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
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

Approximation



trivial for deterministic delays
compositional
Bounded discretization
DTPRG to DTMRC
Two steps:

1.
2.

“Unfolding” of timed delays
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
Unfold & Aggregate
Geometrize & Aggregate
Discrete-time Markov
reward chain
Transient analysis
Transient metrics
Discrete-time Markov
reward chain
Long-run analysis
Long-run analysis
Long-run metrics
Current Verification and Performance
Analysis Environment of Chi
Hybrid
Automata
hybrid model
checking
CTMRC
performance
analysis
Chi
model
checking
simulation
SPIN
muCRL
UPPAAL
Chi simulator

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
Reward Process
Chi
specification
(with hiding)
state space
generation
Timed transition
system (irrelevant
actions are τ‘s)
branching bisimulation
reduction
Probabilities
branching bisimulation
reduction
Minimized timed
transition system
(no τ‘s left)
direct insertion
Discrete-time
probabilistic reward
graph
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