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




Delayed system, shifted distributions, Laplace transform, Lindley integral equation, spectral decomposition method


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 two 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.
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.

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


Kleinrock L. Teoriya massovogo obsluzhivaniya. Moscow, Mashinostroeinie Publ, 1979, 432 p.

Tarasov V. N., Bakhareva N. F., Blatov I. A. Analysis and calculation of queuing system with delay, Automation and

Remote Control, 2015, Vol. 52, 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.

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.

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.

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.,

–1 2019, Vol. 18, No. 1, pp. 279–311. DOI:https://doi.org/10.1137/18M1170637.

Tarasov V. N., Bahareva N. F., Gorelov G. A., Malakhov S.V. Analiz vhodiaschego trafika na urovne treh momentov raspredeleniy, Informacionnye technologii, 2014, No. 9, pp.54–59.

Tarasov V. N., Bahareva N. F. Obobshchennaya dvumepnaya diffuzionnaya model’ massovogo obsluzhivaniya tipa GI/G/1, Telekommunikacii, 2009, No. 7, pp. 2–8.

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, Nauchnotekhnicheskij

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.




How to Cite

Tarasov, V. N. (2019). QUEUEING SYSTEMS WITH DELAY. Radio Electronics, Computer Science, Control, (3), 55–63. https://doi.org/10.15588/1607-3274-2019-3-7



Mathematical and computer modelling

Most read articles by the same author(s)

> >>