Tenth International Workshop onRetrial Queues

10th WRQ

July 24-26, 2014Tokyo Institute of Technology,

Ookayama Campus, West Bldg. 9,Tokyo, Japan

Workshop Schedule

Table of Contents

1. Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. Program Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3. Local Organizing Committee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4. Workshops on Retrial Queues: A Brief History . . . . . . . . . . . . . . . . . . 5

5. General Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

6. Program Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

7. Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

8. List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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Prologue

This meeting is the 10th in a series of workshops to promote research andencourage interaction in the community of retrial queues. The series of work-shops on retrial queues was initiated in 1998 in Madrid as a forum for thecommunication and cooperation among the people that work in this subareaof queueing. The workshops are open to Applied Probabilists, Operation Re-searchers, Engineers, Computer Scientists and Statisticians with a main orside interest in retrial queues. The scope of the conference includes theoret-ical papers with advances in mathematical techniques that can be useful foranalyzing retrial queueing models, as well as applications of retrial queueingsystems. More specifically, the topics of the 10th WRQ include analyticaltechniques, computational methods, optimization, control, statistical infer-ence and applications of retrial queues.

The 10th WRQ has gathered together about 25 participants from 10countries all over the world. We would like to express our deep and sinceregratitude to all the participants for their contributions.

Tuan Phung-Duc

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Program Committee

Chairman

Tuan Phung-Duc (Tokyo Institute of Technology, Japan)

Co-chairman

Antonio Gomez-Corral (Complutense University of Madrid, Spain)

Members

Ioannis Dimitriou (FORTH: Institute of Computer Science, Greece)Antonis Economou (University of Athens, Greece)Bara Kim (Korea University, Korea)A. Krishnamoorthy (Cochin University of Science and Technology, India)Rein D. Nobel (Vrije University of Amsterdam, The Netherlands)Yang Woo Shin (Changwon National University, Korea)Jinting Wang (Beijing Jiaotong University, China)

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Local Organizing Committee

Chairman

Tuan Phung-Duc (Tokyo Institute of Technology)

Co-chairman

Naoto Miyoshi (Tokyo Institute of Technology)

Members

Atsushi Inoie (Kanagawa Institute of Technology)Ken’ichi Kawanishi (Gunma University)Masahiro Kobayashi (Tokyo University of Science)Hiroyuki Masuyama (Kyoto University)Yutaka Sakuma (National Defense Academy)

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Workshops on Retrial Queues:A Brief History

The series of international workshops on Retrial Queues started in Madrid,September 22-24, 1998.

The 1st International Workshop on Retrial Queues (WRQ’98) gathered25 participants from 12 countries. The Chairman was Professor Jesus R.Artalejo. The Proceedings were published in a special issue of Top (Volume7, Number 2, 1999).

Professor Alexander N. Dudin chaired the 2nd International Workshopon Retrial Queues (WRQ’99), Minsk, June 22-24, 1999. The workshop wasconducted jointly with the 15th Belarusian Workshop on Queueing Theory.The Proceedings were published in the monograph ‘Queues, Flows, Systemsand Networks’, Belarus State University Publications.

The 3rd International Workshop on Retrial Queues (WRQ’00) was heldin Amsterdam, March 13-15, 2000, at the Tinbergen Institute. The ProgramChair was Professor Henk C. Tijms from Vrije University of Amsterdam.About 23 participants from both Western and Eastern European countries,USA, Canada, Israel, Sweden and Japan attended the meeting. The Euro-pean Commission gave support to the above conferences through the INTASproject 96-0828 entitled ‘Advances in Retrial Queueing Theory’.

The 4th International Workshop on Retrial Queues (WRQ’02) was heldin Cochin, December 17-21, 2002, at the Cochin University of Science andTechnology. The conference Chairman was Professor A. Krishnamoorthy.The Proceedings were published in the book ‘Advances in Stochastic Mod-elling’, Notable Publications, Inc., New Jersey.

Professor B.D. Choi chaired the 5th International Workshop on RetrialQueues (WRQ’04) at the Telecommunication Mathematics Research Cen-ter, Korea University. This meeting was combined with a Workshop on

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Performance Evaluation of Telecommunication Systems.

The 6th International Workshop on Retrial Queues (WRQ’06) was heldat ‘La Cristalera’, Miraflores de la Sierra, Madrid, July 8-10, 2006. TheProgram Chair was Professor A. Gomez-Corral. That edition gathered 23participants from 9 countries. A selection of original, high quality contribu-tions was published at the special issue ‘Advances in Retrial Queues’ of theEuropean Journal of Operational Research.

The 7th International Workshop on Retrial Queues (WRQ’08) was heldin Athens, July 17-19, 2008. The Conference Chairman was Professor A.Economou. 31 participants from 12 countries attended this workshop. Aselection of the best papers was published in the special issue ”Algorithmicand Computational Methods in Retrial Queues” of the journal Computers& Operations Research (Volume 37, Number 7, 2010).

