随机进程代数新技术：以PEPA语言为中心的综述
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"这篇论文是2011年的一篇关于随机进程代数新技术的综述，重点关注如何解决状态空间爆炸问题。文章以PEPA（Performance Equation Process Algebra，性能方程过程代数）语言作为讨论的例子，作者来自南京工程学院和扬州大学的信息工程学院。论文探讨了用于并发系统行为分析的形式化描述技术，并着重介绍了处理状态空间爆炸问题的新方法。"
在并发系统的研究中，过程代数是一种重要的形式化描述技术，它通过代数手段来分析系统的并行或分布式行为。PEPA是其中一种特殊的过程代数，专门用于建模和分析系统的性能特性。然而，在处理复杂系统时，由于可能的状态组合数量巨大，会出现“状态空间爆炸”问题，这使得模型的分析变得极其困难。
论文中提到的新技术旨在缓解这一问题，它们通常包括但不限于以下方面：
1. **状态归约**：通过识别并消除等价状态，减少状态空间的大小。例如，使用重写规则或等价关系来简化系统模型，降低分析复杂性。
2. **概率抽象**：通过对概率分布进行近似，将精细的概率模型转换为更粗粒度的表示，从而降低计算需求。
3. **分层建模**：将系统分解为可独立分析的层次，每个层次只考虑一部分状态和行为，逐层构建和分析。
4. **动态剪枝**：在分析过程中动态地剔除不会影响最终结果的状态，以减小计算负担。
5. **符号计算和近似算法**：利用符号计算方法，避免直接计算所有状态的转移概率，或者开发能够处理大规模系统的近似算法。
6. **模型检查优化**：改进模型检查技术，比如采用部分订单探索或基于约束的分析，以减少状态空间的探索。
7. **并行计算和分布式计算**：利用多核处理器或分布式计算资源，将分析任务分解，加速状态空间的遍历。
8. **动态调整和自适应方法**：根据分析过程中的信息动态调整建模和分析策略，以适应不同阶段的需求。
这些新技术的应用对于理解和验证复杂的并发系统具有重要意义，特别是在性能评估、可靠性分析和安全性验证等领域。通过这些方法，工程师可以更好地理解和控制系统的复杂行为，提高软件和硬件设计的质量与效率。
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relate, giving rise to labelled activities:
Denition 3:
(
Labelled Activity
).
1) For any individual activity
α,for each
( ), ( , )P pre Q post P
α α
∈ ∈
( ), ( , )P pre Q post P
α α
∈ ∈
,label α
as
Q
P
α
→
.
2) For a shared activity
α
,
for each
(
Q
1
,
Q
2
,… ,
Q
k
) in
post(
pre
(
α
)[1],
α
)
×
post(
pre
(
α
)[2],
α
)
×…×
post
(pre
(
α
)[k],
α
),
label
α
as
α
ω
,
where
ω
represents
(
pre
(
α
)[1] →
Q
1
,
pre
(
α
)[2] →
Q
2
,…，
pre
(
α
)[
k
] →
Q
k
).
Each α
P→Q
or α
ω
is called a labelled activity. The set of
all labelled activities is denoted by A
label
.For the above la-
belled activities α
P→Q
and α
ω
, their respective pre and post
se ts a r e de f i n e d as
pre
(
α
P→ Q
)={P },
po st
(
α
P→ Q
)={
Q
},
pre(
α
ω
)
＝pre
(
α
),
post
(
α
ω
)={
Q
1
,
Q
2
,… ,
Q
k
}.
In the numerical representation scheme, the transitions be-
tween states of the model are represented by a matrix, termed the
activity matrix — this records the impact of the labeled activities
on the local derivatives.
Definition 4:
(
Activity Matrix, Pre Activity Matrix,
Post Activity Matrix
).
For a model with
label
A
N
labelled activi-
ties and
D
N
distinct local derivatives, the activity matrix C is
an
label
D A
N N×
matrix, and the entries are defined as following
1 ( )
( , ) 1 ( )
0
i j
i j i j
if P post
C P if P pre
otherwise
α
α α
+ ∈
= − ∈
where α
j
is a labelled activity. The pre activity matrix C
pre
and
post acti
vity matrix C
post
are defined as
Pr
P
1 ( )
( , ) ,
0 .
