pith. sign in

arxiv: 1906.09560 · v1 · pith:3MQNEVZEnew · submitted 2019-06-23 · 💻 cs.PF · math.PR

Retrial Queueing Models: A Survey on Theory and Applications

Tuan Phung-Duc This is my paper

Pith reviewed 2026-05-25 18:08 UTC · model grok-4.3

classification 💻 cs.PF math.PR
keywords retrial queuesqueueing theorycall centerscellular networksrandom access protocolsstability analysisasymptotic analysismultidimensional models
0
0 comments X

The pith

A survey compiles exact solutions, stability results and applications for retrial queues.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper surveys the theory and applications of retrial queueing models that arise when customers retry after finding a busy server. It examines exact solutions, stability conditions, asymptotic analyses, and multidimensional models in the theoretical research. The survey also covers models from applications in call centers, cellular networks, and random access protocols, and points to open problems. Readers care because these models help analyze and improve performance in communication and service systems where retries are common.

Core claim

This survey compiles the state of the art in retrial queueing theory, covering exact solutions for performance metrics, conditions for stability, asymptotic analyses for large-scale systems, and multidimensional models, while providing an overview of models from real-world applications in call centers, cellular networks and random access protocols, and discussing open problems and research directions.

What carries the argument

Retrial queue, a system where arriving customers who cannot be served immediately leave temporarily and return after a random time to retry.

If this is right

  • Exact solutions allow computation of performance measures such as waiting times in retrial systems.
  • Stability conditions determine when the system reaches a steady state.
  • Asymptotic analyses provide approximations for performance in large or overloaded systems.
  • Multidimensional models handle interactions among multiple customer classes or servers.
  • Application-specific models guide the design and dimensioning of real systems like networks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identified open problems could motivate new work on numerical methods when exact solutions are unavailable.
  • Retrial models from random access protocols may extend to emerging wireless or distributed computing settings.
  • The survey structure could serve as a template for reviewing related queueing variants such as those with vacations or priorities.

Load-bearing premise

The selection of papers in the survey represents the full current state of research on retrial queues without significant omissions.

What would settle it

Identification of an important theoretical result or application in retrial queues published before the survey that is missing from the review.

Figures

Figures reproduced from arXiv: 1906.09560 by Tuan Phung-Duc.

Figure 1
Figure 1. Figure 1: A general multiserver retrial queue. Classical retrial policy In a retrial queue with the classical retrial policy, each blocked customer stays in the orbit for an exponentially distributed time independently of other customers. As a result, the retrial rate is proportional to the number of customers in the orbit. The classical retrial policy naturally arises from applications such as call center and telep… view at source ↗
Figure 2
Figure 2. Figure 2: A retrial queueing model for call centers. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Two adjacent cells in cellular networks. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A model of cellular networks with overlapping cells. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
read the original abstract

Retrial phenomenon naturally arises in various systems such as call centers, cellular networks and random access protocols in local area networks. This paper gives a comprehensive survey on theory and applications of retrial queues in these systems. We investigate the state of the art of the theoretical researches including exact solutions, stability, asymptotic analyses and multidimensional models. We present an overview on retrial models arising from real world applications. Some open problems and promising research directions are also discussed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. This paper claims to provide a comprehensive survey on retrial queueing models in systems such as call centers, cellular networks, and random access protocols. It reviews the state of the art of theoretical research including exact solutions, stability, asymptotic analyses, and multidimensional models; presents an overview of retrial models from real-world applications; and discusses open problems and promising research directions.

Significance. If the literature coverage is accurate and representative without major omissions, the survey would be a useful reference consolidating key theoretical advances and applications in retrial queueing, helping researchers navigate the field and identify open problems.

major comments (1)
  1. [Abstract] Abstract: The central claim of a 'comprehensive survey' on the state of the art (including exact solutions, stability, asymptotic analyses, and multidimensional models) is not supported by any explicit literature search protocol, inclusion/exclusion criteria, or coverage metrics, leaving open the possibility of systematic omissions or selection bias in the reviewed works.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their feedback. We address the single major comment below regarding the abstract's claim and literature coverage.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim of a 'comprehensive survey' on the state of the art (including exact solutions, stability, asymptotic analyses, and multidimensional models) is not supported by any explicit literature search protocol, inclusion/exclusion criteria, or coverage metrics, leaving open the possibility of systematic omissions or selection bias in the reviewed works.

