## A Probabilistic Framework for Schedulability Analysis (2003)

Venue: | Proceedings of the Third International Conference on Embedded Software (EMSOFT 2003 |

Citations: | 19 - 1 self |

### BibTeX

@INPROCEEDINGS{Burns03aprobabilistic,

author = {A. Burns and G. Bernat and I. Broster},

title = {A Probabilistic Framework for Schedulability Analysis},

booktitle = {Proceedings of the Third International Conference on Embedded Software (EMSOFT 2003},

year = {2003},

pages = {1--15}

}

### OpenURL

### Abstract

The limitations of the deterministic formulation of scheduling are outlined and a probabilistic approach is motivated. A number of models are reviewed with one being chosen as a basic framework. Response-time analysis is extended to incorporate a probabilistic characterisation of task arrivals and execution times.

### Citations

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Citation Context .../no guarantee is obtained while faults below a threshold of likelihood are ignored. 4 Probabilistic Guarantees with Non-Periodic Work Initially scheduling analysis assumed a purely periodic work flow =-=[28]-=-. The sporadic jobs were incorporated by assuming a minimum arrival time, that in the worst case was exhibited by the system. In effect a sporadic job behaved exactly the same as a periodic one. Respo... |

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Citation Context ...ill induce a short one next time). 5.1 Use of Copulas Copulas are a general mathematical tool to construct multivariate distributions and to investigate dependence structures between random variables =-=[32]-=-. A copula is basically a joint distribution function with uniform marginals. The main feature is that they allow one to separate the marginal distributions from the dependency between the two random ... |

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Citation Context ...n (5) to find the minimum value of Tf that leads to the system being just schedulable. As indicated earlier, let this value be denoted as TF (it is the threshold fault interval). Sensitivity analysis =-=[39, 24, 23, 34]-=- is used with fixed priority systems to investigate the relationship between values of key task parameters and schedulability. For an unschedulable system it can easily generate (using simple branch a... |

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Citation Context .... We assume a single processor platform and restrict the model to tasks with D ≤ T . For this restriction, an optimal set of priorities can be derived such that Di < Dj ⇒ Pi > Pj for all tasks τi=-=, τj [26]-=-. Tasks may be periodic or sporadic (as long as two consecutive releases are separated by at least T ). Once released, a task is not suspended other than by the possible action of a concurrency contro... |

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Citation Context ...s of delay queues etc. can easily be incorporated into the model [21][10] but are ignored here. The worst-case response time (completion time) Ri for each task (τi) is obtained from the following [20=-=][2]: Ri = Ci-=- + Bi + � � � Ri (1) j∈hp(i) where hp(i) is the set of higher priority tasks (than τi), and Bi is the maximum blocking time caused by a concurrency control protocol protecting shared data. To... |

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Citation Context ...tions of delay queues etc. can easily be incorporated into the model [21][10] but are ignored here. The worst-case response time (completion time) Ri for each task (τi) is obtained from the following=-= [20][2]: Ri =-=- Ci + Bi + � � � Ri (1) j∈hp(i) where hp(i) is the set of higher priority tasks (than τi), and Bi is the maximum blocking time caused by a concurrency control protocol protecting shared data.... |

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Citation Context ...n (5) to find the minimum value of Tf that leads to the system being just schedulable. As indicated earlier, let this value be denoted as TF (it is the threshold fault interval). Sensitivity analysis =-=[39, 24, 23, 34]-=- is used with fixed priority systems to investigate the relationship between values of key task parameters and schedulability. For an unschedulable system it can easily generate (using simple branch a... |

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Citation Context ...surprising that this single parameter approach is becoming limited in its application. Even a two (average and worst-case) or three (add minimum) parameter scheme is far from adequate. In other works =-=[8, 16, 4]-=- we have argued that it is now inadequate to use analysis alone to obtain a single worst-case execution time (WCET) value. Rather a combination of analysis and measurement must be used to obtain a pro... |

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Citation Context ...ask, however, may be preempted at any time by a higher priority task. System overheads such as context switches and kernel manipulations of delay queues etc. can easily be incorporated into the model =-=[21][10] but -=-are ignored here. The worst-case response time (completion time) Ri for each task (τi) is obtained from the following [20][2]: Ri = Ci + Bi + � � � Ri (1) j∈hp(i) where hp(i) is the set of hi... |

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Citation Context ...o the most stringent hard real-time application. Nevertheless, the early work on probabilistic scheduling analysis has been driven by a wish to devise effective QoS control for soft real-time systems =-=[1, 17, 18, 38]-=-. In this paper we consider four interlinked themes: 1 This paper is approximately the same as the one at EmSoft2003s1. Probabilistic guarantees for fault-tolerant systems 2. Representing non-periodic... |

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Citation Context ...sed in Sections 3 to 5 of this paper. Before that we give a short review of standard schedulability analysis using a fixed priority scheme as the underlying dispatching policy (see Burns and Wellings =-=[11] f-=-or a detailed discussion of this analysis). We restrict our consideration to the scheduling of single resources – processors or networks. In Section 6 we bring the discussion together and draw some ... |