1 ( )
( , )
0 .
i j
e
i j
i j
ost
i j
if P pre
C P
otherwise
if P post
C P
otherwise
α
α
α
α
+ ∈
=
+ ∈
=
From Denitions 3 and 4, each column of the activity matrix
corresponds to a system transition and each transition can be rep-
resented by a column of the activity matrix. The activity matrix
equals the difference between the pre and post activity matrices,
i.e.
Pr
.
Post e
C C C= −
The rate of the transition between states is speci-
ed by a transition rate function, but we omit this detail here since
we are concerned with qualitative analysis. See [7] for details. We
rst give the denition of the apparent rate of an activity in a local
derivative.
Definition 5:
(
Apparent Rate of α in P
)
Suppose α
is an
activity of a PEPA model and P is a local derivative enabling α (i.e.
( )P pre
α
∈
). Let
( , )post P
α
be the set of all the local derivatives
derived from P by ring α,i.e.
{ }
( , )
( , ) | .
P Q
r
post P Q P Q
α
α
α
→
=
uuuuuuuuuur
Let
( , )
( ) .
P Q
Q post P
r P r
α α
α
→
∈
=
∑
(1)
The apparent rate of α in P in state x, denoted by
(x, ),r P
α
is
dened as
(x, ) x[ ] ( ).r P P r P
α α
=
(2)
The above denition is used to dene the following transition
rate function.
Denition 6:
(
Transition Rate Function) Suppose α is an ac-
tivity of a PEPA model and x denotes a state vector.
1) If α is individual, then for each
( , )
P Q
r
Q
P
α
→
uuuuuuuuuur
, the transi-
tion rate function of labelled activity
P Q
α
→
in state x is dened
as
(x, ) x[ ] .
P Q P Q
f P r
α
α
→ →
=
(3)
2 ) I f α is s y n c h r o n i s e d , w i t h p o s t ( α ) = { P
1
, P
2
, … ,
P
k
},then for each Q
1
,Q
2
,…,Q
k
in post(α)×post(P
2
,α)×…
×post(P
k
,α),let
1 1 2 2
( , ,..., ).
k k
P Q P Q P Q
ω
= → → →
Then the
transition rate function of labelled activity α
ω
in state x is dened as
{ }
{ }
min
1
1,...,
(x, ) (x, ) ,
( )
i i
P Q
k
i
i
i
i k
r
f r P
r P
ω
α
α
α
α
→
=
∈
=
∏
where
(x, ) x[ ] ( )
i i i
r P P r P
α α
=
is the apparent rate of α in P
i
in state x. So
{ }
{ }
min
1
1,...,
(x, ) x[ ] ( ) .
( )
i i
P Q
k
i i
i
i
i k
r
f P r P
r P
ω
α
α
α
α
→
=
∈
=
∏
(4)
Note that Denition 6 accommodates the passive or unspeci-
ed rate T. An algorithm for automatically deriving the numerical
representation of a PEPA model was presented in [7].
III
New Representation
A
Numerical representation
The syntactic nature of stochastic process algebras leads to
models which are easily understood by humans, but which are not
so convenient for automated analysis directly employing these
tools and techniques. Furthermore, the compositionality of the
formalisms allows a model to be locally dened, but the analysis
of the model often via the underlying CTMC must be carried out
globally. The syntactical and compositional qualities of the sto-
chastic process algebras, which are advantages in model construc-
tion, turn to be disadvantages in model analysis. Ding proposed a
new numerical representation schema in his thesis [7] for the for-
malism PEPA in order to overcome the obstacles to the direct and
convenient application of the mathematical tools. These concepts
numerically describe and represent a PEPA model, and provide a
platform which straightforwardly supports simulating the underly-
ing CTMC, deriving the uid approximation, and in addition leads
to an underlying Place/Transition (P/T) structure (see Theorem 1 in
the next section). These denitions are consistent with the original
semantics of PEPA, and a PEPA model can be recovered from its
numerical representation. An algorithm for automatically deriving
the schema from any given PEPA model has been provided. Some
characteristics of this numerical representation are revealed. For
example, using numerical vector forms the exponential increase of
戴慧 等：随机进程代数新技术综述
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