Optimizing Distributed Resource Allocation using Epistemic GameTheory: A Model-driven Engineering Approach

Fazle Rabbi1,2, Lars Michael Kristensen1 and Yngve Lamo1

1Bergen University College, Bergen, Norway2University of Oslo, Oslo, Norway

Keywords: Metamodelling, Epistemic Game Theory, Model Transformation, Optimization, Distributed Systems.

Abstract: Distributed systems modelling often involves a set of heterogeneous models where each model specifies aset of local constraints capturing a specific view of the system. In real life, distributed systems are oftenloosely connected and interdependencies are not defined into their software models. This limits the scope ofoptimization of distributed resources. In this paper, we merge heterogeneous models of distributed systemsand articulate distributed resource constraints via inter-metamodel constraints. We apply model-driven en-gineering and use model transformation rules to construct an epistemic game theory model for the purposeof optimizing distributed resource allocation. Since the application of transformation rules normally do notguarantee the satisfaction of constraints when applied on a model, it requires a conformance checking whichis an expensive operation. To overcome this problem, we introduce the concept of compliant rule for efficientmodel transformation.

1 INTRODUCTION

Distributed systems are organized in a modular,loosely coupled fashion such as in healthcare organi-zations, where separate departments deliver differenttypes of healthcare. Each unit in a healthcare orga-nization deals with a different set of operational con-straints (Pinelle and Gutwin, 2006). Patients often re-quire a diverse set of services which are provided bydifferent health facilities. However, the medical careprovided by various health facilities are not neces-sarily optimized to reduce patients’ waiting time andother important aspects of getting healthcare services.

While optimizing the resource allocation of ahealthcare system, game theory is more suitablechoice as there are many participants and it is requiredto reason about optimality respecting patients prefer-ences and priorities. Healthcare systems need to beequipped with uncertainty management as many pa-tients attending do not have an obvious diagnosis atpresentation; also there are varieties of uncertainty inhealthcare (Han et al., 2011). We propose to use epis-temic game theory for optimizing the use of health-care resources with the aim of improving the qualityand efficiency of healthcare services. Game theory isthe discipline of science that models strategic situa-tions where individuals reason about others choices

for decision making. Epistemic game theory (Perea,2012; Perea, 2014) can be viewed as rational deci-sion making under uncertainty. It can play an im-portant role in coordinating distributed systems or inthe design of interconnected systems. A model drivenapproach for meta-modelling epistemic game theorywas introduced in (Rabbi et al., 2016a). We inves-tigated how distributed systems may be coordinatedto optimize their resources respecting a distributedset of local constraints (Rabbi et al., 2016a). How-ever, there are situations where loosely coupled dis-tributed systems have to deal with a set of global con-straints. For instance, patients scheduled appointmenttime for getting a healthcare service should not con-flict with other scheduled appointments for gettingother healthcare services. In this paper, we enhancethe approach with inter-metamodel (i.e., global) con-straints.

We present our approach with a running examplefrom the healthcare domain. Let us consider that youand Barbara are two orthopedic patients at the samehospital. Both you and Barbara are asked to visit thedepartment of orthopedic surgery on the same day at9am. When you report to the front desk of the or-thopedic department you are asked to take an X-rayfrom the radiology department which is located on thesame floor as the orthopedic department. The ortho-

Rabbi F., Kristensen L. and Lamo Y.Optimizing Distributed Resource Allocation using Epistemic Game Theory: A Model-driven Engineering Approach.DOI: 10.5220/0006121400410052In Proceedings of the 5th International Conference on Model-Driven Engineering and Software Development (MODELSWARD 2017), pages 41-52ISBN: 978-989-758-210-3Copyright c© 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

41

pedic department sends an imaging order electroni-cally to the radiology department which includes pa-tients information and an instruction about the X-ray.It took some time for you to reach to the radiology de-partment as you have a long leg cast. Barbara reportedto the front desk right after you, but she reached theradiology department before you. The radiology de-partment is equipped with a queue automation andtherefore you will be served after Barbara. While youare waiting for your turn at the radiology departmentyou do not know that Dr. Logan (your doctor) will beout soon for his round at the inpatient medical wards.Barbara also does not know that she will not be servedat the orthopedic department before 10.30am becausethere are more patients waiting to see her doctor Dr.Bryan. When you are done with the X-ray and returnto the orthopedic department, you find that Dr. Lo-gan is out for the rounds and you will have to waituntil 12pm. It could have been an optimal decisionto have your X-ray done before Barbara as Barbarawill have to wait until 10.30am anyway. Even thoughthe orthopedic department and the radiology depart-ment are located on the same floor they do not sharetheir appointment schedules and as a result, the over-all wait time for patients to get health services are in-creased. We propose to use epistemic game theory foroptimizing the use of distributed resources and add re-source constraints to game theory. This paper presentsa formal treatment of distributed resource constraintsvia inter-metamodel constraints. For the efficient con-struction of game theory models we introduce the ideaof compliant rules.

The paper is organized as follows: In section 2, wepresent a formal model for distributed resource con-straints. Section 3 presents a model transformationtechnique for resource allocation. Section 4 presentsa game theory model for optimizing resource alloca-tion. Section 5 introduces compliant rules and co-ordination of transformation rules, section 6 presentssome related works and section 7 concludes the paper.

