Veliko Ivanov

Monika Tzaneva
email: monika_tzaneva@yahoo.com

Alexandra Murdjeva
email: amurjeva@abv.bg

Valentin Kisimov
email: vkisimov@gmail.com

Securing the Core University Business Processes

Keywords: Securing business process, Extended Certification Authority, Secure eDucument Management Architecture

In the paper are presented solutions for securing the core University Business Processes. A Method for identification which Business processes are critical for security point of view, on which is required to pay more attention for its securing. For securing of the elected Business processes is developed a new security system -Extended Certification Authority. Special Secure eDocument Management Architecture is developed, on which base are developed the solutions for securing the following University Business processes -Delegation of exam permissions to lecturers, Recording exam marks, and Exchange management documents.

Introduction

There are few definitions of the term "Business process", which are not too different each other. In our research we have accepted the definition of [START_REF] Kisimov | Dynamic Business-managed Information systems[END_REF], which under Business process understands a set of interrelated tasks leading to create a product or a service. Each Business process needs to have required security of its execution. This means the process should provide the security triangle of parameters Availability, Confidentiality, and Integrity, in an appropriate level.

University is like any other enterprise -with a set of business processes, from which some are more critical from security point of view. The criticality is coming from the generation of the university end result -the degree document, where all exam marks make its content. The processes leading to forming the university end result defines the set of core processes. Based on the mentioned criteria for the criticality, it is not too difficult to list the university core business processes.

The main goal of the current research is to identify what level of security the core business processes need, e.g. which the core university business processes needing higher level of security are, what is the needed security level, and how to provide that security.

Method for selection of core business processes needing high security

There is not standard way for selection of core business processes which need some level of security. There are some practical approaches - [START_REF] Garfamy | Supplier selection and Business process improvement[END_REF] and [START_REF] Lee | Process selection for Business Process Management in a mobile telecommunications company[END_REF], which resolve particular private needs, but no one exists for university business processes security evaluation. For this reason we have developed our "Method for selection of core business processes needing high security".

The proposed Method works with number of University Business processes, which core set consists of 11 processes, identified in general, and more specifically for the case for our particular research -University of National and World Economy (UNWE), Sofia, Bulgaria. These 11 University Business Processes are:

• Delegation of exam permissions;

• Recoding exam marks;

• Exchange Academic Counsel (AC) reports;

• Exchange Rectors' Counsel (RC) reports;

• Exchange Faculty Board reports;

• Exchange Departmental Meeting reports;

• Review Public statements;

• Change the Educational Curriculums;

• Admittance of new students;

• Admittance students from 2nd to 3rd year;

• Change student status.

These business processes are evaluated from the needed security point of view via 9 criteria, grouped in 5 groups. The evaluation is provided with metrics 1 to 3, where 1 means needed low security requirement, 2 means medium and 3 means high. The following criteria are identified with the specified value and security impact: a) Business Continuity -the criterion identifies what are the requirements for the Business continuity from the University business process; b) Disaster Recovery -the criterion identifies the need for available recovery and the time for recovery of the University business process; c) Event management -the criterion identifies how critical are the events for security violation for the business process; d) Availability -the criterion identifies how critical is the business process to be available. Normally the availability is measured in percentage, but for the purpose of the current Method, these percentages are transformed into value 1 to 3 (from 0 to 33% -value 1, from 33% to 66% -value 2, and up to 99%value 3); e) Confidentiality -the criterion identifies how critical is to have encryption of the information for the business process and the level of encryption. This is an integrated criterion, where the percentage of the encrypted messages is evaluated, the type of encryption (symmetrical / asymmetrical), the key length, and the period of key refreshment; f) Integrity -the criterion identifies the need of information integrity provided via security means; g) Business impact -the criterion identifies what is the Business impact of the Business process to the entire University business security. This criteria is formed from a few sub-criteria:

• Relative number of beneficiaries (students, professors, staff) impacted from the Business process; • Relative number of providers (external and internal) impacted from the Business process; • Level of cost of this Business process compared with the total cost of university processes; • Level of monthly transactions executed from the Business process;

• Political sensibility of the Business process, which relates to the view of the University to the country and to the entire world, the effect of the Business process to the reputation of the University, the effect of the Business process to the high-education process in Bulgaria and Europe; • What is the overall impact to the University from the specific Business process; • What is the overall impact to professors (lecturers) and students from the specific Business process;

The collection of values from the above mentioned sub-criteria for Business impact is proposed to be done via a table -Table 1; Each sub-criterion is also evaluated via the measures from 1 (low level) to 3 (high level). For each Business process a Total average score is created, which is rounded to the values 1, 2 and 3. This value is the defined value for Business Impact for the specific Business process. h) Risk level -the criterion identifies the level of which the risk of the entire University business can raise, based on the current business process. The University Business strategy and University Security strategy define levels of risk (there are no many universities in the world which have developed Business strategy and Security strategy, like the big corporation. In the current case of UNWE, the research team has identified the appropriate risks, based on interviews with the university management);

i) Minimum acceptance level -the criterion identifies which is the minimum security level, which is needed for the specified Business process. This criteria is formed from a few sub-criteria: Eligibility -inquires and responses of the Business process; Enrollment and Disenrollment of data for/to the Business process; Authorization -requests and responses to the Business process; Claims -receipts and adjudications from the execution of the Business process; Claims status -requests and responses from the execution of the Business process.

The collection of values from the above mentioned sub-criteria for Minimum acceptance level is proposed to be done via a table -Table 2;

BP #1 BP #2 etc.
Each sub-criterion is also evaluated via the measures from 1 (low level) to 3 (high level). For each Business process a Total average score is created, which is rounded to the values 1, 2 and 3. This value is the defined value for Minimum acceptance level for the specific Business process.

The proposed Method for selection of business processes and their level of needed security uses a table, shown in Table 3 below.

3 3 2 3 3 3 3 2 3 3 Exchange AC reports 3 3 1 2 1 1 3 1 3 4 Exchange RC reports 3 2 1 2 1 1 3 1 3 5 Exchange Faculty Board reports 2 1 1 1 1 1 2 1 2 6 Exchange Departm.Meet. reports 2 1 1 1 1 1 1 1 1 7 Review Public statements 3 1 1 3 1 1 3 2 3 8 Change the Educ. Curriculums 1 1 1 1 1 1 1 1 1 9 Admittance of new students 3 3 2 3 1 1 3 2 2 10 Admittance students 2nd to 3rg year 3 2 2 3 1 1 2 1 1 11 Change student status 2 2 1 2 2 3 1 1 1
Participating in: Degrade level Business process # For the target object -UNWE, particular values are resulted into the table. From those results we have concluded, that the core university business processes from security point of view are:

• Delegation of exam permissions (Protocols) to lecturers;

• Recording exam marks; and

• Exchange management documents o Exchange AC reports; o Exchange RC reports; The Business process "Admittance of new students" has also relatively high score -20, but because it is with relatively small infrastructure part of the University IS infrastructure, we will exclude it from our research activities. The Business processes Exchange Academic Counsel (AC) reports and Exchange Rectors' Counsel (RC) reports we can combine for the future presentation of research results, because their security requirements are closed. In this way we receive an aggregated business process called "Exchange management documents". For this reason, the mentioned 3 core business processes will continue to be the focus of the further research, presented in the paper.

Extended Certification Authority

For securing the core university business process is required a special solution with involvement of Certification Authority (CA) and Public Key Infrastructure (PKI). The required functions for securing the core business processes are logically between the functions of CA and functions of PKI.

Certification Authority generally is either independent system or it is an agent of a PKI. The user can do the main certification functions through its keys and certificateauthentication, encryption, digital signature, data integrity, non-repudiation, etc., while the certificate can be used also for reputable identification, timeframe for validity and specification of possible security functions. The security credentials are linked to the CA via the digital signature of the certificate, where the Public key is included, and the Public key recognizes the pair connection with its Private key. CA is an entity that issues security keys -Private and Public and Digital Certificates for use by other parties. The functions of the CA can be summarized into:

• Has its own Root certificate; • Cross Certification -used to establish a trusted relationship between separate PKI's. This allows for a distributed and decentralized infrastructure; • Support for Non-Repudiation -prevents a certificate owner from denying that data was sent using the owner's certificate; • Time stamping -certifies that the time stamp on the secured data is set and it is accurate and valid; • Client API -A means for an application to use the services offered by a PKI.