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Citation Context ...ent faults. In some applications this frequency can be quite large; one experiment on a satellite system observed 35 transient faults in a 15 minute interval due to cosmic ray ions [12]. Hou and Shin =-=[19]-=- have studied the probability of meeting deadlines when tasks are replicated in a hardware-redundant system. However, they only consider permanent faults without repair or recovery. A similar problem ... |

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Citation Context ...ecution of some routine, the execution of an exception handler or a recovery block. Various algorithms have been published whichsattempt to maximise the available resources for this extra computation =-=[37, 35, 3]-=-. Here we consider the nature of the guarantee that these algorithms provide. Most approaches make the common homogeneous Poisson process (HPP) assumptions that the fault arrival rate is constant and ... |

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Citation Context ...τi or any higher priority task. We assume that any extra computation for a task will be executed at the task’s (fixed) priority 3 . Hence if there is just a single fault, equation (1) will become [=-=33][7] 4 : Ri = Ci +-=- Bi + � � � Ri Cj + max k∈hep(i) (Fk) (4) j∈hp(i) where hep(i) is the set of tasks with priority equal or higher than τi, that is hep(i) = hp(i) ∪ {τi}. This equation can again be solved... |

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Citation Context ... the maximum fault arrival rate that can be tolerated [9] – represented by TF , the minimum fault arrival interval. – Calculate the maximum number of faults each task can tolerate before its deadl=-=ine [31]-=-. The first approach is more straightforward (there is only a single parameter) and is reviewed in the following section. The basic form of the analysis is to obtain TF from the task set, and then to ... |

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Citation Context ...ry large (for example LCM could be 200ms, and L fifteen years!). 3 Probabilistic Guarantees for Fault-tolerant Systems In this review we restrict our consideration to transient faults. Castillo at al =-=[13]-=- in their study of several systems indicate that the occurrences of transient faults are 10 to 50 times more frequent than permanent faults. In some applications this frequency can be quite large; one... |

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Citation Context ...surprising that this single parameter approach is becoming limited in its application. Even a two (average and worst-case) or three (add minimum) parameter scheme is far from adequate. In other works =-=[8, 16, 4]-=- we have argued that it is now inadequate to use analysis alone to obtain a single worst-case execution time (WCET) value. Rather a combination of analysis and measurement must be used to obtain a pro... |

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Citation Context ... C3 + .. + CN with probability bound ρ. 6 Other Relevant Work on Probabilistic Analysis There have been some other approaches using probabilistic methods in real-time systems. The work of Diaz et.al.=-= [15]-=- computes probability distributions of the response times of entirely periodic (fixed release times) task systems with random execution times. The work relies on the independence of the execution time... |

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Citation Context ...stead. More importantly, if the dependency is not known, then it is possible to find upper and lower bounds of the distribution function for any possible dependency between the marginal distributions =-=[5, 29, 14]-=-. Some generalisations of these results allow to tighten even more these bounds if partial knowledge of the dependence is known. 5.2 Representing Extreme Execution Times It was noted earlier that for ... |

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Citation Context ...em, and then use the system fault model to assign a probability to that behaviour. Two ways of doing this have been studied in detail. – Calculate the maximum fault arrival rate that can be tolerate=-=d [9] –-=- represented by TF , the minimum fault arrival interval. – Calculate the maximum number of faults each task can tolerate before its deadline [31]. The first approach is more straightforward (there i... |

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Citation Context ...o the most stringent hard real-time application. Nevertheless, the early work on probabilistic scheduling analysis has been driven by a wish to devise effective QoS control for soft real-time systems =-=[1, 17, 18, 38]-=-. In this paper we consider four interlinked themes: 1 This paper is approximately the same as the one at EmSoft2003s1. Probabilistic guarantees for fault-tolerant systems 2. Representing non-periodic... |

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Citation Context ...ecution of some routine, the execution of an exception handler or a recovery block. Various algorithms have been published whichsattempt to maximise the available resources for this extra computation =-=[37, 35, 3]-=-. Here we consider the nature of the guarantee that these algorithms provide. Most approaches make the common homogeneous Poisson process (HPP) assumptions that the fault arrival rate is constant and ... |

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Citation Context ...om execution times. The work relies on the independence of the execution times of the different tasks. The work improves on an earlier work by Gardner et.al. [18]. The works of Nissanke [25] and Eles =-=[30]-=- also tackle this problem. However, none of these approaches address the issue of extreme distributions or dependencies between execution times or task arrivals.s7 Conclusion: A Probabilistic Framewor... |

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Citation Context ...n (5) to find the minimum value of Tf that leads to the system being just schedulable. As indicated earlier, let this value be denoted as TF (it is the threshold fault interval). Sensitivity analysis =-=[39, 24, 23, 34]-=- is used with fixed priority systems to investigate the relationship between values of key task parameters and schedulability. For an unschedulable system it can easily generate (using simple branch a... |