2 MODELLING RESOURCECONSTRAINTS

Our approach is based on category theory (Barr andWells, 1995), model transformations (Ehrig et al.,2006) and epistemic game theory (Perea, 2012). Weuse Diagrammatic Logic (Diskin and Wolter, 2008)and the Diagram Predicate Framework (DPF) (Rutle,2010) for the formal development of metamodel spec-ifications. DPF is suitable for developing domainspecific modelling languages and has support for anunbounded number of metalevels. In DPF, a model

is represented by a diagrammatic specification S =

(S ,CS : Σ) consisting of an underlying graph S to-gether with a set of atomic constraints CS specifiedby a predicate signature Σ. A predicate signature con-sists of a collection of predicates, each having a name,an arity (or shape graph), a visualization, and a se-mantic interpretation (see Table 1). A predicate isused to specify a constraint in a model by means ofgraph homomorphisms. An atomic constraint (p, δ)is added to a graph S by a predicate symbol p anda graph homomorphism δ : αΣ(p)→ S where αΣ(p)represents the shape graph of predicate p. DPF pro-vides a formalization of multi level meta-modellingby defining the conformance relation between mod-els at adjacent levels of a meta-modelling hierarchy.There are two kinds of conformance: typed by andsatisfaction of constraints. Figure 1(i) and (ii) showsthe DPF metamodel specifications of an orthopedicdepartment (S1) and a radiology department (S2), re-spectively. These specifications are constrained bypredicates from Table 1.

Constraints are added into the specifications bygraph homomorphisms from the shape graph of thepredicates to the model elements. Below is a list ofconstraints specified in S1 and S2:

• C1. A patient must have exactly one birthdate inan instance of S1 (specified by <mult(1,1)> )

• C2. A person must have exactly one birthdate inan instance of S2 (specified by <mult(1,1)> )

• C3. An appointment time-slot (i.e., TS@Dept)allocated to a patient in an instance of S1 mustbelong to that patients assigned doctor (specifiedby <composite>)

• C4. A person can only check-in for an exami-nation in an instance of S2 if the person has animaging order (specified by <pre-condition>)

• C5. Only registered persons are allocated withexamination time-slots (i.e., TS@Lab) in an in-stance of S2 (specified by <pre-condition>)

• C6. An appointment time-slot in an instance ofS1 must not be allocated to more than one patient(specified by <injective>)

• C7. An exam time-slot in an instance of S2 mustnot be allocated to more than one person (speci-fied by <injective>)

2.1 Distributed Resource Constraints

Usually orthopedic doctors need to inspect the X-rayreports of the patient during consultation. Therefore

MODELSWARD 2017 - 5th International Conference on Model-Driven Engineering and Software Development

42

Ф1

TS@Dept

assigned

Doctor

[1..1]

belongsTo

birthdate

Imaging

Exam

hasOrder

admittedTo

Doctor

Patient DepartmentDate

Prescription

[comp]

1TS@Lab

checkin

[1..1]

birthdate

Imaging

Exam

hasOrder

Registration

Person DeptDate

Report

[Pcond]

[Pcond]

2

ѱ1

ѱ2

birthdate

Imaging

Exam

hasOrder

Patient DeptDate

0

Ф2

( i ) ( iii ) ( ii )

TS@Dept

assigned

Doctor

[1..1]

belongsTo

checkin

admittedTo

orderingDept

Doctor

Patient

Dept

Registration

Imaging

Exam

[comp]

Prescription

Datebirthdate

Report+

[Pcond]

[Pcond]

TS@Lab

( iv )

[1..1]

Figure 1: (i) Metamodel specification of an orthopedic department, (ii) Metamodel specification of a radiology department,(iii) overlap specification, (iv) merged metamodel specification with an inter-metamodel constraint.

Table 1: Predicates of a signature, Σ.

p Arity Visualization Semantic interpretation

f must have at least n and

at most m instances for

each instance of X

For each instance of f there exists an instance of g with the same source node

For each composition of instances f;g, there exists an instance of h

If there are instances of f and gwith the same source node, then there exists an ordering of values between instances of Y and Z

Instances of f never maps distinct elements of its domain to the same element of its codomain

<m

ult

(n,m

)>

<pre

-

con

dit

ion

><com

posit

e>

<pre

cede>

1

1 2

1 2

3

1 2

3

f

gh

f

g

f

h

X Y

Z

f

g

X Y

Z

f

g[Pcond]

Xf

Y[n..m]

[comp]

X Y

Z

f

g[prcd]

3

f

g

2

1 2f

Xf

Y[inj]

<in

jecti

ve>

α (p)∑

orthopedic patients’ time-slot at the radiology depart-ment must be preceded by the time-slot at the orthope-dic department. We will refer to this constraint as C8.We use the concept ‘time-slot’ for an appointment oran examination to represent the time assigned for ascheduled job. A time-slot t2 which include informa-tion about the start-time and end-time of a scheduledjob is preceded by a time-slot t1 if the end-time of t2are less than the end-time of t1. To model this dis-tributed resource constraint, we need to merge speci-fications S1 and S2.

While merging specifications, atomic constraintsspecified in the component metamodel specificationsneed to be preserved. We merge specifications in sucha way that for any component metamodel specifica-tion Si there exists a specification morphism from Si

to the merged metamodel specification. The defini-tion of specification morphism is given below (Rutle,2010):

Definition 1 (Specification Morphism). Given twospecifications S = (S ,CS : Σ) and S′ = (S ′,CS′ : Σ),a specification morphism φ : S → S′ is a graphhomomorphism φ : S → S ′ such that for any atomicconstraint (p, δ) ∈ CS implies (p, δ;φ) ∈ CS

′as

illustrated in the following diagram.