    Authors: We agree that the abstract does not specify a formal search protocol or quantitative coverage metrics, which is a valid observation. Surveys in queueing theory are conventionally narrative overviews drawing on the authors' expertise rather than PRISMA-style systematic reviews; however, to address the concern we will revise the abstract to replace 'comprehensive survey' with 'extensive survey of key results' and add a short paragraph in the introduction describing the scope (focus on models admitting exact solutions or stability/asymptotic analysis, drawn from prominent journals and conferences up to the submission date). This clarifies selection without introducing new material. revision: yes

Circularity Check

0 steps flagged

No derivation chain present; survey contains no predictions or fitted claims

full rationale

This is a literature survey paper whose abstract and structure consist solely of overviews of prior work on retrial queues, stability, asymptotics, and applications. No original equations, derivations, parameter fittings, or predictions are advanced. The patterns for circularity (self-definitional claims, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems, ansatz smuggling, or renaming known results) have no instances because there is no derivation chain at all. The comprehensiveness claim is a standard survey assertion and does not reduce to any self-referential construction within the paper's own content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

As a survey paper there are no free parameters, axioms, or invented entities introduced by the authors. The work relies entirely on summarizing existing literature in the field.

pith-pipeline@v0.9.0 · 5588 in / 960 out tokens · 36329 ms · 2026-05-25T18:08:00.949344+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

131 extracted references · 131 canonical work pages

  1. [1]

    Aguir, F

    S. Aguir, F. Karaesmen, O. Z. Aksin and F. Chauvet. The impact o f retrials on call center performance. OR Spectrum, 26 (2004), 353-376

    work page 2004

  2. [2]

    A. S. Alfa and W. Li. PCS networks with correlated arrival proces s and retrial phenomenon. IEEE Transactions on Wireless Communications , 1 (2002), 630–637

    work page 2002

  3. [3]

    Avrachenkov and E

    K. Avrachenkov and E. Morozov. Stability analysis of GI/GI/c/K retrial queue with constant retrial rate. Mathematical Methods of Operations Research , 79 (2014), 273-291

    work page 2014

  4. [4]

    Avrachenkov, E

    K. Avrachenkov, E. Morozov and B. Steyaert. Sufficient stabilit y conditions for multi-class constant retrial rate systems. Queueing Systems , 82 (2016), 149-171

    work page 2016

  5. [5]

    Amador and J.R

    J. Amador and J.R. Artalejo. On the distribution of the successf ul and blocked events in the M/M/c retrial queue: A computational approach. Applied Mathematics and Computation, 190 (2007), 1612-1626. 19

    work page 2007

  6. [6]

    Amador and J.R

    J. Amador and J.R. Artalejo. The M/G/1 retrial queue: New desc riptors of the cus- tomer’s behavior. Journal of Computational and Applied Mathematics , 223 (2009), 15-26

    work page 2009

  7. [7]

    Amador and J.R

    J. Amador and J.R. Artalejo. Transient analysis of the successf ul and blocked events in retrial queues. Telecommunications Systems, 41 (2009), 255-265

    work page 2009

  8. [8]

    V. V. Anisimov and J. R. Artalejo. Approximation of multiserver re trial queues by means of generalized truncated models. Top, 10 (2002), 51–66

    work page 2002

  9. [9]

    Artalejo, A

    J.R. Artalejo, A. Economou and M.J. Lopez-Herrero. Algorithmic analysis of the maximum queue length in a busy period for the M/M/c retrial queue. INFORMS Journal on Computing , 19 (2007), 121-126

    work page 2007

  10. [10]

