MODEL DRIVEN TESTING WITH TIME AUGMENTED MARKOV

CHAIN USAGE MODELS

Computations and Test Case Generation Algorithms for Time Augmented Markov

Chain Usage Models

Sebastian Siegl, Vladimir Entin, Reinhard German

Department of Computer Science, FAU Erlangen-N

urnberg, Martensstr. 3, 91058, Erlangen, Germany

Gerhard Kiffe

AUDI AG, 85045 Ingolstadt, Germany

Keywords:

Software tests, Model based testing, Automated test case generation, Statistical testing, Usage modeling,

Testing strategies, Test coverage, Automotive systems, Embedded systems.

Abstract:

In statistical testing Markov Chain Usage Models (MCUM) are employed to describe the possible usage of

the System-under-test (SUT) and to automatically derive test cases. However, MCUMs, as they are applied

today, do not provide a universal concept for the integration of time. The estimated or known delay time of

usage states and execution time of actions can only be considered by auxiliary means. But the importance

of this aspect grows continously with the integration of increasingly complex and time sensitive systems in

the automotive environment, such as hybrid vehicles. To overcome this drawback we present an intuitive

extension of MCUMs that allows the consistent integration of time in MCUMs. We discuss how computations

for MCUMs can be easily adapted to Time Augmented MCUMs and present algorithms for automatic test

case generation from Time Augmented MCUMs together with a case study that has been conducted in the

automotive domain.

1 INTRODUCTION

In many application domains, e.g. avionic and au-

tomotive systems, software has become the method

of choice for innovation. This development is inten-

siﬁed by the customer demand for increasingly safe,

comfortable, and environment friendly systems like

hybrid or electric vehicles. To satisfy this demand

the development of highly complex and distributed

systems has become inevitable. These systems pose

new challenges to testing. The approach presented

in this paper is based on model based testing (Dalal

et al., 1999), and can be further categorized into us-

age model based black box testing (Beyer and Dulz,

2005). Nowadays the focus in usage oriented test-

ing based on Markov chain usage models (MCUM)

lies in testing various sampled test cases and in test-

ing that comes close to a known or estimated usage

proﬁle (Prowell, 2004). A broad spectrum of differ-

ent types of algorithms for the generation of test cases

exists (Siegl, 2008). Only the variance in actions is

currently considered in a consistent manner, however.

Time and especially the variance in usage time, i.e.

the time between actions and the duration of actions

themselves, is not integrated coherently. However, the

timing of different actions has an important impact on

the behavior of a system and thus on the ability of

test cases to discover failures. Timing in this context

stands for the duration of the execution of stimuli and

the duration of usage states, i.e. the time between the

execution of stimuli. In this contribution we present

a temporal extension to classical MCUMs that offers

the following advantages:

• A realistic, usage time considering, representation

of system usage

• Integration of usage timing into computations

• Integration of variance in time of usage into test

case generation

Our enhancement is based on the concept of semi-

Markov processes as they are used in the perfor-

mance analysis of systems with non-exponential tim-

202

Siegl S., Entin V., Ger man R. and Kiffe G. (2009).

MODEL DRIVEN TESTING WITH TIME AUGMENTED MARKOV CHAIN USAGE MODELS - Computations and Test Case Generation Algorithms for

Time Augmented Markov Chain Usage Models.

In Proceedings of the 4th International Conference on Software and Data Technologies, pages 202-207

DOI: 10.5220/0002255902020207

 SciTePress

ings (German, 2000). They are applied in the ﬁeld of

system-modeling and analysis. In the context of us-

age modeling no application of this concept is known

to the authors.

The paper is structured as follows. In the next sec-

tion model driven testing based on Markov chain us-

age models is brieﬂy introduced, followed by a pre-

sentation of the testing requirements in the automo-

tive domain. Next we present the extension, calcula-

tions that are possible with this extension and algo-

rithms for an automatic generation of test cases from

our enhanced MCUMs that we call Time Augmented

Markov chain usage models. Before the conclusion

and outlook, we present a case study that has been

conducted in the automotive domain.