α (p)∑

S S’Фδ

δ;Ф

=

Diskin et al. presented an approach of mergingmetamodels by making metamodels explicit and byintroducing a correspondence span between compo-nent metamodels (Diskin et al., 2010) (Diskin, 2011).We adapt the approach with specification morphismsin order to preserve constraints into the merged meta-model specifications. Details of merging heteroge-neous metamodels and specifying inter-metamodelconstraints can be found in (Diskin et al., 2010). Un-like (Diskin et al., 2010), in our approach the headof the span is an overlap specification given in DPFwhich specifies the common concepts of both meta-models and the legs of the span are specification

Optimizing Distributed Resource Allocation using Epistemic Game Theory: A Model-driven Engineering Approach

43

morphisms. Figure 1(iii) shows the overlap specifi-cation S0 = (S 0,CS

0: Σ) (highlighted in grey) and

Figure 1(iv) shows the merged metamodel specifica-tion S+ = (S +,CS

+: Σ). S+ is constructed by merg-

ing component metamodel specifications S1 and S2

modulo their overlap S0 (i.e., common elements ap-pear only once in the merge). Overlaps are specified

by two specification morphisms S1 ψ1←−− S0 ψ2−−→ S2

where S0 contains all common concepts, S 1 and S 2

are the underlying graphs of specifications S1 andS2. Any pair of elements x1 ∈ S 1 and x2 ∈ S 2 areconsidered to be the same if there is an elementx ∈ S 0 such that ψ1(x) = x1 and ψ2(x) = x2. Fol-lowing (Diskin et al., 2010), we call this configu-ration of metamodel specifications and mappings amulti-metamodel specification M, and write M =

(S1,S2,S0,ψ1,ψ2). The multi-metamodel specifi-cation M declares the unification of concepts Date,birthdate, hasOrder, ImagingExam, orderingDeptand unifies also the following concepts:

• Patient and Person since ψ1(Patient) = Patientand ψ2(Patient) = Person

• Department and Dept since ψ1(Dept) =

Department and ψ2(Dept) = Dept

The mappings φ1 : S1 → S+ and φ2 : S2 → S+

in Figure 1 maps elements fromS1 andS2 to the cor-responding elements in S+.

Definition 2 (Merged Metamodel Specification).Given two (component) metamodel specificationsS1 = (S 1,CS

1: Σ) and S2 = (S 2,CS

2: Σ), a merged

metamodel specification S+ = (S +,CS+

: Σ) of S1

and S2 is given by two specification morphismsφ1 :S1→S+, φ2 :S2→S+ and a multi-metamodelspecification M = (S1,S2,S0,ψ1,ψ2) such that thegraph S + is obtained by the pushout of ψ1 and ψ2.This is illustrated in the following diagram:

Ф1

ѱ1 ѱ2

0S

Ф2

1S 2S

+S

PO

∑α (p ) ....1

δ1

δ ; 1 Ф1

∑... α (q )1

δ'1

δ‘ ; 1Ф2

∑α (d ) ...

1δd

δd2

δd1

δ ; = δ ;d1 Ф1 d2 Ф2

The binary case can be generalized to the N-arycase by constructing S + as a so-called colimit andpreserving all the atomic constraints specified in thecomponent metamodels. Constraints of componentmetamodel specifications are preserved in a merged

metamodel specification. Additional constraints maybe added into a merged metamodel specification.

The distributed resource constraints are added intothe merged metamodel specification by means ofinter-metamodel constraints. The distributed resourceconstraint C8 is added into S+ by using the predicate<precede>. In general, merged metamodel specifica-tions have the following properties:

Theorem 1. Let S+ be a merged metamodel spec-ification obtained from component metamodel spec-ifications S1, . . .Sn. For any component metamodelspecification Si(1 ≤ i ≤ n), there exists a specificationmorphism from Si to S+.

Proof. This follows directly from the definition ofmerged metamodel specification. �

A model (I, ι) of a metamodel specification S =

(S ,CS : Σ) is a graph I together with a typing graphhomomorphism ι : I→ S . In order to be a valid modelof a metamodel specification S the typing morphismmust satisfy the atomic constraints specified in CS.Here we define the merged model of two componentmodels.

Definition 3. Given two (component) models(I1, ι1), (I2, ι2) of metamodel specifications S1,S2

and an overlap model I1 g1←−− I0 g2−−→ I2 typed by theoverlap specification of S1 and S2. The mergedmodel is a graph I+ together with a typing graphhomomorphism ι+ : I+ → S + where S + is the under-lying graph of the merged metamodel specificationsof S1,S2 such that the graph I+ is obtained by thepushout of g1 and g2 and the top face constitute avan Kampen square in the category of Graph asillustrated in the following diagram:

(top)

S0

S1

S2

S+

I0

I1

I2

I+

ѱ1

ѱ2

φ1

φ2

ι1ι0

ι+ι2

g1

h2h1

g2

The reader interested in the property of van Kam-pen square may wish to consult (Mantz, 2014). It ispossible that the atomic constraints when preserved ina merged metamodel specification may conflict withother constraints or may become redundant. Thereexists some research for detecting these problems au-tomatically such as the alloy analyzer which may beused to detect these problems by specification anal-ysis (Wang et al., 2015). An atomic constraint speci-fied in a component metamodel and satisfied in a com-

MODELSWARD 2017 - 5th International Conference on Model-Driven Engineering and Software Development

44

ponent model may become unsatisfiable in a mergedmodel or vice versa. Figure 2 shows an example ofa model merge where a component model (I1, ι1) istyped by the underlying graph S 1 of the metamodelspecification S1. The predicate p is used to con-straint S1 by a graph homomorphism δ : αΣ(p)→ S 1.