Professor Q.L. Li was the Chairman of the 8th International Workshopon Retrial Queues (WRQ’10) which was held at the Tshinghua University,Beijing, July 27-29, 2010. This edition gathered together about 40 partici-pants from 12 countries. A selection of papers was published in OperationalResearch: An International Journal (Vol. 12, No. 2, 2012).

Professor P. Moreno was the Chairwoman of the 9th International Work-shop on Retrial Queues (WRQ’12) which was held at the Universidad Pablode Olavide, Seville, Spain, July 28-30, 2012. This edition gathered togetherabout 20 participants from 13 countries. A selection of papers was publishedsoon in Asia-Pacific Journal of Operational Research (Vol. 31, No. 2, 2014).

The 10th International Workshop on Retrial Queues is held on 24-26July 2014, in Tokyo Institute of Technology, Tokyo (Japan). The chairman isProf. T. Phung-Duc. This workshop is dedicated to the memory of ProfessorJ. R. Artalejo. A selection of high quality papers will be published in aSpecial Issue on Retrial Queues and Related Models in Annals of OperationsResearch.

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General Information

Dates

July 24 (Thursday) – July 26 (Saturday), 2014

Conference Venue

The meeting will take place at Collaboration Room in West Bldg. 9of Ookayama Campus, Tokyo Institute of Technology. The campus is a1-minute walk from Ookayama Station on the Tokyu-Meguro or Tokyu-Oimachi lines. The West Building is situated within 5 minutes walk fromthe main gate of the campus.

Website

http://www.is.titech.ac.jp/~tuan/10th_WRQ/index.html

Social Program

Conference Meal: Lunches and Dinners will be provided at a restaurantinside Tokyo Institute of Technology.

Excursion: The main event of the social program of the 10th WRQ willtake place the last day of the workshop (26 July 2014). More concretely,an excursion to Edo-Tokyo Museum is scheduled. We will take the bus at10:30 in the morning and we will be back at about 18:00 in the afternoon.A buffet-style lunch will take place during the excursion.

Proceedings

There will not be a proceedings volume for this workshop. However,the participants are encouraged to submit their work for possible publica-tion in the special issue ”Retrial Queues and Related Models” of Annals ofOperations Research. The papers should be submitted via the electronicsubmission system of this journal. For the deadline, the submission proce-dure and further details please see:http://rutcor.rutgers.edu/CfP%20retrial%20Queues.pdf

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Program Schedule

July 24, 2014

Keynote TalkSession 1: Advance in Retrial QueuesSession 2: Retrial Queues with Finite SourcesSession 3: Retrial Queues with Call Center Applications

09:00 – 10:00 Registration

10:00 – 10:15 Opening address

Keynote Talk

10:15 – 11:00 Title: Performance, availability & power analysis forInfrastructure-as-a-Service cloudAuthor: K. Trivedi (Duke University, USA), R. Ghosh (IBM, Durham,NC, USA) and F. Longo (University of Messina, Italy)

11:00 – 11:30 Coffee break

Session 1 (Chair: B. Kim)

11:30 – 12:00 Title: On a single server batch arrival retrial queueAuthors: B. Kim (Korea University, Korea) and J. Kim (ChungbukNational University, Korea)

12:00 – 12:30 Title: Performance analysis of a multiprogramming – multi-processor retrial queueing system with orderly reattemptsAuthors: B. Krishna Kumar (Anna University, India), A. Thanikacha-lam (Anna University, India) and V. Kanakasabapathi (Anna Univer-sity, India)

12:30 – 14:00 Lunch

Session 2 (Chair: R. Nobel)

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14:00 – 14:30 Title: On memoriam of Prof. Jesus R. ArtalejoAuthor: A. Gomez-Corral (ICMAT–Institute of Mathematical Sci-ences, Spain)

14:30 – 15:00 Title: On a queueing-inventory with reservation, cancellationand retrial: A one period modelAuthor: A. Krishnamoorthy (Cochin University of Science and Tech-nology, India)

15:00 – 15:30 Title: On two finite queueing systems: an M/G/1 queue withlosses and an M/G/1 queue with retrialsAuthor: V. Dragieva (Sofia University of Forestry, Bulgaria)

15:30 – 16:00 Coffee break

Session 3 (Chair: Krishnamoorthy)

16:00 – 16:30 Title: A functional approximation for retrial queues with twoway communicationAuthors: S. Ouazine (University of Bejaia, Algeria) and K. Abbas(University of Bejaia, Algeria)

16:30 – 17:00 Title: A mixed retrial/delay queueing model in discrete timewith high and low priority customers and a tolerant server for the highpriority customersAuthor: R.D. Nobel (Vrije University of Amsterdam, The Nether-lands)

17:00 – 17:30 Title: Infinite level-dependent QBD models for call centerswith multi-class customers under balking, impatience, and retrialsAuthors: T. Dayar (Bilkent University, Turkey) and M. Can Orhan(Bilkent University, Turkey)

