1. Performance Analysis of Chi Models 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 zero in a probabilistic state uniquely specified by the outgoing transition in a timed state

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

7. DTPRG to DTMRC Two steps: 1. “Unfolding” of timed delays 2. 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 duration of 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