The edge 1e1−−→ 2 is mapped to A

f−→ B and the edge2

e2−−→ 1 is mapped to Bg−→ A. The overlap specifi-

cation is given by (S1,S2,S0,ψ1,ψ2) which unifiesof the following concepts: A and AA, B and BB. Tocheck if (I1, ι1) satisfies the constraint (p, δ) we extractthe fragment (O∗) of the component model that is af-fected by the constraint. Again, to check if the mergedmodel (I+, ιm) satisfies the constraint (p, δ;φ1) we ex-tract the fragment (O∗m) of the merged model that is af-fected by the constraint. The object O∗ is obtained by

the pullback of αΣ(p)δ−→ S 1 ι1←− I1 and the object O∗m

is obtained by the pullback of αΣ(p)δ;φ1−−−−→ S +

ιm←−− I+.Since we obtain two different objects by the pullbackoperation, it is possible that the satisfaction of the lo-cal constraint will vary.

Typing, l

1 2e

α (p)∑

δ

PB(1)

m

l*

Af

B

δ*

1

e2 gAA BB AA BB

g

a b:f a

:fb AA BB

f

g

a:g

b

a:g

b:f

a:g

b:f

δ*

l*

1

m

m

l1

l2

ѱ1 ѱ2

Ф1

PB(2)

Ф2

Ф*1 Ф*

2

O* I1 I2S+

S1

S0

S2

O*m I+

δ;Ф1

Figure 2: Construction of pullback for a component modelI1 and merged model I+.

3 RESOURCE ALLOCATION

Our example introduced earlier consists of two com-ponent metamodels representing two distributed sys-tems and an inter-metamodel constraint C8. Figure 3shows a model (I, ιI) of the merged metamodel S+

of Figure 1. The typing ιI of the modelling elementsare not explicitly represented in the figure. In the fig-ure, You and Barbara are instances of Patient, yourassigned doctor is Dr.Logan and Barbara’s assigneddoctor is Dr.Bryan. Dr. Logan has two available time-slots: 0950−1010@Logan and 1200−1220@Loganand Dr. Bryan has one available time-slot: 1030−1050@Bryan. There are two available time-slotsat the radiology department: 0945− 1000@Lab and0930−0945@Lab. In time-slot 0950−1010@Logan,09:50 is the start time and 10:10 is the end time.

Model (I, ιI) need to be completed with resource al-location for you and Barbara at the radiology and or-thopedic department. One can find that there are fourpossible choices of resource allocation for you andtwo possible choices of resource allocation for Bar-bara with the available resources.

We use a transformation approach where transfor-mation rules have a set of negative application condi-tions proposed by Lambers et al., in (Lambers et al.,2008) for allocating resources to the patients. Figure 4illustrates a rule r1 typed by the merged metamodelS+. The typing information of a modelling elementin r1 appears after a colon (:). The rule specifies thata patient pt1 is allocated with an appointment time-slot t1 and an exam time-slot t2 if t1 belongs to pt1’sassigned doctor d and t1, t2 are not allocated to anyother patient. The green color is used to indicate el-ements that the rule is going to produce. Negativeapplication conditions (NACs) are typically used ingraph transformation to prohibit an infinite number ofrule applications.

The rule can be applied over the model (I, ιI) insix different ways since there are six matches. Fig-ure 3 shows briefly the choices of resource allocationfor you in y1−y4 and for Barbara in b1,b2. y1 shows aresource allocation where you are assigned with examtime-slot 0930− 0945@Lab at the radiology depart-ment and appointment time-slot 0950−1010@Loganat the orthopedics department with Dr. Logan. Thisresource distribution is valid since it is not violatingany constraint. Similarly, other valid resource distri-butions are y2, y3, b1 and b2. However y4 is not validas it is violating the distributed resource constraintC8: your exam time-slot 0945 − 1000@Lab mustbe preceded by your appointment time-slot 0950 −1010@Logan.

We apply epistemic game theory for the optimiza-tion of resources over the valid choices of resourceallocation. Epistemic game theory is a broad area ofresearch that formalizes the assumptions about ratio-nality and mutual beliefs in a formal language to an-alyze games. It introduces a Bayesian perspective ondecision-making in strategic situations. In a strate-gic game, the outcome does not only depend on thechoice of a single player but also on the choices ofother players. The combination of all the players’choices is called a strategy profile. From an epistemicpoint of view, the strategy profile should include thebeliefs players have about each other’s possible ac-tions (and beliefs). This model of interdependent de-cision making can essentially represent a wide arrayof social interactions. A metamodel for modellingepistemic aspects of a system as presented in (Rabbiet al., 2016a) is shown in Figure 5. Players in an epis-

Optimizing Distributed Resource Allocation using Epistemic Game Theory: A Model-driven Engineering Approach