July 25, 2014

Session 4: Retrial Queues Unreliable ServersSession 5: Statistics of Retrial QueuesSession 6: Methodological Advances in Retrial Queues

9:30 – 10:00 Coffee break

Session 4 (Chair: A. Gomez-Corral)

10:00 – 10:30 Title: Light tailed approximation in an unreliable retrialqueue with impatience and preventive maintenance

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Author: A. Aissani (University of Science & Technology USTHB, Al-geria)

10:30 – 11:00 Title: Probabilistic behaviour before absorption: Quasi-stationary and ratio of expectations distributionsAuthor: M.J. Lopez-Herrero (Complutense University of Madrid, Spain)

11:00 – 11:30 Title: M/PH/1 Retrial Queue with Postponed work underN -policyAuthors: A. Krishnamoorthy (Cochin University of Science and Tech-nology, India) and C.B. Ajayakumar (College of Engineering, Kidan-goor, India)

11:30 – 12:00 Title: A discrete-time unreliable retrial queue with BernoullivacationsAuthors: F. Zhang (Yanshan University, China), D. Yue (YanshanUniversity, China) and Y. Qin (Yanshan University, China)

12:00 – 14:00 Lunch

Session 5 (Chair: Y. W. Shin)

14:00 – 14:30 Title: Preventive maintenance in an unreliable M/G/1 retrialqueue with persistent and impatient customersAuthor: S. Taleb (University of Science and Technology USTHB, Al-gelia) and A. Aissani (University of Science and Technology USTHB,Algelia)

14:30 – 15:00 Title: Bayesian analysis of M/M/1 retrial queueAuthors: J. Wang (Beijing Jiaotong University, China) and Y. Zhang(Beijing Jiaotong University, China)

15:00 – 15:30 Title: The chaos of propagation in a retrial supermarketmodelAuthors: Q.-L. Li (Yanshan University, China), M. Wang (YanshanUniversity, China), J.C.S. Lui (The Chinese University of Hong Kong,Hong Kong) and Y. Wang (Peking University, China)

15:30 – 16:00 Coffee break

Session 6 (Chair: A. Aissani)

16:00 – 16:30 Title: Approximation of retrial queues with PH retrial timesand server vacationsAuthors: Y.W. Shin (Changwon National University, Korea) and D.H.Moon (Changwon National University, Korea)

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16:30 – 17:00 Title: Asymptotic analysis of single server retrial queues withoutgoing callsAuthors: H. Sakurai (Tokyo Institute of Technology, Japan) andT. Phung-Duc (Tokyo Institute of Technology, Japan)

17:00 – 17:15 Closing session

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Abstracts

In this section, abstracts are listed in alphabetical order of the presentingauthor.

Light tailed approximation in an unreliable retrial queue with im-patience and preventive maintenance

A. Aissani – University of Science & Technology USTHB , Algeria

Abstract.– We consider an M/G/1 retrial queue with unreliable serverand two types of primary customers, persistent and impatient under linearretrial policy.

The server is subject to two types of interruptions. The first type cor-responds to failure breakdowns which can occur in passive of active server’sstate and are followed by corrective maintenance actions. The second onecorresponds to preventive maintenance actions. If a failure breakdown oc-curs when a service is in course, the interrupted customer joins the orbitand retry his call according to the retrial linear policy.

We show that the evolution of the system can be described by a linearMarkovian stochastic process in the sense of Gnedenko and Kovalenko. Thenwe obtain the stationary distribution of the joint distribution of the serverstate and the number of orbiting customers.

This distribution is obtained in terms of Generating functions and Laplace-Stieltjes transforms. If these expressions allow us to obtain the mean per-formance characteristics (mean waiting time,...), the distribution itself isdifficult to obtain except perhaps numerically.

In this talk, we consider the approximation of the queue size of abovementioned model under light tailed assumptions. The method will be il-lustrated on the standard Unreliable Retrial Queue studied by Aissani andArtalejo (1998).

Preventive maintenance in an unreliable M/G/1 retrial queue withpersistent and impatient customers

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S. Taleb – – University of Science & Technology USTHB , AlgeriaA. Aissani – University of Science & Technology USTHB , Algeria

Abstract.– In this paper we study a new version of an unreliable retrialqueue with persistent and impatient customers. The model considered takesinto account corrective and preventive maintenances. If a preventive actionoccurs in a busy period, then it is postponed to an ulterior date. We give thenecessary and sufficient condition for the system to be stable and obtain thejoint distribution of the server state and the number of orbiting customersin the system in terms of generating function. Some performance measuresare derived. From the reliability view point, we analyze the time to the firstfailure of the server.