PKI is an expensive and difficult to operate system. Only limited number of companies has the luxury to own a PKI. University generally does not operate with budgets of the big companies, for which reason it is not practical to expect that a University will create or buy a PKI. The world has established and easier approach to the PKI, defining "Small PKI". Small PKI was born out of a joint effort to overcome the over complication and scalability problems of traditional PKI, decreasing the role of RA, Repository, and some of the functions. The functions of the Small PKI cover all functions of the CA, with incorporation of some functions of the PKI. In summary, the functions of Small PKI can be listed as:

• Full CA functions;

• Mechanisms to support security in a wide range of Internet applications, including IPSec protocol; • Keys renewal and key management; • Encrypt electronic mail and WWW documents;

• Range of Secure application functions requiring use of Public key

Certificates as ePayment and B2B;

• Support a range of trust models.

The analysis of the security needs for securing the core university business processes shows that the functionality of the Small PKI is too rich from security point of view and a university does not need of such a system. Here we have to add that the Small PKI is also an expensive system for the university budget and to securing the core University Business Processes we need a system, which functions are between those of CA and Small PKI. We call such a system "Extended CA" and position it in the functional axe of crypto functionality in a way, shown in figure 1.

Fig. 1. An axe of crypto functional systems, where the place of "Extended CA" is defined to be between CA and Small PKI The authors have analyzed the functions needed for securing the core university business processes, providing cost-functions analysis. On this base a crypto system with the name "Extended CA" was designed, which is relatively cheap from university budget point of view and functionally reach for security point of view. We have found that the Extended CA has to be a CA with LDAP, Web server for communications, and supporting some PKI services such as digital signature, keys exchange, email encryption, and secure token support. After that analysis, we have concluded that the Extended CA has to have the following functions: There are two standards (RFC) for certificates -for Public Key Certificates [START_REF]Internet X.509 Public Key Infrastructure Certificate and CRL Profile[END_REF] and for Attribute Certificates [START_REF]An Internet Attribute Certificate Profile for Authorization[END_REF]. The most used certificates in the world are the Public Key Certificates. At the same time, the Attribute Certificates provides some futures, which can be used for securing in a good way the university business processes. Generally, attributes in Attribute Certificate offer user's short-lived information and dedication, such as user's roles and access permissions, which suite the university short term access permissions, like giving access of a professor to exam marks for a subject, valid for a block or semester. For this reason the capabilities of the attributes in Attribute Certificate is preferably to use for the university purposes. For this reason we have analyzed the suitability of incorporation both certificates values into a single certificate, covering the international standards. We went to a conclusion, that the Extended CA can provide certificates, which combine the usability of both certificates.

Our proposed solution for a Certificate, issued by the Extended CA is to include the necessary attributes, which normally are part of the Attribute Certificate, in the Private Extensions of the Public Key Certificate. Applying the proposed approach, the certification issued by the Extended CA will have the following value-added functionality:

• Secure identification and authentication using pure SSL protocol + LDAP server;

Secure eDocument Management Architecture

The elected core University Business Processes for security operate predominantly with electronic documents. Like in any corporate Information System, solutions for a few problems can be a single integrated architectural solution. For this reason we have developed a special Security architecture, which will be the base for the securing the elected core University Business Processes. The developed architecture we called "Secure eDocument Management Architecture". The purpose of the offered in the paper architecture is to manage in a secure different management documents, providing for them the necessary security features, which do not exist in general electronic documents. The Secure eDocument Management Architecture has two focuses of security features: a) Securing the elements of the electronic document (eDocument); b) Securing the transactions with the eDocuments.

The Secure eDocument Management architecture operates with Lecturer's PC, where the Secured document is processed, and with Web server, where the University Business processes have centralized for treatment. As part of the Web site is the explained above Extended Certification Authority. The Secured document operates with 3 security features:

• Marcos, activated on loading the document;

• eButton with integrated macros, activated on pressing the Button;

• Security fields, keeping encrypted information, which is decrypted on Lecturer's PC.

Graphically, the Secure eDocument Management Architecture can be presented in figure 2.

Fig.2. Conceptual architecture of Secure eDocument Management Architecture

The mentioned 3 security features are part of the eDocument. Having at least one of those security features, the eDocument is converted to a Secure eDocument. Each one of those 3 security features can exist in many instances in the Secure eDocument. The first security feature -Macros are programs, which are executed during the transaction (transferring) of the eDocuments from the Web site to the Lecturer's PC. For example, when a document is loaded from the University Web site to a Lecturer's PC, one of a few Macros can be executed automatically. The second security feature -Buttons are also programs, but activated by the user of the eDocument. For example, when a lecturer wants to digitally sign a document, he has to press a button, which keeps virtually a program code executing the process of digital signature. The third security feature -Security fields are part of the eDocument, which can be treated with different security algorithms, for example the field can be encrypted and only with the wish of the end-user (lecturer) the field would be decrypted and represented in a plain text (via pressing of a button or executing of a Macros).

The developed Secure eDocument Management Architecture is used in securing of the presented below 3 core University Business processes.

Secure Process: Delegation of exam permissions to lecturers

The exam permission is a university document called Protocol, which specify which students to which subject have rights to be examined, by which lecturer. Not all student studying appropriate subject have rights to be examined at the end of given semester -they are different university procedures permitting under which conditions a student can go to be examined, for example if he finalized all assignments during the semester.

The current process can be secured using the explained in the previous section Secure eDocument Management architecture. We have designed, the Macros embedded in the eDocument -Protocol to provide the following functions:

• Full document protection via Password (via the lecturer's ID);

• Partial Confidentiality -some fields are only readable by the specified lecturer -Protocol name, Protocol number, semester validity; • Document data integrity -Protocol checking on receiving;

• Checking the document time period validity: Document data creation -Document date receiving; • Document digital signature verification. The Macros also provide 3 additional security functions, related to the entire eDocument:

• Protocol cannot be read and used by non-dedicated lecturer;

• Protocol cannot be operated by non-dedicated lecturer;

• Lecturer cannot take a Protocol, which is not created by the University Information Security system.

The eDocument-Protocol is developed with 2 buttons:

• Accepting the document after visual control;

• Non-repudiation of receiving the eDocument.

Secure Process: Recording exam marks

The process of Recording the exam marks is associated with moving the exam marks from the eDocument-Protocol to the Information system "Student", where all exam marks are recording. Based on those exam marks are generated the degree documents. The manual process is related to many paper records, and using many clerks. At the end, the responsibility for the exam results is to the lecturer, but many other people participate between the lecturer and the computer record of the exam marks. To eliminate all those interim people who do not keep responsibilities, is developed the current security process.

The securing of that business process uses a few security features:

• Secure eDocument-Protocol;

• Digital signing of the Protocol by the lecturer, before entering it into the system "Student"; • Using of One Time Password (OTP) for additional authentication of the lecturer;

The sequence diagram of the securing of the specified University business process is presented in figure 3. c) During the signature verification, the major elements of the lecturer's certificate are checked, like whether that lecturer has right to enter the Protocol in the system "Student", what are the credibility of the lecturer in its certificate according to the subject for which the exams are entering, etc.; d) The complex signature verification includes also the check whether the certificate is part of the Certification Revocation List (CRL). In the proposed solution CRL is not web accessible, but via an internal interface, because the check is provided inside the University Web site, where the Extended CA is also part of; e) The OTP has a special solution which ignores its major security weaknessman-in-the-middle. This solution implement encryption and digitally signing the received password by the lecturer and then send it to the Web site. These both technologies identify and authenticate exactly the lecturer and the University Web site, working with entering the exam marks.

Secure Process: Exchange management documents

The Business process of Exchange management documents is designed as Peer-topeer exchange of encrypted documents. This peer-to-peer link is a logical link, using the University Web site as an interim and control station. The main message moving mechanism used in the proposed solution is the emailing. The idea is to use exactly two interim stations -Web site and email server. The Web site is part of the Secure eDocument Management Architecture, which use as subsequent messaging mechanism for emailing. The conceptual architecture of that security architecture is presented in figure 4.

It is important to mention that the email services which are involved in the solution can be a corporate Email server or a Cloud Computing email services.

The developed solution provides as additional security features:

• Detail logging of all transactions -in the Web server;

• Auditing of all transactions -also in the Web server;

Conclusions

The presented security solutions for securing of the core University Business Processes are based on electronic document incorporating security program code in a few ways, integrating existing security technologies, but executed in a secure ways, using new security system -Extended Certification authority. The prototyping of the proposed solution uses documents based on MS Office, while the security technologies for them have been developed in two platforms -Java and VB-.Net.

Fig. 3 .

 3 Fig.3. Control flow of the secure technologies used to secure the Business process "Recording the exam marks" There are few special security technologies used in that diagram: a) The program for Digital signing of the eDocument-Protocol is always loaded from the University Web site, just before the signing. This ensure correct digital signing; b) The lecturer uses an USB Crypto device, which stores the Private key and the Certificate of the lecturer. To access those security credentials, the lecturer should enter the PIN code of the USB Crypto device;

Fig. 4 .

