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Efficient Probabilistic Model Checking of Smart Building Maintenance using Fault Maintenance Trees

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Cyber-physical systems, like Smart Buildings, power plants and data centers have to meet high standards, both in terms of reliability and availability. Such metrics are typically evaluated using Fault trees (FTs) and do not consider maintenance strategies which can significantly improve lifespan and reliability. Fault Maintenance trees (FMTs) -- an extension of FTs that also incorporate maintenance and degradation models, are a novel technique that serve as a good planning platform for balancing total costs and dependability of a system. In this work, we apply the FMT formalism to a Smart Building application. We propose a framework for modelling FMTs using probabilistic model checking and present an algorithm for performing abstraction of the FMT in order to reduce the size of its equivalent Continuous Time Markov Chain. This allows us to apply the probabilistic model checking more efficiently. We demonstrate the applicability of our proposed approach by evaluating various dependability metrics and maintenance strategies of a Heating, Ventilation and Air-Conditioning system's FMT.
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Eicient Probabilistic Model Checking of Smart Building
Maintenance using Fault Maintenance Trees
Nathalie Cauchi
Department of Computer Science, University of Oxford
Oxford, United Kingdom
nathalie.cauchi@cs.ox.ac.uk
Khaza Anuarul Hoque
Department of Computer Science, University of Oxford
Oxford, United Kingdom
khaza.hoque@cs.ox.ac.uk
Alessandro Abate
Department of Computer Science, University of Oxford
Oxford, United Kingdom
aabate@cs.ox.ac.uk
Mari¨
elle Stoelinga
FMT Group, University of Twente
Twente, e Netherlands
marielle@cs.utwente.nl
ABSTRACT
Cyber-physical systems, like Smart Buildings and power plants,
have to meet high standards, both in terms of reliability and avail-
ability. Such metrics are typically evaluated using Fault trees (FTs)
and do not consider maintenance strategies which can signicantly
improve lifespan and reliability. Fault Maintenance trees (FMTs) –
an extension of FTs that also incorporate maintenance and degra-
dation models, are a novel technique that serve as a good planning
platform for balancing total costs and dependability of a system.
In this work, we apply the FMT formalism to a Smart Building
application. We propose a framework for modelling FMTs using
probabilistic model checking and present an algorithm for per-
forming abstraction of the FMT in order to reduce the size of its
equivalent Continuous Time Markov Chain. is allows us to apply
the probabilistic model checking more eciently. We demonstrate
the applicability of our proposed approach by evaluating various
dependability metrics and maintenance strategies of a Heating,
Ventilation and Air-Conditioning system’s FMT.
CCS CONCEPTS
Computer systems organization
Maintainability and main-
tenance;
KEYWORDS
Fault Maintenance Trees, Formal modelling, Probabilistic Model
checking, Reliability, Building Automation Systems, PRISM
ACM Reference format:
Nathalie Cauchi, Khaza Anuarul Hoque, Alessandro Abate, and Mari
¨
elle
Stoelinga. 2017. Ecient Probabilistic Model Checking of Smart Building
Maintenance using Fault Maintenance Trees. In Proceedings of BuildSys ’17,
Del, Netherlands, November 8–9, 2017, 10 pages.
DOI: 10.1145/3137133.3137138
e corresponding author
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BuildSys ’17, Del, Netherlands
©2017 ACM. 978-1-4503-5544-5/17/11.. . $15.00
DOI: 10.1145/3137133.3137138
1 INTRODUCTION
Worldwide, buildings account for approximately 40% of the total
energy consumption and 20% of the total
CO2
emissions, annu-
ally [
5
]. Ecient Building Automation Systems (BAS) can reduce
energy consumption by up to 30% through their optimal operation,
continuous commissioning and maintenance [
5
]. Constructions
employing such technologies are termed Smart Buildings. High
standards have to be adhered by such technologies, both in terms of
reliability and availability. One way of achieving this is by employ-
ing methods to perform preventative and predictive maintenance
actions. Diagnostic and fault detection techniques for Smart Build-
ing applications have been developed in [
2
,
15
]. Predictive and
preventative maintenance strategies are devised in [
1
,
4
]. However,
these techniques preclude availability and reliability measurements
and focus only on synthesis of maintenance policies in the pres-
ence of degradation and fault nding. Reliability and availability
are typically tackled using Fault Trees (FTs), where the focus is
on nding the root causes of a system failure using a top-down
approach. FTs do not include maintenance strategies in the analysis
– a key element in reducing component failures. [
14
] presents the
Fault Maintenance Tree (FMT) as an extension of FT encompassing
both degradation and maintenance models. e degradation models
represent the dierent levels of component degradation and are
known as Extended Basic Events (EBE). e maintenance models
incorporate the undertaken maintenance policy which includes
both inspections and repairs. ese are modelled using Repair and
Inspection modules in the FMT framework.
In literature, FMTs are analysed using Statistical Model Checking
technique (SMC) [
14
] and provide statistical guarantees. In contrast,
Probabilistic Model Checking (PMC), based on numerical analysis,
provide formal guarantees with higher accuracy when compared
with SMC [
17
]. However, numerical methods are far more memory
intensive and may result in a state space explosion. is limitation
of PMC oen leaves SMC as the last resort [
17
]. In this paper
we tackle the FMT analysis using PMC. Our contributions can be
summarised as follows:
(1)
We formalise the FMT framework using Continuous Time
Markov Chain (CTMCs).
(2)
We formalise the dependability metrics using the extended
Continuous Stochastic Logic (CSL) formalism such that
they can be computed using the PRISM model checker [
12
].
BuildSys ’17, November 8–9, 2017, Del, Netherlands N. Cauchi et al.
(3)
To mitigate the state space explosion problem, we present
an FMT abstraction technique which decomposes a large
FMT into an equivalent abstract FMT based on our pro-
posed graph decomposition algorithm. Using our frame-
work, we are able to achieve a 67% reduction in the state
space size.
(4)
Finally, we construct a FMT that identies failure of a Heat-
ing, Ventilation and Air-conditioning system (HVAC). We
apply the developed framework to the built FMT and evalu-
ate relevant dependability metrics, together with dierent
maintenance strategies using the PRISM model checker.
