# Performance Modeling of a Quorum Pattern in Layered Service Systems

## Summary (2 min read)

### 1. Introduction

- The π-calculus [15] is a process algebra for modelling concurrency and mobility.
- For both classes of systems, probability is often also a key ingredient.
- For efficiency reasons, however, the authors take a compositional approach, applying MMCsp to each parallel component of a system, processing the results to produce a high-level description in the modelling language of PRISM and then performing probabilistic verification.
- Various tools exist for automatic verification of the (non-probabilistic) π-calculus.
- In [5], a variant which is essentially the same as that used in this paper was presented and probabilistic testing equivalences were defined to reason about randomised security protocols.

### 2. The simple probabilistic π-calculus

- The π-calculus is a process algebra for modelling concurrency and mobility.
- Any process not satisfying this condition can easily be converted to an structurally congruent one that does (through renaming of bound names).
- The operational semantics for probabilistic extensions of the π-calculus are typically expressed in terms of MDPs or, equivalently, probabilistic automata [20], which allow both probabilistic and nondeterministic behaviour.
- This allows a compositional approach to be adopted: given a parallel composition of several processes, the semantics of each can be computed in full separately, and then composed afterwards.
- More specifically, it encodes the set of terms derivable from Q by substitutions applied to its input-bound names.

### 3. Generating PSTGs using MMC

- In this section the authors describe the automatic generation of the probabilistic symbolic transition graph (PSTG) for an arbitrary process expressed in the simple probabilistic π-calculus.
- Firstly it gives a clear and intuitive implementation; secondly, and more importantly, this encoding is provably correct [24], also known as This has several benefits.
- The authors then adapt MMC’s predicate trans to represent the symbolic semantics of πsp.
- This is effectively a depth-first traversal of the PSTG and enumeration of all states and probabilis- tic symbolic transitions found.
- All bound names are given unique names (e.g. h417) and displayed on lines beginning >.

### 4. Translating PSTGs into PRISM

- The scheme described in the previous section can be used to translate an arbitrary process described in the simple probabilistic π-calculus into its probabilistic symbolic transition graph (PSTG).
- At the level of PSTGs, their restricted form ensures that there are no bounded output transitions x̄(y).
- In brief, (1) is handled by enumerating the set of all free names (which is known since the system is input-closed), assigning each an (identically named) integer constant to represent it, and (2) is handled by introducing a synchronous action label for each required combination of process sender/receiver pair, channel and name.
- The full semantics of the PRISM language can be found at [18].

### 5. Implementation and results

- Firstly, the authors consider the dining cryptographers protocol (DCP) [6], Chaum’s randomised solution to the classic anonymity problem in which a group of N parties collectively establish whether either one of the group or an independent party has to make a payment.
- Thirdly, the authors constructed a πsp model of mobile communication network (MCN), based on the (non-probabilistic) π-calculus model in [17].
- The mobile station roams between the base stations.
- Finally, the authors give the time to check a single PCTL property for each using PRISM (with the MTBDD engine).

### 6. Conclusions

- In this paper the authors have demonstrated the feasibility of implementing model checking for the probabilistic πcalculus.
- The variant of the calculus (with blind probabilistic choice) to which their techniques are applicable has proved to be expressive enough for the appropriate application domains (probabilistic algorithms for security and dynamic communication protocols with failures and/or randomisation) and yet amenable to analysis with extensions and adaptions of existing verifica- tion tools.
- Furthermore the authors have shown, through its application to several large examples, the efficiency of the approach.
- For convenience of modelling, the authors plan to add support for polyadic communication over channels.
- Finally, the authors will investigate ways to further improve the efficiency of their implementation, in particular, with regards to the automatically generated PRISM code.

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "Model checking the probabilistic pi-calculus" ?

The authors would like to extend this work in several directions. For convenience of modelling, the authors plan to add support for polyadic communication over channels. The authors also hope to add support for more flexible property specifications using watchdog processes and to extend their approach to the stochastic π-calculus. Possibilities include optimisations to reduce the resulting symbolic ( MTBDD ) storage in PRISM and bisimulation minimisation techniques.