    Artalejo, V.C

    J.R. Artalejo, V.C. Joshua and A. Krishnamoorthy An M/G/1 ret rial queue with orbital search by the server. In J. R. Artalejo and A. Krishnamoo rthy, Eds. Advances in Stochastic Modeling . Notable publications. Inc. New Jersey. 2002

    work page 2002

  11. [11]

    J. R. Artalejo. A classified bibliography of research on retrial q ueues: Progress in 1990-1999. Top, 7, (1999), 187-211

    work page 1990

  12. [12]

    J. R. Artalejo, A. Gomez-Corral and M. F. Neuts. Analysis of m ultiserver queues with constant retrial rate. European Journal of Operational Research , 135 (2001), 569-581

    work page 2001

  13. [13]

    Artalejo and A

    J.R. Artalejo and A. Gomez-Corral. Channel idle periods in compu ter and telecom- munication systems with customer retrials. Telecommunication Systems, 24 (2003), 29-46

    work page 2003

  14. [14]

    J. R. Artalejo and M. Pozo. Numerical calculation of the station ary distribution of the main multiserver retrial queue. Annals of Operations Research , 116 (2002), 41-56

    work page 2002

  15. [15]

    Artalejo and G

    J. Artalejo and G. Falin. Standard and retrial queueing system s: a comparative analysis. Revista matematica complutense , XV (2002), 101-129. 20

    work page 2002

  16. [16]

    Artalejo and A

    J.R. Artalejo and A. Economou. On the non-existence of produ ct-form solutions for queueing networks with retrials. Electronic Modeling, 27 (2005), 13-19

    work page 2005

  17. [17]

    J. R. Artalejo and M. J. Lopez-Herrero. On the distribution of the number of retrials. Applied mathematical modelling , 31 (2007), 478-489

    work page 2007

  18. [18]

    J. R. Artalejo. Accessible bibliography on retrial queues: Prog ress in 2000-2009. Mathematical and computer modelling , 51 (2010), 1071-1081

    work page 2000

  19. [19]

    J. R. Artalejo and A. Gomez-Corral. Retrial queueing systems : a computational approach. Berlin Heidelberg: Springer-Verlag, 2008

    work page 2008

  20. [20]

    J. R. Artalejo and M. J. Lopez-Herrero. Cellular mobile network s with repeated calls operating in random environment. Computers & Operations Research , 37 (2010), 1158-1166

    work page 2010

  21. [21]

    J. R. Artalejo and T. Phung-Duc. Markovian retrial queues wit h two way communi- cation. Journal of Industrial and Management Optimization , 8 (2012), 781-806

    work page 2012

  22. [22]

    J. R. Artalejo and T. Phung-Duc. Single server retrial queues with two way commu- nication. Applied Mathematical Modelling , 37 (2013), 1811-1822

    work page 2013

  23. [23]

    Atencia and P

    I. Atencia and P. Moreno. A single-server retrial queue with ge neral retrial times and Bernoulli schedule. Applied Mathematics and Computation , 162 (2005), 855-880

    work page 2005

  24. [24]

    Avrachenkov and U

    K. Avrachenkov and U. Yechiali. Retrial networks with finite buff ers and their application to internet data traffic. Probability in the Engineering and Informational Sciences, 22 (2008), 519-536

    work page 2008

  25. [25]

    Avrachenkov and U

    K. Avrachenkov and U. Yechiali. On tandem blocking queues with a common retrial queue. Computers & Operations Research , 37 (2010), 1174-1180

    work page 2010

  26. [26]

    Avrachenkov, P

    K. Avrachenkov, P. Nain and U. Yechiali. A retrial system with tw o input streams and two orbit queues. Queueing Systems , 77 (2014), 1-31

    work page 2014

  27. [27]

    Avram, A

    F. Avram, A. J. E. M. Janssen and J. S. H. Van Leeuwaarden. L oss systems with slow retrials in the Halfin-Whitt regime. Advances in Applied Probability , 45 (2013), 274-294. 21

    work page 2013

  28. [28]

