# Performance Evaluation of Distributed Systems Based on a Discrete Real- and Stochastic-Time Process Algebra

## Summary (2 min read)

### 1. Introduction

- Over the past decade stochastic process algebras have emerged as compo iti nal modeling formalisms for systems that not only require functional verification, but performance analysis as well.
- Similarly, the semantics of stochastic process algebras is given using clocks that represent the stochastic delays at the symbolic level.
- For the sampling of the clock two execution policies can be adopted: (1) racecondition [26, 20, 31, 10], which enables the action transitions guarded by the clocks that expirefirst, and (2) pre-selection policy [13, 12], which preselects the clocks by a probabilistic choice.
- The algebra also providesthe possibility of specifying a partial race of stochastic delays, e.g., that one delay has always a shorter, equal, or longer sample than the other delay.
- For analysis of the concurrent alternating bit protocol the authors depend on the toolse of theχ-language [8, 38, 11, 2].

### 2. Timed and Stochastic Delays

- The authors refer the interested reader for more technical detail to [32].
- Therefore, the names of the losing delays must be protected inp, i.e., they become dependent.
- Byσ∅ X , the authors denote the event where the delay does not expire in one time unit, i.e., the stochastic delayX loses the race to a unit time delay and there are no additional winners.
- Also, the authors favor weak choice between immediate actions and passage of time, i.e., they impose a nondeterministic choice on the immediate actions andthe passage of time in the vein of the timed process algebras of [4].

### 3. Process Theory

- In this section the authors introduce the process theoryTCPdst of communicating processes with discrete real and stochastic time for race-complete process specifications that induce races with all possible outcomes.
- The general idea of having both dependent and independent delays available is the following:.
- To denote that after a delay[WL ], the same time that passed for the winnersW has also passed for the losersL, the authors use an environmentα : V → N. For eachX ∈ V, α(X) represents the amount of time thatX has raced.
- Axiom A17 states that if the losers of the first timed delay have acommon delay with the winners of the second, then all delays of the second delay are losers in the resulting delay.
- Finally, the axioms A19–A21 give the standard axioms for the encapsulation operator that suppresses the actions inH.

### 4. Performance Evaluation

- For the purpose of performance analysis, the authors choose the framework ofthe languageχ.
- This provides for a better expressivity and modeling convenience [33].
- The discrete-time probabilistic reward graph is represented as an equivalent discrete-time Markov reward chain, which is then analyzed, and the results are interpreted back in the discrete-time probabilistic reward graph setting.
- The aggregation eliminates the probabilistic states4 and5 and splits the incoming timed transitions from the states6 and3.
- The multiplication of the transition matrix ofM with its folding collector produces the accumulative probability of residing in each unfolded timed state ofM per unfolding set.

### 5. The Concurrent Alternating Bit Protocol

- The authors specify the concurrent alternating bit protocol both in the process theoryTCPdst and in the specification languageχ.
- If the acknowledgement is received before the timeout expires, the process flips the alternating bit, packs the new data intp time units, and sends it again via channelc3.
- The authors takea similar approach as for the absence of a probabilistic choice, and add rewards by manipulating theχ specification (again side-stepping changes inχ), see Figure 6 below.
- Figure 9 gives the utilization of the data channelK, when the distribution of the delay of the data channel is uniform between2 and10 and the distribution of the delay of the acknowledgement channel is uniform between1 and4.
- The Markovian analysis always underestimates the performance because the expected value of the maximum of two exponential d lays is greater than maximum of the expected values of both delays.

### 6. Conclusion

- The authors proposed a performance evaluation framework that is based on a process theory that enables specification of distributed systems with discrete timed and stochastic delays.
- The authors provided expansion laws for the parallel composition and the maximal progress operat r.
- The authors gave transient analysis of these models by translating them to discrete-time Markov reward chains.
- This should pave the way for bigger case studies on Internet protocol verification and analysis as detailed performance specification becomes viable by using both generally-distributed stochastic delays and standard timeou s.
- Many thanks to Jos Baeten for fruitful discussions on the topic.

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### Cites background from "Performance Evaluation of Distribut..."

...There is a recent work by Markovski and de Vink [10], where a SPA with discrete time is defined, providing for it an interleaving semantics, but in this work immediate actions are I....

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...There is a recent work by Markovski and de Vink [10], where a SPA with discrete time is defined, providing for it an interleaving semantics, but in this work immediate actions are not considered....

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6 citations

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### Cites background or methods from "Performance Evaluation of Distribut..."

...This approach modifies the syntax of the pi-calculus and extends it to a stochastic pi-calculus [6-14]....

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...In the former approach, a family of stochastic process algebra [6-14] has been introduced that extends the standard process algebra with timing or probability information....

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##### References

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### "Performance Evaluation of Distribut..." refers methods in this paper

...Many Markovian process algebras are developed like EMPA [9], PEPA [27], IMC [25], etc. exploiting the memoryless Address for correspondence: J. Markovski, TU/e, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands property of the exponential distribution....

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "Performance evaluation of distributed systems based on a discrete real- and stochastic-time process algebra" ?

As future work, the authors plan to introduce the hiding operator that produces internal transitions and to develop a notion of branching or weak bisimulation in that setting. The authors can also exploit existing real-time specification as the theory is sufficiently flexible to allow extension of real-time with stochastic time while retaining any imposed ordering of the original delays. The authors are indebted to the reviewers for their constructive comments and suggestions.