2 TEST CASE GENERATION

FROM MARKOV CHAIN

USAGE MODELS

A MCUM is usually described by a directed graph

that consists of (Prowell, 2000):

• A set of states S = {s

, . . . , s

}, that represent pos-

sible states of usage.

• A set of arcs A, representing state transitions. An

arc from state i to state j is denoted by a

i j

, mul-

tiple arcs between s

and s

are not allowed. A

stimulus on the SUT is assigned to each arc.

• The transition probability from state i to state j,

denoted by p

i, j

for an existing arc a

i j

. Otherwise

the transition probability p

i, j

= 0. The transition

probabilities obey the conditions 0 ≤ p

i, j

≤ 1 and

∑

j=1

i, j

= 1 ∀i = 1, . . . , n (1)

that states that the probabilities of all outgoing

arcs from a certain state s

must sum up to one.

Two states have special characteristics, that are s

which is the sole initial state (also: start state) and s

which is the ﬁnal state (also: end state). The values

i, j

can be stored in a state-transition-matrix P. El-

ement describes p

i j

the probability of the transition

from state s

to state s

The representation of a MCUM as directed graph

is depicted in Figure 1.

Each path from the start to the ﬁnal state is a valid

test case. Test cases can be generated straightforward

by random walk starting from the initial state, this

way obtaining statistical sampled test cases. Differ-

ent usage proﬁles, representing e.g. different types of

Figure 1: MCUM representation as directed graph.

users, can be applied to direct the statistical derivation

of test cases. Therefore, distinct user characteristics

or testing targets can be considered during test case

generation.

For a usage model based on a Markov chain many

computations can be performed that provide valuable

information for test management (cp. section 4.2).

3 TEST REQUIREMENTS AND

CONSTRAINTS IN THE

AUTOMOTIVE DOMAIN

Test requirements and constraints in the automotive

domain are composed of the

• SUT test requirements

• test automation

• test bench

In the following, the test automation Extended Au-

tomation Method (EXAM), applied within the Volk-

swagen AG, and the test benches employed in the con-

text of this paper are brieﬂy introduced.

EXAM Testing Process. Test automation in the

scope of EXAM means the automated generation of

platform dependent code and the execution of the

derived test suite without human interaction. The

majority of test cases is automatically executed on

Hardware-in-the-loop (HIL) systems, that are intro-

duced in the next paragraph. However, it is hardly

possible to systematically derive test suites that meet

the requirements in the automotive domain in an op-

timal manner.

HIL Testing. Hardware-in-the-loop (HIL) simula-

tion provides a platform for testing embedded systems

composed of Electronic Control Units (ECUs) under

conditions so that the ECUs feel like being in a real

car. This is achieved by including the ECUs in a sim-

ulated environment.

The main issue is currently how to use the HIL test

platform in an efﬁcient manner, since testing time on

HIL systems is expensive.

MODEL DRIVEN TESTING WITH TIME AUGMENTED MARKOV CHAIN USAGE MODELS - Computations and

Test Case Generation Algorithms for Time Augmented Markov Chain Usage Models

203

Challenges in Automated Test Case Generation.

As test cases until now are created without systemati-

cally considerung time and timing constraints the de-

cision which test cases to create and to execute within

time constraints remained in the hands of the test de-

signer. Thus, this decision has not been done system-

atically on the basis of an unambiguous model. One

drawback of the existing procedure is that valuable

testing time may not be used in an optimal manner.

Strategies have been proposed in (Siegl, 2008) to

derive test cases from MCUMs under consideration

of time. However, the existing methods for test case

generation based on MCUMs lack the systematic in-

tegration of usage time in the MCUMs themselves,

since they are discrete time Markov chains and each

state transition corresponds to one time step on an ab-

stract time base. Until now concepts are known to in-

tegrate timing aspects provisionally (Beyer and Dulz,

2005), but not coherently and in a manner that the us-

age time is integrated by keeping the semantics of the

MCUM. Therefore, we present in this paper an ex-

tension of MCUMs that keeps the semantic and that

supports automated test case generation for embedded

systems in test environments as they are established in

the automotive industry.