    G. Bianchi. Performance analysis of the IEEE 802.11 distributed coordination func- tion. IEEE Journal on selected areas in communications , 18 (2000), 535-547

    work page 2000

  29. [29]

    Murtuza Ali, O

    A. Murtuza Ali, O. J. Boxma J. Resing. Analysis and optimization of vacation and polling models with retrials. Performance Evaluation , 98 (2016), 52-69

    work page 2016

  30. [30]

    Breuer, A

    L. Breuer, A. Dudin and V. Klimenok. A retrial BMAP/PH/ N system. Queueing Systems, 40 (2002), 433-457

    work page 2002

  31. [31]

    Bright and P.G

    L. Bright and P.G. Taylor. Calculating the equilibrium distribution in le vel dependent quasi-birth-and-death processes. Stochastic Models, 11 (1995), 497-525

    work page 1995

  32. [32]

    Bhulai and G

    S. Bhulai and G. Koole. A queueing model for call blending in call ce nters. IEEE Transactions on Automatic Control , 48 (2003), 1434-1438

    work page 2003

  33. [33]

    Chakravarthy, A

    S.R. Chakravarthy, A. Krishnamoorthy, and V.C. Joshua, Ana lysis of a multi-server retrial queue with search of customers from the orbit, Performance Evaluation , 63 (2006), 776-798

    work page 2006

  34. [34]

    S. R. Chakravarthy. Analysis of MAP/PH/c retrial queue with p hase type retrials - simulation approach. In Modern Probabilistic Methods for Analysis of Telecommuni- cation Networks, (2013), Springer Berlin Heidelberg, 37-49

    work page 2013

  35. [35]

    B. D. Choi, Y. W. Shin and W. C. Ahn. Retrial queues with collision ar ising from unslotted CSMA/CD protocol. Queueing Systems , 11 (1992), 335-356

    work page 1992

  36. [36]

    B. D. Choi and Y. Chang. Single server retrial queues with prior ity calls. Mathematical and Computer Modelling , 30 (1999), 7-32

    work page 1999

  37. [37]

    B. D. Choi, Y. Chang and B. Kim. MAP1, MAP2/M/ c retrial queue with guard channels and its application to cellular networks. Top, 7 (1999), 231-248

    work page 1999

  38. [38]

    J.W. Cohen. Basic problems of telephone traffic theory and the in fluence of repeated calls, Philips Telecommunication Review, 18 (1957), 49-100

    work page 1957

  39. [39]

    Dayar and M

    T. Dayar and M. Can Orhan. Steady-state analysis of a multiclas s MAP/PH/c queue with acyclic PH retrials. Journal of Applied Probability . 53 (2016), 1098-1110. 22

    work page 2016

  40. [40]

    T. G. Deepak, A. N. Dudin, V. C. Joshua and A. Krishnamoorthy . On an M (X)/G/1 retrial system with two types of search of customers from the or bit. Stochastic Analysis and Applications , 31 (2003), 92-107

    work page 2003

  41. [41]

    J. E. Diamond and A. S. Alfa. The MAP/PH/1 retrial queue. Stochastic Models , 14 (1998), 1151–1177

    work page 1998

  42. [42]

    Dimitriou

    I. Dimitriou. A queueing model with two types of retrial custome rs and paired services. Annals of Operations Research , 238 (2016), 123-143

    work page 2016

  43. [43]

    Dimitriou

    I. Dimitriou. A two class retrial system with coupled orbit queues . Probability in the Engineering and Informational Sciences

    work page

  44. [44]

    T. V. Do and R. Chakka. An efficient method to compute the rate matrix for retrial queues with large number of servers. Applied Mathematics Letters, 23 (2010), 638-643

    work page 2010

  45. [45]

    T. V. Do. Solution for a retrial queueing problem in cellular networ ks with the fractional guard channel policy. Mathematical and Computer Modelling , 53 (2010), 2059-2066

    work page 2010

  46. [46]

