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# 数学代写|理论计算机代写THEORETICAL COMPUTER SCIENCE代写|CS154 Extended Markovian Process Algebra

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## 数学代写|理论计算机代写THEORETICAL COMPUTER SCIENCE代写|Extended Markovian Process Algebra

In this section we revisit EMPA $\mathrm{gr}_{\text {[4] }}$, a Markovian process algebra extended with prioritized/weighted immediate actions à la GSPN [1] in which interprocess communication is based on a multiway generative-reactive [5] synchronization mechanism.

In $\mathrm{EMPA}_{\mathrm{gr}}$ every action is durational, hence it is represented as a pair $$, where a is the action name and \tilde{\lambda} is the action rate. There are three kinds of actions: exponentially timed, immediate, and passive. Exponentially timed actions are of the form$$ with $\lambda \in \mathbf{R}_{>0}$. The duration of each such action is exponentially distributed with parameter equal to the action rate (hence its average duration is the inverse of its rate). Whenever several exponentially timed actions are enabled, the race policy is adopted, hence the fastest action is the one that is executed. As a consequence, the execution probability of an exponentially timed action is proportional to its rate.

Immediate actions are of the form $\left\langle a, \infty_{l, w}>\right.$, where $l \in \mathbf{N}{>0}$ is the priority level and $w \in \mathbb{R}{>0}$ is the weight. Each immediate action has duration zero and takes precedence over exponentially timed actions (which are assumed to have priority level 0 ). Whenever several immediate actions are enabled, the generative preselection policy is adopted, i.e. each action is given an execution probability proportional to its weight provided that it has the highest priority level within the considered action set.

## 数学代写|理论计算机代写THEORETICAL COMPUTER SCIENCE代写|Weak Extended Markovian Bisimilarity

A behavioral equivalence typically used for reasoning about Markovian process terms is Markovian bisimilarity [7]. In particular, an extension of it has been defined in Refs. 3,4 in order to deal with prioritized/weighted immediate actions. The limitation of extended Markovian bisimilarity is that it does not provide any abstraction mechanism. In fact, actions of the form $<\tau, \infty_{l, w}>$ are invisible and take no time, hence they are unimportant both from the functional viewpoint and from the performance viewpoint. Thus they should not be considered when comparing the behavior of two terms.

After recalling the definition of extended Markovian bisimilarity, we introduce a weak variant of it that abstracts from internal immediate actions. Then we show that this weak variant is a congruence and admits a sound and complete axiomatization for the class of well-prioritized process terms.

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