45

:assigned

Doctor

:belongsTo

:checkin

:admittedTo

:orderingDept

Dr. Logan

You

Orthopedics

:Registration

:Imaging

Exam01.01.1990

0945-1000

@Lab

:assigned

Doctor

:belongsTo

:checkin

:admittedTo

:orderingDept

Dr. Bryan

Barbara

Orthopedics

:Registration

:Imaging

Exam02.11.1991

0930-0945

@Lab

0950-1010

@Logan

1200-1220

@Logan

:belongsTo

1030-1050

@Bryan

I

:assigned

Doctor

:belongsTo

Dr. Logan

You

0945-1000

@Lab

0930-0945

@Lab

0950-1010

@Logan

1200-1220

@Logan

:belongsTo

:examTime

:apptTime

:assigned

Doctor

:belongsTo

Dr. Logan

You

0945-1000

@Lab

0930-0945

@Lab

0950-1010

@Logan

1200-1220

@Logan

:belongsTo

:assigned

Doctor

:belongsTo

Dr. Logan

You

0945-1000

@Lab

:assigned

Doctor

:belongsTo

Dr. Bryan

Barbara

0950-1010

@Logan

1200-1220

@Logan

:belongsTo

1030-1050

@Bryan

:apptTime

:assigned

Doctor

:belongsTo

Dr. Logan

You

0945-1000

@Lab

:assigned

Doctor

:belongsTo

Dr. Bryan

Barbara

0950-1010

@Logan

1200-1220

@Logan

:belongsTo

1030-1050

@Bryan

:apptTime

:examTime

:apptTime

y1

y3

y2

y40930-0945

@Lab

0930-0945

@Lab

:examTime

0945-1000

@Lab

0930-0945

@Lab

0945-1000

@Lab

0930-0945

@Lab

b1

b2

Figure 3: A model (I, ιI) of S+ (top) and individual resource allocation.

pt :Patient

t1:TS@Dept pt :Patientf2:apptTime

pt :Patient t2:TS@Lab

f1:apptTime

b:belongsTo

t1:TS@Dept

g1:examTime

d:Doctor

:assigned

Doctor

f1:apptTime

g1:examTimept :Patient t2:TS@Lab

pt :Patient

g2:examTime

Transformation rule r1

NAC

NAC

NACr1

b:belongsTo

d:Doctor

:assigned

Doctor

b:belongsTo

d:Doctor

:assigned

Doctor

t1:TS@Dept

t2:TS@Lab

1

1

1

2

3

Figure 4: Transformation rule r1 for individual resource al-location of patients.

temic game may have choices. The surjective con-straint imposed on the hasChoice relation ensures thatinstances of Choice must be associated with players.An instance of Belie f connects the choices of a playerwith other players’ choices denoting the choice com-bination of players. In game theory, utility representsthe motivation or satisfaction of players. The higherthe utility the more the outcome is preferred. We use

the <utility(n)> predicate to assign utility to an in-stance of a belief. A utility assigned to an instance ofa belief denotes the utility that a player derives fromthe outcome induced by the choice combination. Weuse the <prob(r)> predicate to annotate an instanceof a belief with probability. Of course in a healthcaresetting, patients do not have to know about each oth-ers choices; they are not expected to make decisionsfor mutual benefits. We propose to construct a strat-egy profile for them indirectly with an aim to optimizeresource allocation respecting patients’ preferences.

Given an initial epistemic model ξ of the meta-model in Figure 5 with two instances of Player: Youand Barbara, we use coupled model transformationrules to automatically produce choices for you andBarbara. For each valid choice of resource alloca-tion of individual patients from Figure 3, we createchoices in ξ. The special kind of coupled transfor-mation rule with coordinating edges creates choices ξand establishes links from choices to instances of themerged metamodel S+. Coordinating edges are usedto co-ordinate between heterogeneous models and ormodelling elements (Rabbi et al., 2014) (Rabbi et al.,2015). Figure 6 shows a transformation rule that iden-tifies a player and produces choices for the player.The rule specifies that if we have an instance x of typePlayer in the epistemic model and x is an instance ofPatient in a model (I, ιI) ofS+, such that x is allocatedwith an appointment time-slot z and an exam time-sloty, then we create an epistemic choice ω for player x

MODELSWARD 2017 - 5th International Conference on Model-Driven Engineering and Software Development

46

Player Choice Belief

source

[1..*]

hasChoice

[surj]

Predicate, p Arity Visualization

<mult(n,m)>

<surjective>

<prob(r)>

<utility(n)>

X Y[n..m]

X Y[surj]

1 2

1 2

target

[1..*]

1 2 X Y(r)

constraint

1 2 X Yn

Figure 5: Metamodel for modelling epistemic aspects.

Typing

Patient TS@Dept

x z:apptTime

apptTimePlayer Choice

x ω:hasChoice

hasChoice

c

Iy

:examTime

TS@LabexamTime

Typing, ι

Model (I, ι )I

I

Figure 6: Transformation rule r2 for producing epistemicchoices.

in the epistemic model and create a coordinating edgec linking the choice ω to the model (I, ιI).

Figure 7 illustrates the result of the applicationof this transformation rule over the possible choicesof resource allocation from Figure 3. Figure 7(left)shows an epistemic model ξ where You and Barbaraare instances of Player. Choices for you and Bar-bara are produced in ξ where the choices are linkedto the models of the merged metamodel S+ in Fig-ure 7(right) by coordinating edges shown in blue.Note that we only produce epistemic choices for validmodels of the entity metamodel. Since y4 is not avalid model, we did not produce any epistemic choicefor y4.

Figure 8 briefly presents the coordinating meta-model used for coordinating heterogeneous metamod-elling hierarchies. The coordinating metamodel spec-ifies that we may have coordinating-edges betweennodes. The coordinating-edge may be instantiatedin an arbitrary level of a metamodelling hierarchy.Rossini et al. presented a formal way of specify-ing this requirement in DPF using deep metamod-elling (Rossini et al., 2012). A reference with multi-potency at a metalevel i can be instantiated in allmetalevels below i. Since the coordinating-edge hasmulti-potency, it can be instantiated in an arbitrarymetalevel. In Figure 8, S+, y1, y2 are placeholdernodes representing the identity of the (meta)models.The instance of the coordinating-edge c1 establishesa link between an epistemic choice instance to the in-stance y1 by referring to the node y1.