    V. I. Dragieva. A finite source retrial queue: number of retria ls. Communications in Statistics-Theory and Methods , 42 (2013), 812-829

    work page 2013

  47. [47]

    V. I. Dragieva. Number of retrials in a finite source retrial queu e with unreliable server. Asia-Pacific Journal of Operational Research , 31 (2014), 1440005

    work page 2014

  48. [48]

    Dragieva and T

    V. Dragieva and T. Phung-Duc. Two-way communication M/M/1 r etrial queue with server-orbit interaction. Proceedings of The 11th International Conference on Queue- ing Theory and Network Applications (QTNA2016) , (2016) 7 pages. ACM Digital Library

    work page 2016

  49. [49]

    A. N. Dudin, A. Krishnamoorthy, V. C. Joshua and G. V. Tsaren kov (2004). Analysis of the BMAP/G/1 retrial system with search of customers from th e orbit. European Journal of Operational Research , 157(2004), 169-179. 23

    work page 2004

  50. [50]

    Dudin, M

    A. Dudin, M. Lee, O. Dudina and S. Lee. Analysis of priority retria l queue with many types of customers and servers reservation as a model of cognit ive radio system. IEEE Transactions on Communications , 65, (2017), 186-199

    work page 2017

  51. [51]

    Economou and S

    A. Economou and S. Kanta. Equilibrium balking strategies in the ob servable single- server queue with breakdowns and repairs. Operations Research Letters , 36 (2008), 696-699

    work page 2008

  52. [52]

    Economou and M

    A. Economou and M. J. Lopez]Herrero. Performance analysis o f a cellular mobile net- work with retrials and guard channels using waiting and first passage time measures. European Transactions on Telecommunications, 20 (2009), 389-401

    work page 2009

  53. [53]

    Economou and S

    A. Economou and S. Kanta. Equilibrium customer strategies and social-profit max- imization in the single]server constant retrial queue. Naval Research Logistics , 58 (2011), 107-122

    work page 2011

  54. [54]

    G. I. Falin. On the waiting-time process in a single-line queue with re peated calls. Journal of Applied Probability , 23 (1986), 185-192

    work page 1986

  55. [55]

    G. I. Falin. On a multiclass batch arrival retrial queue. Advances in Applied Proba- bility, 20 (1988), 483-487

    work page 1988

  56. [56]

    G. I. Falin. A survey of retrial queues. Queueing Systems , 7 (1990), 127-168

    work page 1990

  57. [57]

    G. I. Falin and J. G. C. Templeton. Retrial Queues. Chapman & Ha ll, London, 1997

    work page 1997

  58. [58]

    G. I. Falin, J. R. Artalejo and M. Martin. On the single server ret rial queue with priority customers. Queueing systems , 14 (1993), 439-455

    work page 1993

  59. [59]

    Fedorova

    E. Fedorova. The second order asymptotic analysis under hea vy load condition for retrial queueing system MMPP/M/1. Communications in Computer and Information Science, 564 (2015), 344-357

    work page 2015

  60. [60]

    Fiems and T

    D. Fiems and T. Phung-Duc. Light-traffic analysis of queues with limited heteroge- nous retrials. In Proceedings of 11th International Conference on Queueing T heory and Network Application (QTNA2016) , 2016. 24

    work page 2016

  61. [61]

    Fiems, J

    D. Fiems, J. P. L. Dorsman and W. Rogiest. Analysing queueing be haviour in void- avoiding fibre-loop optical buffers. Performance Evaluation, 103 (2016), 23-40

    work page 2016

  62. [62]

    N. Gans, G. Koole and A. Mandelbaum. Telephone call centers: T utorial, review, and research prospects. Manufacturing & Service Operations Management , 5 (2003), 79-141

    work page 2003

  63. [63]

    Gao, X Niu and T

    S. Gao, X Niu and T. Li. Analysis of a constant retrial queue with j oining strategy and impatient retrial customers. Mathematical Problems in Engineering , (2017), Article ID 9618215

    work page 2017

  64. [64]