 4 Fig.4. Architecture for secure exchange of management documents

Table 1 .

 1 Forming of Business Impact criteria

	Business processes	Relative number of benefi-ciaries impacted	Relative number of providers impacted	Level of costs of this business process compared with total cost of university	Level of of monthly transac-tions	Political sensitivity	What's the overall impact on the University	What's the impact on professors	What's the impact on students	Total average score
	BP #1
	BP #2
	etc.

Table 2 .

 2 Forming of Minimum acceptance level criteria

						Claims
	Business process	Eligibility -inquiries & responses	Enrolments & Disenrollments	Authorization -requests & responses	Claims -receipts & adjudications	Status -inquiries &	Total average score
						responses

Table 3 .

 3 Business processes selection for needed security.

	Bus. Risk	Min	Total

  From other side, the PKI is set of hardware, software, policies and procedures needed to issue and maintain Asymmetrical and Symmetrical keys and Public Key Certificates incorporating different user's identities, and also to order, issue, register, store, distribute, renew, revoke and manage Public Key Certificates. PKI serves as a trusted third party between many end-users. The functions of the PKI are much bigger and complex than the CA, and they can be summarized into:• Operate with CA, Registration Authority (to verify and accept requests for certificates) and Repository (repository for certificates and CR List); • Provide Backup and Recovery for the purpose to restore lost or damaged Certificates; • Update Key History -at any certificate change, to update the history logs (because of expiration or a name change). Any data secured using the older keys would not be accessible unless the older keys are kept in an archive; • Revoke Certificates, when the Certificate is no anymore valid or it is discredited; • Automatic Key Renewing and Certificate renewing -after the expire of the certificate, new Private key and Public key has to be issued, on which base a new Certificate has to be issued for the same user, with a process of automatic renewal of them to the end-used repository -smart cards or USB devices. Automated key recertification can update the certificate with a new expiration date when necessary, without manual intervention;

	• Play a role for the trust party; • Maintain Certification revocation (CR) process, CR List and
	Repository
	• Use OCSP protocol (On-line Certificate Status Protocol) for access
	to CR List

• Verify the identity of entities asking to issue certificate; • Generate Private and Public keys; • Issue digital certificate attesting to the identity; • Digitally sign the certificate via its Root certificate; • Store certificate and keys in the secure tokens;

•

  Separation access control based on SSL managed certificates;• Add complementary security credentials, e.g. second PW,

	Biometric identification, etc.;
	• Easy to manage Private information in Corporate (in our case -
	University) systems;
	• Public Key Certificate (PKC) is per user, the attributes are "per
	group", and integration of both is for relatively medium-term
	certificate, e.g. per semester;
	• Provide "Revocation" for attributes;
	• Security policy defined per medium-term (per semester);
	• Provide trust from different CA/PKI via incorporated additional
	PKC (another way for cross certification);
	• Incorporate Privilege attributes as Copy rights, Patents, Trademarks,
	access to management documents (levels and specific);
	• Internet and Intranet specific rights;
	• Role based control using attributes.

  Both platforms showed equal flexibility for operation and security quality. The proposed research has few open research problems, like a Framework for development of Extended Certification Authority, and well securing the University Web site (with integration with Extended CA), if it is not in the University Intranet, but on open Internet.


Takahiro Saitoh

Sohichi Hirose

Takuo Fukui

Convolution Quadrature Time-domain Boundary Element Method and Acceleration by the Fast Multipole Method in 2-D Viscoelastic Wave Propagation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Since the boundary element method (BEM) is known as a suitable numerical approach for wave analysis, time-domain transient problems have been solved by many researchers using the BEM formulated by Mansur and BrebbiaCll, and HiroseC 2 J. However, it is known that the use of a direct time-domain BEM sometimes causes numerical instability in time-stepping procedures.

Recently, the convolution quadrature boundary element method (CQ-BEM) has been researched by SchanzO! and Abreu et al.C [START_REF] Abreu | Scalar wave propagation in 2D: a BEM formulation based on the operational quadrature method[END_REF] for transient analysis, following the proposal of the convolution quadrature method (CQM) by LubichC 5 l in 1988. In the formulation of the CQ-BEM, convolution integrals in the time-domain boundary element formulation are numerically approximated by the CQM. Application of the CQM to the time-domain BEM not only improves the numerical stability of the time-domain BEM, but also expands its range of engineering applications, because the CQ-BEM utilizes the Lap]acedomain, and not time-domain, fundamental solutions derived from the governing equations of a problem. In fact, the conventional time-domain BEM approach cannot be used for viscoelastic problems because no time-domain fundamental solution is known. Therefore, using the CQ-BEM is particularly helpful for viscoelastic problems.

Thus, the CQ-BEM is elegant. However, it shares a critical disadvantage with the conventional timedomain BEM: these methods cannot solve large-scale problems because of low computational efficiency. If this shortcoming is resolved, the CQ-BEM should become a more powerful technique for transient analysis of wave propagation problems.

To overcome the difficulty, in this paper we propose an efficient CQ-BEM using the fast multipole method (FMM)< 6 >< 7 l developed by Rokhlin and Greengard. In the following, after a review of the CQM, the conventional time-domain BEM and CQ-BEM formulations for 2-D viscoelastic wave propagation are presented. The CQ-BEM using the FMM is then discussed. Finally, numerical examples are shown to validate the proposed method.

CONVOLUTION QUADRATURE METHOD (CQM)

In this section, the convolution quadrature method (CQM) is described briefly. The convolution quadrature method (CQM), first proposed by Lubich, approximates the convolution f * g(t) by a dis- crete convolution using the Lap lace transform of the time-dependent function f ( t -T). In general, the convolution integral is defined as follows:

f * g(t) = [ f(t-r)g(r)dr, t ::0: 0 (1)
where * denotes the convolution. The convolution integral defined by Eq.(l) is approximated by the CQM as follows:

n f * g(n6.t) ~ L Wn-j(6.t)g(j6.t), n = 0, 1, ... , N (2)
j=O where timet was divided into N equal steps 6.t. Moreover, wi(6.t) denotes the quadrature weights, which are determined by the coefficients of the following power series with complex variable z, namely, 8(()

~ n F( 6.t ) = L.-Wn(6.t)z . n=O (3)
In Eq.(3), F is the La place transform of the time-dependent function f and 8 ( () is the quotient of the generating polynomials of a linear multistep method. The power series defined in Eq.( 3) can be calculated by Cauchy's integral formula. Considering a polar coordinate transformation, Cauchy's integral is approximated by a trapezoidal rule with L equal steps 27r / L as follows:

(4

)
where i is the imaginary unit and (t is given by ( 1 = Re 2 1rill L. In addition, R is the radius of a circle in the domain of analyticity of F.

CQ•BEM FORMULATION FOR 2•D VISCOELASTIC WAVE PROPAGATION

We consider 2-D visco~lastic wave scattering by scatterer iJ in an exterior viscoelastic medium D. When the incident wave um hits the boundary surfaceS of scatterer iJ, scattered waves are generated by the interaction between scatterer D and the incident wave. Assuming zero initial conditions, i.e., ui(x, t = 0) = 0 and aui(x, t = 0)/8t = 0, the governing equations and boundary conditions are written as f'(t) • U;J;(;z;, t) + ( K(t) + ~l'(t)) • U;,;;(X, t) = pii;(X, t) in D ui = ili on 81, ti = li on 82, 52= S \51.

(5) (6) In Eq.(S) and Eq.( 6), J.L(t) and K(t) are the shear relaxation and bulk relaxation functions, respectively, which satisfy the axiom of nonretroactivity, i.e., J.L(t) = K(t) = 0, -oo < t < 0. In addition, pis the density of elastic medium D. ui(x, t) and ti(x, t) are the displacement and traction components of the displacement and traction vector in Cartesian coordinate system at x and time t, respectively. Moreover, ui and fi are the given boundary conditions, ( ) ,i denotes space derivative, and ( • ) indicats the partial derivative with respect to time t. Summation over repeated subscripts is implied and indices range from 1 to 2 in this paper.

In general, the time-domain fundamental solution UiJ(x, y, t) and its double layer kernel TiJ(X, y, t) cannot be obtained analytically for a linear viscoelastic problem as mentioned above. However, assuming the exsistence of an explicit form of the time-domain fundamental solutions, the time-domain boundary integral equation in 2-D viscoelastodynamics can be expressed by where CiJ is the free term. Normally, the boundary integral equation is discretized by using the appropriate interpolation functions for the unknown values and solved by a time-stepping algorithm. The boundary integral equation ( 7), however, cannot be solved using such a scheme because there is no explicit time-domain fundamental solution. Therefore, the CQM is applied to Eq.( 7) to overcome the difficulty.