To the best of our knowledge, this is the rst aempt to anal-
yse FMTs using Probabilistic Model Checking and also the rst
application to Smart Building systems.
is article has the following structure: Section 2 introduces the
fault maintenance trees and probabilistic model checking frame-
works. is is followed by the developed methodology for mod-
elling FMT using CTMCs and performing model checking in Sec-
tion 3. e framework is applied to a heating, ventilation and
air-conditioning (HVAC) case study which is presented in Section 4.
2 PRELIMINARIES
2.1 Fault maintenance trees framework
Fault trees are directed acyclic graphs (DAG) describing the com-
binations of component failures that lead to system failures. e
leaves in the fault trees are called basic events and denote the sys-
tem failures. e internal nodes of the graph are called gates and
describe the dierent ways that failures can interact to cause other
components to fail. e gates in a fault tree can be of several types
and these include the AND gate, OR gate, k/N-gate [14].
Fault maintenance trees (FMT) extend fault trees by including
maintenance (all the standard FT gates are also employed by the
FMTs). is is achieved by making use of:
(1)
Extended Basic Events - e basic events are modied to
incorporate degradation models of the component the leaf
represents. e degradation models represent dierent
discrete levels of degradations the components can be in
and are a function of time.
(2)
Rate Dependency Events - A new gate introduced in [
14
],
labelled as RDEP that accelerates the degradation rates of
dependent child nodes and is depicted in Figure 1. When
the component connected to the input of the RDEP fails,
the degradation rate of the dependent components is ac-
celerated with an acceleration factor γ.
RDEP
in
Children (n)
Figure 1: RDEP gate with 1 input and dependent components also known
as children.
(3)
Repair and Inspection modules - e repair module (RM) per-
forms cleaning or replacements actions. ese actions can
be either carried out using xed time schedules or when
enabled by the inspection module (IM). e IM performs
periodic inspections and when components fall below a cer-
tain degradation threshold a repair or partial replacement
is initiated by the IM to be performed by the RM.
2.2 Probabilistic model checking
Model checking is a well-established formal verication technique
used to verify the correctness of nite-state systems. Given a for-
mal model of the system to be veried in terms of labelled state
transitions and the properties to be veried in terms of temporal
logic, the model checking algorithm exhaustively and automatically
explores all the possible states in a system to verify if the property
is satisable or not. Probabilistic model checking deals with systems
that exhibit stochastic behaviour and is based on the construction
and analysis of a probabilistic model of the system. We make use
of CTMCs, having both transition and state labels, to perform sto-
chastic modelling. Properties are expressed in the form of extended
Continuous Stochastic Logic (CSL) [11].
Denition 2.1. e tuple
C=(S,s0,TL,AP,L,R)
denes a CTMC
which is composed of a set of states
S
, the initial state
s0
, a nite
set of transition labels
TL
, a nite set of atomic propositions
AP
,
a labelling function
L
:
S
2
AP
and the transition rate matrix
R
:
S×SR0
. e rate
R(s,s0)
denes the delay before which
a transition between states
s
and
s0
takes place. If
R(s,s0),
0
then the probability that a transition between the states
s
and
s0
is
dened as 1
eR(s,s0)t
where
t
is time. No transitions will trigger
if R(s,s0)=0.
e logic of CSL species state-based properties for CTMCs,
built out of propositional logic, a steady-state operator that refers
to the stationary probabilities, and a probabilistic operator for rea-
soning about transient state probabilities. e state formulas are
interpreted over states of a CTMC, whereas the path formulas are
interpreted over paths in a CTMC. For detail about the syntax and
semantics of CSL (which also includes reward formulae), we refer
the interested readers to [
11
]. Examples of a CSL property with
its natural language translation are: (i) P
0.95
[F
complet e
] - “e
probability of the system eventually completing its execution suc-
cessfully is at least 0.95”. (ii) R
=?
[F
success
] - “What is the expected
reward accumulated before the system successfully terminates?”
3 FORMALIZING FMTS USING CTMCS
In this section, we rst formalise the FMT framework by presenting
the formal syntax and semantics for modelling FMTs using CTMCs.
Next, we list the set of metrics used to analyse the FMT. Finally, we
present the developed framework which allows us to analyse large
FMTs using probabilistic model checking (PMC).
3.1 FMT Syntax
To formalise the syntax of FMTs using CTMCs, we rst dene the
set
F
, characterizing each FMT element by type, inputs and rates.
We introduce a new element called DELAY which will be used to
model the deterministic time delays required by the extended basic
events (EBE), repair module (RM) and inspection module (IM). We
restrict the set
F
to contain the EBE, RDEP gate, OR gate, DELAY,
RM and IM modules since these will be the components used in the
case study presented in Section 4.
Eicient PMC of Smart Building Maintenance using FMTs BuildSys ’17, November 8–9, 2017, Del, Netherlands
Denition 3.1. e set
F
of FMT elements consists of the follow-
ing tuples. Here,
n,NN
are natural numbers,
thresh,in,trig
{
0
,
1
}
take binary values,
Tde д,Tcl n
,
Trp lc ,Tr e p ,Toh
,
Tinsp R0
are deterministic delays, and γR0is a rate.
(EBE ,Td eд,Tc l n ,Tr pl c ,N)
represent the extended basic
events with
N
discrete degradation levels, each of which
degrade with a time delay equal to
Tde д
. It also takes as
inputs the time taken to restore the EBE to the previous
degradation level
Tcl n
when cleaning is performed and
the time taken to restore the EBE to its initial state
Trp l c
following a replacement action.
(RDE P ,n,γ,in,Td e д)
represents the RDEP gate with
n
de-
pendent children, acceleration rate
γ
, the input
in
which
activates the gate and
Tde д
the degradation rate of the
dependent children.
(OR,n)represents the OR gate with ninputs.