Infinite level-dependent QBD models for call centers with multi-class customers under balking, impatience, and retrials

T. Dayar – Bilkent University, TurkeyM. Can Orhan – Bilkent University, Turkey

Abstract.–With the help of Kronecker products [4], infinite level-dependentquasi-birth-and-death (LDQBD) processes [3, 5] can be used to model Marko-vian systems with countably infinite multi-dimensional state spaces. Re-cently, infinite LDQBDs have been used to model call center systems inwhich there are multiple classes of customers [2]. In this work, we extendthe infinite LDQBD models using Kronecker products for these call centersto systems, where customers exhibit balking, impatience, and retrials [1].REFERENCES:[1] Artalejo, J.R., Pla, V.: On the impact of customer balking, impatienceand retrials in telecommunication systems. Computers & Mathematics withApplications. 57, 217–229 (2009)[2] Baumann, H., Dayar, T., Orhan, M.C., Sandmann, W.: On the numericalsolution of Kronecker-based infinite level-dependent QBD processes. Perfor-mance Evaluation. 70, 663–681 (2013)[3] Bright, L., Taylor, P.G.: Calculating the equilibrium distribution in leveldependent quasi-birth-and-death processes. Stochastic Models. 11, 497–525(1995)[4] Dayar, T., Orhan, M.C.: Kronecker-based infinite level-dependent QBDs.Journal of Applied Probability. 49, 1166–1187 (2012)[5] Ramaswami, V., Taylor, P.G.: Some properties of the rate operators inlevel dependent quasi-birth-and-death processes with a countable number ofphases. Stochastic Models 12, 143–164 (1996)

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On two finite queueing systems: an M/G/1 queue with losses andan M/G/1 queue with retrials

V. Dragieva – Sofia University of Forestry, Bulgaria

Abstract.– One of the system under consideration is a finite-source M/G/1queue in which the unsuccessful customers are not allowed to repeat theirattempts for service. Instead, for an exponentially distributed time intervalthey are blocked in the orbit of inactive customers. This system can beconsidered as an extended variant of the Engset loss model. We carry outan analysis of the system and on the basis of numerical results compare itsmain characteristics with those in the corresponding finite M/G/1 queuewith retrials. We further discuss some problems for optimization the retrialmodel performance via temporarily forbidding repeated attempts.

On a single server batch arrival retrial queue

B. Kim – Korea University, KoreaJ. Kim – Chungbuk National University, Korea

Abstract.–We consider a single server batch arrivalMX/G/1 retrial queue.The single server batch arrival retrial queues are characterized by the fol-lowing features: If the server is busy at the arrival epoch, then the wholebatch joins the retrial group, whereas if the server is free, then one of thearriving customers starts its service while the other customers (if any) jointhe retrial group.

First, we study the queue length (i.e., the number of customers in theorbit) distribution. We find a necessary and sufficient condition for the ex-istence of the moments of the queue length distribution. This condition isexpressed in terms of the moment condition for the batch size and servicetime distributions. We provide recursive formulas for the higher order mo-ments of the queue length distribution. As a corollary, we obtain an explicitexpression for the variance of the queue size distribution, in addition to thewell-known expression for the first moment.

Next, we study the waiting time distribution. To do this we derive anequation for the joint transform of the queue length and waiting time distri-butions. Then we find a necessary and sufficient condition for the existenceof the moments of the waiting time distribution. The moments of the wait-ing time distribution are expressed in terms of the solution of the system oflinear equations. We obtain an explicit expression for the mean and varianceof the waiting time distribution, in addition to the well-known expressionfor the first moment.

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Performance analysis of a multiprogramming – multiprocessor re-trial queueing system with orderly reattempts

B. Krishna Kumar – Anna University, IndiaA. Thanikachalam – Anna University, IndiaV. Kanakasabapathi – Anna University, India

Abstract.– In this paper, we analyze a multiprocessor network retrialqueueing system with constant retrial rate consisting of identical multi-server central processor units (CPU) and identical multi-server input-output(I/O) devices. It is assumed that the system operates under light-trafficcondition. The arriving customers (programs) belong either to orbit whichmay be empty or to inner multiprocessor network consisting of identicalmulti-server CPU queue and identical multi-server I/O queue for requestingCPU operations and I/O operations respectively if they are allowed accessto the resources of the system. The maximum number of customers calledmulti-programs in the multiprocessor network is assumed to be finite. UnderMarkovian assumptions, the system is investigated in the steady-state situ-ation using matrix geometric method. The steady-state probabilities of thenumber of customers in the system, expressions for the Laplace transformof the waiting time as well as its mean, the mean number of retrials madeby a tagged retrial customer in the stationary regime are obtained alongwith other interesting and important performance measures of the system.Finally, extensive numerical results are presented to reveal the impact of theparameters on the performance of the system.