If we discretize the boundary surface S into M elements by a piecewise constant approximation of the unknown displacement ui and traction ti, taking the limit of x E D --+ X E S, and finally using CQM for the convolutions in Eq.( 7), we arrive at the following discretized boundary integral equation for time increment ~t and n steps as follows:

M n ~ui(x, n~t) = u~n(x, n~t) + L L [Aij-k(x, ya)tj(k~t) -B;j-k(x, ya)uj(k~t)] . ( 8
)
a=l k=l

Here, Ai and Bf' are influence functions defined by

(9) (10)
where St is given by s1 = 8((t)/ ~t. In addition, Uij(X, y, s) and 'fiJ(x, y, s) are the Laplace-domain fundamental solutions, which can be analytically obtained as follows:

(11

)
where c;, and c1 are the transverse and longitudinal wave velocities given by c;, = J Jl* ( s) j p and c£ = vf(K*(s) + (4/3)J.L*(s))jp, respectively. In addition, Jl*(s) and K*(s) are defined by Jl*(s) = sfl(s)

and K* ( s) = sk ( s) using the Lap lace-transformed relaxation functions fl( s) and k ( s), respectively.

Also, OiJ is the Kronecker delta, r is given by r = IX-Yl and Kn is the modified Bessel function of the second kind. Note that sL and sr are defined by sL = 8(()/(c1~t) and sr = 8(()/(c;.,~t) due to the simple expression in Eq.(ll). The calculations of Eq.( 9) are identical to the discrete Fourier transform. Therefore, Eq.( 9) can be evaluated rapidly by means of the fast Fourier transform (FFf) algorithm. Arranging Eq.( 8) according to the boundary conditions, we obtain

M ~ui(x, n~t)+ 2: [B?j(x, Y 0 )uj(n~t)-A~j(X, y 0 )tj(n~t)] a=l M n-l =u}n(x, n~t) + :L 2: [Aij-k(x, Y 0 )tj(k~t)-B:_j-k(x, y 0 )uj(k~t)]. (13)
For the n-th time step, all the quantities on the right-hand side of Eq.( 13) are known. Therefore, the unknown values uf(n~t) and tf(n~t) in the n-th time step can be obtained by solving Eq.(l3). Unfortunately, we cannot solve a large-scale problem with a large number of boundary elements M and time steps N by the CQ-BEM because the required computational complexity and memory become O(LA1 2 N) and 0(1\1 2 N) in Eq.(l3), respectively. Therefore, the CQ-BEM is accelerated by the FMM in this research.

APPLICATION OF FMM TO CQ-BEM

The FMM, developed by Rokhlin and Greengard< 6 >< 7 l, is a technique for reducing computational time and memory requirements for a large-scale problem. Since the FMM and its application to the BEM have been described in detail in other published papers, we will summarize only the essential formulas here.

MULTIPOLE EXPANSION AND LOCAL EXPANSION

To apply the FMM to the CQ-BEM, we need to separate the variables of the fundamental solutions (11) and (12). For example, the fundamental solution in Eq.(ll) is transformed into the equation:

A 1 [ u U] Uii(x, y, s) = J..L*(s)s} <I>,i + e3iiw,j . (14)
Here <I>u and wu are the displacement potentials, which are defined by (

u 1 u 1 <I> = 2 7T Ko(sLix-Y\),k, w = e3kl 2 1T Ko(sriX-Yl),l•
) ~ ~ 17
In Eq.( 17), In shows the modified Bessel function of the first kind. Similarly, multi pole moments Af'{: and NJ' for the double layer kernel defined in Eq.( 12) are derived, as well as J.\1~1 and Ni{ for the fundamental solution defined in Eq.(ll).

In addition, local expansions are also obtained corresponding to the multipole expansions defined in Eq.( 16) as follows:

TRANSLATION FORMULAS FOR FMM

Once the multi pole moments are obtained, we can quickly evaluate the matrix-vector products of the discretized boundary integral equation (13) using the fast multipole algorithm. The translation formulas for the fast multipole algorithm, M2M, M2L, and L2L, are also derived from Graf's addition theorem, and are defined as follows: This fact sometimes causes instability of the translation formulas ( 19)-( 21) when cell size is large. To overcome this problem, we introduce a scaling of the multi pole moments ( 17) using the scaled modified Bessel functions ln(z) = e-z In(z) and Kn(z) = ez Kn(z).

NUMERICAL EXAMPLES

In the fo11owing practical applications, the three-element standard linear model< 8 l is considered as a typical elastic model.

VISCOELASTIC WAVE SCATTERING BY A CAVITY

We first consider the problem of an incident plane SV-wave scattered by a cavity with radius a as shown in Fig. 2. In this analysis, we use. a Ricker wavelet modified for a viscoelastic wave, as the incident wave. The modified Ricker wavelet u~n(x, t) is defined by a Fourier transform of the original Ricker wavelet as (23) where t0 corresponds to the peak frequency 27!' /t 0 , ts corresponds to the minimum peak in the time do- main and u 0 is the reference displacement. The difference from the elastic wave is that the wavenumber 000 000 000 at x1/a = 0.0, 5.0, 10.0 and 15.0.

kr is complex and depends on the angular frequency w. Figs. 345show the time histories of the displacements u 2 fu 0 at points A, B, and Con the cavity as shown in Fig. 2. Solutions using two different time increments, CLoi:!J.tf a = 0.125 and C£0/:ltf a = 0.0625, an elastic solution, and a numerical solution obtained by Fukui et. al. < 9 ) are shown in Figs. 345for comparison. We solved this problem solely through the CQ-BEM without using the FMM. The boundary• of the cavity is discretized into 72 boundary elements using piecewise constant approximation, and we assumed a traction-free condition: ti = 0. The CQM parameters, N and L, are given by N = L = 128. In addition, 'R, is assumed to be 'R, = 0.91398170 (e = 10-10 ). The material parameters of the three- element standard linear model are J-LR/ J-Lo = 0.5, K / J-Lo = 1.0, ru = T0, and re = 0.5T0, where J-LR is the relaxed shear modulus, J-Lo is the initial shear modulus, ru is the stress relaxation time, and r 6 is the strain relaxation time. Moreover, the time To is defined by T0 = 2af C£0, which is the time that an elastic wave with initial wave velocity C£o = ..j(K + (4/3)J.Lo)/ p passes through a cavity with radius a as shown in Fig. 2. We set time tjT 0 = 0.0 at the moment the incident wave hits point A on the boundary of the cavity in Fig. 2. We can see that the results obtained by the CQ-BEM are in good agreement with the analytical numerical results calculated by Fukui et.al.

VISCOELASTIC WAVE SCAITERING BY MANY CAVITIES

Next, viscoelastic wave scattering by many cavities, as shown in Fig. 6, is solved using the CQ-BEM accelerated by the FMM. In this analysis, we use a desktop computer with an Intel(R) Xeon(R) 1.60GHz CPU 5310 and 2GB of memory. The components of an incident viscoelastic plane wave with initial wave velocity CLo are defined as follows:

.

[ 2(1

iwTo) l m _ . -I Wo -e i/cL:z:1 ui (re, t) -uo8z1:F 2 . ( 2
2 ) e lW W -W 0 ( 24
)
where w 0 = 21r JT 0 • Fig. 7 shows the time variations of utfu0 of the incident viscoelastic waves cal- culated by Eq.( 24) at xtfa = 0, 5, 10, and 15. The parameters of the three-element standard model are J.LR/ J.Lo = 0.85, K/ J-Lo = 5/3, ru = 0.5T0, and r6 = 17T0/40. For comparison, incident elastic waves with wave velocity CLo corresponding to the incident viscoelastic waves are plotted by solid lines in Fig. 7. We can see that the wave-form& of the incident viscoelastic wave decrease while the wave is propagating.

-10 -5

-10

-5 We set the time tjT 0 = 0.0 at the moment the incident wave hits the cavities in the far left column in Fig. 6. The parameters for the CQM are taken as N = L = 256, p = 0.95609320 (E = 10-10 ). The time increment is cL06tja = 0.125, the number ofDOF in each time step is 8192, and the truncation number for the summation of multi pole expansion and local expansion such as Eq.(l6) and Eq.( 18) is chosen asp= ±10. This problem cannot be solved by the use of the CQ-BEM alone because of memory restrictions. Therefore, the FMM is applied to accelerate the calculation of the matrix-vector products of the discretized boundary integral equation and to conserve memory. In addition, OpenMP with eight threads is used to parallelize this analysis. We can see that scattered waves are generated by the interaction of the incident wave with each cavity. The effect of attenuation from the viscoelastic model can be seen by comparing Fig. 8 and Fig. 9. Thus, the CQ-BEM using the FMM is very effective for large-scale problems.

