / 30
Multiprocessor
Multiprocessor-based design has been used widely in the present-day
people and machines, and signal-processing systems such as synthetic-aperture. In this case, the previous theories on uniprocessor are insufficient.
Soft real-time system
Then, not all real-time systems require that the deadline of job has to be
tolerant tardiness as long as it is bounded with limitation.
Non-preemptive
Finally, this paper derives a tardiness bound under non-preemptive global EDF, which provides a theory foundation for the case in practice that tasks cannot be paused before completion.
Amo Guangmo Tong (UTD) January 24, 2014 13 / 30
An O(m) Analysis Technique for Supporting Real-Time Self-Suspending Task Systems Liu Cong and James H. Anderson
Amo Guangmo Tong (UTD) January 24, 2014 14 / 30
Abstact
This paper considers soft real-time self-suspending task systems under the preemptive global EDF and derives a new schedulability test with O(m) suspension-related utilization loss. Similar to the first paper, this paper also analyze the tardiness bound by using the allocation difference between PS schedule and EDF schedule
Amo Guangmo Tong (UTD) January 24, 2014 15 / 30
Soft real-time Self-Suspending system T Self-suspending: jobs alternate between computation and suspension phase n sporadic tasks Tn m identical processors ,m > 1 minimum inter-arrival time or period, pi > 0 execution cost ei; suspension costsi; ei + si < pi relative deadline Di = pi utilization ui = ei/pi; suspension ratio vi = si/pi; u = ui + vi Usum = ui ≤ m
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Processor Sharing(PS) schedule In a PS schedule, each job of Ti is allocated a fraction ui of a processor at each instant. Every job will complete executing exactly at its deadline. As long as Usum = ui ≤ m , a valid PS schedule exists. Earliest-Deadline-First(EDF) schedule In a EDF schedule, the job with earlier deadline has higher priority.
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In the first paper, the critical point is if t is non-busy time instant there are at most m-1, tasks have tardy jobs at t. Otherwise, t cannot be non-busy. However, when it comes to self-suspending system, there could be at most n tasks have tardy jobs at a non-busy instant t.
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Convert all the suspension into execution? Then, e
′
i = ei + si. To guarantee a valid PS schedule, we need
U
′
sum = Usum + i≤n si pi ≤ m
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If ,by converting at most m tasks’ suspensions into execution at any instant t in PS, we can guarantee that any non-busy instant t
′ in
EDF there are at most m − 1 tasks have tardy jobs. In this case, to guarantee a valid PS schedule, we need Usum +
Vmax(m) vi ≤ m
Amo Guangmo Tong (UTD) January 24, 2014 20 / 30
Consider processor Mk. Let tf be the finish time of the system. From tf to the left, let th denote the first encountered non-busy time instant on Mk where at least one task Ti has an tardy job Ti,v suspending at th. Let v − c denote the minimum job index of Ti such that all job from Ti,v−c to Ti,v are tardy. Then [A1
k, B1 k] = [ri,v−c, th + 1]. In this way, from ri,v−c we can get
[A2
k, B2 k]. Until time 0, we will get [Aq k, Bq k ] transformation intervals
respect to Mk. Note that each job in a transformation interval has its release time and deadline time within the interval. Move Switch Convert Do the same things to all processors.
Amo Guangmo Tong (UTD) January 24, 2014 21 / 30
Check At any non-busy time instant t ∈ [0, tf ] in the transformed schedule, at most m − 1 tasks can have enabled tardy jobs with deadlines at or before t. A valid PS schedule exists when Usum +
Vmax(m) vi ≤ m.
We denote the new schedule as PS and EDF
Amo Guangmo Tong (UTD) January 24, 2014 22 / 30
For any given soft real-time system T A(PS, Ti, t) denotes the allocation of resource to task Ti within the interval [0, t] under PS schedule; A(EDF, Ti, t) denotes the allocation
lag(Ti, t) = A(PS, Ti, t) − A(EDF, Ti, t) LAG(T, t) =
i≤n lag(Ti, t)
Amo Guangmo Tong (UTD) January 24, 2014 23 / 30
Step 1 Let td = di,j denotes the deadline of Ti,j. Notice that jobs with lower priority have no influence on the schedule of jobs with higher priority. We ignore the jobs with priority lower than Ti,j. In this case, LAG(T, td) represents the workload remaining to be done after td in order to finish all the execution. Intuitively, the tardiness bound
Amo Guangmo Tong (UTD) January 24, 2014 24 / 30
Step 2 Lemma 1 Let the tardiness of Tk with a deadline less than td be at most x + ek + sk, where x ≥ 0, for all 1 ≤ i ≤ n under the EDF schedule for system T on m processors. If LAG(T, td) ≤ mx . Then, the tardiness of Ti,j is at most x + ei + si. We denote mx as Lower Bound.
Amo Guangmo Tong (UTD) January 24, 2014 25 / 30
Step 3 Lemma 2 Let the tardiness of Tk with a deadline less than td be at most x + ek + sk, where x ≥ 0, for all 1 ≤ i ≤ n under the EDF schedule for system T on m processors. And Usum ≤ m. Let E denotes the largest value of the expression
τi∈τ(ei + si) + τi∈ψ u · ei, where ψ denotes
any set of m − 1 tasks. Let Um−1 be the sum of the m − 1 largest ui Then LAG(T, td) ≥ x · Um−1 + E We denote x · Um−1 + E as Upper Bound.
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Step 4 Let the UpperBound not more than LowerBound x · Um−1 + E ≤ (mx + ei + si). Solve this inequation. Then, x ≥ [E − ei − si m − Um−1 ], in which case, the condition in Lemma 1 can be satisfied.
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The tardiness bounds under global EDF for task Ti in a sporadic real-time systems on multiprocessors is [E − ei − si m − Um−1 ] + ei, when the total utilization of a task system is not more than the number of processors, m, which means the resource can be fully utilized in the long run. The tardiness bound under non-preemptive global EDF can be derived by the similar framework with slight difference in the prove
Amo Guangmo Tong (UTD) January 24, 2014 28 / 30
Suspension
As the paper says, suspension delays may occur in many systems due to the cases such as the interaction with external devices and resource
number of tasks is very large then the O(n) utilization loss can be highly significant.
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