4 OPTIMIZATION OFRESOURCE ALLOCATION

In this section we discuss how resource allocationof distributed systems may be optimized using epis-temic game theory. In our example, the patients wereasked to come to the hospital at 9am and the clinicat the orthopedic department remains open until 5pm.Therefore, the maximum time spent by the patientsto get healthcare services could be 8 hours (480 min-utes). We calculate the utility based on the time spentat the hospital by the patients to get healthcare ser-vices. If a patient spends m minutes in total to getservices, then her utitity is (480−m). Therefore, thelonger the patients are waiting, the lower utility theyget. Our target is to optimize the resource alloca-tion for the patients so that their wait time will bereduced. Consider the choice y1 for you, which rep-resents the resource allocation of 0930− 0945@Laband 0950 − 1010@Logan. This gives a utility of480− 70 = 410 as you are spending only 70 minutesat the hospital to get your services. Similarly, we cancalculate Barbara and your utilities and complete theepistemic model ξ with belief instances by perform-ing a Cartesian product of Barbara and your choicesas shown in Figure 9. Curved arrows in Figure 9 arerepresenting belief instances which will be referred toas belief arrows. The belief arrow from y1 to b1 withan attribute 410 is representing an instance of Belie fwith utility 410.

An essential element of epistemic game theoryanalysis is the notion of belief hierarchies which areused to characterize solution concepts. We obtain be-lief hierarchies from an epistemic model by followingbelief arrows starting from any choice of a player. Inour epistemic model we are only considering simplebelief hierarchies. The idea of a simple belief hier-archy states that the whole belief hierarchy may besummarized by a combination of beliefs about play-ers’ choices. A belief hierarchy is simple if it con-tains at most one belief of each player. The epistemicmodel in Figure 9 represents six simple belief hier-

archies of which three are: (i) y1410−−−→ b1

370−−−→ y1; (ii)

y1410−−−→ b2

370−−−→ y1; (iii) y2280−−−→ b1

370−−−→ y2. Typically abelief hierarchy specifies a player’s belief over otherplayers’ beliefs but in the healthcare context we use abelief hierarchy to specify a systems belief about pa-

tient’s choices. The belief hierarchy y1410−−−→ b1

370−−−→ y1can be read as: the system believes that you will con-sider making appointments represented in y1 if Bar-bara considers making appointments represented inb1; and Barbara considers making appointments rep-resented in b1 if you consider making appointments

Optimizing Distributed Resource Allocation using Epistemic Game Theory: A Model-driven Engineering Approach

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:assigned

Doctor

:belongsTo

Dr. Bryan

Barbara

1030-1050

@Dept

:examTime

:apptTime

0945-1000

@Lab

0930-0945

@Labb2

:assigned

Doctor

:belongsTo

Dr. Bryan

Barbara

1030-1050

@Dept

:apptTime

0945-1000

@Lab

0930-0945

@Labb1

:assigned

Doctor

:belongsTo

Dr. Logan

You

0945-1000

@Lab

0930-0945

@Lab

0950-1010

@Dept

1200-1220

@Dept

:belongsTo

y3

:assigned

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:belongsTo

Dr. Logan

You

0945-1000

@Lab

0950-1010

@Dept

1200-1220

@Dept

:belongsTo

y2

0930-0945

@Lab

:examTime

Entity models (Instances of the merged

metamodel shown in Figure 1)Epistemic model

Barbara

:hasChoice

:hasChoice

y1

:assigned

Doctor

:belongsTo

Dr. Logan

You

0945-1000

@Lab

0930-0945

@Lab

0950-1010

@Logan

1200-1220

@Logan

:belongsTo

:examTime

:apptTime

You :hasChoice

:hasChoice

:hasChoice

y ( )0930-0945@Lab

0950-1010@Logan1

y ( )0930-0945@Lab

1200-1220@Logan2

y ( )0945-1000@Lab

1200-1220@Logan3

b ( )0945-1000@Lab

1030-1050@Bryan1

b ( )0930-0945@Lab

1030-1050@Bryan2

+

Figure 7: Epistemic choices produced by the application of model transformation rules.

+

0950-1010

@Logan

:belongsToy2

:assign

Doctor

Dr. Logan

You

0950-1010

@Logan

120

@

:belongsTo

:apptTime

0950-1010

@Logan

:belongsToy1

:assign

Doctor

:be

Dr. Logan

You

0950-1010

@Logan

120

@

:belongsTo

:e

:apptTimeYou

:hasChoice

:hasChoice y ( )0930-0945@Lab

0950-1010@Logan1

y ( )0930-0945@Lab

1200-1220@Logan2

Node

coordinating-edge

arrow

TS@Dept

belongsTo

checkinRegistration

[comp]

[Pcond]

TS@Lab

Player Choice

[1..*]

hasChoice

[surj]

[1..*]

c1

c2

Figure 8: Heterogeneous metamodelling with Coordinatingmetamodel.

Barbara

You

:hasChoice

y ( )0930-0945@Lab

0950-1010@Logan1

y ( )0930-0945@Lab

1200-1220@Logan2

y ( )0945-1000@Lab

1200-1220@Logan3

b ( )0945-1000@Lab

1030-1050@Bryan1

b ( )0930-0945@Lab

1030-1050@Bryan2

410370

410370

280370

280370

280370

280370

Figure 9: Epistemic model with combination of beliefs.

represented in y1. In our approach, uncertainty canbe modelled by assigning probability to the belief ar-rows.

