×

Numerical investigation of a multiserver retrial model. (English) Zbl 0711.60094

To obtain numerical results for multi-channel retrial queues, the initial queueing system is replaced by a similar system where the number of sources of repeated calls is bounded by some sufficiently large constant. The truncated system is convenient since it has a finite set of equations for the stationary probabilities. However, in the case of heavy traffic or low intensity of repetition this set of equations can be of extremely high dimensionality.
To overcome this difficulty, the authors introduce a new approximating queueing system where the intensity of the total flow of repeated calls is limited. This queueing system has matrix-geometric stationary distribution which can be efficiently computed even in the case of heavy traffic or long intervals between retrials. The new method provides better accuracy than the above mentioned classical method.
Reviewer: G.Falin

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
65F30 Other matrix algorithms (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A.M. Alexsandrov, A queueing system with repeated orders, Eng. Cybernet. 12 (1974) 1-4.
[2] J.W. Cohen, Basic problems of telephone traffic theory and the influence of repeated calls, Philips Telecomm. Rev. 18 (1957) 49-99.
[3] G.I. Falin, A single-line system with secondary orders, Eng. Cybernet. 17 (1979) 76-83. · Zbl 0437.60073
[4] G.I. Falin, Double channel queueing system with repeated calls, Paper #4221-84, All-Union Institute for Scientific and Technical Information, Moscow, USSR (1984).
[5] G.I. Falin, On sufficient conditions for ergodicity of multichannel queueing systems with repeated calls, Adv. Appl. Probab. 16 (1984) 447-448. · Zbl 0535.60087
[6] G.I. Falin, Multichannel queueing systems with repeated calls under high intensity of repetition, J. Inform. Proc. Cybernet. 1 (1987) 37-47. · Zbl 0616.90020
[7] G.I. Falin, Error estimates in the approximation of countable Markov chains connected with the models of repeated calls, Vestnik Moskov Univ. Ser. 1, Mat. Mekh. 2 (1987) 12-15 (in Russian). · Zbl 0664.60104
[8] A.A. Fredericks and G.A. Reisner, Approximations to stochastic service systems, with an application to a retrial model, Bell Syst. Techn. J. 58 (1979) 557-576. · Zbl 0402.90043
[9] B. Greenberg, An upper bound on the performance of queues with returning customers, J. Appl. Probab. 24 (1987) 466-475. · Zbl 0626.60092
[10] T. Hanschke, Explicit formulas for the characteristics of theM/M/2/2 queue with repeated attempts, J. Appl. Probab. 24 (1987) 486-494. · Zbl 0624.60110
[11] G.L. Jonin and J.J. Sedol, Telephone systems with repeated calls,6th Int. Teletraffic Congress, Munich (1970) pp. 435.1-453.5.
[12] L. Kaufman, Matrix methods for queueing systems, SIAM J. Sci. Statist. Comput. 4 (1983) 525-552. · Zbl 0551.65096
[13] J. Keilson, J. Cozzolino and M. Young, A service with unfilled requests repeated, Oper. Res. 16 (1968) 1126-1137. · Zbl 0165.52703
[14] V.G. Kulkarni, Letter to the Editor, J. Appl. Probab. 19 (1982) 901-904.
[15] V.G. Kulkarni, On queueing systems with retrials, J. Appl. Probab. 20 (1983) 380-389. · Zbl 0518.90023
[16] M.F. Neuts,Matrix Geometric Solutions in Stochastic Models (The Johns Hopkins University Press, Baltimore, MD 1981). · Zbl 0469.60002
[17] M.F. Neuts and M.F. Ramalhoto, A service model in which the server is required to search for customers, J. Appl. Probab. 21 (1984) 157-166. · Zbl 0531.60089
[18] M.F. Neuts, The caudal characteristic curve of queues, Adv. Appl. Probab. 18 (1986) 221-254. · Zbl 0588.60094
[19] C.E.M. Pearce, On the problem of re-attempted calls in teletraffic, Stochastic Models 3 (1987) 393-407. · Zbl 0641.90041
[20] J. Riordan,Stochastic Service Systems (Wiley, New York, 1962). · Zbl 0106.33601
[21] R.I. Wilkinson, Theories for toll traffic engineering in the USA, Bell Syst. Techn. J. 35 (1956) 421-514.
[22] T. Yang and J.G.C. Templeton, A survey on retrial queues, Queueing Systems 2 (1987) 201-233. · Zbl 0658.60124
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.