ANALYSIS OF H2/E2/1 SYSTEM AND HER OF THE ANALOG WITH SHIFTED INPUT DISTRIBUTIONS
DOI:
https://doi.org/10.15588/1607-3274-2020-1-10Keywords:
Hyperexponential and erlangian distribution laws, Lindley integral equation, spectral decomposition method, Laplace transform.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 finding a solution for the average waiting time in queue in a closed form for two systems with ordinary and shifted hyperexponential and erlangian input distributions is considered.
Objective. Obtaining a solution for the main system characteristic – the average waiting time in queue for two queuing systems of type G/G/1 with ordinary and shifted hyperexponential and erlangian input distributions.
Method. 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 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. The spectral decompositions of the solution of the Lindley integral equation for a pair of dual systems are for the first time received, with the help of which the formulas for the average waiting time in a closed form are derived.
Conclusions. The spectral expansions of the solution of the Lindley integral equation for the systems under consideration are obtained and with their help the formulas for the average waiting time in the queue for these systems in a closed form are derived. It is shown that in systems with a time lag, the average waiting time is less than in conventional 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 with arbitrary laws of the input flow distribution and service time. This approach allows us to calculate the average latency for these systems in mathematical packages for a wide range of traffic parameters. All other characteristics of the systems are derived from the waiting time. 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. The results are published for the first time.
References
Tarasov V. N., Bakhareva N. F., Blatov I. A. Analysis and calculation of queuing system with delay, Automation and Remote Control, 2015, Vol. 76, No. 11, pp. 1945–1951. DOI: 10.1134/S0005117915110041.
Tarasov V. N. Extension of the Class of Queueing Systems with Delay, Automation and Remote Control, 2018, Vol. 79, No. 12, pp. 2147–2157. DOI: 10.1134/S0005117918120056.
Kleinrock L. Teoriya massovogo obsluzhivaniya. Moscow, Mashinostroeinie Publ, 1979, 432 p.
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.
Bocharov P. P., Pechinkin A. V. Teoriya massovogo obsluzhivaniya. Moscow, Publishing House of Peoples' Friendship University, 1995, 529 p.
Tarasov V. N. Analysis of queues with hyperexponential arrival distributions, Problems of Information Transmission, 2016, Vol. 52, No. 1, pp. 14–23. DOI:10.1134/S0032946016010038.
Tarasov V. N. Analysis and comparison of two queueing systems with hypererlangian input distributions, Radio Electronics, Computer Science, Control, 2018, Vol. 47, No. 4, pp. 61–70. DOI 10.15588/1607-3274-2018-4-6.
Tarasov V. N., Bakhareva N. F. Research of queueing systems with shifted erlangian and exponential input distributions, Radio Electronics, Computer Science, Control, 2019, Vol. 48, No. 1, pp.67–76. DOI 10.15588/1607-3274-2019-1-7.
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.
Aliev T. I. Approksimaciya veroyatnostnyh raspredelenij v modelyah massovogo obsluzhivaniya. Nauchno-tekhnicheskij vestnik informacionnyh tekhnologij, mekhaniki i optiki, 2013, Vol. 84, No. 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/s11134-016-9493-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
Kruglikov V.K., Tarasov V.N. Analysis and calculation of queuingnetworks using the two-dimensional diffusion-approximation, Automation and Remote Control, 1983, Vol. 44, No. 8, pp. 1026– 1034.
Downloads
How to Cite
Issue
Section
License
Copyright (c) 2020 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.