An important concern regarding the belief hierar-

chies is their consistency. A consistent belief hier-archy represents a solution concept that satisfies allthe constraints specified in the metamodel specifica-tions. Following (Rabbi et al., 2016a) we determinehow inconsistent combination of beliefs can be au-tomatically removed from an epistemic model. If acombination of beliefs includes choices from a ho-mogeneous metamodel, then the models are mergedinto a single model. For instance if we wish to checkif the combination of beliefs represented by the be-

lief hierarchy y1410−−−→ b1

370−−−→ y1 is consistent, we needto merge the models linked to y1 and b1 since theyare typed by the merged metamodel S+. The mergedmodel is then checked for consistency and if it vi-olates any constraint specified in its metamodel, weconclude that the combination of beliefs is incon-sistent and should be discarded from the epistemicmodel.

The combination of beliefs represented by the be-

lief hierarchy y1410−−−→ b2

370−−−→ y1 is inconsistent as themerged instance is violating constraint C7 (see sec-tion 2). The reason is that according to the injec-tive constraint imposed on the reference examTime,patients cannot have the same time-slot. Therefore,assigning time-slot 0930−0945@Lab to both of youand Barbara is making the merged instance inconsis-tent. To check that a constraint is satisfied in a givenmodel of a metamodel, we inspect the part of a modelwhich is affected by the constraint. This is checkedby projecting out the part of the model which is af-fected by the constraint. This is formally defined as apullback operation in category theory.

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Similarly we can show that the combination of

beliefs represented by the belief hierarchies y2280−−−→

b2370−−−→ y2, y3

280−−−→ b1370−−−→ y3 are inconsistent. Fig-

ure 10 shows an epistemic model with only consistentcombination of beliefs.

Barbara

You

:hasChoice

y ( )0930-0945@Lab

0950-1010@Logan1

y ( )0930-0945@Lab

1200-1220@Logan2

y ( )0945-1000@Lab

1200-1220@Logan3

b ( )0945-1000@Lab

1030-1050@Bryan1

b ( )0930-0945@Lab

1030-1050@Bryan2

410370

280370

280370

Figure 10: Epistemic model with consistent combination ofbeliefs.

Barbara

You

:hasChoice

y ( )0930-0945@Lab

0950-1010@Logan1

y ( )0930-0945@Lab

1200-1220@Logan2

y ( )0945-1000@Lab

1200-1220@Logan3

b ( )0945-1000@Lab

1030-1050@Bryan1

b ( )0930-0945@Lab

1030-1050@Bryan2

410370

280370

Figure 11: Epistemic model with rational combination ofbeliefs.

The epistemic choice y2 which represents the re-source allocation of 0930 − 0945@Lab and 1200 −1220@Logan is not a rational choice since this is notoptimal for any of the epistemic choices for Barbara.If you are allocated with time-slot 0930−0945@Labat the radiology department, it will be optimal foryou to see the doctor in appointment time-slot 0950−1010@Logan. Therefore, the epistemic choice y2 isdominated by y1. The epistemic choice y3 is a ratio-nal choice since this is optimal if Barbara is allocatedwith time-slot 0930−0945@Lab at the radiology de-partment.

Figure 11 shows belief combinations with rationalchoices. The combination of beliefs is rational as thechoices consist of the belief combinations are all ra-tional choices with respect to the choice combinationrepresented by the combination of beliefs.

Following (Perea, 2012) (Rabbi et al., 2016a), asimple belief hierarchy generated by a combinationof beliefs leads to a Nash equilibrium if the combi-nation of beliefs is rational. Therefore, we obtain twoNash equilibria from the epistemic model representedin Figure 11:

• Allocating time-slot 0930 − 0945@Lab and0950− 1010@Logan for you is rational if time-slot 0945− 1000@Lab and 1030− 1050@Bryan

are allocated to Barbara gives a total utility of(410 + 370) = 780;

• Allocating time-slot 0945 − 1000@Lab and1200− 1220@Logan for you is rational if time-slot 0930− 0945@Lab and 1030− 1050@Bryanare allocated to Barbara gives a total utility of(280 + 370) = 650.However, the first equilibrium has a higher total

utility for the system.

5 COORDINATION OFTRANSFORMATION RULES

Scalability is an important issue regarding the appli-cation of optimization techniques in real-life scenario.In our approach, we proposed to use model transfor-mation rules for allocating resources. In general, theresult of the application of a model transformationrule may become inconsistent i.e., may violate theconstraints specified in a metamodel. Therefore, theapplication of a model transformation rule requires aconsistency checking which is a time consuming op-eration. To tackle this problem, we need techniquesto reduce the complexity of consistency checking. Inthis section, we introduce the idea of a compliant rulethat preserves the validity of a model.

The transformation rule r1 presented in Figure 4is not compliant with the set of atomic constraintsspecified in the merged metamodel specification S+

of Figure 1. If the rule is applied over a valid model,it does not guarantee that the result will be a validmodel conforming to S+. The rule can be enhancedso that while matching with a model it makes sure thatthe result will be a valid model.Definition 4 (Compliant Rule). Given a metamodelspecification S = (S ,CS : Σ). A transformation ruler is compliant with a set of atomic constraints fromCS if the application of r over any valid model of Sresults in a valid model of S.

Figure 12 shows a transformation rule r3 whichis compliant with the set of atomic constraints CS

+

specified in S+ (Figure 1). The addition of new in-stances of examTime and apptT ime in a valid modelofS+ can possibly violate atomic constraints C3, C5,C6, C7 and C8. The transformation rule r3 makessure that when applied on a valid model of S+, theaddition of new elements does not violate any of theabove mentioned constraints. The additional match-ing condition endTime(t2)< endTime(t1) specified inr3 makes sure that C8 is not violated. Therefore theapplication of r3 will not require any further consis-tency checking.