    Gomez-Corral

    A. Gomez-Corral. Stochastic analysis of a single server retrial queue with general retrial times. Naval Research Logistics, 46 (1999), 561-581

    work page 1999

  65. [65]

    Gomez-Corral

    A. Gomez-Corral. A matrix-geometric approximation for tande m queues with block- ing and repeated attempts. Operations Research Letters, 30 (2002), 360-374

    work page 2002

  66. [66]

    Gomez-Corral

    A. Gomez-Corral. A bibliographical guide to the analysis of retria l queues through matrix analytic techniques. Annals of Operations Research , 141 (2006), 163-191

    work page 2006

  67. [67]

    Gomez-Corral

    A. Gomez-Corral. On the applicability of the number of collisions in p -persistent CSMA/CD protocols. Computers & Operations Research , 37 (2010), 1199-1211

    work page 2010

  68. [68]

    Gomez-Corral and M.L

    A. Gomez-Corral and M.L. Garcia. Maximum queue lengths during a fixed time interval in the M/M/c retrial queue. Applied Mathematics and Computation , 235 (2014), 124-136

    work page 2014

  69. [69]

    Gomez-Corral and T

    A. Gomez-Corral and T. Phung-Duc (Eds.) Retrial queues and related models, Annals of Operations Research, 247 (2016)

    work page 2016

  70. [70]

    A.Grishechkin

    S. A.Grishechkin. Multiclass batch arrival retrial queues analyz ed as branching processes with immigration Queueing Systems , 11 (1992), 395-418

    work page 1992

  71. [71]

    Hanschke

    T. Hanschke. Explicit formulas for the characteristics of the M /M/2/2 queue with repeated attempts. Journal of Applied Probability , 24 (1987), 486-494

    work page 1987

  72. [72]

    Kajiwara and T

    K. Kajiwara and T. Phung-Duc. Multiserver queue with guard ch annel for priority and retrial customers. International Journal of Stochastic Analysis , 2016. 25

    work page 2016

  73. [73]

    C. S. Kim, S. H. Park, A. Dudin, V. Klimenok and G. Tsarenkov Inv estigation of the BMAP/G/1 → · /PH/1/M tandem queue with retrials and losses. Applied Mathematical Modelling 34 (2010), 2926-2940

    work page 2010

  74. [74]

    C. S. Kim, V. Klimenok and O. Taramin. A tandem retrial queueing s ystem with two Markovian flows and reservation of channels. Computer & Operations Research , 37 (2010), 1238-1246

    work page 2010

  75. [75]

    C. Kim, V. I. Klimenok and A. N. Dudin. Analysis and optimzation of G uard Channel Policy in cellular mobile networks with account of retrials. Computers & Operations Research, 43 (2014), 181-190

    work page 2014

  76. [76]

    J. Kim, J. Kim and B. Kim. Tail asymptotics of the queue size distrib ution in the M/M/m retrial queue. Journal of Computational and Applied Mathematics , 236 (2012), 3445-3460

    work page 2012

  77. [77]

    Kim and B

    J. Kim and B. Kim. Exact tail asymptotics for the M/M/m retrial q ueue with nonpersistent customers. Operations Research Letters, 40 (2012), 537-540

    work page 2012

  78. [78]

    J. Kim, B. Kim and S. S. Ko. Tail asymptotics for the queue size dis tribution in an M/G/1 retrial queue. Journal of Applied Probability , 44 (2007), 1111-1118

    work page 2007

  79. [79]

    Klimenok and A

    V. Klimenok and A. Dudin. Multi-dimensional asymptotically quasi-T oeplitz Markov chains and their application in queueing theory. Queueing Systems , 54 (2006), 245– 259

    work page 2006

  80. [80]

    Konishi, H

    Y. Konishi, H. Masuyama, S. Kasahara and Y. Takahashi. Perfo rmance analysis of dynamic spectrum handoff scheme with variable bandwidth demand of secondary users for cognitive radio networks. Wireless Networks , 19 (2013), 607-617

    work page 2013

Showing first 80 references.