COMPARISON OF TWO FORMS OF ERLANGIAN DISTRIBUTION LAW IN QUEUING THEORY
DOI:
https://doi.org/10.15588/1607-3274-2021-3-5Keywords:
Ordinary and normalized Erlangian distribution laws, Lindley integral equation, spectral decomposition method, Laplace transform.Abstract
Context. For modeling various data transmission systems, queuing systems G/G/1 are in demand, this is especially important because there is no final solution for them in the general case. The problem of the derivation in closed form of the solution for the average waiting time in the queue for ordinary system with erlangian input distributions of the second order and for the same system with shifted to the right distributions is considered.
Objective. Obtaining a solution for the main system characteristic – the average waiting time for queue requirements for three types of queuing systems of type G/G/1 with usual and shifted erlangian input distributions.
Method. To solve this problem, we used the classical method of spectral decomposition of the solution of Lindley integral equation, which allows one to obtain a solution for average the waiting time for systems under consideration in a closed form. For the practical application of the results obtained, the well-known method of moments of the theory of probability was used.
Results. For the first time, spectral expansions of the solution of the Lindley integral equation for systems with ordinary and shifted Erlang distributions are obtained, with the help of which the calculation formulas for the average waiting time in the queue for the above systems in closed form are derived.
Conclusions. The difference between the usual and normalized distribution is that the normalized distribution has a mathematical expectation independent of the order of the distribution k, therefore, the normalized and normal Erlang distributions differ in numerical characteristics. The introduction of the time shift parameter in the laws of input flow distribution and service time for the systems under consideration turns them into systems with a delay with a shorter waiting time. This is because the time shift operation reduces the coefficient of variation in the intervals between the receipts of the requirements and their service time, and as is known from queuing theory, the average wait time of requirements is related to these coefficients of variation by a quadratic dependence. The system with usual erlangian input distributions of the second order is applicable only at a certain point value of the coefficients of variation of the intervals between the receipts of the requirements and their service time. The same system with shifted distributions allows us to operate with interval values of coefficients of variations, which expands the scope of these systems. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters.
References
Kleinrock L. Queueing Systems, Vol. I. Theory. New York, Wiley, 1975, 417 p.
Tarasov V. N., Bakhareva N. F. Research of queueing systems with shifted erlangian and exponential input distributions, Radio Electronics, Computer Science, Control, 2019, No. 1, pp. 67–76. DOI: 10.15588/1607-3274-2019-1-7
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. 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., 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
Kruglikov V. K., Tarasov V. N. Analysis and calculation of queuing-networks using the two-dimensional diffusionapproximation, Automation and Remote Control, 1983, No. 8, pp. 1026–1034.
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. Osnovy modelirovaniya diskretnyh system. SPb, SPbGU ITMO, 2009, 363 p.
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
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2021 V. N. Tarasov
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.