CONCLUSION

In this paper, the CQ-BEM using the FMM was developed for 2-D viscoelastic wave propagation. In the proposed method, the convolution integrals in a time-domain boundary integral equation are discretized using the CQM, and the Laplace-domain fundamental solutions are used to calculate influence functions. Therefore, the proposed method has the potential to tackle problems where no time-domain fundamental solutions are available, e.g., poroelastodynamics or 3-D viscoelastodynamics. Moreover, the FMM was applied to accelerate the calculations of matrix-vector products for the retarded potential and to reduce the memory required in the CQ-BEM. As numerical examples, scattering problems of incident waves by cavities were demonstrated to validate the proposed method. In the near future, we will extend the CQ-BEM using the FMM to 3-D viscoelastic wave propagation.

Figure 1 .

 1 Figure 1. Multipole expansion and Graf's addition theorem.

  Yo near a source point y as shown in Fig.l, and using Graf's addition theorem for Eq.( 15), we can obtain the multipole expansions as follows: q,U(x) = 2~ E Af;!Kn(sLr)ein(), wU(x) = 2~ E N;[Kn(srr)einO (16) n=-oo n=-oo where r and (} are the polar coordinate components of a field point X originated at Yo as shown in Fig. I. In Eq .( 16), the coefficients A1;( and N;( do not depend on the field point X. These coefficients are called the multipole moment on Yo. and are given by Af;( = -8 8 [In(sLp)e-in<P], N;[ = -e3kt 8 8 [In(srp)e-in<P].

  (18), r and (} are the polar coordinate components of a field point X originated at the local expansion point Xo. In addition, Ln and Hn are local expansion coefficients corresponding to multipole moments 1\!ln and Nn, respectively.

Figure 2 .Figure 3 .Figure 4 . 2 OFigure 5 .

 23425 Figure 2. Viscoelastic wave scattering by a cavity.Figure3. Vertical total displacement u 2 ju 0 at point A

Figure 6 .

 6 Figure 6. Multiple scattering model.

Figure 7 .

 7 Figure 7. Time variations of uduo of the incident viscoelastic wave calculated by Eq.(24)

Figure 8 .

 8 Figure 8. Time histories of the displacements ul/uo Figure 9. around the cavities in the viscoelastic case.

(a) tjT 0 = 1 .

 1 25, (b) t/To = 6.25, (c) t/To 11.25.

5 10

 5 Time histories of the displacements uifu0 around the cavities in the elastic case. (a) t/To -1.2:5, (b) t/To -6.25, (c) t/To = 11.25.

Fig. 8 -

 8 Fig.8-(a), (b), and (c) show the time histories of displacement uduo around cavities at t/To = 1.25,


Christophe Chesneau
email: christophe.chesneau@unicaen.fr

A new family of distributions based on the hypoexponential distribution with fitting reliability data

Keywords: Hypoexponential distribution, Generalized mixture, Moments, Maximum likelihood estimation. 2000 MSC: 60E05, 62E15

In this paper, a new general family of distributions using the hypoexponential distribution is introduced and studied. A special case of this family is explored in detail, corresponding to a new finite generalized mixture of generalized exponential distributions. Some of their mathematical properties are provided. We investigate maximum likelihood estimation of the model parameters. Three real data sets are used to prove the potential of this distribution among some recent extensions of the exponential distribution.

Introduction

Over the past decades, considerable efforts have been made to develop flexible distributions for modeling lifetime data. In many situations, the weighted exponential distributions reach this goal with a great success. They are characterized by probability density functions of the form: f (x) = w(x)g(x), where g(x) denotes a pdf of the exponential distribution with parameter λ > 0 and w(x) denotes a weight function. A suitable choice for w(x), based on practical or theoretical consideration, can lead a proper statistical model for analysis specific lifetime data. The literature covering weighted exponential distributions is vast and is growing fast. We refer to the review of [START_REF] Saghir | Weighted Distributions: A Brief Review, Perspective and Characterizations[END_REF], and the references therein.

In this paper, we introduce a new general family of distributions constructed from the so-called hypoexponential distribution and a transformation proposed by [START_REF] Al-Hussaini | Composition of cumulative distribution functions[END_REF]. Let us recall that the hypoexponential distribution is characterized by the distribution of a sum of several independent exponential random variables with different parameters (see [START_REF] Amari | Closed-Form Expression for Distribution of the Sum of Independent E Random Variables[END_REF] and [START_REF] Akkouchi | On the convolution of exponential distributions[END_REF]). It is used in many domains of application, including tele-traffic engineering and queueing systems. Several tuning parameters are thus necessary for a perfect model. Our new family of distributions profits of this flexibility to open the door of new applications.

By considering a specific configuration of this family, we exhibit a new weighted exponential distribution, called the EH(a, λ) distribution, having some interesting properties. The corresponding cumulative distribution function can also be viewed a special linear combination of several cumulative distribution functions associated to the well known exponentiated exponential distribution introduced by Gupta andKundu (1999, 2001). Since some coefficients of the combination can be negative, it also belongs to the family of finite generalized mixture distributions (see [START_REF] Bartholomew | Sufficient conditions for a mixture of exponentials to be a probability density function[END_REF], [START_REF] Franco | Generalized mixtures of Weibull components[END_REF] and [START_REF] Bakouch | Lindley first-order autoregressive model with applications[END_REF]). Mathematical properties of our distribution are studied, including moments, conditional moments, reversed conditional moments, moment generating function, Lorenz curve, Bonferroni curve, mean deviations and order statistics. Estimation of the parameters is determined using the method of maximum likelihood. Considering three real data sets of different natures, we show the superiority of our distribution in terms of some goodness-of-fit statistics in comparison to the well-known distributions: the Lindley distribution (see [START_REF] Ghitany | Lindley distribution and its applications[END_REF]), the exponential distribution and the exponentiated exponential distribution (see Gupta andKundu (1999, 2001)).

The rest of the paper is organized as follows: Section 2 presents our new distribution. Some of its mathematical properties are described in Section 3. Distributions of order statistics following our distribution are investigated in Section 4. Estimation of the parameters via the maximum likelihood is studied in Section 5. Applications to real data sets are performed in Section 6. Some concluding remarks are outlined in Section 7.

The EH(a, λ) distribution

On the hypoexponential distribution

We now briefly present the so-called hypoexponential distribution. Let m ≥ 2 be a positive integer and α 1 , α 2 , . . . , α m be m different real numbers. Let X 1 , X 2 , . . . , X m be m independent random variables, with X i following the exponential distribution of parameter α i > 0; the probability distribution function (pdf) of X i is given by

f Xi (x) = α i e -αix ,
x > 0.

Then the random variable S = m i=1 X i follows the hypoexponential distribution with a parameter vector a = (α 1 , . . . , α m ); a pdf of S is given by

f S (x) = m i=1 α i P i e -αix , x > 0,
where

P i = m j=1 j =i 1 - α i α j .
Note that P i is completely determined by a and can be positive or negative according to the sign of m j=1 j =i (α j -α i ), offering a great richness in the construction of f S (x). The corresponding cumulative distribution function (cdf) is given by

F S (x) = 1 - m i=1 1 P i e -αix , x > 0. ( 1
)
Further details on the hypoexponential distribution can be found in [START_REF] Amari | Closed-Form Expression for Distribution of the Sum of Independent E Random Variables[END_REF], [START_REF] Akkouchi | On the convolution of exponential distributions[END_REF] and [START_REF] Smaili | Hypoexponential distribution with different parameters[END_REF]. Before introducing the EH(a, λ) distribution, we present a more general family of distributions based on a general transformation of cdf and the hypoexponential distribution.

A family of distributions based on the hypoexponential distribution

Lemma 1 below presents an idea of construction of a new cdf using two existing cdfs.

Lemma 1 (AL-Hussaini ( 2012)). Let G(x) and H(x) be two cdfs of continuous distributions with G(x) of support (0, +∞). Let K(x) be the function defined by

K(x) = 1 -G(-ln(H(x))), x ∈ R.
Then K(x) is a cdf of a continuous distribution.

Remark 1. In the context of Lemma 1, one can observe that K(x) is the cdf of the random variable H -1 (e -U ) where U denotes a random variable having the cdf G(x). This idea was also used with specific distributions on U in [START_REF] Zografos | On families of beta-and generalized gamma-generated distributions and associated inference[END_REF] and [START_REF] Ristić | The gamma-exponentiated exponential distribution[END_REF].