Figure 2: Hardware in the loop simulation.

4 TIME AUGMENTED MARKOV

CHAIN USAGE MODELS

This section is structured as follows: First of all the

enhancement of MCUMs is described. Next Markov

Computations are introduced. These can be made be-

fore and after test case execution and provide valu-

able test management information, e.g. the estimated

number of test cases required to achieve a coverage

level. The enhancement of standard algorithms for

test case generation is described in the following para-

graph and ﬁnally the issue of coverage criteria for

Time Augmented MCUMs is presented.

4.1 Enhancement

The enhancement of MCUMs is structured as follows:

• A probability density function (pdf) of the resi-

dence time is assigned to each state s

. We denote

it as ts

• A pdf of the stimulus time ta

i, j

is assigned to each

arc a

i j

. This pdf describes the duration of the ex-

ecution of a stimulus on the SUT.

Besides, the transition probabilities p

i, j

as well as

the values ts

and ta

i, j

can be stored and exchanged

by means of a matrix P. The single values ∀ p

i j

are

replaced by the two elements p

i, j

and ta

i, j

. An addi-

tional row or column that extends the matrix by ele-

ments p

in+1

or p

n+1 j

is used to store the pdf ts

as-

signed to the usage state s

. This information could

also be stored in a one-dimensional vector related to a

matrix P. The extensions introduced by our approach

are presented in Figure 3.

Figure 3: MCUM extended by usage time pdfs.

The pdfs of time we used for this approach are:

• Deterministic time as a step function.

• Uniform distribution (discrete uniform distribu-

tion)

• Gaussian distribution (binomial distribution)

• Exponential distribution (geometric distribution)

In brackets the discrete approximation of the con-

tinuous functions is given.

By assigning these distributions it is possible to

describe usage that

• is always exactly predictable in time.

• is characterizable that it is equally likely within

time boundaries.

• happens typically after a certain time. The stan-

dard deviation is known.

• whose occurrence probability does not change

over time.

Therefore not only the variance of stimuli them-

selves, but also the variance of the timing, i.e. the ex-

ecution time of stimuli and the time between them can

be described by a MCUM. There is one issue that has

to be considered when creating such a model. Multi-

ple arcs a

j, that have their origin in the same state s

may be assigned the same stimulus. In this case the

range of values of the pdfs that describe the duration

of the arc traversal must be disjoint.

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4.2 Computations

Markov Computations for MCUMs are (Prowell,

2000):

• Expected number of occurrences of a state in a

test case

• Long-run state probabilities

• Long-run arc probabilities

• State occurrence probability

• Arc occurrence probability

• Expected length of a test case

Up to now these computations have produced ab-

stract and relative numbers with respect to the other

elements of the MCUM, i.e. describing characteris-

tics of the model. An example is the long-run state

probability, that is computed for each state in the fol-

lowing way: The expected number of occurrences of

the considered state is divided by the number of oc-

currences of all states. This is quite an abstract mea-

sure that does not represent the testing time spent in

each state. With our extension this computation be-

comes more valuable, because we can take the expec-

tation value of each ts

and determine in this manner

the fraction of time spent in each state during test-

ing. This result might differ considerably from the

relative occurrence rates and can thus be a better indi-

cator for the decision making which test cases should

be executed. Also, the other computations gain im-

portance, e.g. the number of occurrences of arc and

state weighted by the pdfs of time give an important

indicator for test planning. To complete the list, the

following computations are now possible with Time

Augmented MCUMs:

• mean time consumed in transitions

• mean time consumed by execution of each stimu-

lus

• Expected state residence time in a test case

• Long-run state mean residence time

• Long-run arc mean execution time

4.3 Algorithms

The existing algorithms for the generation of test

cases can still be used, however, they do not make use

of the timing information (Prowell, 2004). For this

reason we propose new algorithms that incorporate

the knowledge of time for test generation strategies.

The variance in time is considered during test gener-

ation and can this way also be found in the test cases.