(RM,n,Tr ep ,Toh ,Tin sp ,Tc l n ,Tr pl c ,thresh,trig)
represents
the RM module which acts on
n
EBEs (in our case, this cor-
responds to all the EBEs in the FMT). e RM can either be
triggered periodically to perform a cleaning action, every
Tr ep
delay, or a replacement action, every
Toh
delay, or by
the IM when the delay
Tinsp
has elapsed and the
thresh
condition is met. e time to perform a cleaning action
is
Tcl n
, while the time taken to perform a replacement is
Trp l c
. e
trig
signal ensures that when the component is
not in the degraded states, no unnecessary maintenance
actions are carried out.
(IM,n,Ti ns p ,Tcl n ,Tr pl c ,thresh)
represents the IM module
which acts on
n
EBEs (in our case, this corresponds to all the
EBEs in the FMT). e IM initiates a repair depending on
the current state of the EBE. Inspections are performed in a
periodic manner, every
Tinsp
. If during an inspection, the
current state of the EBE does not correspond to the new or
failed state (i.e. the degradation level of the inspected EBE
is below a certain threshold), the
thr esh
signal is activated
and is sent to the RM. Once a repair action is performed the
IM moves back to the initial state with a delay equal to
Tcl n
or
Trp l c
depending on the maintenance action performed.
(DE LAY ,T,N)
represents the DELAY module which takes
two inputs representing the deterministic delay
T∈ {Tdeд,
Tcl n ,Tr pl c ,Tr ep ,Toh ,Tins p }
to be approximated using an
Erlang distribution with
N
number of states. is DELAY
module can be extended by inclusion of a reset transition
label, which when triggered restarts the approximation of
the deterministic delay before it has elapsed. e extended
DELAY module is referred to as (DELAY ,T,N)ex t .
e FMT is dened as a special type of directed acyclic graph
G=(V,E)
where the vertices
V
represent the gates and the events
which represent an occurrence within the system, typically the
failure of a subsystem down to an individual component level, and
the edges
E
which represent the connections between vertices.
Events can either represent the EBEs or intermediate events which
are caused by one or more other events. e event at the top
of the FMT is the top event (TE) and corresponds to the event
being analysed - modelling the failure of the (sub)system under
consideration. e EBE are the leaves of the DAG. For
G
to be a
well-formed FMT, we take the following assumptions (i) vertices
are composed of the OR, RDEP gates, (ii) there is only one top event,
(iii) RDEP can only be triggered by EBEs and (iv) RM and IM are
not part of the DAG tree but are modelled separately
1
. is DAG
formulation allows us to propose a framework in Subsection 3.5
such that we can eciently perform probabilistic model checking.
Denition 3.2. A fault maintenance tree is a directed acyclic
graph G=(V,E)composed of vertices Vand edges E.
3.2 Semantics of FMT elements
Next, we provide the CTMC semantics for each FMT element
f∈ F
.
ese elements are then instantiated based on the underlying FMT
structure to form the semantics of the whole FMT in CTMC form.
DELAY
.We dene the semantics for the
(DE LAY ,T,N)
element
using Figure 2(a) and describe the corresponding CTMC using
the set of states given by
D={d0,d1, . . . , dN+1}
, the initial state
d0
, the set of transitions labels
TL ={trigger,move}
, the set of
atomic propositions
AP ={T}
with
L(d0)=· · · =L(dN)=
, and
L(dN+1)={T}
. e rate matrix
R
becomes clear from Figure 2(a)
and
Rij =
µi=0j=1
N
T((i1i<N+1)j=i+1)
(i=N+1j=1)
0 otherwise,
(1)
with
i
representing the current state,
j
is the next state and
µ
is
a xed large value corresponding to introducing a negligible de-
lay, which is used to trigger all the DELAY modules at the same
time (cf. Denition 2.1). In Figure 2(b) we dene the semantics of
(DE LAY ,T,N)ex t
. is results in the CTMC described using the
state space
D={d0,d1, . . . , dN+1}
, the initial state
d0
, the set of
transition labels
TL ={trigger,move,reset}
, the set of atomic
propositions
AP ={T}
, the labelling function
L(d0)=L(d1)=
· · · =L(dN)=, and L(dN+1)={T}and the rate matrix Rwhere
Rij =
µi=0j=1
1(i2i<N+1)j=1
N
T((i1i<N+1)j=i+1)
(i=N+1j=1)
0 otherwise,
(2)
with
i
representing the current state and
j
is the next state. In both
instances, the deterministic delays is approximated using an Erlang
distribution [
8
] and all DELAY modules are synchronised to start
together using the trigger transition label. e extended DELAY
module have the transition labels
reset
which restarts the Erlang
distribution approximation whenever the guard condition is met
at a rate of 1
×Rsync
where
Rsync
is the rate coming from the
use of synchronisation with other modules causing the reset to
occur ( as explained in Subsection 3.3). is is required when a
maintenance action is performed which restores the EBE’s state
back to the original state and thus restart the degradation process,
before the degradation time has elapsed.
1
Note, for dierent FMT structure same RM and IM modules are used, thus RM and
IM modules are independent of FMT structure
BuildSys ’17, November 8–9, 2017, Del, Netherlands N. Cauchi et al.
Remark 1. e basic properties of an Erlang distribution: A ran-
dom variable
ZR+
has an Erlang distribution with
kN
stages
and a rate
λR+,ZErl anд(k,λ)
, if
Z=Y1+Y2+. . . Yk
where
each
Yi
is exponentially distributed with rate
λ
. e cumulative den-
sity function of the Erlang distribution is characterised using,
f(t;k,λ)=1
k1
X
n=0
1
n!exp(λt)(λt)nfor t,λ0 (3)
and for
k=
1, the Erlang distribution simplies to the exponential
distribution. In particular, the sequence
ZkErl anд(k,λk)
converges
to the deterministic value
1
λ
for large
k
. us, we can approximate a
deterministic delay
T
with a random variable
ZkErl anд(k,k
T)
[
3
].