Let us now introduce a particular family of distributions which will be at the heart of this study. Let H(x) be a cdf of a continuous distribution and G(x) be the cdf associated to the hypoexponential distribution with a parameter vector a = (α 1 , . . . , α m ) (1). It follows from Lemma 1 that the following function K(x) is a cdf:

K(x) = 1 -G(-ln(H(x))) = 1 -1 - m i=1 1 P i e -αi(-ln(H(x))) = m i=1 1 P i [H(x)] αi .
From this expression, new distributions arise by taking specific cdf H(x) (uniform distribution, normal distribution, Cauchy distribution, Pareto distribution . . . ). We will call the associated distribution the GH(H; a) distribution (for general hypoexponential distribution with a parameter vector a = (α 1 , . . . , α m ) using a cdf H(x)) for frequently used purpose in the present study or elsewhere.

Let us now consider a random variable Y following the GH(H; a) distribution. Then the cdf of Y is given by

F Y (x) = m i=1 1 P i [H(x)] αi , x ∈ R.
Using the equality

m i=1 1 Pi = 1, the survival function (sf) of Y is given by S Y (x) = 1 -F Y (x) = 1 - m i=1 1 P i [H(x)] αi = m i=1 1 P i [1 -[H(x)] αi ], x ∈ R.
Denoting by h(x) a pdf associated to H(x), a pdf of Y is given by

f Y (x) = h(x) m i=1 α i P i [H(x)] αi-1 , x ∈ R.
The hazard rate function (hrf) of Y is given by

h Y (x) = f Y (x) S Y (x) = h(x) m i=1 αi Pi [H(x)] αi-1 m i=1 1 Pi [1 -[H(x)] αi ] , x ∈ R.
Remark 2. The GH(H; a) distribution belongs to the family of weighted distributions; we can express

f Y (x) as: f Y (x) = w(x)g(x)
, where g(x) denotes a pdf of the exponential distribution with parameter

λ > 0 and w(x) = m i=1 αi Pi [H(x)] αi-1 .
Remark 3. Since the coefficients 1 P1 , 1 P2 , . . . , 1 Pm have possible negative and positive values with m i=1 1 Pi = 1, the GH(H; a) distribution also belongs to the family of finite generalized mixture of exponentiated baseline cdf [H(x)] α (also known as Lehmann type-I distribution) (see [START_REF] Bartholomew | Sufficient conditions for a mixture of exponentials to be a probability density function[END_REF], [START_REF] Franco | Generalized mixtures of Weibull components[END_REF] and [START_REF] Bakouch | Lindley first-order autoregressive model with applications[END_REF]).

Remark 4. In the particular case

m = 2, a = (α 1 , α 2 ), α 1 , α 2 > 0, α 1 = α 2 , we have P 1 = α2-α1 α2 and P 2 = α1-α2 α1 , F Y (x) = α 2 α 2 -α 1 [H(x)] α1 + α 1 α 1 -α 2 [H(x)] α2 , x ∈ R, and
f Y (x) = α 1 α 2 α 2 -α 1 h(x) [H(x)] α1-1 -[H(x)] α2-1 , x ∈ R.
Assuming that α 2 > α 1 , note that P 1 > 0 and P 2 < 0.

The distribution of a sum of two independent GH(H; a) distributions can be characterized. Let Y and Z be two independent random variables, Y follows the GH(H 1 ; a) distribution with a = (α 1 , α 2 , . . . , α m ) and Z follows the GH(H 2 ; b) distribution with b = (β 1 , β 2 , . . . , β m ). Let h 1 (x) be a pdf associated to the cdf H 1 (x) and h 2 (x) be a pdf associated to the cdf

H 2 (x). Set Q = m j=1 j = 1 -β βj .
Then a pdf for Y + Z is given by

f Y +Z (x) = m i=1 m =1 α i β P i Q +∞ -∞ h 1 (t)h 2 (x -t)[H 1 (t)] αi-1 [H 2 (x -t)] β -1 dt, x ∈ R.
This result is an immediate consequence of the continuous convolution formula.

The EH(a, λ) distribution

Let us now consider the GH(H; a) distribution defined with the cdf H(x) associated to the exponential distribution of parameter λ > 0: H(x) = 1 -e -λx , x > 0. We will call the associated distribution the EH(a, λ) distribution (for exponential hypoexponential distribution with a parameter vector a = (α 1 , . . . , α m ) and λ) for frequently used purpose in the present study or elsewhere. Let us now consider a random variable Y following the EH(a, λ) distribution. Then the cdf of Y given by

F Y (x) = m i=1 1 P i (1 -e -λx ) αi , x > 0.
This cdf can be viewed as a finite generalized mixture of generalized exponential distribution baseline cdf introduced by Gupta and [START_REF] Gupta | Generalized Exponential Distributions[END_REF][START_REF] Gupta | Exponentiated exponential distribution: an alternative to gamma and Weibull distributions[END_REF], with different parameters for the power (the standard cdf of the generalized exponential distribution is obtained by taking m = 1). Thanks to its several tuning parameters, the EH(a, λ) distribution has a great flexibility which can be useful for determine a nice statistical model. The sf of Y is given by

S Y (x) = m i=1 1 P i [1 -(1 -e -λx ) αi ], x > 0.
A pdf of Y is given by

f Y (x) = λe -λx m i=1 α i P i (1 -e -λx ) αi-1 , x > 0.
The hrf of Y is given by

h Y (x) = λe -λx m i=1 αi Pi (1 -e -λx ) αi-1 m i=1 1 Pi [1 -(1 -e -λx ) αi ] , x > 0.
Some asymptotic properties of the previous functions are given below: when x → 0, we have

F Y (x) ∼ m i=1 1 P i (λx) αi , f Y (x) ∼ λ m i=1 α i P i (λx) αi-1 , h Y (x) ∼ λ m i=1 α i P i (λx) αi-1 . Note that, if inf(α 1 , α 2 , . . . , α m ) > 1, we have lim x→0 f Y (x) = 0. If inf(α 1 , α 2 , .
. . , α m ) < 1, the limit depends on the signs of P 1 , P 2 , . . . , P m ; we can have lim Remark 6. We now adapt Remark 1 to the EH(a, λ) distribution. Let us observe that, for a random variable U following the hypoexponential distribution with parameter a, the random variable W = -1 λ ln(1 -e -U ) follows the EH(a, λ) distribution. Remark 7. In the particular case m = 2, a = (α 1 , α 2 ), α 1 , α 2 > 0, α 1 = α 2 , we have P 1 = α2-α1 α2 and P 2 = α1-α2 α1 ,

F Y (x) = α 2 α 2 -α 1 [1 -e -λx ] α1 + α 1 α 1 -α 2 [1 -e -λx ] α2 , x > 0, and f Y (x) = λα 1 α 2 α 2 -α 1 e -λx [1 -e -λx ] α1-1 -[1 -e -λx ] α2-1 , x > 0.
Figures 1, 2 and 3 show the graphical features of pdfs, cdfs and hrfs of the EH(a, λ) distribution with a = (α 1 , α 2 ), α 1 = α and α 2 = α + 0.1, for several choices of parameters (α, λ). In this particular case, let us precise that

F Y (x) = (1 + 10α)(1 -e -λx ) α -10α(1 -e -λx ) α+0.1 , (2)
a pdf is given by

f Y (x) = λα(1 + 10α)e -λx (1 -e -λx ) α-1 [1 -(1 -e -λx ) 0.1 ] (3)
and the hrf is given by

h Y (x) = λα(1 + 10α)e -λx (1 -e -λx ) α-1 [1 -(1 -e -λx ) 0.1 ] 1 -(1 + 10α)(1 -e -λx ) α + 10α(1 -e -λx ) α+0.1 . ( 4
)
We observe that the plots of the pdfs and hrfs of the EH(a, λ) distribution are very flexible; different shapes and curves can be of interest for modeling lifetime data.

Remark 8. The considered pdf f Y (x) (3) is a very special case of the EH(a, λ) distribution and the value 0.1 is subjective. It can also be viewed as a particular two parameters pdf of the McDonald modified weibull distribution introduced in [START_REF] Merovci | The McDonald Modified Weibull Distribution: Properties and Applications[END_REF] or the Beta exponentiated Weibull distribution developed by [START_REF] Cordeiro | The beta exponentiated Weibull distribution[END_REF]. Note that the main differences between f Y (x) (3) and similar looking pdfs as those associated to the weighted exponentiated exponential distribution introduced by [START_REF] Mahdavi | Two weighted distributions generated by exponential distribution[END_REF] or the transmuted generalized exponential distribution proposed by [START_REF] Khan | Transmuted generalized exponential distribution: A generalization of the exponential distribution with applications to survival data[END_REF], is the last term [1 -(1 -e -λx ) 0.1 ] which only depends on λ, not α.

