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