Note, there is a trade-o between the accuracy and the resulting blow-
up in size of the CTMC model for larger values of
k
(a factor of
k
increase in the model size) [
8
,
9
]. In this work, the Erlang distribution
will be used to model the xed degradation rates, the maintenance
and inspection signals. is is a similar approach taken in [
14
] where
degradation phases are approximated by an (k,
λ
)-Erlang distribution.
d0
start
d1d2d3. . . dN+1
trigger,µ
move,N
Tmove,N
Tmove,N
Tmove,N
T
move,N
T
(a) CTMC representing DELAY with Nstates used to approximate a de-
lay equal to Tapproximated using Er l anд(N,N
T). e transition labels
TL ={trigger,move}are shown on each of the transitions. e state la-
bels are not shown and the initial state of the CTMC is pointed to using
an arrow labelled with start.
d0
start
d1d2d3. . . dN+1
trigger ,µ
move ,N
Tmove ,N
Tmove ,N
Tmove ,N
T
reset,1
reset,1
reset,1
reset,1
(b) CTMC representing the extended DELAY with Nstates used to ap-
proximate a delay equal to T. Delay approximated using Er l anд(N,N
T).
e transition labels TL ={trigger,move,reset}are shown on each of
the state transitions, while the state labels are not shown.
Figure 2: CTMC for (a) DELAY and (b) DELAY with reset guard.
Extended Basic Events (EBE)
.e EBE are the leaves of the
FMT and incorporate the component’s degradation model. EBE are
a function of the total number of degradation steps
N
considered.
Figure 3 shows the semantics of the
(EBE ,Td eд,Tc l n ,Tr ep ,N=
3
)
.
e corresponding CTMC is described by the tuple
({s0,s1,s2,s3},
s0,TLE BE ,APE B E ,LEB E ,RE BE )where s0is the initial state ,
TLEBE ={degradei {0, . .., N},perform clean,perform replace},
the atomic propositions
APEBE ={new,thresh,failed}
, the la-
belling function
L(s0)={new},L(s1)=L(s2)={thresh},L(s3)=
{f ailed}
and
REB E ="0 1 0 0
1 0 1 0
1 1 0 1
1 0 1 0 #.
e deterministic time delays
taken as inputs are modelled using three dierent DELAY modules:
s0
start
s1s2s3
degrade1,1degrade2,1degrade3,1
perform clean,1perform clean,1
perform clean,1
perform replace,1
perform replace,1
perform replace,1
Figure 3: CTMC representing the EBE with N=3with the transition labels
TLE BE ={degradei {1,2,3},perform clean,perform replace}on each of the
state transitions. e state labels are not shown and the initial state is pointed
to by the arrow labelled with start.
(1)
an extended DELAY module approximating
Tde д
with the
transition label
move
replaced with
degradeN
such that
synchronisation between the two CTMCs is performed
(explained in Subsection 3.3). When
Tde д
has elapsed the
transition labelled with
degradeN
is triggered and the EBE
moves to the next state at a rate equal to
N
Tde д×
1
2
. e
reset
transition label and corresponding transitions are
replicated in extended DELAY module and replaced with
perform clean
and
perform replace
. When the corre-
sponding maintenance action is performed one of the tran-
sition label is triggered and the state of the EBE moves to
previous state (if cleaning action is carried out) or to the
initial state (if replace action is performed).
(2)
a DELAY module approximating
Tcl n
with the transition
label
move
replaced with
perform clean
. When
Tcl n
has
elapsed the transition with transition label
perform clean
is triggered and the EBE moves to the previous state at a
rate equal to N
Tcl n .
(3)
a DELAY module approximating
Trp l c
with the transi-
tion label
move
replaced with
perform replace
. When
Trp l c
has elapsed the transition having the transition label
perform replace
is triggered and the EBE moves to the
initial state at a rate equal to N
Trp l c .
e transition labels
perform clean
and
perform replace
cannot
be triggered at the same time and it is assumed that
Tcl n ,Tr pl c
.
is is a realistic assumption as only one maintenance action is
performed at the same time.
RDEP gate
.e RDEP gate has static semantics and is used in
combination with the semantics of its
n
dependent EBEs. When trig-
gered (
in =
1), the associated EBE reaches the state labelled
failed
,
the degradation rate of the
n
dependent children is accelerated by
a factor γ. We model the in signal using,
in =
1L(s)=failed,
0 otherwise,(4)
where
L(s)
is the label of the current state of the associated EBE.
Similarly, we map the RDEP gate function using,
RA =
γTde д1, . . . , γTde дnin =1,
Tde д1, . . . , Tde дnotherwise,(5)
2
is is a direct consequence of synchronisation and corresponds to
R×REB E
. Refer
to Subsection 3.3
Eicient PMC of Smart Building Maintenance using FMTs BuildSys ’17, November 8–9, 2017, Del, Netherlands
where
Tde дi,i1, . . . n
corresponds to the degradation rate of the
ndependent children. 3
OR gate
.e OR gate indicates a failure when either of its input
nodes have failed and also does not have semantics itself but is used
in combination with the semantics of its
n
dependent input events
(EBEs or intermediate events). We use,
FAIL =
0E1=1∧ · · · En=1
1 otherwise (6)
where
Ei=
1
,i
1
. . . n
corresponds to when the
n
events, con-
nected to the OR gate, represent a failure in the system. In the case
of EBEs, E1=1 occurs when the EBE reaches the failed state .
Repair module (RM)
.Figure 4 (a) shows the semantics of
(RM,n,
Tr ep ,Toh ,Ti ns p ,Tcl n ,Tr pl c ,Tr pl c ,thresh,trig)
. e CTMC is de-
scribed using the state space
{rm0,rm1}
, the initial state
rm0
, the
transition labels
TLRM ={inspect,check clean,check replace,
trigger clean,trigger replace}
, the atomic propositions
AP =
{maintenance }
, the labelling function
L(rm0)={∅},L(rm1)=
{maintenance}
and with
RI M =f1 1
1 0 g
. For the sake of clarity in
Figure 4 (a), we used the transition labels
check maintenance
and
trigger maintenance
. e transition label
check maintenance
and corresponding transitions are replicated and the transition
labels replaced by
check clean
or
check replace
to allow for
both type of maintenance checks. Similarly, the transition la-
bel
trigger maintenance
and corresponding transitions are du-
plicated and the transition labels replaced by
trigger clean
or
trigger replace
to allow the initiation of both type of main-
tenance actions to be performed. Due to synchronisation, only
one of the transitions may trigger at any time instance (as ex-
plained in Subsection 3.3). e transition labels
trigger clean
or
trigger replace
correspond to the transition label
trigger
within the DELAY module approximating the deterministic delays
Tcl n
and
Trp l c
respectively. e deterministic delays which trig-
ger
inspect
,
check clean
or
check replace
correspond to when
the time delays
Tin sp ,Tr e p
and
Toh
respectively, have elapsed. All
these signals are generated using individual DELAY modules with
the
move
transition label for each module replaced using
inspect
,
check clean
or
check replace
respectively. e
thresh
signal is
modelled using,
thresh =
1L(sj,1)=thresh ∨ · · · ∨ L(sj,n)=thresh,
0 otherwise,(7)
where
L(sj,i),j
0
. . . N,i
1
. . . n
correspond to the label of the
current state
j
of each of the
n
EBE. Similarly, we model the
trig
signal using
trig =
1L(sj,1),new ∨ · · · ∨ L(sj,n),new,
0 otherwise.(8)
Both signals act as guards which when triggered determine which
transition to perform (cf. Fig. 4 (a)).