Mathematical properties related to the EH(a, λ) distribution

We now present some mathematical properties of our distribution.

The distribution of a sum of two independent EH(a, λ) distributions can be characterized. Let Y and Z be two independent random variables, Y follows the EH(a, λ 1 ) distribution with a = (α 1 , α 2 , . . . , α m ) and Z follows the EH(b, λ 2 ) distribution with b

= (β 1 , β 2 , . . . , β m ). Set Q = m j=1 j = 1 -β βj .
Then a pdf for Y + Z is given by

f Y +Z (x) = λ 1 λ 2 m i=1 m =1 α i β P i Q x 0 e -λ1t e -λ1(x-t) [1 -e -λ1t ] αi-1 [1 -e -λ2(x-t) ] β -1 dt, x > 0.
Also, note that

P(Y ≤ Z) = m i=1 m =1 β P i Q (α i + β ) .
This probability can be of interest in the context of reliability the stress-strength model (see [START_REF] Kundu | Estimation of P (Y < X) for Generalized Exponential Distribution[END_REF]).

Let us work with only a random variable Y following the EH(a, λ) distribution.

Owing to the binomial series, we have the following expansion for F Y (x):

F Y (x) = m i=1 1 P i (1 -e -λx ) αi = m i=1 +∞ k=0 α i k (-1) k 1 P i e -kλx .
Using again the binomial series, we have the following expansion for f Y (x):

f Y (x) = λe -λx m i=1 α i P i (1 -e -λx ) αi-1 = m i=1 +∞ k=0 η i,k e -(k+1)λx ,
where

η i,k = λ α i -1 k (-1) k α i P i .
We can obtain the quantile function Q Y (x) via the nonlinear equation:

F Y (Q Y (x)) = x ⇔ m i=1 1 P i (1 -e -λQ Y (x) ) αi = x.
The EH(a, λ) distribution can be simulated by using Y = Q Y (U ) where U is a random variable having the uniform distribution on [0, 1]. Let us consider the Gamma function: Γ(ν) = +∞ 0

x ν-1 e -x dx, ν > 0. The r-th moment of Y is given by

E(Y r ) = +∞ -∞ x r f Y (x)dx = m i=1 +∞ k=0 η i,k +∞ 0 x r e -(k+1)λx dx = Γ(r + 1) λ r+1 m i=1 +∞ k=0 η i,k 1 (k + 1) r+1 .
The moment generating function of Y is given by, for t < λ,

M Y (t) = E(e tY ) = +∞ -∞ e tx f Y (x)dx = m i=1 +∞ k=0 η i,k +∞ 0 e (t-(k+1)λ)x dx = m i=1 +∞ k=0 η i,k 1 (k + 1)λ -t .
Proceeding as in (Gupta and Kundu, 2001, Equation (2.3)), with the change of variable y = e -λx , we also have, for t < λ,

M Y (t) = m i=1 1 P i Γ(α i + 1)Γ 1 -t λ Γ α i -t λ + 1 .
Let us consider the lower incomplete Gamma function: Γ(t, ν) = t 0 x ν-1 e -x dx. Then we have

t 0 x r f Y (x)dx = m i=1 +∞ k=0 η i,k t 0 x r e -(k+1)λx dx = 1 λ r+1 m i=1 +∞ k=0 η i,k Γ((k + 1)λt, r + 1) (k + 1) r+1 .
This equality will be useful in the next. For t > 0, the conditional r-th moment of Y is given by

E(Y r | Y > t) = 1 1 -F Y (t) +∞ t x r f Y (x)dx = 1 1 -F Y (t) E(Y r ) - t 0 x r f Y (x)dx = 1 λ r+1 (1 -F Y (t)) m i=1 +∞ k=0 η i,k Γ(r + 1) -Γ((k + 1)λt, r + 1) (k + 1) r+1 .
For t > 0, the reversed conditional r-th moment of Y is given by

E(Y r | Y ≤ t) = 1 F Y (t) t 0 x r f Y (x)dx = 1 F Y (t)λ r+1 m i=1 +∞ k=0 η i,k Γ((k + 1)λt, r + 1) (k + 1) r+1 . Let µ = E(Y ). The Lorenz curve L(F Y (t)) is given by L(F Y (t)) = 1 µ t 0 xf Y (x)dx = 1 µλ r+1 m i=1 +∞ k=0 η i,k Γ((k + 1)λt, 2) (k + 1) r+1 . The Bonferroni curve B(F Y (t)) is given by B(F Y (t)) = 1 µF Y (t) t 0 xf Y (x)dx = 1 µF Y (x)λ r+1 m i=1 +∞ k=0 η i,k Γ((k + 1)λt, 2) (k + 1) r+1 .
The mean deviation of Y about the mean µ can be expressed as

δ 1 (Y ) = E(|Y -µ|) = 2µF Y (µ) -2 µ 0 xf Y (x)dx = 2µF Y (µ) - 2 λ r+1 m i=1 +∞ k=0 η i,k Γ((k + 1)λµ, 2) (k + 1) r+1 .
The mean deviation of Y about the median M can be expressed as

δ 2 (Y ) = E(|Y -M |) = µ -2 M 0 xf Y (x)dx = µ - 2 λ r+1 m i=1 +∞ k=0 η i,k Γ((k + 1)λM, 2) (k + 1) r+1 .

Order statistics

Order statistics are essential in many areas of statistics. They naturally appear in the probabilistic analysis of reliability of a system. We explicit here the distributions of these order statistics in the context of our new distribution. Let Y 1 , Y 2 , . . . , Y n be n independent and identically distributed random variables following the EH(a, λ) distribution. Let us consider its order statistics is Y 1:n , Y 2:n , . . . , Y n:n . A pdf of the i-th order statistic Y i:n is given by

f i:n (x) = n! (i -1)! (n -i)! [F Y (x)] i-1 [1 -F Y (x)] n-i f Y (x) = n! (i -1)! (n -i)! m i=1 1 P i (1 -e -λx ) αi i-1 1 - m i=1 1 P i (1 -e -λx ) αi n-i × λe -λx m i=1 α i P i (1 -e -λx ) αi-1 , x > 0.
In particular, a pdf of

Y 1:n = inf(Y 1 , Y 2 , . . . , Y n ) is given by f 1:n (x) = λne -λx 1 - m i=1 1 P i (1 -e -λx ) αi n-1 m i=1 α i P i (1 -e -λx ) αi-1
and a pdf of

Y n:n = sup(Y 1 , Y 2 , . . . , Y n ) is given by f n:n (x) = λne -λx m i=1 1 P i (1 -e -λx ) αi n-1 m i=1 α i P i (1 -e -λx ) αi-1 .
Consequently, the cdf of the i-th order statistic Y i:n is given by

F i:n (x) = x -∞ f i:n (t)dt = n! (i -1)! (n -i)! n-i k=0 n -i k (-1) k i + k [F Y (x)] i+k = n! (i -1)! (n -i)! n-i k=0 n -i k (-1) k i + k m i=1 1 P i (1 -e -λx ) αi i+k , x > 0.
Further, for j < k, a joint pdf of (Y j:n , Y k:n ) is given by

f (j:n,k:n) (x j , x k ) = n! (j -1)! (n -k)! (k -j -1) [F Y (x j )] j-1 [F Y (x k ) -F Y (x j )] k-j-1 [1 -F Y (x k )] n-k f Y (x j )f Y (x k ) = n! (j -1)! (n -k)! (k -j -1) m i=1 1 P i (1 -e -λxj ) αi j-1 m i=1 1 P i [(1 -e -λx k ) αi -(1 -e -λxj ) αi ] k-j-1 × 1 - m i=1 1 P i (1 -e -λx k ) αi n-k λ 2 e -λ(xj +x k ) m i=1 α i P i (1 -e -λxj ) αi-1 m i=1 α i P i (1 -e -λx k ) αi-1 , 0 < x j < x k .

Maximum likelihood estimation

Let Y 1 , Y 2 , . . . , Y n be a random sample from the EH(a, λ) distribution with unknown parameters α 1 , . . . , α m , λ. We consider the maximum likelihood estimation providing the maximum likelihood estimators (MLEs) α1 , . . . , αm , λ for α 1 , . . . , α m , λ. Let us recall that the MLEs have some statistical desirable properties (under regularity conditions) as the sufficiency, invariance, consistency, efficiency and asymptotic normality. Using the observed information matrix, asymptotic confidence interval for α 1 , . . . , α m , λ can be constructed. Details can be found in [START_REF] Larsen | An Introduction to Mathematical Statistics and Its Applications[END_REF].

