## Probabilistic duration calculus for continuous time (1994)

Venue: | Formal Aspects of Computing |

Citations: | 8 - 3 self |

### BibTeX

@INPROCEEDINGS{Hung94probabilisticduration,

author = {Dang Van Hung and Zhou Chaochen},

title = {Probabilistic duration calculus for continuous time},

booktitle = {Formal Aspects of Computing},

year = {1994},

pages = {21--44}

}

### OpenURL

### Abstract

Abstract. This paper deals with dependability of imperfect implementations concerning given requirements. The requirements are assumed to be written as formulas in Duration Calculus. Implementations are modelled by continuous semi-Markov processes with finite state space, which are expressed in the paper as finite automata with stochastic delays of state transitions. A probabilistic model for Duration Calculus formulas is introduced, so that the satisfaction probabilities of Duration Calculus formulas with respect to semi-Markov processes can be defined, reasoned about and calculated through a set of axioms and rules of the model. 1.

### Citations

384 |
The theory of stochastic processes
- Cox, Miller
- 1965
(Show Context)
Citation Context ..., are the unique solutions of the forward equation d dt fs(t) = � s ′ ∈S fs ′(t)cs,s ′, s ∈ S. Hence, fs(t)(s ∈ S) define the probability distribution of a time-homogeneous honest Markov process (see =-=[CoM90]-=-). Note that fs(t) is the probability that M is in s at time t given that M started in s0 at time 0. From the equation, many interesting properties of fs(t) (see e.g. [CoM90]) can be derived. Theorem ... |

254 | A Really Temporal Logic
- Alur, Henzinger
- 1994
(Show Context)
Citation Context ...ca, Beijing, P.R. China.s2 D. V. Hung and C. Zhou ✗ f ✔ ✗✔ ✗✔ ❄ O ✖✕ ✻✖ r F ✖✕ ✕ Fig. 1. A system model embedded hard real-time systems, many formal tools have been proposed, such as Real-Time Logics =-=[AlH89]-=-, Timed CSP [Sch91], Metric Temporal Logics [Koy90], timed transition systems [HMP92], etc. Among them, Duration Calculus (DC) [ZHR92] has proved to be a promising tool. One of the main features of DC... |

205 |
Specifying real-time properties with metric temporal logic
- Koymans
- 1990
(Show Context)
Citation Context ... f ✔ ✗✔ ✗✔ ❄ O ✖✕ ✻✖ r F ✖✕ ✕ Fig. 1. A system model embedded hard real-time systems, many formal tools have been proposed, such as Real-Time Logics [AlH89], Timed CSP [Sch91], Metric Temporal Logics =-=[Koy90]-=-, timed transition systems [HMP92], etc. Among them, Duration Calculus (DC) [ZHR92] has proved to be a promising tool. One of the main features of DC is that it can handle continuous time without expl... |

79 | Timed transition systems
- TA, Manna, et al.
- 1992
(Show Context)
Citation Context ... 1. A system model embedded hard real-time systems, many formal tools have been proposed, such as Real-Time Logics [AlH89], Timed CSP [Sch91], Metric Temporal Logics [Koy90], timed transition systems =-=[HMP92]-=-, etc. Among them, Duration Calculus (DC) [ZHR92] has proved to be a promising tool. One of the main features of DC is that it can handle continuous time without explicitly referring to absolute time.... |

63 |
A calculus of durations
- Zhou, Hoare, et al.
- 1991
(Show Context)
Citation Context ...s, many formal tools have been proposed, such as Real-Time Logics [AlH89], Timed CSP [Sch91], Metric Temporal Logics [Koy90], timed transition systems [HMP92], etc. Among them, Duration Calculus (DC) =-=[ZHR92]-=- has proved to be a promising tool. One of the main features of DC is that it can handle continuous time without explicitly referring to absolute time. For dealing with the dependability requirements,... |

42 |
Semantics and completeness of the Duration Calculus
- Hansen, Chaochen
- 1992
(Show Context)
Citation Context ...l iff D holds for some subinterval of it, and ✷D is true for an interval iff D holds for all subintervals of it. DC has a set of axioms about states and rules which is sound and (relatively) complete =-=[HaZ91]-=-. These axioms and rules are listed below. DA 1. � 0 = 0. DA 2. For an arbitrary state P , � P ≥ 0. The additivity rule of durations is described as DA 3. For arbitrary states P and Q, � P + � Q = � (... |

34 |
Timed CSP: Theory and practice
- Schneider, Davies, et al.
- 1991
(Show Context)
Citation Context ...hina.s2 D. V. Hung and C. Zhou ✗ f ✔ ✗✔ ✗✔ ❄ O ✖✕ ✻✖ r F ✖✕ ✕ Fig. 1. A system model embedded hard real-time systems, many formal tools have been proposed, such as Real-Time Logics [AlH89], Timed CSP =-=[Sch91]-=-, Metric Temporal Logics [Koy90], timed transition systems [HMP92], etc. Among them, Duration Calculus (DC) [ZHR92] has proved to be a promising tool. One of the main features of DC is that it can han... |

31 |
Continuity of generalized semi-Markov processes
- Whitt
- 1980
(Show Context)
Citation Context ... the time τ at which f becomes enabled, but depends only upon t, the length of time since f became enabled. This assumption simplifies the model, and defines a so-called semi-Markov process (see e.g. =-=[Whi80]-=-). The untimed behaviour of the system in Figure 1 can be described by transition sequences such as fr, frf, frfr, etc. In order to represent real-time system behaviour, we use transition sequences wi... |

4 |
From CSP Models to Markov Models
- Sorensen, Nordahl, et al.
- 1993
(Show Context)
Citation Context ...take place at discrete time points represented by integers. The discrete time model is not suitable for many practical applications because physical components work in continuous time. As inspired by =-=[SNH93]-=-, the present paper makes another attempt in this direction. It uses probabilistic automata with transitions occurring in continuous time to model implementations, and then establishes a probabilistic... |

3 | Towards a calculus of systems dependability - Liu, Ravn, et al. - 1994 |

1 |
Evaluation techniques, ch. 4 in Design and Analysis of Fault Tolerant Digital Systems
- Johnson
- 1989
(Show Context)
Citation Context ...t can handle continuous time without explicitly referring to absolute time. For dealing with the dependability requirements, the methods based on principles from the fields of reliability engineering =-=[Joh90]-=- are used. The mathematical foundation of these methods is the theory of probability and stochastic processes (e.g. Markov processes). Clearly, a combined calculus capable of coping with both kinds of... |

1 | C.: A Probabilistic Duration Calculus, presented - Liu, P |