Performance Analysis of Chi Models using Discrete-Time Probabilistic Reward Graphs

1. Performance Analysis of ChiModels using Discrete-Time Probabilistic Reward Graphs N. Trčka, S. Georgievska, J. Markovski, S. Andova, and E.P. de Vink Formal Methods Group Eindhoven University of Technology

2. Overview • Stochastic models • Discrete-time Markov reward chains • Continuous-time Markov reward chains • Our model: Discrete-time probabilistic reward graphs • Analysis of discrete-time probabilistic reward graphs • Transformation to discrete-time Markov reward chains • Optimization by geometrization • Introduction to Chi language and environment • Generation of discrete-time probabilistic reward graphs from Chi • Case study: Performance analysis of a turntable drilling machine

3. Discrete-Time Markov Reward Chains (DTMRCs) • Semantics • Spend one time unit in a state • Gain a reward • Jump to next state probabilistically • Performance metrics • expected reward rate • at time t or in the long-run • can express: throughput, utilization, etc.

4. Continuous-Time Markov Reward Chains (CTMRCs) • Sojourn time • exponentially distributed • determined by the minimum of all outgoing transitions • reward gained with the given rate • Same performance metrics • Phase-type approximation of general distributions

5. Our model: Discrete-Time Probabilistic Reward Graphs (DTPRGs) • Two types of states • timed and probabilistic • Sojourn times • deterministic and discrete • zeroin a probabilistic state • uniquely specified by the outgoing transition in a timed state

6. Discrete phase-types Bounded discretization Approximating General Distributions using DTPRGs • Approximation • trivial for deterministic delays • compositional

7. DTPRG to DTMRC • Two steps: • “Unfolding” of timed delays • Elimination of (zero-time) probabilistic states • Weakness: A delay of n units introduces n-1 new states (at most)!

8. Alternative Way: Geometrization of a DTPRG • Replace deterministic delays by geometric delays • Expected sojourn time in the long run is the durationof the timed delay • Works only for long-run analysis

9. Performance Analysis of DTPRGs

10. Current Verification and Performance Analysis Environment of Chi • CTMRC analysis: • only exponential delays • large state space (full interleaving of time transitions)

11. The Language Chi by an Example proc B(chan a?, b!:[nat]) = |[ var xs,ys:[nat] = [] :: *( a?ys; xs:= xs ++ ys | len(xs) > 0 -> b!take(xs); xs:= drop(xs) ) ]| proc M(chan a?,b!:[nat]) = |[ xs:[nat] :: *( a?xs; delay 2.5; b!xs) ]| model L(var ta: real) = |[ chan a,b,c:[nat] :: B(a,b) || M(b,c) ]|

12. Chi to DTPRG

13. Case Study: Turntable Drilling Machine • Performance metrics • Throughput • Utilization of the drill • Average number of products • Parameters: • Drill reliability • Product availability

14. Throughput

15. Comparing Results

16. Conclusion • DTPRGs are a powerful formalism for modeling stochastic aspects in systems • By translating DTPRGs to DTMRCs one obtains all kinds of performance metrics fast • Chi is a suitable high-level specification formalism for generation of DTPRGs • proper extension needed