# TWO PAIRS OF DUAL QUEUEING SYSTEMS WITH CONVENTIONAL AND SHIFTED DISTRIBUTION LAWS

## Authors

• V. N. Tarasov Volga State University of Telecommunications and Informatics, Samara, Russian Federation, Russian Federation
• N. F. Bakhareva Volga State University of Telecommunications and Informatics, Samara, Russian Federation, Russian Federation

## Keywords:

Erlang and exponential distribution laws, Lindley integral equation, spectral expansion solution method, Laplace transform

## Abstract

Context. The relevance of studies of G/G/1 systems is associated with the fact that they are in demand for modeling data transmission systems for various purposes, as well as with the fact that for them there is no final solution in the general case. We consider the problem of deriving a solution for the average delay of requests in a queue in a closed form for ordinary systems with Erlang and exponential input distributions and for the same systems with distributions shifted to the right.

Objective. Obtaining a solution for the main characteristic of the system – the average delay of requests in a queue for two pairs of queuing systems with ordinary and shifted Erlang and exponential input distributions, as well as comparing the results for systems with normalized Erlang distributions.

Methods. To solve the problem posed, the method of spectral solution of the Lindley integral equation was used, which allows one to obtain a solution for the average delay for the systems under consideration in a closed form. For the practical application of the results obtained, the method of moments of the theory of probability was used.

Results. Spectral solutions of the Lindley integral equation for two pairs of systems are obtained, with the help of which calculation formulas are derived for the average delay of requests in the queue in a closed form. Comparison of the results obtained with the data for systems with normalized Erlang distributions confirms their identity.

Conclusions. The introduction of the time shift parameter into the distribution laws of the input flow and service time for the systems under consideration transforms them into systems with a delay with a shorter waiting time. This is because the time shift operation reduces the value of the variation coefficients of the intervals between the arrivals of claims and their service time, and as is known from the queuing theory, the average delay of requests is related to these variation coefficients by a quadratic dependence. If a system with Erlang and exponential input distributions works only for one fixed pair of values of the coefficients of variation of the intervals between arrivals and their service time, then the same system with shifted distributions allows operating with interval values of the coefficients of variations, which expands the scope of these systems. The situation is similar with shifted exponential distributions. In addition, the shifted exponential distribution contains two parameters and allows one to approximate arbitrary distribution laws using the first two moments. This approach makes it possible to calculate the average latency and higher-order moments for the specified systems in mathematical packets for a wide range of changes in traffic parameters. The method of spectral solution of the Lindley integral equation for the systems under consideration has made it possible to obtain a solution in closed form, and these obtained solutions are published for the first time.

## Author Biographies

### V. N. Tarasov, Volga State University of Telecommunications and Informatics, Samara, Russian Federation

Dr. Sc., Professor, Head of Department of Software and Management in Technical Systems

### N. F. Bakhareva, Volga State University of Telecommunications and Informatics, Samara, Russian Federation

Dr. Sc., Professor, Head of Department of Informatics and Computer Engineering

## References

Kleinrock L. Queueing Systems, Vol. I: Theory. New York: Wiley, 1975, 417 p.

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

Tarasov V. N. Queuing 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. Analysis and comparison of two queueing systems with hypererlangian input distributions, Radio Electronics, Computer Science, Control, 2018, No.4, pp. 61–70. DOI: 10.15588/1607-3274-2018-4-6

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

Do T. V., Chakka R., Sztrik J. Spectral Expansion Solution Methodology for QBD-M Processes and Applications in Future Internet Engineering, ICCSAMA, 2016, SCI 479, pp. 131–142. DOI: 10.1007/978-3-319-00293-4-11

Ma X., Wang Y., Zhu X., Liu W., Lan Q., Xiao W. A Spectral Method for Two-Dimensional Ocean Acoustic Propagation, J. Mar. Sci. Eng., 2021, No. 9, pp. 1–19. DOI: https://doi.org/ 10.3390/jmse9080892

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.

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, No. 2, pp. 88–93.

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.

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).

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

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,Vol. 93, 2019, pp. 153–190. DOI: 10.1007/s11134-019-09628-9

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

2022-04-05

## How to Cite

Tarasov, V. N., & Bakhareva, N. F. (2022). TWO PAIRS OF DUAL QUEUEING SYSTEMS WITH CONVENTIONAL AND SHIFTED DISTRIBUTION LAWS. Radio Electronics, Computer Science, Control, (1), 66. https://doi.org/10.15588/1607-3274-2022-1-8

## Section

Mathematical and computer modelling