On a queueing inventory with reservation, cancellation and retrial:A one period model

A. Krishnamoorthy – Cochin University of Science and Technology, India

Abstract.– We consider the following problem in queueing-inventory (i.e.,inventory with positive service time). There are S items in the inventorywhich have a common lifetime, exponentially distributed with parameter α.Customers arrive according to a Poisson process of rate λ demanding oneitem each. They are served by a single server according to an exponentialdistribution with parameter µ. Cancellation of item sold (i.e., returning theitem) before its expiry is permitted. This takes place at rate βn when thereare n inventory sold. Unsatisfied customers first queue up in a finite waitingspace of capacity K. When it overflows an arrival goes to an orbit of infinitecapacity with probability p and is lost forever with probability 1− p. Once

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all items are sold, through cancellation of purchases, inventory gets addeduntil the expiry time.

We collect together the assumptions/notations.(i) Common life time of inventory ∼ exp(α)(ii) Demand Arrival :- Poisson process of rate λ(iii) Service time ∼ exp(µ)(iv) Maximum inventory available :- S(v) Maximum Buffer size(varies with number of items in inventory):- S(vi) Additional waiting room size:- K(vii) Inventory return/cancellation of purchase ∼ exp(βn), when n items aresold.(viii) Cancellation time assumed to be negligible.

The aim of this analysis is to compute the optimal value of S underthe assumption that any item not sold involves heavy cost to the system.Further, customers in the additional waiting space, if forced to leave withoutgetting inventory, then a penalty is imposed on the system. The optimalvalue of K is also investigated.

M/PH/1 retrial queue with postponed work under N-policy

A. Krishnamoorthy – Cochin University, IndiaC.B. Ajayakumar – College of Engineering, Kidangoor, India

Abstract.– In this paper we consider a continuous time M/PH/1 retrialqueue with postponed work under N -policy. There is a finite buffer ofcapacity K1 and a finite pool of postponed work of capacity K2 . If thebuffer contains less than K1 customers including the one at server, newlyarriving customers will join it. When the buffer is full with K1 customers,newly arriving jobs are offered the choice of leaving the system immediatelywith probability 1 − γ or of being sent to a pool of postponed work withprobability γ (0 < γ < 1).

If the pool is also full, the customer will join an orbit of infinite capacity.From the orbit, the head of the queue retries according to an exponentialdistribution with parameter θ. If the buffer is full at the instant of retrial,the customer may leave the system for ever with probability 1− δ or sent tothe pool with probability δ (0 < δ < 1). If the pool is also full, the customerwill again join the orbit.

When at the end of a service, if there are postponed customers, thesystem operates as follows. If the buffer is empty, the one ahead of allwaiting in the pool gets transferred to the buffer for immediate service. Ifthe buffer contains y jobs, where 1 ≤ y ≤ L− 1, 2 ≤ L ≤ K − 1 at a servicecompletion epoch, then again the job at the head of the buffer starts service

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and with probability p, the head of the queue in pool is transferred (we callthis a p-transfer) to the finite buffer and positioned as the last among thewaiting customers in the buffer. With probability q = 1−p, no such transfertakes place. No such transfer takes place at a service completion epoch , ifthere is at least L customers in the buffer.

If the pool contains at least one postponed job, the continuously servedcustomers from the buffer is counted, at each service completion epoch.When it reaches N , (N > 0) then the one ahead of all waiting in the poolgets transferred to the buffer for immediate service. At this time, systemdoes not consider the p-transfer. The N -policy introduced here differs fromthe classical N -policy. Whereas in the classical case, N customers are toqueue up to start the new service cycle once the system becomes empty.However in the present case N -policy is applied to determine a priorityservice to be given to a customer from the pool. We study it as a quasi birth-death process and a solution of classical matrix geometric type is obtained.We study its long run behaviour. Several system performance measures areprovided.

The chaos of propagation in a retrial supermarket model

Q.-L. Li – Yanshan University, ChinaM. Wang – Yanshan University, ChinaJ.C.S. Lui – The Chinese University of Hong Kong, Hong KongY. Wang – Peking University, China

Abstract.– When decomposing the total orbit into N sub-orbits (or simplyorbits) related to each of N servers and through comparing the numbers ofcustomers in these orbits, we introduce a retrial supermarket model of Nidentical servers, where two probing-server choice numbers are respectivelydesigned for dynamically allocating each primary arrival and each retrialarrival into these orbits when the chosen servers are all busy. Note thatthe designed purpose of the two choice numbers can effectively improveperformance measures of this retrial supermarket model.

This paper analyzes a simple and basic retrial supermarket model of Nidentical servers, that is, Poisson arrivals, exponential service and retrialtimes. To this end, we first provide a detailed probability computation toset up an infinite-dimensional system of differential equations (or mean-fieldequations) satisfied by the expected fraction vector. Then, as N → ∞, weapply the operator semigroup to obtaining the mean-field limit (or chaos ofpropagation) for the sequence of Markov processes which express the stateof this retrial supermarket model. Specifically, some simple and basic con-ditions for the mean-field limit as well as for the Lipschitz condition areestablished through the first two moments of the queue length in any orbit.