Optimizing Distributed Resource Allocation using Epistemic Game Theory: A Model-driven Engineering Approach

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pt :Patient

t1:TS@Dept pt :Patientf2:apptTime

pt :Patient t2:TS@Lab

f1:apptTime

b:belongsTo

t1:TS@Dept

g1:examTime

d:Doctor

:assigned

Doctor

f1:apptTime

g1:examTimept :Patient t2:TS@Lab

pt :Patient

g2:examTime

Transformation rule r3

NAC

NAC

NACr

e:Imaging

Exam h:hasOrder

Additional Matching Condition, MC =

3

r3{ ‘endTime(t2) < endTime(t1)’ }

t2:TS@Lab

b:belongsTot1:TS@Dept

d:Doctor

:assigned

Doctor

e:Imaging

Exam h:hasOrder

b:belongsTo

d:Doctor

:assigned

Doctor

e:Imaging

Exam h:hasOrder

1

1

1

2

3

Figure 12: Transformation rule r3 for individual resourceallocation of patients.

Further enhancement of the rule is performed byamalgamation of rule r3 (Figure 12) and r2 (Fig-ure 6). Rule amalgamation is a model transformationtechnique commonly used for coordinating rules withoverlapping parts (Lamo et al., 2013). To apply a ruleover a source model requires to perform an injectivematching. In order to reduce the number of match-ing we coordinate rule r3 and r2. Figure 13 shows atransformation rule r4 that we formulate by combin-ing r3 and r2. While allocating resources for individ-ual patients the rule r4 produces epistemic choices atthe same time. The rule is also compliant with theset of constraints specified in the merged metamodelspecification and therefore if the rule is applied over avalid model, it guarantees consistency.

6 RELATED WORK

The necessity of merging different formalisms of soft-ware specifications is intrinsic to the discipline ofsoftware engineering. Category theory has been con-sidered to be a viable solution to accommodate the di-versity of formalisms (Fiadeiro and Maibaum, 1995;Diskin et al., 2010; Rutle, 2010). Diskin et al.,(Diskin et al., 2010) introduced the concept of inter-metamodel constraints to specify constraints over in-terdependent models and later on Konig et al., (Konigand Diskin, 2016) proposed an advanced consistencychecking algorithm for inter-metamodel constraints.A slight different but related idea was presented byRabbi et al., in (Rabbi et al., 2014) which formalizesthe co-ordination among metamodels, using a linguis-

tic extension of the metamodelling hierarchy. Theproposed linguistic extension is based on an addedmetalevel which models the integration of two ormore different aspects of a system.

Bak et al. presented a language called Clafer thatunifies features and class modelling (Bak et al., 2016).The language is supported by a tool that supportsmodel based analyses such as consistency checking,instance completion, single and multi-objective opti-mization. The tool utilizes existing model based an-alyzers such as Alloy (relational logic solver) (Jack-son, 2002), Z3 (SMT solver) (De Moura and Bjørner,2008), Choco 3 (constraint satisfaction problems andconstraint programming solver) (Jussien et al., 2008)by translating Clafer models into the backend solvers.While Clafer provides a concise notation for featuremodeling it does not support modelling distributedsystems with inter-metamodel constraints. Also theoptimization technique is different in contrast to theoptimization technique of epistemic game theory.

Alanen and Porres investigated an approach forautomatic discovery of the difference and union ofmodels in the context of a version control system(Alanen and Porres, 2003). They presented algo-rithms to calculate the difference between models forMOF-based models. This work can be adapted in or-der to identify overlaps between component modelsfor merging DPF (meta)model specifications.

7 CONCLUSION

This paper is based on the metamodelling epistemicgame theory approach introduced in (Rabbi et al.,2016a). Epistemic games are relevant for modelingdistributed systems in order to optimize the use ofshared resources. We proposed several enhancementin this paper by incorporating model merge with inter-metamodel constraints and compliant rules.

In this paper, we adopted merged metamodelspecifications and specified distributed resource con-straints via inter-metamodel constraints. We appliedmodel transformations to produce epistemic gametheory models for the purpose of optimizing the useof distributed resources. Currently we are investigat-ing the Clafer (Bak et al., 2016) and WebDPF tool(Rabbi et al., 2016b) for reasoning over DPF meta-models. In order to apply the proposed method effi-ciently, we presented the concept of compliant rules.An automatic resolution of compliant rules is out ofscope of this paper and it is currently under investiga-tion.

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pt :Patient

t1:TS@Dept pt :Patientf2:apptTime

pt :Patient t2:TS@Lab

f1:apptTime

b:belongsTo

t1:TS@Dept

g1:examTime

d:Doctor

:assigned

Doctor

f1:apptTime

g1:examTimept :Patient t2:TS@Lab

pt :Patient

g2:examTime

NAC

NAC

NACr

e:Imaging

Exam h:hasOrder

Additional Matching Condition, MC =

4

r4{ ‘endTime(t2) < endTime(t1)’ }

t2:TS@Lab

b:belongsTot1:TS@Dept

d:Doctor

:assigned

Doctor

e:Imaging

Exam h:hasOrder

b:belongsTo

d:Doctor

:assigned

Doctor

e:Imaging

Exam h:hasOrder

Typing

Player Choice

pt ω:hasChoice

hasChoice

cInstance, (I, ι )I

1

1

1

2

3

1

Figure 13: Transformation rule r4 for resource allocation and producing epistemic choices.

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