Difference between revisions of "Solving Models"
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# All timed actions are exponentially distributed (Markov processes). | # All timed actions are exponentially distributed (Markov processes). | ||
| − | : 2. All timed actions are deterministic or exponentially distributed, with at most one deterministic action enabled at any time. Furthermore, the firing delay of the deterministic actions may not be state-dependent. | + | |
| + | : 2. All timed actions are deterministic or exponentially distributed, with at most one deterministic action enabled at any time. Furthermore, the firing delay of the deterministic actions may not be state-dependent. | ||
The only restrictions on the use of instantaneous (zero-timed) actions are that the model must begin in a stable state, and the model must be ''stabilizing'' and ''well-specified''<ref name=Sanders:phd:88>W. H. Sanders. ''Construction and Solution of Performability Models Based on Stochastic Activity Networks''. PhD thesis, University of Michigan, Ann Arbor, Michigan, 1988.</ref>. It should also be noted that the reactivation predicates (see Section [[Building_Models#SAN_primitives|4.1.1]] of Building Models) must preserve the Markov property. In other words, the timed actions in the model must be reactivated so that the firing time distributions depend only on the current state, and not on any past state. That rule pertains only to timed actions with firing delays that are state-dependent. | The only restrictions on the use of instantaneous (zero-timed) actions are that the model must begin in a stable state, and the model must be ''stabilizing'' and ''well-specified''<ref name=Sanders:phd:88>W. H. Sanders. ''Construction and Solution of Performability Models Based on Stochastic Activity Networks''. PhD thesis, University of Michigan, Ann Arbor, Michigan, 1988.</ref>. It should also be noted that the reactivation predicates (see Section [[Building_Models#SAN_primitives|4.1.1]] of Building Models) must preserve the Markov property. In other words, the timed actions in the model must be reactivated so that the firing time distributions depend only on the current state, and not on any past state. That rule pertains only to timed actions with firing delays that are state-dependent. | ||
Revision as of 19:33, 7 March 2014
Contents
How to pick the solver
Möbius provides two types of solvers for obtaining solutions on measures of interest: simulation and numerical solvers. The choice of which type of solvers to use depends on a number of factors. More details on these factors are provided in the sections on simulation (Section 3) and numerical solvers (Section 4).
In general, the simulation solver can be used to solve all models that were built in Möbius, whereas numerical solvers can be used on only those modes that have only exponentially and deterministically distributed actions. In addition, simulation may be used on models that have arbitrarily large state-space descriptions, whereas numerical solvers are limited to models that have finite, small state-space description (that may be held in core memory). Furthermore, simulation may be more useful than numerical solvers for stiff models.
On the other hand, all numerical solvers in Möbius are capable of providing exact solutions (up to machine precision), whereas simulation provides statistically accurate solutions within some user-specifiable confidence interval. The desired accuracy of the computed solutions can be increased without excessive increase in computation time for most numerical solvers, while an increase in accuracy may be quite expensive for simulation. Additionally, full distributions may be computed for results from the numerical solvers, but usually not for results from simulation. Furthermore, for models in which numerical solvers are applicable, detection of rare events incurs no extra costs and requires no special techniques, whereas such computation by simulation is extremely expensive and uses the statistical technique of importance sampling.
Transformers
Introduction
Some of the solution techniques within Möbius, such as the simulator, operate directly on the model representation defined using the Atomic and Composed editors described in earlier chapters of the manual. These solvers operator on the model using the Möbius model-level abstract functional interface. There are other solution techniques, specifically the numerical solvers described in the next chapter, which require a different representation of the model as an input. Instead of operating on the high-level model description, numerical solution techniques use a lower-level, state space representation, namely the Markov chain.
Flat State Space Generator
The flat state space generator1 is used to generate the state space of the discrete-state stochastic process inherent in a model. The state space consists of the set of all states and the transitions between them. Once the state space has been generated, an appropriate analytical solver can be chosen to solve the model, as explained in Section 4.
- 1 This was the only state space generator (SSG) available prior to version 1.6.0 and it was simply called the state space generator. From that version on, it is called the flat state space generator.
While simulation can be applied to models with any underlying stochastic process, numerical solution requires that the model satisfy one of the following properties:
- All timed actions are exponentially distributed (Markov processes).
- 2. All timed actions are deterministic or exponentially distributed, with at most one deterministic action enabled at any time. Furthermore, the firing delay of the deterministic actions may not be state-dependent.