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Finally, we show that the fixed point satisfies a system of nonlinear equationswhich is an interesting networking generalization of the tail equations givenin theM/M/1 retrial queue, and also use the fixed point to give performanceanalysis of this retrial supermarket model through numerical computation.Noting that there are few available works on the analysis of retrial queueingnetworks in the current literature, we believe the mean-field method given inthis paper can open a new avenue in the future study of retrial supermarketmodels, and more generally, of retrial queueing networks.

Probabilistic behaviour before absorption: Quasi-stationary andratio of expectations distributions

M.J. Lopez-Herrero – Complutense University of Madrid, Spain

Abstract.– In the mathematical modeling of dynamic situations wherethere is a subset which is absorbing a Markov chain with absorbing statesis involved. This set can be viewed as a hole, a trap, when the absorption iscertain. For studying the situation before the absorption then it is needed toseek for analogs of the stationary distribution of an irreducible chain. Theobjectives of this talk is firstly to describe the quasi-stationary and ratioof expectations distributions as two different approaches for understandingthe long time behaviour of the process conditioned that absorption has notoccurred yet. For many applications, when it takes a very long time forthe absorption, the quasi-stationary distribution gives an excellent measureof the long term behaviour of the system, but due to nonlinear structureof the quasi-stationary equations it is usually impossible to obtain explicitexpressions for the quasi-stationary distribution. The ratio of expectationsdistribution gives an alternative, despite of how long the absorption time isand it can be evaluated more simply since the distribution is governed by aset of linear equations. A second objective is to investigate the possibilityof using the ratio of expectations distribution as an approximation to thequasi-stationary distribution. Some applications to stochastic populationmodels illustrate the interest of both distributions.

A mixed retrial/delay queueing model in discrete time with highand low priority customers and a tolerant server for the high pri-ority customers

R.D. Nobel – Vrije University Amsterdam, Netherlands

Abstract.– We consider a one-server queueing model in discrete time withtwo types of customers. So time is divided in time slots, and all events [ar-

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rivals, start of a service and departures] are considered to occur at the slotboundaries. The type 1 [low-priority] customers arrive in batches followinga general probability distribution. Upon arrival a batch of low-priority cus-tomers is put in a waiting line from which the customers are served one byone on a first come first served basis [when no customers from previouslyarrived batches are present the first customer of the newly arrived batchstarts service the next time slot at the earliest]. The type 2 [high-prioritycustomers] also arrive in batches, possibly following a different probabilitydistribution, and when upon arrival a batch of high-priority customers seesthe server busy, all incoming high-priority customers are sent into orbit, avirtual waiting space from which they will try to reenter the system individ-ually some random time later. The service times of the low-priority and thehigh-priority customers are all independent and follow [possibly] a differentgeneral distribution. The modeling assumption is made of a late arrival setup with delayed access, i.e. arrivals have precedence over departures and aservice of newly arrived customers can only start at the time slot followingthe slot of the arrival at the earliest. To give priority to the high-prioritycustomers the time slot after a departure epoch the server always stays idle,also when low-priority customers are present waiting in line. A low-prioritycustomer will start service only when no high-priority customers will havearrived during this idle slot after a departure, either from a newly arrivedbatch or from the orbit. The server stays idle when neither high-prioritycustomers will have arrived nor low-priority customers are present at thebeginning of the next time slot. When high-priority customers have arrivedthe server accepts all these customers [a mixed batch of newly arrived cus-tomers and customers from the orbit] and starts serving them one by oneuninterruptedly, i.e. the server is tolerant for the high-priority customers,but not for the low-priority customers. The server becomes idle again onlyafter the service of this mixed batch of high-priority customers will havebeen completed. The low-priority customers are also served one by one, butafter each departure of a low-priority customer the server stays idle again,to enable the uninterrupted service of a next mixed batch of high-prioritycustomers,

The generating function of the joint equilibrium distribution of the num-ber of high priority customers in orbit, the number of low priority customerswaiting in queue to be served and the residual service time of the (batchof high priority) customer(s) in service is calculated. From the generatingfunction several performance measures are deduced, like the average num-ber of high-priority customers in orbit, the average number of low-prioritycustomers waiting in line, among others.

A functional approximation for retrial queues with two way com-munication

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S. Ouazine – University of Bejaia, AlgeriaK. Abbas – University of Bejaia, Algeria

Abstract.– Queueing systems with repeated attempts have been widelyused to model various real situations in computer and communication sys-tems. The stochastic models underlying these systems are generally veryhard to analyze by analytical approaches. Consequently, it is important todevelop numerical methods for computing performance measures for suchsystems. In this paper we use two different methods to the numerical ap-proximation of the M1,M2/G1, G2/1 queue with two way communication.Specifically, we will develop a functional approximation of the measures per-formance of this queue where the parameter of interest is the rate of outgoingcall. The sensitivity analysis is also performed to explore the effect of theoutgoing call on the system performance. Therefore, we use the strong sta-bility method to obtain a perturbation bounds for the infinite capacity ofthe considered queueing model. Besides, we use the Taylor series expansionmethod to study the finite version of the same model. These methods avoidthe use of Laplace transforms and/or numerical inversion techniques, whichare predominantly used in the literature. Several numerical examples areconsidered to illustrate the performance of the both approaches.

