Retrial queueing systems. A computational approach.

*(English)*Zbl 1161.60033
Berlin: Springer (ISBN 978-3-540-78724-2/hbk; 978-3-642-09748-5/pbk; 978-3-540-78725-9/ebook). xiii, 318 p. (2008).

Retrial queueing systems have found much interest in application oriented research papers in the telecommunications and engineering literature as well as in the area of mathematical queueing theory, resp. in applied probability. The fundamental problem setting is easily described: When customers arriving at a service system cannot receive service (immediately or in due time) they often leave the system due to impatience, balking, overflow due to finite capacity constraints, etc., but are not completely lost to the system because of returning later on in a way, that can be incorporated into a structured model.

So retrial queues can be described in a first approach as consisting of any standard queueing system with an additional feature which organizes the retries by the customers.

The challenging problem is that even for easy to tackle standard queues the additional retrial component of the system poses notoriously heavy burden on researches investigating the theory and on practitioners which cannot neglect the effects originating the reattempts of customers in their system description. Consequently, classical investigations provide the standard performance indices (under equilibrium conditions) of retrial systems in involved transform expressions which are often not easy to invert. Nevertheless, there is an overwhelming amount of papers with results of this form. A taste of the techniques used and the results obtained can be found in Part II of the book under review. For making the book useful to applications these results are accompanied by approximations presented in the literature. A different approach which to a certain extent has revived research especially in the area of retrial queues is elaborated on in Part III, where matrix-analytical methods are applied to provide algorithmic schemes to compute performance measures. So, the promised “Computational approach” is the main part of the monograph. The contents are:

Part I - An introduction to retrial queueing systems: Introduction and motivating examples; A general overview.

Part II - Computational analysis of performance descriptors: Limiting distribution of the system state; Busy period; Waiting time; Other descriptors.

Part III - Retrial queueing systems analyzed through the matrix-analytic formalism: The matrix-analytic formalism; Selected retrial queues with QBD structure; Selected retrial queues with GI/M/1 and M/G1 structures.

It should be noticed, that Part II (and the introductory Part I, too) is restricted to mainly discussing M/G/1/0 and M/M/c/0, with a short excursion to discrete time Geo/G/1/0. In Part III the versatile class of Markovian arrival processes to the systems is introduced, where matrix-analytical methods are tailored for. So the restriction to Poissonian Arrivals can be weakened. Also, the Geo/Geo/c queues is investigated here with the help of matrix-analytical methods. I personally like the Section 2.3 “Short description of some advanced retrial queueing systems” which provides the readers with information and references to “Batch arrivals, finite population; generalized retrial policies; multiclass queues; negative arrivals and disasters; nonpersistent customers; priorities; retrials due to balking, impatience and breakdowns”, in any case with references to the literature. My impression is that this section, possibly produced en passant, is rather valuable to researchers and practitioners encountered with such non-standard problems - and looking for fast help with these systems which they have to assess or to investigate in more detail.

In my opinion the book is an interesting supplement to classical books on queueing theory, for researchers in the field probably a necessary completion of their book shelf. For those, not active in the field it seems to be advisable to maintain a least a pointer to the book: In case they will be encountered with the feature of retrials in some modeling process, the book will surely support their investigation. The book contains a reference list of 706 items, up to 2007 and (partly in press) 2008.

So retrial queues can be described in a first approach as consisting of any standard queueing system with an additional feature which organizes the retries by the customers.

The challenging problem is that even for easy to tackle standard queues the additional retrial component of the system poses notoriously heavy burden on researches investigating the theory and on practitioners which cannot neglect the effects originating the reattempts of customers in their system description. Consequently, classical investigations provide the standard performance indices (under equilibrium conditions) of retrial systems in involved transform expressions which are often not easy to invert. Nevertheless, there is an overwhelming amount of papers with results of this form. A taste of the techniques used and the results obtained can be found in Part II of the book under review. For making the book useful to applications these results are accompanied by approximations presented in the literature. A different approach which to a certain extent has revived research especially in the area of retrial queues is elaborated on in Part III, where matrix-analytical methods are applied to provide algorithmic schemes to compute performance measures. So, the promised “Computational approach” is the main part of the monograph. The contents are:

Part I - An introduction to retrial queueing systems: Introduction and motivating examples; A general overview.

Part II - Computational analysis of performance descriptors: Limiting distribution of the system state; Busy period; Waiting time; Other descriptors.

Part III - Retrial queueing systems analyzed through the matrix-analytic formalism: The matrix-analytic formalism; Selected retrial queues with QBD structure; Selected retrial queues with GI/M/1 and M/G1 structures.

It should be noticed, that Part II (and the introductory Part I, too) is restricted to mainly discussing M/G/1/0 and M/M/c/0, with a short excursion to discrete time Geo/G/1/0. In Part III the versatile class of Markovian arrival processes to the systems is introduced, where matrix-analytical methods are tailored for. So the restriction to Poissonian Arrivals can be weakened. Also, the Geo/Geo/c queues is investigated here with the help of matrix-analytical methods. I personally like the Section 2.3 “Short description of some advanced retrial queueing systems” which provides the readers with information and references to “Batch arrivals, finite population; generalized retrial policies; multiclass queues; negative arrivals and disasters; nonpersistent customers; priorities; retrials due to balking, impatience and breakdowns”, in any case with references to the literature. My impression is that this section, possibly produced en passant, is rather valuable to researchers and practitioners encountered with such non-standard problems - and looking for fast help with these systems which they have to assess or to investigate in more detail.

In my opinion the book is an interesting supplement to classical books on queueing theory, for researchers in the field probably a necessary completion of their book shelf. For those, not active in the field it seems to be advisable to maintain a least a pointer to the book: In case they will be encountered with the feature of retrials in some modeling process, the book will surely support their investigation. The book contains a reference list of 706 items, up to 2007 and (partly in press) 2008.

Reviewer: Hans Daduna (Hamburg)

##### MSC:

60K25 | Queueing theory (aspects of probability theory) |

90B22 | Queues and service in operations research |

60K20 | Applications of Markov renewal processes (reliability, queueing networks, etc.) |

65C40 | Numerical analysis or methods applied to Markov chains |