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Citation Context ...alternative formulation is to start with a required guarantee (for example, probability of fault per task release of 10 −6 ) and to then test for schedulability. This is the approach of Broster et a=-=l [6]-=- and is outlined in Section 3.2. 2 MTBF usually stands for mean time between failures, but as the systems of interest are fault tolerant many faults will not cause system failure. Hence we use the ter... |

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Citation Context ...requent than permanent faults. In some applications this frequency can be quite large; one experiment on a satellite system observed 35 transient faults in a 15 minute interval due to cosmic ray ions =-=[12]-=-. Hou and Shin [19] have studied the probability of meeting deadlines when tasks are replicated in a hardware-redundant system. However, they only consider permanent faults without repair or recovery.... |

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Citation Context ...o the most stringent hard real-time application. Nevertheless, the early work on probabilistic scheduling analysis has been driven by a wish to devise effective QoS control for soft real-time systems =-=[1, 17, 18, 38]-=-. In this paper we consider four interlinked themes: 1 This paper is approximately the same as the one at EmSoft2003s1. Probabilistic guarantees for fault-tolerant systems 2. Representing non-periodic... |

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Citation Context ...stead. More importantly, if the dependency is not known, then it is possible to find upper and lower bounds of the distribution function for any possible dependency between the marginal distributions =-=[5, 29, 14]-=-. Some generalisations of these results allow to tighten even more these bounds if partial knowledge of the dependence is known. 5.2 Representing Extreme Execution Times It was noted earlier that for ... |

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Citation Context ... however, may be preempted at any time by a higher priority task. System overheads such as context switches and kernel manipulations of delay queues etc. can easily be incorporated into the model [21]=-=[10] but are -=-ignored here. The worst-case response time (completion time) Ri for each task (τi) is obtained from the following [20][2]: Ri = Ci + Bi + � � � Ri (1) j∈hp(i) where hp(i) is the set of higher... |

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Citation Context ... in τi or any higher priority task. We assume that any extra computation for a task will be executed at the task’s (fixed) priority 3 . Hence if there is just a single fault, equation (1) will beco=-=me [33][7] 4 : Ri = C-=-i + Bi + � � � Ri Cj + max k∈hep(i) (Fk) (4) j∈hp(i) where hep(i) is the set of tasks with priority equal or higher than τi, that is hep(i) = hp(i) ∪ {τi}. This equation can again be sol... |

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Citation Context ...ity of meeting deadlines when tasks are replicated in a hardware-redundant system. However, they only consider permanent faults without repair or recovery. A similar problem was studied by Shin et al =-=[36]-=-. Kim et al [22] consider another related problem: the probability of a real-time controller meeting a deadline when subject to permanent faults with repair. To tolerate transient faults at the task l... |

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Citation Context ...s long as faults are rare). This is easily accommodated into the analysis. Tj 3 Recent results had improved the following analysis by allowing the recovery actions to be executed at a higher priority =-=[27]. 4-=- We assume that in the absence of faults, the task set is schedulable. Tj TfsLimits to Schedulability Task P T C D F R R Tf = 300 Tf = 200 τ1 1 100 30 100 30 60 60 τ2 2 175 35 175 35 100 100 τ3 3 2... |

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Citation Context ...surprising that this single parameter approach is becoming limited in its application. Even a two (average and worst-case) or three (add minimum) parameter scheme is far from adequate. In other works =-=[8, 16, 4]-=- we have argued that it is now inadequate to use analysis alone to obtain a single worst-case execution time (WCET) value. Rather a combination of analysis and measurement must be used to obtain a pro... |

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Citation Context ...ecution of some routine, the execution of an exception handler or a recovery block. Various algorithms have been published whichsattempt to maximise the available resources for this extra computation =-=[37, 35, 3]-=-. Here we consider the nature of the guarantee that these algorithms provide. Most approaches make the common homogeneous Poisson process (HPP) assumptions that the fault arrival rate is constant and ... |

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Citation Context ...eadlines when tasks are replicated in a hardware-redundant system. However, they only consider permanent faults without repair or recovery. A similar problem was studied by Shin et al [36]. Kim et al =-=[22]-=- consider another related problem: the probability of a real-time controller meeting a deadline when subject to permanent faults with repair. To tolerate transient faults at the task level will requir... |

3 |
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Citation Context ...stead. More importantly, if the dependency is not known, then it is possible to find upper and lower bounds of the distribution function for any possible dependency between the marginal distributions =-=[5, 29, 14]-=-. Some generalisations of these results allow to tighten even more these bounds if partial knowledge of the dependence is known. 5.2 Representing Extreme Execution Times It was noted earlier that for ... |

1 |
Stochastic analysis of periodic real-time systems
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Citation Context ...tems with random execution times. The work relies on the independence of the execution times of the different tasks. The work improves on an earlier work by Gardner et.al. [18]. The works of Nissanke =-=[25]-=- and Eles [30] also tackle this problem. However, none of these approaches address the issue of extreme distributions or dependencies between execution times or task arrivals.s7 Conclusion: A Probabil... |

1 |
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