Asymptotic analysis of single server retrial queues with outgoingcalls

H. Sakurai – Tokyo Institute of Technology, JapanT. Phung-Duc – Tokyo Institute of Technology, Japan

Abstract.– This paper studies the asymptotic behaviour of the number ofcustomers in an M/G/1-type retrial queue in which there are two flows ofarrivals namely ingoing calls made by regular customers and outgoing callsmade by the server when it is idle. The stationary analysis of this systemhas been carried out in a recent paper by Artalejo and Phung-Duc. In thispaper, we obtain the asymptotic distribution of the number of customersin the system under the following conditions: i) heavy traffic, ii) slow re-trials and iii) fast connect to outgoing calls. Furthermore, we also presenta decomposition property where we prove that the number of customers inthe system is decomposed into the sum of three random variables with clearphysical meaning.

Approximation of retrial queues with PH retrial times and servervacations

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Y.W. Shin – Changwon National University, KoreaD.H. Moon – Changwon National University, Korea

Abstract.– Retrial queues and vacation queues have been studied sepa-rately for last several decades. However, there are many practical situationswith two features retrials and vacations e.g. call centers. Recently, theinterests for retrial queues with vacations are growing rapidly. However, al-most all the literature deals with the system with single-server and constantretrial policy that only one customer in orbit can retry and there are fewliterature about multi-server retrial queue with vacations that the customersin orbit repeat their attempts independently.

We consider the M/M/c retrial queues that consists of an orbit of infi-nite size. When an arriving customer finds that the service facility is full,the customer joins orbit and repeats its request after a random amount oftime whose distribution is of phase type (PH) until the customer gets intothe service facility. The customers in orbit retry independently. A part ofservers takes vacation. If any a (a < c) or more servers are idle at a servicecompletion, then b (b ≤ a) servers among idle servers take a vacation andthe remaining c − b servers are available. The vacation time is assumed tohave a phase type distribution. In this paper, an approximation method ofthe distribution of the number of customers in service facility and the meannumber of customers in orbit is presented.

Performance, availability & power analysis for Infrastructure-as-a-Service cloud

K. Trivedi – Duke University, USAR. Ghosh – IBM, Durham, NC, USAF. Longo – University of Messina, Italy

Abstract.– Handling diverse client demands and managing unexpected fail-ures without degrading performance are two key promises of a cloud deliv-ered service. However, evaluation of cloud service quality becomes difficultas the scale and complexity of cloud system increases. In a cloud envi-ronment, a service request from a user goes through a variety of providerspecific processing steps from the instant it is submitted until the serviceis fully delivered. Measurement-based evaluation is expensive especially ifmany configurations, workload scenarios, and management methods are tobe analyzed. To overcome these difficulties, in this talk we propose a generalanalytic model based approach for the performance, availability and powerconsumption analysis of a cloud service. We illustrate our approach usingInfrastructure-as-a-Service (IaaS) cloud, where service availability, power

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consumption and provisioning delays are key QoS metrics. A novelty of ourapproach is in reducing the complexity of analysis by dividing the overallmodel into multiple interacting stochastic process models and then obtain-ing the overall solution by iteration over individual sub-model solutions. Incontrast to a single one-level monolithic model, our approach yields a highfidelity model that is tractable and scalable. Our approach and underlyingmodels can be readily extended to other types of cloud services and are ap-plicable to public, private and hybrid clouds.

Bayesian analysis of M/M/1 retrial queue

J. Wang – Beijing Jiaotong University, ChinaY. Zhang – Beijing Jiaotong University, China

Abstract.– In this paper, we consider the M/M/1 retrial queueing systemfrom a Bayesian inference perspective. Cases that the parameters of the sys-tem are independent or dependent are considered. Monte Carlo and Markovchain Monte Carlo methods are used to estimate the performance measuresof interests under the assumption that the queue is in equilibrium.

A discrete-time unreliable retrial queue with Bernoulli vacations

F. Zhang – Yanshan University, ChinaD. Yue – Yanshan University, ChinaY. Qin – Yanshan University, China

Abstract.– This paper is concerned with a discrete-time unreliableGeo/G/1retrial queueing system with Bernoulli vacations. If the orbit is empty, theserver takes a vacation randomly after each service completion. It is assumedthat the server is subject to breakdowns while serving the customers. If theserver breaks down, it is sent immediately for repair. For this model, the au-thors focus on the steady-state analysis for the considered queueing system.Firstly, the generating functions of the number of customers in the orbit andin the system are obtained. Then, the authors obtain the closed-form expres-sions of some performance measures of the system and also give a stochasticdecomposition result for the system size. Besides, the relationship betweenthis discrete-time model and the corresponding continuous-time model isalso investigated. Finally, some numerical results are provided to illustratethe effect of vacations and breakdowns on several performance measures ofthe system.