The only restrictions on the use of instantaneous (zero-timed) actions are that the model must begin in a stable state, and the model must be stabilizing and well-specified[1]. It should also be noted that the reactivation predicates (see Section 4.1.1 of Building Models) must preserve the Markov property. In other words, the timed actions in the model must be reactivated so that the firing time distributions depend only on the current state, and not on any past state. That rule pertains only to timed actions with firing delays that are state-dependent.
The flat state space generator consists of a window with two tabs, SSG Info and SSG Output, which are discussed below.
Möbius
Möbius
Motivation
Solution
Graph
Edit Möbius Documentation
“” –
<equation id="eqn:binom" shownumber>
</equation>
Sort of like <xr id="eqn:binom" />, but not really.- ↑ W. H. Sanders. Construction and Solution of Performability Models Based on Stochastic Activity Networks. PhD thesis, University of Michigan, Ann Arbor, Michigan, 1988.
Contents
How to pick the solver[edit]
Möbius provides two types of solvers for obtaining solutions on measures of interest: simulation and numerical solvers. The choice of which type of solvers to use depends on a number of factors. More details on these factors are provided in the sections on simulation (Section 3) and numerical solvers (Section 4).
In general, the simulation solver can be used to solve all models that were built in Möbius, whereas numerical solvers can be used on only those modes that have only exponentially and deterministically distributed actions. In addition, simulation may be used on models that have arbitrarily large state-space descriptions, whereas numerical solvers are limited to models that have finite, small state-space description (that may be held in core memory). Furthermore, simulation may be more useful than numerical solvers for stiff models.
On the other hand, all numerical solvers in Möbius are capable of providing exact solutions (up to machine precision), whereas simulation provides statistically accurate solutions within some user-specifiable confidence interval. The desired accuracy of the computed solutions can be increased without excessive increase in computation time for most numerical solvers, while an increase in accuracy may be quite expensive for simulation. Additionally, full distributions may be computed for results from the numerical solvers, but usually not for results from simulation. Furthermore, for models in which numerical solvers are applicable, detection of rare events incurs no extra costs and requires no special techniques, whereas such computation by simulation is extremely expensive and uses the statistical technique of importance sampling.
Transformers[edit]
Introduction[edit]
Some of the solution techniques within Möbius, such as the simulator, operate directly on the model representation defined using the Atomic and Composed editors described in earlier chapters of the manual. These solvers operator on the model using the Möbius model-level abstract functional interface. There are other solution techniques, specifically the numerical solvers described in the next chapter, which require a different representation of the model as an input. Instead of operating on the high-level model description, numerical solution techniques use a lower-level, state space representation, namely the Markov chain.
Flat State Space Generator[edit]
The flat state space generator1 is used to generate the state space of the discrete-state stochastic process inherent in a model. The state space consists of the set of all states and the transitions between them. Once the state space has been generated, an appropriate analytical solver can be chosen to solve the model, as explained in Section 4.
- 1 This was the only state space generator (SSG) available prior to version 1.6.0 and it was simply called the state space generator. From that version on, it is called the flat state space generator.
While simulation can be applied to models with any underlying stochastic process, numerical solution requires that the model satisfy one of the following properties:
- All timed actions are exponentially distributed (Markov processes).
- 2. All timed actions are deterministic or exponentially distributed, with at most one deterministic action enabled at any time. Furthermore, the firing delay of the deterministic actions may not be state-dependent.
The only restrictions on the use of instantaneous (zero-timed) actions are that the model must begin in a stable state, and the model must be stabilizing and well-specified[1]. It should also be noted that the reactivation predicates (see Section 4.1.1 of Building Models) must preserve the Markov property. In other words, the timed actions in the model must be reactivated so that the firing time distributions depend only on the current state, and not on any past state. That rule pertains only to timed actions with firing delays that are state-dependent.
The flat state space generator consists of a window with two tabs, SSG Info and SSG Output, which are discussed below.
Möbius
Möbius[edit]
Motivation[edit]
Solution[edit]
Graph
Edit Möbius Documentation
“” –
<equation id="eqn:binom" shownumber>
</equation>
Sort of like <xr id="eqn:binom" />, but not really.- ↑ W. H. Sanders. Construction and Solution of Performability Models Based on Stochastic Activity Networks. PhD thesis, University of Michigan, Ann Arbor, Michigan, 1988.