3
Note, this eectively results in changing the deterministic delay being modelled by
the DELAY module to a new value if in =1.
Inspection module (IM)
.e semantics of the
(IM,n,Ti ns p ,
Tcl n ,Tr pl c ,thresh)
is depicted in Figure 4 (b). e CTMC is dened
using the tuple
({im0,im1},im0,TLI M ,API M ,LI M ,RI M )
. Here,
TLI M ={inspect,perform clean,perform replace}
,
API M =
{∅}
,
L(s0)=L(s1)=
and
RI M =f1 1
1 0 g
. e
thresh
signal corre-
sponds to same signal used by the RM, given using
(7)
. In Figure 4
(b), for clarity, we use the transition label
perform maintenance
.
is transition label and corresponding transitions are duplicated
and the transition labels are replaced by either perform clean or
perform replace
to allow for both type of maintenance actions to
be performed when one of them is triggered using synchronisation.
e same DELAY modules used in the RM and EBE to represent
the deterministic delays are used by the IM. e DELAY module
used to represent the deterministic delays
Tcl n
and
Trp l c
triggers
the transition labels
perform clean
or
perform replace
. is
represents that the maintenance action has completed.
rm0
start rm1
inspect,thresh =0,1
check maintenance, trig =0,1
check maintenance, trig=1,1
inspect, thresh =1,1
trigger maintenance,1
(a) CTMC representing the RM with TLRM =
{inspect,check maintenance,perform maintenance}shown on the
state transitions. e guard condition trig =0/1or thresh =0/1must be
satised for the corresponding transition to trigger when it is activated
via synchronisation with the transition label.
im0
start im1
inspect, thresh =0,1
inspect, thresh =1,1
perform maintenance ,1
(b) CTMC representing the IM with TLI M =
{inspect,perform maintenance}shown on the state transitions.
e guard condition trig =0and thresh =1must be satised for the cor-
responding transition to trigger when it is activated via synchronisation
with the transition label.
Figure 4: CTMC for (a) RM and (b) IM.
3.3 Semantics of FMT
Next, we show how to obtain the semantics of a FMT from the
semantics of its elements using the FMT syntax introduced in Sub-
section 3.1. We dene the DAG
G
by dening the vertices
V
and
the corresponding events
E
. e leaves of the DAG are the events
corresponding to the EBE. e events
E
are connected to the ver-
tices
V
, which trigger the corresponding auxiliary function used to
represent the semantics of the gates. e
Events
connected to the
RM and IM are initiated by triggering the auxiliary functions
thresh
and
trig
given using
(7)
and
(8)
respectively. Based on the structure
of
G
, we compute the corresponding CTMC by applying parallel
composition of the individual CTMCs representing the elements of
the FMT. e parallel composition formulae are derived from [
7
]
and dened as follows,
BuildSys ’17, November 8–9, 2017, Del, Netherlands N. Cauchi et al.
Denition 3.3 (Interleaving Synchronization). e interleaving
synchronous product of
C1=(S1,s01,TL1,AP1,L1,R1)
and
C2=
(S2,s02,TL2,AP2,L2,R2)
is
C1||C2=(S1×S2,(s01,s02 ),TL1TL2,AP1
AP2,L1L2,R)where Ris given by:
s1
α1,λ1
s0
1
(s1,s2)α1,λ1
(s0
1,s2)
,and s2
α2,λ2
s0
2
(s1,s2)α2,λ2
(s1,s0
2)
,
and
s1,s0
1S1
,
α1TL1
,
R1(s1,s0
1)=λ1
,
s2,s0
2S2
,
α2TL2
,
R2(s2,s0
2)=λ2.
Denition 3.4 (Full Synchronization). e full synchronous prod-
uct of
C1=(S1,s01,TL1,AP1,L1,R1)
and
C2=(S2,s02,TL2,AP2,L2,R2)
is
C1||C2=(S1×S2,(s01,s02),TL1TL2,AP1AP2,L1L2,R)
where Ris given by:
s1
α,λ1
s0
1and s2
α,λ2
s0
2
(s1,s2)α,λ1×λ2
(s0
1,s0
2)
and
s1,s0
1S1
,
αTL1TL2
,
R1(s1,s0
1)=λ1
,
s2,s0
2S2
,
α2TL2
,
R2(s2,s0
2)=λ2.
For any pair of states, synchronisation is performed either using
interleaving or full synchronisation. For full synchronisation, as
in Denitions 3.3, the rate of a synchronous transition is dened
as the product of the rates for each transition. e intended rate
is specied in one transition and the rate of other transition(s)
is specied as 1. For instance, the RM synchronises using full
synchronisation with the DELAY modules representing
Tinsp
,
Tr ep
and
Trp l c
and therefore, to perform synchronisation between the
RM and the DELAY modules, the rates of all the transitions of RM
should have a value of 1 (cf. Fig. 4 (a)), while the rate of the DELAY
modules represent the actual rates (cf. Fig 2). e same principle
holds for the EBEs and the IM. We refer the reader to Table 1 to
further elucidate the synchronisation between the FMT components
and the method employed during the parallel composition.
