# RESEARCH OF TWO SYSTEMS E2/H2/1 WITH ORDINARY AND SHIFTED DISTRIBUTIONS BY THE SPECTRAL DECOMPOSITION METHOD

## Authors

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

## Keywords:

Delayed system, E2/H2/1 system, the average waiting time, Laplace transform, the spectral decomposition method.

## Abstract

ABSTRACT Context. In the queueing theory, the studies of G/G/1 systems are relevant because it is impossible to obtain solutions for the waiting time in the final form in the general case with arbitrary laws of distributions of the input flow and of the service time. Therefore, the study of such systems for particular cases of input distributions is important. The problem of deriving a solution for the average waiting time in a queue in closed form for a pair of systems with ordinary and with shifted Erlang and hyperexponential 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 conventional and with shifted second-order Erlang and Hyperexponential input distributions.

Method. To solve this problem, we used the classical spectral decomposition method for solving the Lindley integral equation, which plays an important role in the theory of G/G/1 systems. This method allows obtaining a solution for the average waiting time for the considered systems in a closed form. For the practical application of the obtained results, the well-known probability theory moments method is used.

Results. For the first time, spectral expansions of the solution of the Lindley integral equation are obtained for two systems, with the help of which the formulas for the average waiting time in the queue are derived in closed form.

Conclusions. Spectral expansions of the solution of the Lindley integral equation for the systems under consideration are obtained and their complete coincidence is proved. Consequently, the formulas for the average waiting time in the queue for these systems are the same, but with modified parameters. It is shown that in the system with a delay in time, the average waiting time is less than in a conventional system. The resulting for waiting time formulas expand and supplement the known queuing theory incomplete formula for the average waiting time for G/G/1 systems with arbitrary laws distributions of input flow 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.

## Author Biography

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

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

## References

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

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. 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 10.15588/1607-32742019-1-7.

Tarasov V.N. The analysis of two queuing systems HE2/M/1 with ordinary and shifted input distributions, Radio Electronics, Computer Science, Control, 2019, Vol. 49, 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, Vol. 50, No. 3, pp. 71–79. DOI: 10.15588/1607-3274-2019-4-5.

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, Vol. 51, No. 4, pp. 50–58. DOI 10.15588/1607-3274-2019-4-5.

Bocharov P. P., Pechinkin A. V. Teoriya massovogo obsluzhivaniya. Moscow, Publishing House of Peoples' Friendship University, 1995, 529 p.

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) [Electronic resource]. Available at: https://tools.ietf.org/html/rfc3393.

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.

Liu X. Diffusion approximations for double-ended queues with reneging in heavy traffic, Queueing Systems: Theory and Applications, Springer, 2019, Vol. 91, No. 1, pp. 49–87. DOI: 10.1007/s11134-018-9589-7.

Poojary S., Sharma V. An asymptotic approximation for TCP CUBIC, Queueing Systems: Theory and Applications, 2019, Vol. 91, No. 1, pp. 171–203. DOI: 10.1007/s11134018-9594-x.

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. 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/s11134-019-096289.

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.