QUEUEING SYSTEMS WITH TIME LAG
DOI:
https://doi.org/10.15588/1607-3274-2021-4-5Keywords:
delayed system, shifted distributions, Laplace transform, Lindley integral equation, spectral decomposition method.Abstract
Context. In the queuing theory of a research of the G/G/1 systems are relevant because it is impossible to receive decisions for the average waiting time in queue in a final form in case of arbitrary laws of distributions of an input flow and service time. Therefore, the study of such systems for particular cases of input distributions is important. The problem of deriving solutions for the average waiting time in a queue in closed form for systems with distributions shifted to the right from the zero point is considered.
Objective. Getting solutions for the main characteristics of the systems – the average waiting time of requirements in the queue for queuing systems (QS) of type G/G/1 with shifted input distributions.
Methods. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation. This method allows to obtaining a solution for the average waiting time for the systems under consideration in a closed form. The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used.
Results. For the first time, spectral expansions are obtained for the solution of the Lindley integral equation for systems with delay, which are used to derive formulas for the average waiting time in a queue in closed form. The paper presents the final studies for the remaining eight delay systems.
Conclusions. It is shown that in systems with delay, the average waiting time is less than in in the usual systems. The obtained formula for the average waiting time expands and complements the well-known queuing theory incomplete formula for the average waiting time for G/G/1 systems. This approach allows us to calculate the average latency for these systems in mathematical packages for a wide range of traffic parameters. In addition to the average waiting time, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the waiting time from its average value, the jitter can be determined through the variance of the waiting time.
References
Kleinrock L. Queueing Systems, Vol. I. Theory. New York, Wiley, 1975, 417 p.
Tarasov V. N. Queueing systems with delay, Radio Electronics, Computer Science, Control, 2019, No. 3, pp. 55–63. DOI: 10.15588/1607-3274-2019-3-7
Tarasov V. N. The analysis of two queuing systems HE2/M/1 with ordinary and shifted input distributions, Radio Electronics, Computer Science, Control, 2019, No. 2, pp. 71–79. DOI: 10.15588/1607-3274-2019-2-8
Tarasov V. N., Bakhareva N. F. Comparative analysis of two queuing systems M/HE2/1 with ordinary and with the shifted input distributions, Radio Electronics, Computer Science, Control, 2019, No. 4, pp. 50–58. DOI: 10.15588/1607-3274-2019-4-5
Tarasov V. N. Analysis of H-2/E-2/1 system and her of the analog with shifted input distributions, Radio Electronics, Computer Science, Control, 2020, No. 1, pp. 90–97. DOI: 10.15588/1607-3274-2020-1-10
Tarasov V. N., Bakhareva N. F. Model of teletraffic based on queueing systems E-2/HE2/1 with ordinary and shifted input distributions, Radio Electronics, Computer Science, Control, 2020, No. 2, pp. 65–74. DOI: 10.15588/1607-32742020-2-7
Tarasov V. N. Research of two systems E-2/H-2/1 with ordinary and shifted distributions by the spectral decomposition method, Radio Electronics, Computer Science, Control, 2020, №3, pp. 69–77. DOI 10.15588/1607-3274-2020-3-6
Brannstrom N. A. Queueing Theory analysis of wireless radio systems. Appllied to HS-DSCH. Lulea university of technology, 2004, 79 p.
Whitt W. Approximating a point process by a renewal process: two basic methods, Operation Research, 1982, Vol. 30, No. 1, pp. 125–147.
Novitzky S., Pender J., Rand R.H., Wesson E. Limiting the oscillations in queues with delayed information through a novel type of delay announcement, Queueing Systems, 2020, Vol. 95, pp. 281–330. DOI: https://doi.org/10.1007/s11134-020-09657-9
Novitzky S., Pender J., Rand R. H., Wesson E. Nonlinear Dynamics in Queueing Theory: Determining the Size of Oscillations in Queues with Delay, SIAM J. Appl. Dyn. Syst., 18–1 2019, Vol. 18, No. 1, pp. 279–311. DOI: https://doi.org/10.1137/18M1170637
RFC 3393 [IP Packet Delay Variation Metric for IP Performance Metrics (IPPM)] Available at: https://tools.ietf.org/html/rfc3393. (accessed: 26.02.2016).
Myskja A. An improved heuristic approximation for the GI/GI/1 queue with bursty arrivals. Teletraffic and datatraffic in a Period of Change. ITC-13. Elsevier Science Publishers, 1991, pp. 683–688.
Aliev T. I. Approximation of Probability Distributions in Queuing Models, Scientific and technical bulletin of information technologies, mechanics and optics, 2013, № 2, pp. 88–93.
Aras A. K., Chen X. & Liu Y. Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment, Queueing Systems, 2018, Vol. 89, No. 1, pp. 81–125. DOI: https://doi.org/10.1007/s11134-018-9575-0
Jennings O.B. & Pender J. Comparisons of ticket and standard queues, Queueing Systems, 2016, Vol. 84, No. 1, pp. 145–202. DOI: https://doi.org/10.1007/s11134-0169493-y
Gromoll H.C., Terwilliger B. & Zwart B. Heavy traffic limit for a tandem queue with identical service times, Queueing Systems, 2018, Vol. 89, No. 3, pp. 213–241. DOI: https://doi.org/10.1007/s11134-017-9560-z
Legros B. M/G/1 queue with event-dependent arrival rates, Queueing Systems, 2018, Vol. 89, No. 3, pp. 269– 301. DOI: https://doi.org/10.1007 /s11134-017-9557-7
Bazhba M., Blanchet J., Rhee CH., et al. Queue with heavy-tailed Weibull service times, Queueing Systems, 2019, Vol. 93, No. 11, pp. 1–32. DOI: https://doi.org/10.1007/s11134-019-09640-z/
Adan I., D’Auria B., Kella O. Special volume on ‘Recent Developments in Queueing Theory’ of the third ECQT conference. Queueing Systems, 2019, Vol. 93, No. 1, pp. 1– 190. DOI: https://doi.org/10.1007/s11134-019-09630-1
Adan I., D’Auria B., Kella O. Special volume on ‘Recent Developments in Queueing Theory’ of the third ECQT conference: part 2, Queueing Systems, 2019, pp. 1–2. DOI: https://doi.org/10.1007/s11134-019-09637-8
Tibi D. Martingales and buffer overflow for the symmetric shortest queue model. Queueing Systems, 2019, Vol. 93, pp. 153–190. DOI: 10.1007/s11134-01909628-9
Jacobovic R., Kella O. Asymptotic independence of regenerative processes with a special dependence structure. Queueing Systems, 2019, Vol. 93, pp. 139–152. DOI: 10.1007/s11134-019-09606-1
Wang L., Kulkarni V. Fluid and diffusion models for a system of taxis and customers with delayed matching. Queueing Systems, 2020, Vol. 96, pp. 101–131. DOI: 10.1007/s11134-020-09659-7
Kruglikov V. K., Tarasov V. N. Analysis and calculation of queuing-networks using the two-dimensional diffusionapproximation, Automation and Remote Control, 1983, Vol. 44, No. 8, pp. 1026–1034.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 В. Н. Тарасов, Н. Ф. Бахарева
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Creative Commons Licensing Notifications in the Copyright Notices
The journal allows the authors to hold the copyright without restrictions and to retain publishing rights without restrictions.
The journal allows readers to read, download, copy, distribute, print, search, or link to the full texts of its articles.
The journal allows to reuse and remixing of its content, in accordance with a Creative Commons license СС BY -SA.
Authors who publish with this journal agree to the following terms:
-
Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License CC BY-SA that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
-
Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
-
Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.
