Abstract
It is well-known that an analytical solution of multiserver retrial queues is difficult and does not lead to numerical implementation. Thus, many papers approximate the original intractable system by the so-called generalized truncated systems which are simpler and converge to the original model. Most papers assume heuristically the convergence but do not provide a rigorous mathematical proof. In this paper, we present a proof based on a synchronization procedure. To this end, we concentrate on theM/M/c retrial queue and the approximation developed by Neuts and Rao (1990). However, the methodology can be employed to establish the convergence of several generalized truncated systems and a variety of Markovian multiserver retrial queues.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anisimov V.V. (1999). Switching stochastic models and applications in retrial queues.Top 7, 169–186.
Anisimov V.V. and Artalejo J.R. (2001). Analysis of Markov multiserver retrial queues with negative arrivals.Queueing Systems 39, 157–182.
Artalejo J.R. (1996). Stationary analysis of the characteristics of theM/M/2 queue with constant repeated attempts.Opsearch 33, 83–95.
Artalejo J.R. (1999a). Accessible bibliography on retrial queues.Mathematical and Computer Modelling 30, 1–6.
Artalejo J.R. (1999b). A classified bibliography of research on retrial queues: Progress in 1990–1999.Top 7, 187–211.
Artalejo J.R. and Gomez-Corral A. (1997). Steady state solution of a single-server queue with linear repeated requests.Journal of Applied Probability 34, 223–233.
Artalejo J.R. and Pozo M. (2001). Numerical calculation of the stationary distribution of the main multiserver retrial queue.Annals of Operations Research 111 (to appear).
Choi B.D., Chang Y. and Kim B. (1999).MAP 1,MAP 2/M/c retrial queue with guard channels and its application to cellular networks.Top 7, 231–248.
Cinlar E. (1975).Introduction to Stochastic Processes. Pretince-Hall.
Cohen J.W. (1957). Basic problems of telephone traffic theory and the influence of repeated calls.Phillips Telecommunication Review 18, 49–100.
Falin G.I. (1983). Calculation of probability characteristics of a multiline system with repeat calls.Moscow University Computational Mathematics and Cybernetics 1, 43–49.
Falin G.I. and Templeton J.G.C. (1997).Retrial Queues. Chapman and Hall.
Gomez-Corral A. and Ramalhoto M.F. (1999). The stationary distribution of a Markovian process arising in the theory of multiserver retrial queueing systems.Mathematical and Computer Modelling 30, 141–158.
Laha R.G. and Rohatgi V.K. (1979).Probability Theory. John Wiley.
Li H. and Yang T. (1999). Steady-state queue size distribution of discrete-timePH/Geo/1 retrial queues.Mathematical and Computer Modelling 30, 51–63.
Neuts M.F. and Rao B.M. (1990). Numerical investigation of a multiserver retrial model.Queueing Systems 7, 169–190.
Pearce C.E.M. (1989). Extended continued fraction, recurrence relations and two-dimensional Markov processes.Advances in Applied Probability 21, 357–375.
Stepanov S.N. (1999). Markov models with retrials: the calculation of stationary performance measures based on the concept of truncation.Mathematical and Computer Modelling 30, 207–228.
Stoyan D. (1983).Comparison Methods for Queues and other Stochastic Models. John Wiley.
Wilkinson, R.I. (1956). Theories for toll traffic engineering in the U.S.A.The Bell System Technical Journal 35, 421–514.
Author information
Authors and Affiliations
Additional information
J.R. Artalejo thanks the support received from DGES 98-0837.
Rights and permissions
About this article
Cite this article
Anisimov, V.V., Artalejo, J.R. Approximation of multiserver retrial queues by means of generalized truncated models. Top 10, 51–66 (2002). https://doi.org/10.1007/BF02578940
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02578940
Key Words
- Retrial queues
- stationary distribution
- generalized truncated systems
- synchronization
- stochastic comparability