Example. Consider, a simple example showing the time signals
and synchronisations required for modelling an EBE and the RM
and IM. e EBE has a degradation rate equal to
Tde д
and we limit
the functionality of the RM and IM by allowing only the mainte-
nance action to perform cleaning. We also need the corresponding
DELAY modules generating the degradation rates,
Tde д
and the main-
tenance rates
Tcl n ,Ti ns p ,Tre p
. e resulting CTMC is obtained by
performing a parallel composition of the components
Cal l =CE BE ||
CTde д||CRM | |CI M ||CTcl n ||CTin sp | |CTr ep .
e resulting state space
is then
Sal l =SE BE ×STd e д×SRM ×SI M ×STc l n ×STin sp ×STr e p
.
e synchronisation between the dierent components is shown in
Figure 5 and proceeds as follows:
(1)
All the DELAY modules (except
Tcl n
) start at the same time
using the trigger transition label.
(2)
When the extended DELAY module generating the
Tde д
time
delay elapses, the corresponding EBE moves to the next state
through synchronisation with the transition label
degradeN
.
(3)
e clock signals
Tr ep ,Ti ns p
represent periodic maintenance
and inspection actions and when the deterministic delay
is reached, through synchronisation with the transition la-
bel
check clean
or the
inspect
, the RM or IM modules is
triggered (cf. Fig. 4(a) and 4(b)). If RM triggers a main-
tenance action, the DELAY representing
Tcl n
is triggered
using the synchronisation labels
trigger clean
. Once the
deterministic delay
Tcl n
elapses, the EBE, the extended DE-
LAY module representing
Tde д
(where the
reset
transition
label within the extended DELAY module is replaced with
perform clean
) and the IM are reset using the transition
label perform clean.
Figure 5: Block diagram showing the synchronisation connections be-
tween one component and the other, together with the corresponding tran-
sition label which trigger synchronisation.
Remark 2. One should note that this results in the requirement of
a large state space, which is a function of the number of states used to
approximate the deterministic delays. us, to counteract this eect
we propose an abstraction framework in Subsection 3.5.
3.4 Metrics
We use PRISM to compute the metrics of the model described in
Subsection 2.1. e metrics can be expressed using the extended
Continuous Stochastic Logic (CSL) as follows:
(1)
Reliability : is can be expressed as the complement of the
probability of failure over the time
T
, 1
P
=?
[F
Tf ailed
].
(2)
Availability: is can be expressed as R
=?
[
CT
]
/T
, which
corresponds to the cumulative reward of the total time
spent in states labelled with okay and thresh during the
time T.
(3)
Expected cost: is can be expressed using R
=?
[
CT
], which
corresponds to the cumulative reward of the total costs
(operational, maintenance and failure) within the time T.
(4)
Expected number of failure: is can be expressed using
R
=?
[
CT
], which corresponds to the cumulative transition
reward that counts the number of times the top event enters
the failed state within the time T.
3.5 Decomposition of FMTs
e use of CTMC and deterministic time delays results in the re-
quirement of a large state space for modelling the whole FMT (cf.
Remark 2). We therefore propose an approach which decomposes
the large FMT into an equivalent abstract CTMC which can be
analysed using PRISM. e process involves two transformation
steps. First we convert the FMT into the equivalent directed acyclic
graph (DAG) and split this graph into a set of smaller sub-graphs.
Second, we transform the sub-graphs into the equivalent CTMC
by making use of the developed FMT components semantics (cf.
Subsec. 3.2), and performing parallel composition of the individual
Eicient PMC of Smart Building Maintenance using FMTs BuildSys ’17, November 8–9, 2017, Del, Netherlands
Component Synchronised with component Transition label Synchronisation method
DELAY representing Tde дDELAY modules representingTc l n ,Trp lc ,Ti ns p trigger Full synchronisation
RM DELAY module representingTr e p trigger clean Full synchronisation
RM DELAY module representingTo h trigger replace Full synchronisation
EBE DELAY representing Tde дdegradeNFull synchronisation
DELAY representing Tcl n RM, EBE check clean Full synchronisation
DELAY representing Trp lc RM, EBE check replace Full synchronisation
DELAY representing Tins p RM, IM inspect Full synchronisation
DELAY representing Tre p RM, IM, EBE perform clean Full synchronisation
DELAY representing Toh RM, IM, EBE perform replace Full synchronisation
EBE RM,IM, all DELAY modules, other EBEs - Interleave synchronisation
Table 1: Performing synchronisation between the dierent FMT components and the synchronisation method used.
G3
G2
G1
B4B3
B2
B1 B2
G3
G2
G1
B1
G3G2
B4B3
RDEP
B2
PMC
OF SUB-GRAPHS
GRAPH
DECOMPOSITION
ORIGINAL
FMT
MTTF(G2)
CSL
PROPERTIES
PMC OF FINAL
CTMC
MTTF(G3)
G2
G3
RDEP
G1
B1 B2 B4B3
B2
RDEP
G3G2
G1
EQUIVALENT
GRAPH
LIBRARY OF CTMC MOD ELS PARALLEL
COMPOSITION
DEPENDABILITY,
COSTS TRADE-OFFs
CSL
PROPERTIES
FINAL CTMC
Figure 6: Overall developed framework for decomposition of FMTs into the equivalent abstract CTMCs.
FMT components based on the underlying structure of the sub-
graph. e smaller sub-graphs are then sequentially recomposed to
generate the higher level abstract FMT. Figure 6 depicts a high-level
diagram of the decomposition procedure.
Conversion of original FMT to the equivalent graph
.e
FMT is a DAG (cf. Subsection 3) and in this framework we need to
apply a transformation to the DAG in the presence of an RDEP gate,
such that we can perform the decomposition. e RDEP causes an
acceleration of events on dependent child nodes when the input
node fails. In order to capture this feature in a DAG, we need to
duplicate the input node such that it is connected directly to the
RDEP vertex. is allows us to capture when the failure of the
input occurs and the corresponding acceleration of the the children.
is is reasonable as the same RM and IM are used irrespective of
the underlying FMT structure.
