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 GroupEindhoven University of Technology

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

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.

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

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

Approximating General Distributions using DTPRGs

Discrete phase-types Bounded discretization

Approximation trivial for deterministic delays compositional

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)!

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

Performance Analysis of DTPRGs

Discrete-time probabilistic reward graph

Discrete-time Markov reward chain

Discrete-time Markov reward chain

Unfold & Aggregate Geometrize & Aggregate

Transient analysis Long-run analysis

Long-run metricsTransient metrics

Long-run analysis

Current Verification and Performance Analysis Environment of Chi

modelchecking

hybrid model checking

simulation

Chi simulator

ChiCTMRC

performance analysis

Hybrid Automata

SPIN

muCRL

UPPAAL

CTMRC analysis: only exponential delays large state space (full interleaving of time transitions)

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) ]|

Chi to DTPRG

Timed transition system (irrelevant

actions are τ‘s)

Reward Process

branching bisimulation reduction

Chispecification(with hiding)

state space generation

Minimized timed transition system

(no τ‘s left)branching bisimulation

reduction

Discrete-time probabilistic reward

graphdirect insertion

Probabilities

Case Study: Turntable Drilling Machine

Performance metrics Throughput Utilization of the drill Average number

of products

Parameters: Drill reliability Product availability

Throughput

Comparing Results

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