Furthermore, the estimated execution time for each

generated test case is automatically available, which

is an important information for the test management.

Random Walk. Random walk is the standard sam-

pling technique to derive test cases from MCUMs. In

the following algorithms it is assumed that a certain

usage proﬁle is given from which the values t

, p

c, j

and ta

c, j

are obtained. These values describe the du-

ration of a state t

or of the execution of a stimulus

c j

in the concrete test case that is generated. The ex-

tended random walk for Time Augmented MCUMs is

described in Figure 4.

Data:

Time Augmented MCUM;

state under consideration;

Initialization:

← s

;

while s

not reached do

Get t

by sampling ts

;

Get a

c j

randomly from all p

c, j

;

Get t

c j

by sampling ta

c, j

;

← s

;

end

Figure 4: Random walk for Time Augmented MCUMs.

Most Probable Test Case. Algorithms to derive

the most probable test case usually choose the most

probable arc at each state. For the timing informations

assigned to states and arcs we assume that the expec-

tation value (in the following also: mean) of the pdf

for the time, denoted as E[pd f ] represents the most

probable value for the time.

Data:

Time Augmented MCUM;

state under consideration;

Initialization:

← s

;

while s

not reached do

← E[ts

];

c j

← a

c j

with max p

c, j

;

c j

← E[ta

c, j

];

← s

;

end

Figure 5: Most probable test case from Time Augmented

MCUMs.

Note that the determination of the next arc is left

untouched, i.e. the arc with the highest probability.

The time that is determined for each arc and state is

the expectation value for the pdf that is assigned to the

arc and state respectively.

Minimal Arc Coverage. Minimal arc coverage al-

gorithms can be used without modiﬁcation. However,

MODEL DRIVEN TESTING WITH TIME AUGMENTED MARKOV CHAIN USAGE MODELS - Computations and

Test Case Generation Algorithms for Time Augmented Markov Chain Usage Models

205

the timing information has to be applied depending

on the test strategy that one follows. If testing time

is short, one could take the minimal limiting value of

the pdfs.

Boundary Value Testing. Boundary value testing

is supported by the time information incorporated in

the Time Augmented MCUM. The following four

cases have to be considered:

Deterministic Distribution. In this case there

is no choice of time.

Uniform Distribution. Also in this case the de-

termination of boundary values is straightforward,

as they are given by the limits of the interval.

Gaussian Distribution. In these cases the

range of values is potentially inﬁnite. A rea-

sonable value for boundary value testing has to

be determined. If one prefers to have a unique

deﬁnition of the boundary values the values at

= µ[pd f ] ± 3σ[pd f ] can be taken. These val-

ues encompass 99.7% of the whole set.

Exponential Distribution. In the case of the

exponential distribution, a minimal value is given

by the point of origin. As a maximal boundary

value we suggest to use E[pd f ] = µ[pd f ].

4.4 Coverage Criteria

Graph based coverage criteria rely on the evaluation

of how the test paths belonging to the test cases cover

the graph abstraction (Hierons et al., 2008, p. 119).

This concept is hardly transferable to an inﬁnite do-

main like time, so a coverage criterion considering

timing aspects has to be deﬁned. These deﬁnitions

are based on the assumptions described in section 4.3.

These deﬁnitions, of course, can only give a general

idea for orientation, since the signiﬁcance depends

also on the special characteristics of the system under

test (SUT). That is to say that for testing a low level

non functional property such deﬁnitions can be eas-

ier applied than for testing a high level non functional

property. To give an example hybrid vehicle operat-

ing strategies shall be stated, where beside the oper-

ating strategy (Wallentowitz and Reif, 2006) itself the

potential timing in usage of various participants has to

be considered. A general deﬁnition of coverage con-

cerning the timing aspects is hard to give and cover-

age criteria dependent on characteristics of the tested

functionality (e.g. its safety criticality) is part of our

future work. Some basic deﬁnitions shall be given:

Deterministic Distribution. In this case there

is no range so only the speciﬁed value has to be

considered for coverage under a usage point of

view.

Uniform Distribution. The minimal, mean,

and maximal value could be good choices for min-

imal coverage testing.