Graph decomposition
.We dene modules within the DAG as
sub-trees composed of at least two events which have no inputs
from the rest of the tree and no outputs to the rest except from its
output event [
13
]. We can divide the graph into multiple partitions
based on the number of modules making up the DAG. We dene
the following notations to ease in the description of the algorithm:
Vo
indicates whether the node is the top node of the DAG.
Vдindicates the node where graph split is performed.
Modules correspond to sub-graphs in DAG.
We set
Vo
when we construct the DAG from the FMT and then
proceed with executing Algorithm 1. We rst identify all the sub-
graphs within the whole DAG and label all the top nodes of each sub-
graph
i
as
VTi
. We loop through each sub-graph and its immediate
child (the sub-graph at immediate lower level) and at the point
where the sub-graph and child are connected, the two graphs are
split and a new node
Vд
is introduced. us, executing Algorithm 1
results in a set of sub-graphs linked together by the labelled nodes
Vд
. For each of lower level sub-graphs we now proceed to compute
the mean time to failure (MTTF). is will serve as an input to the
higher-level sub-graphs such that metrics for the abstract equivalent
CTMC can be computed.
PMC of sub-graphs
.We start from the boom level sub-graphs
and perform the conversion to CTMC using the formal models pre-
sented in Subsection 3.2. e formal models have been built into a
library of PRISM modules and based on the underlying components
and structure making up the sub-graph, the corresponding individ-
ual formal models are converted into the sub-graph’s equivalent
CTMC by performing parallel composition (cf. Subsec. 3.3). For
each sub-graph, we compute the probability of failure
De(T)
at time
T
, from which we calculate the MTTF using,
MTTF =ln(1De(T))
T
.
BuildSys ’17, November 8–9, 2017, Del, Netherlands N. Cauchi et al.
Algorithm 1: DAG decomposition algorithm
input : DAG G=(V,E)
output: Set of sub-graphs with one of the end nodes labelled
as Vд.
1Identify sub-graphs using ‘depth-rst’ traversal
2Label all top nodes of each sub-graph ias VTi
3forall the
select the top node of every sub-graph and immediate
child dened at immediate lower level do
4if label VTalready found in one of the leaf nodes of
sub-graph then
5Split sub-graph
6Insert new node Vдwhich will be used as input from
connected sub-graph
e MTTF serves as the input to the higher level sub-graph at time
T
. e new node in the higher-level sub-graph, now degrades with
the a new time delay
Tde д=MTTF
, which is fed into the corre-
sponding DELAY component. is process is repeated for all the
dierent sub-graphs until the top level node Vois reached.
PMC of nal equivalent abstract CTMC
.On reaching the top
level node
Vo
, we compute the metrics for the equivalent abstract
CTMC for a specic time horizon
T
. For dierent horizons, the
previous step of computing the MTTF for the underlying lower
level sub-graphs needs to be repeated. Using this technique, we
can formally verify larger FMTs, while using less memory and
computational time due to signicantly smaller state space of the
underlying CTMCs. Next, we proceed with an illustrative example
comparing the process of directly modelling the large FMT using
CTMCs versus the de-compositional modelling procedure. Figure 7
presents the FMT composed of two modules and the corresponding
abstracted FMT. e abstract FMT is a pictorial representation of
the moel represented by the equivalent abstract CTMC obtained
using the developed decomposition framework (cf. Fig. 6). For
Figure 7: e original FMT and the abstract FMT corresponding to the
equivalent abstract CTMC generated by the developed framework. e MT TF
for the F’ is computed based on the probability of failure of the heating coil.
both the large FMT and the equivalent abstract FMT a comparison
between the total number of states for the resulting CTMC models,
the total time to compute the reliability metric and the resulting
reliability metric is performed. All computations are run on an 2.3
GHz Intel Core i5 processor with 8GB of RAM and the resulting
statistics are listed in Table 2. e original FMT has a state space
with 193543 states, while the equivalent abstract CTMC has a state
space with 63937 states. is corresponds to a 67% reduction in the
state space size. e total time to compute the reliability metric is a
function of the nal time horizon and a maximal 73% reduction in
computation time is achieved. Accuracy in the reliability metric of
the abstract model is a function of the time horizon. e accuracy
of the reliability metric computed by the abstract FMT results in a
maximal reduction of 0.61%.
Time Original FMT Abstracted FMT
Horizon Time to compute Reliability Time to compute Total Reliability
metric MT TF metric Time
(years) (mins) (mins) (mins) (mins)
50.727 0.9842 0.142 0.181 0.223 0.9842
10 1.406 0.8761 0.219 0.309 0.528 0.8769
15 2.489 0.3290 0.292 0.622 0.914 0.3270
Table 2: Comparison between the original large FMT and the abstracted
FMT.
4 CASE STUDY
We apply the FMT framework to a Heating, Ventilation and Air-
conditioning (HVAC) system used to regulate a building’s internal
environment. e HVAC system under consideration for the FMT
analysis is presented in Figure 8. It is composed of two circuits -
the air ow circuitry and the water circuit. e gas boiler heats
up the supply water which is fed into the heat pump. e heat
pump transfers the supply water into two sections - the supply
air heating and cooling coils and the radiators - via the splier.
e rate of water owing in the heating coil is controlled using a
heating coil valve, while the rate of water ow in the radiator is
controlled using a separate valve. e outside air is mixed with the
extracted room air temperature via the mixer. is is fed into the
heating coil, which warms up the input air to the desired supply
air temperature. is air is supplied back, at a rate controlled by
the Air Handling unit (AHU) dampers, into the zone via the supply
fan. e radiators are directly connected to the water circuitry and
transfer the heat from the water into the zone. e return water is
then passed through the collector and is returned back to the boiler.