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List of Participants

Karim AbbasUniversity of BejaiaLAMOS – Operational Research DepartmentLAMOS, University of Bejaia, Campus of Targua Ouzemour, Bejaia, AlgeriaE-mail: karabbas2003@yahoo.fr

Amar AissaniUniversity of Science & Technology USTHBBP 32, El Alia, Bab-Ez-Zouar, Algiers, AlgeriaE-mail: aaissani@usthb.dz

C.B. AjayakumarKerala State Co-operative Academy of Professional EducationDepartment of MathematicsCollege of EngineeringKottayam-686583, Kerala, IndiaE-mail: ajayakumarcb@gmail.com

Velika DragievaSofia University of Forestry10, Kliment Ohridski Blvd., Sofia, BulgariaE-mail: dragievav@yahoo.com

Antonio Gomez-CorralICMAT – Institute of Mathematical SciencesCalle Nicolas Cabrera 13-15Madrid 28049, SpainE-mail: antonio.gomez@icmat.es

Atsushi InoieKanagawa Institute of TechnologyDepartment of Information Network and Communication

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Atsugi-city, Kanagawa 243-0292, JapanE-mail: inoie@nw.kanagawa-it.ac.jp

Ken’ichi KawanishiGunma UniversityDepartment of Computer ScienceKiryu-City, 376-8515, JapanE-mail: kawanisi@cs.gunma-u.ac.jp

Bara KimKorea UniversityDepartment of Mathematics145, Anam-ro, Seongbuk-gu, Seoul, 136-701, KoreaE-mail: bara@korea.ac.kr

Jeongsim KimChungbuk National UniversityDepartment of Mathematics Education52 Naesudong-ro, Heungdeok-gu, Cheongju, Chungbuk, 361-763, KoreaE-mail: jeongsimkim@chungbuk.ac.kr

Masahiro KobayashiTokyo University of ScienceDepartment of Information SciencesNoda-City, 278-8510, JapanE-mail: m kobayashi@is.noda.tus.ac.jp

B. Krishna KumarAnna UniversityDepartment of MathematicsCollege of Engineering600 025 Chennai, IndiaE-mail: drbkkumar@hotmail.com

Achyutha KrishnamoorthyCochin University of Science and TechnologyDepartment of MathematicsCochin-682022, Kerala, IndiaE-mail: achyuthacusat@gmail.com

Quan-Lin LiYanshan UniversitySchool of Economics and Management Sciences

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Qinhuangdao 066004, ChinaE-mail: liquanlin@mail.tsinghua.edu.cn

Mariajesus Lopez-HerreroComplutense University of MadridFaculty of Statistical Studies28045 Madrid, SpainE-mail: Lherrero@ucm.es

Hiroyuki MasuyamaKyoto UniversityDepartment of Systems ScienceGraduate School of InformaticsKyoto 606-8501, JapanE-mail: masuyama@sys.i.kyoto-u.ac.jp

Naoto MiyoshiTokyo Institute of TechnologyDepartment of Mathematical and Computing Sciences2-12-1-W8-52 Ookayama, Tokyo 152-8552, JapanE-mail: miyoshi@is.titech.ac.jp

Rein NobelVrije University AmsterdamDepartment of EconometricsDe Boelelaan 1105, 1081 HV Amsterdam, The NetherlandsE-mail: r.d.nobel@vu.nl

M. Can OrhanBilkent UniversityDepartment of Computer EngineeringTR-06800 Bilkent, Ankara, TurkeyE-mail: morhan@cs.bilkent.edu.tr

Tuan Phung-DucTokyo Institute of TechnologyDepartment of Mathematical and Computing Sciences2-12-1-W8-40 Ookayama, Tokyo 152-8552, JapanE-mail: tuan@is.titech.ac.jp

Yaling QinYanshan UniversityCollege of Science438, Hebei Avenue, Qinhuangdao City, 066004, Hebei Province, P.R. China

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E-mail: 16963906@qq.com

Yutaka SakumaNational Defense AcademyDepartment of Computer ScienceYokosuka-City, Kanagawa 239-8686, JapanE-mail: sakuma@nda.ac.jp

Yang Woo ShinChangwon National UniversityDepartment of StatisticsChangwondaehag-ro 20, Changwon, Gyeongnam, KoreaE-mail: ywshin@changon.ac.kr

Kishor TrivediDuke UniversityDepartment of Electrical and Computer Engineering202 Hudson Hall, Elect. & Comp. Eng. Dept., PO Box 90291, Durham,NC, USAE-mail: ktrivedi@duke.edu

Yu ZhangBeijing Jiaotong UniversityDepartment of Mathematics#3 Shang Yuan Cun, Hai Dian District, Beijing, P.R. ChinaE-mail: 13118393@bjtu.edu.cn