Gaussian Distribution. Mathematical indica-

tors, such as the included fraction of possible re-

alizations of the value domain, or requirements,

can be a sound basis.

Exponential Distribution. The minimal or ex-

pectation value E[pd f ] = µ[pd f ] can be sufﬁcient,

strongly dependent on the functionality.

5 CASE STUDY

For the evaluation of our approach, a test issue in the

automotive domain has been chosen. It concerns an

aspect of the operation of the Electronic Stability Pro-

gram (ESP) switch in modern cars. Different func-

tionalities are invoked depending on the state of the

car and the pressing duration of the switch (AUDI

AG, 2005). Some of these functionalities are directly

desired by the user and some are assumed to represent

the reaction desired by the user.

In some cars, the ESP switch was placed in the

center console, where car passengers often deposit

their hand bag. This led to frequent misactivations

of this switch and as a consequence to undesired de-

activation and activation operations of the ESP func-

tionality. Due to this problem the handbag switching

mechanism is now integrated in the cars. This mech-

anism activates ESP irrevocable for the whole jour-

ney if the ESP Switch is pressed longer than a cer-

tain time, thus preventing the undesired deactivation

of ESP if an object is placed on top of the ESP switch.

In Figure 6 the model is presented that describes

the different activation durations of the ESP switch.

A subset of the computations in the classical way

is shown in table 1.

The probability is the same, that is 0.03844, to

stay during a test case in the start state (S9998), ﬁnal

state (S9999), and in the state in which the ignition

has been turned on (S1). The probability, however,

that the test case is in the state (S234) where the car

drives at 15kmph and the ESP switch has been pressed

for less than 3 seconds is only 0.01443.

Table 1: Computations for Case Study ESP Switch.

State Stationary Visiting Expected number

ID distrib. probab. of visits

S9998 0.03844 1.00000 1.00000

S1 0.03844 1.00000 1.00000

S234 0.01443 0.22232 0.37538

S4 0.07687 1.00000 2.00000

S9999 0.03844 1.00000 1.00000

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Figure 6: Sub-Usage scenario for ESP push button.

Table 2: Computations for Case Study ESP Switch with

Time Augmented MCUM.

State Mean Expected number Expected

ID residence time of visits residence time

S9998 0.00000 1.00000 0.00000

S1 9.00000 1.00000 9.00000

S234 10.0000 0.37538 3.75380

S4 0.01000 2.00000 0.02000

S9999 0.00000 1.00000 0.00000

Now have a look at the computations for a Time

Augmented MCUM, depicted in Table 2. All mea-

sures for the time are given in seconds. Now we can

see that for the given usage proﬁle we would expect

to be in state S1 9 seconds in a typical test case and

3.75 seconds in state S234. Next we see that states

S9998, S9999 and S4 have less impact on the testing

time, though they are visited once or twice in each

test case. This information is valuable for the person

engaged to decide when to test and what to test and

to assess, if the usage proﬁle is suitable to derive test

cases that meet the testing goals within time limits.

6 FUTURE WORK

The future work will focus on computations for the

test planing, integrating coverage aspects, and test

case generation strategies. The elaboration of timing

coverage criteria for functional requirements is neces-

sary. So test case generation strategies are developed

that should optimize the usage of the test benches,

considerung functional-based timing requirements.

7 CONCLUSIONS

In this paper an extension has been presented and

evaluated that integrates usage time in MCUMs. The

case study has been conducted with an example from

the automotive domain. Substantial advantages arised

from the application of our extension, since the inte-

grated time information can be used either for the test

planing or for the test case generation. In the ﬁrst

case indicators such as test case length are not any

more abstract numbers, but provides the test designer

with information about the expected duration of a test

case. Secondly this information can be used by the

test case generation algorithms for one thing to gen-

erate test cases that are also variant in time and not

only in operations. For another thing test case gener-

ation algorithms can make use of this information to

generate test suites that meet test requirements within

time constraints in an optimal manner. In this way the

whole testing process can be supported by the pre-

sented approach.

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