Based on this HVAC system we construct the corresponding FMT
shown in Figure 9. e leaves of the tree are EBE with discrete
degradation rates computed using Table 3, approximated by the
Erlang distribution where
N
is the number of degradation phases
(
k=N
for the Erlang distribution) and MTTF is the expected
time to failure with
MTT F =
1
/λ
(cf. Remark 1). We choose
an acceleration factor
γ=
2 for the RDEP gate. e system is
periodically repaired every 6 months (
Tr ep =
182
days
) and a major
overhaul with a complete replacement of all components is carried
out once every 20 years (
Toh =
20
×
365
days
). Weekly inspections
are performed (
Tinsp =
7
days
) which return the components
back to the previous state. Only cleaning actions are performed
when inspections are carried out. e total time to perform a
cleaning action is 1 day (
Tcl n =
1
day
), while performing a total
replacement of components takes 7 days (
Trp l c =
7
days
). e time
timing signals
{Tre p ,Toh ,Ti nsp ,Tc l n ,Tr plc }
are all approximated
using the Erlang distribution with
N=
3. All maintenance actions
are performed simultaneously on all components.
Eicient PMC of Smart Building Maintenance using FMTs BuildSys ’17, November 8–9, 2017, Del, Netherlands
Dampers
Zone
Heat
Pump
Heating &
cooling coil
Outside Air Intake
Mixer
Splitter Collector
Boiler
Supply Fan
Radiator
Air Input Water Input
Heating
coil valve
Radiator valve
Figure 8: High level schematic ofan HVAC system.
Failure of HVAC component
8
Insufficient
Radiator Pout
Failure in
Heating coil Failure of
Supply Fan
76
RDEP
No heating /
cooling Reduced
Capacity
1 2
93 5
4
Figure 9: FMT for failure in HVAC system with leaves represented using
EBE (associated RM and IM not shown in gure). e EBE are labelled to cor-
respond to the component failure they represent using the fault index pre-
sented in Table 3.
Fault Index Failure Mode N MTTF
(years)
1 Failure in cooling coil 4 20
2 Broken AHU Damper 2 20
3 Fan motor failure 3 35
4 Obstructed supply fan 4 31
5 Fan bearing failure 6 17
6 Radiator failure 4 25
7 Radiator stuck valve 2 10
8 Heater stuck valve 2 10
9 Failure in heat pump 4 20
Table 3: Extended Basic events in FMT with associated degradation rates
(N, MTTF) obtained from [6, 10].
4.1 antitative results
We make use of the developed framework (cf. Subsec. 3.5) and con-
vert the FMT representing the failure of the HVAC system (cf. Fig.
9) into the equivalent abstract CTMC. e abstracted CTMC has a
state space of 62779 states. Using our current computing set-up, the
complex CTMC representing the whole FMT was not computable
as it results in a state space explosion. Highlighting, the advantage
of the developed framework. e process is performed over six
time horizons
Nr={
0
,
5
,
10
,
15
,
20
,
25
}
years with the maintenance
policy consisting of periodic cleaning every 6 months, a major over-
haul every 20 years and inspections on a weekly basis. For this
set-up, the metrics corresponding to the reliability and availabil-
ity of the HVAC systems over the time horizon are computed and
are shown in Figure 10(b). e maximal time taken to compute a
metric using the abstract FMT is 1.47 minutes. It is deduced that
both the reliability and availability reduce over time and there is
a saturation in the number of maintenance actions which one can
perform before the system no longer achieves higher performance
in reliability and availability. Next, we compare the total cost of
0 10 20
0.96
0.97
0.98
0.99
1
Time (years)
Reliability
(a) Reliability of HVAC system.
0 10 20
0.96
0.97
0.98
0.99
1
Time (years)
Availability
(b) Availability of HVAC system.
Figure 10: Reliability and availability of HVAC over time horizon Nr.
maintenance and the expected number of failures over the time
horizon
Nr={
0
,
5
,
10
,
15
,
20
,
25
}
years when considering dierent
maintenance strategies, such that we can identify the maintenance
strategy that minimises cost and the number of failures over time.
We consider six dierent maintenance strategies which are listed
in Table 4. e total maintenance cost to perform a repair is 100
[GBP], while a replacement costs 5000 [GBP]. We now compute the
total expected maintenance costs and the total expected number of
failures for each strategy. ese are shown in Figure 11. e most ef-
fective strategy which oers a good trade-o between maintenance
costs and the expected number of failures is achieved when repairs
are carried out on a yearly basis, replacements are carried out every
20 years and inspections are carried out weekly (corresponding to
BuildSys ’17, November 8–9, 2017, Del, Netherlands N. Cauchi et al.
strategy
M1
). Furthermore, it can be seen that the frequency of
inspections has a large eect on the total number of failures. When
the frequency of inspection is low (as in
M4
and
M5
), the expected
number of component failures increases signicantly. Note that
reducing the periodicity of repairs, as in the case of maintenance
strategy
M2
also results in an increase in the expected number of
failures.
Strategy index Tre p Toh Ti ns p
M06 months 20 years 1 Week
M112 months 20 years 1 Week
M248 months 20 years 1 Week
M36 months 10 years 1 Week
M46 months 20 years 2 years
M56 months 20 years 5 years
Table 4: Implemented maintenance strategies
M0M1M2M3M4M5
0 10 20
0
0.5
1
1.5
·104
Time (years)
Maintenance cost
(a) Maintenance Costs.
5 10 15 20 25
0
0.05
0.1
0.15
0.2
Time (years)
Expected number of failures
(b) Expected number of failures.
Figure 11: Comparison between dierent number of maintenance strate-
gies for an HVAC systems.
5 CONCLUSION AND FUTURE WORKS
e paper has presented a methodology for applying probabilistic
model checking to FMTs. e FMTs are modelled in the form of
CTMCs which simplies the transformation of FMT into formal
models that can be analysed using PRISM. A novel technique for
abstracting the equivalent CTMC model is also presented. e novel
decomposition procedure tackles the issue of state space explosion
and results in a signicant reduction in both the state space size
and the total time required to compute metrics. e framework
has been applied to an HVAC system and the eect of applying
dierent maintenance strategies has been presented. e presented
framework can be further enhanced by adding more gates to the
PRISM modules library which include the Priority-AND, INHIBIT,
k/N gates and to incorporate lumping of states as in [
16
], such that
the state space can be further reduced.
ACKNOWLEDGMENTS
is work has been funded by the AMBI project under Grant No.:
324432, by the Alan Turing Institute, UK, post-doctoral research
grant from Fonds de Recherche du ebec - Nature et Technologies
(FRQNT) and Malta’s ENDEAVOUR Scholarships Scheme.
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