Let Θ = (α 1 , . . . , α m , λ) and y 1 , y 2 , . . . , y n be the observed values. The likelihood function is given by

L(Θ) = λ n e -λ n u=1 yu n u=1 m i=1 α i P i (1 -e -λyu ) αi-1 .
The log-likelihood function is given by

(Θ) = log(L(Θ)) = n log(λ) -λ n u=1 y u + n u=1 log m i=1 α i P i (1 -e -λyu ) αi-1 .
The nonlinear log-likelihood equations given by ∂ (Θ) ∂Θ = 0 are listed below

∂ (Θ) ∂λ = n λ - n u=1 y u + n u=1 y u e -λyu m i=1 αi(αi-1) Pi (1 -e -λyu ) αi-2 m i=1 αi Pi (1 -e -λyu ) αi-1 = 0 (5)
and, for any q ∈ {1, . . . , m},

∂ (Θ) ∂α q = n u=1 ∂ ∂αq m i=1 αi Pi (1 -e -λyu ) αi-1 m i=1 αi Pi (1 -e -λyu ) αi-1 = 0. (6)
Let us now investigate the numerator. We have

m i=1 α i P i (1 -e -λyu ) αi-1 = α q P q (1 -e -λyu ) αq-1 + m i=1 i =q α i P i (1 -e -λyu ) αi-1 . Hence ∂ ∂α q m i=1 α i P i (1 -e -λyu ) αi-1 = ∂ ∂α q α q P q (1 -e -λyu ) αq-1 + α q P q (1 -e -λyu ) αq-1 log(1 -e -λyu ) + m i=1 i =q α i (1 -e -λyu ) αi-1 ∂ ∂α q 1 P i .
Observe that

∂ ∂α q α q P q = ∂ ∂α q   αq m j=1 j =q α j α j -α q    = m j=1 j =q α j α j -α q + α q    m j=1 j =q α j α j -α q       m v=1 v =q 1 α v -α q    .
On the other hand, for i = q, we have

∂ ∂α q 1 P i = ∂ ∂α q    m j=1 j =i α j α j -α i    =     m j=1 j ={i,q} α j α j -α i     ∂ ∂α q α q α q -α i = -     m j=1 j ={i,q} α j α j -α i     α i (α q -α i ) 2 .
Putting these equalities together, we obtain an unified equation for (6). The MLEs are solutions of ( 5) and ( 6). These equations are not solvable analytically, but some numerical iterative methods, as Newton-Raphson method, can be used. The solutions can be approximate numerically by using software such as MATHEMATICA, MAPLE and R. Here we work with MATHEMATICA.

Study of a particular case. Let us now consider a simple two parameters EH(a, λ) distribution where a = (α 1 , . . . , α m ), for any i ∈ {1, . . . , m}, α i = α + i , with 1 , . . . , m denote different fixed positive real numbers. Then α > 0 and λ > 0 can be estimated via the maximum likelihood method. It is enough to set Θ = (α, λ), using (5) with α i = α + i and consider the new equation:

∂ (Θ) ∂α = n u=1 ∂ ∂α m i=1 α+ i Pi (1 -e -λyu ) α+ i-1 m i=1 α+ i Pi (1 -e -λyu ) α+ i-1 = 0. ( 7
)
We can explicit the numerator by observing that

∂ ∂α m i=1 α + i P i (1 -e -λyu ) α+ i-1 = m i=1 ∂ ∂α α + i P i (1 -e -λyu ) α+ i-1 + m i=1 α + i P i (1 -e -λyu ) α+ i-1 ln(1 -e -λyu ), with ∂ ∂α α + i P i =    m j=1 j =i 1 j -i    ∂ ∂α   m j=1 (α + j )   =    m j=1 j =i 1 j -i      m j=1 (α + j )     m j=1 1 α + j   .
Again, these equations can be solve numerically. Thanks to its simplicity, this particular EH(a, λ) distribution will be considered in the applications presented in the next section.

Illustrative real data examples

In this section, we analysis real data sets to show that the EH(a, λ) distribution can be a better model than other existing distributions. We consider the following well-known weighted exponential distributions for comparison purpose: Lindley distribution: A pdf associated to the Lindley distribution of parameter θ > 0 is given by

f (x) = θ 2 θ + 1 (1 + x)e -θx , x > 0.
Theory and applications related to this distribution can be found in [START_REF] Ghitany | Lindley distribution and its applications[END_REF].

Exponential distribution: A pdf associated to the exponential distribution of parameter λ > 0 is given by f (x) = λe -λx , x > 0.

Exponentiated exponential distribution (E Exponential): A pdf associated to the Exponentiated exponential distribution of parameters α, λ > 0 is given by

f (x) = λαe -λx 1 -e -λx α-1 , x > 0.
Details about this distribution can be found in [START_REF] Gupta | Generalized Exponential Distributions[END_REF].

Exponential Hypoexponential distribution (Exp Hypo): To simplify the situation, we consider a particular simple two parameters EH(a, λ) distribution with m = 2, a = (α 1 , α 2 ), α 1 = α and α 2 = α + 0.1, α > 0 and λ > 0 are the parameters to be estimated; the value 0.1 is subjective. The corresponding pdf is given by

f (x) = λα(1 + 10α)e -λx (1 -e -λx ) α-1 [1 -(1 -e -λx ) 0.1 ],
x > 0.

Remark 9. Another two parameters EH(a, λ) distribution can be used, as the one defined with m = 3, say a = (α 1 , α 2 , α 3 ), α 1 = α, α 2 = α + and α 3 = α + υ, for fixed and υ with = υ, the parameters to be estimated are α > 0 and λ > 0.

The three considered real data sets are described as follows:

Data set 1: This data set contains n = 31 measures of the strength data of glass of the aircraft window. It is extracted from Fuller et al. (1994): 18.83, 20.8, 21.657, 23.03, 23.23, 24.05, 24.321, 25.5, 25.52, 25.8, 26.69, 26.77, 26.78, 27.05, 27.67, 29.9, 31.11, 33.2, 33.73, 33.76, 33.89, 34.76, 35.75, 35.91, 36.98, 37.08, 37.09, 39.58, 44.045, 45.29 Data set 2: The data set contains n = 63 measures related to the strength of 1.5cm glass fibers. It is reported in Smith and Naylor (1987): 0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.61, 1.64, 1.68, 1.73, 1.81, 2, 0.74, 1.04, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76, 1.82, 2.01, 0.77, 1.11, 1.28, 1.42, 1.5, 1.54, 1.6, 1.62, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.48, 1.5, 1.55, 1.61, 1.62, 1.66, 1.7, 1.77, 1.84, 0.84, 1.24, 1.3, 1.48, 1.51, 1.55, 1.61, 1.63, 1.67, 1.7, 1.78, 1.89 Data set 3: The data set contains n = 128 measures on the remission times in months of bladder cancer patients. It is extracted from Lee and Wang (2003): 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69, 5.49 For each data set, we compare the fitted distributions using the four criteria: -2 log(L), AIC (Akaike Information Criterion), AICC (Akaike Information Criterion Corrected) and BIC (Bayesian Information Criterion). Let us precise that log(L) is the log-likelihood taking with the estimate values, AIC=-2 log(L) + 2k, AICC=AIC+ 2k(k+1) n-k-1 and BIC=-2 log(L) + k log(n), where k denotes the number of estimated parameters and n denotes the sample size. The best fitted distribution corresponds to lower -2 log(L), AIC, AICC and BIC.

We see in Table 1 that the EH(a, λ) distribution has the smallest -2 log(L), AIC, AICC and BIC for the three data sets, indicating that it is a serious competitor to the other considered distributions.

Conclusion

A new weighted exponential distribution based on the hypoexponential distribution is introduced. Some of its structural properties are studied. The analysis of three real data sets shows that the fit of the model related to our new distribution can be superior to other models.

As future work, we plan to study the GH(H; a) distribution with another cdf H(x) and also to provide more applications for the EH(a, λ) distribution with m ≥ 3. Inverted or transmuted transformations of EH(a, λ) can be of interest too for some applications.

  x→0 f (x) = +∞ in some configurations. We have lim x→+∞ f (x) = 0. The limit for h(x) when x → +∞ depends only on λ and P 1 , P 2 , . . . , P m . Remark 5. As already observed in Remark 3, the EH(a, λ) distribution belongs to the family of weighted exponential distributions; we can express f Y (x) as: f Y (x) = w(x)g(x), where g(x) denotes a pdf of the exponential distribution with parameter λ > 0 and w(x) = m i=1 αi Pi (1 -e -λx ) αi-1 .

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Some cdfs F (x) = F (x; (α, λ)) (2) with various values for α and λ.

Table 1 :

 1 -2 log(L), AIC, AICC and BIC of the fitted distributions of Data sets 1, 2 and 3.

	,