DOI: https://doi.org/10.15588/1607-3274-2019-1-7

RESEARCH OF QUEUEING SYSTEMS WITH SHIFTED ERLANGIAN AND EXPONENTIAL INPUT DISTRIBUTIONS

V.N. Tarasov, N. F. Bakhareva

Abstract


Context. In queuing theory, the study of G/G/1 systems is particularly relevant due to the fact that until now there is no solution in the final form
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 systems with
erlangian and exponential input distributions and for the same systems with shifted 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 conventional and shifted erlangian and exponential 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. The method of spectral decomposition of
the solution of 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 the three kinds of systems were first obtained with the
help of which the calculated expressions for the average waiting time in the queue for the above systems in a closed form were derived.
Conclusions. 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 due to the fact that 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 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. Similarly the situation and with the shifted exponential distributions is. In addition, the shifted exponential distribution contains two
parameters and allows one to approximate arbitrary distribution laws using the first two moments. 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. The method of spectral decomposition of the solution of the Lindley integral equation for the systems under consideration makes it
possible to obtain a solution in a closed form and these solutions are published for the first time.


Keywords


Erlangian and exponential distribution laws; Lindley integral equation; spectral decomposition method; Laplace transform.

References


Tarasov V. N., Bakhareva N. F., Blatov I. A. Analysis and calculation

of queuing system with delay, Automation and Remote Control,

, No. 11, pp. 1945–1951. DOI: 10.1134/S0005117915110041

Kleinrock L. Teoriya massovogo obsluzhivaniya. Moscow, Mashinostroeinie

Publ, 1979, 432p.

Brannstrom N. A Queueing Theory analysis of wireless radio systems.

Appllied to HS-DSCH. Lulea university of technology, 2004,

p.

Whitt W. Approximating a point process by a renewal process: two

basic methods. Operation Research, 1982, Vol. 30, No. 1, pp. 125–

Bocharov P. P., Pechinkin A. V. Teoriya massovogo obsluzhivaniya.

Moscow, Publishing House of Peoples’ Friendship University,

, 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., Kartashevskiy I. V. Opredelenie srednego vremeni

ozhidaniya trebovaniy v upravliaemoi sisteme massovogo obsluzhiavaniya

Н2/Н2/1, Sistemy upravleniya i informatsionniye

tehnologii, 2014, No. 3, pp. 92–95.

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.

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


GOST Style Citations


1. Тарасов В. Н. Анализ и расчет системы массового обслужива-
ния с запаздыванием / В. Н. Тарасов, Н. Ф. Бахарева, И. А. Бла-
тов // Автоматика и телемеханика. – 2015. – № 11. – С. 51–59.
2. Клейнрок Л. Теория массового обслуживания ; пер. с англ. под
редакцией В. И. Неймана / Л. Клейнрок. – М. : Машинострое-
ние, 1979. – 432 с.
3. Brannstrom N. A Queueing Theory analysis of wireless radio systems
/ N. Brannstrom. – Appllied to HS-DSCH, Lulea university of
technology, 2004. –79 p.
4. Whitt W. Approximating a point process by a renewal process: two
basic methods / W. Whitt // Operation Research. – 1982. – № 1. –
P. 125–147.
5. Бочаров П. П. Теория массового обслуживания / П. П. Бочаров,
А. В. Печинкин. – М. : Изд-во РУДН, 1995. – 529 c.
6. Тарасов В.Н. Исследование систем массового обслуживания с
гиперэкспоненциальными входными распределениями /
В. Н. Тарасов // Проблемы передачи информации. – 2016. –
№ 1. – С. 16–26.
7. Тарасов В. Н. Определение среднего времени ожидания требо-
ваний в управляемой системе массового обслуживания H2/H2/1
/ В. Н. Тарасов, И. В. Карташевский // Системы управления и
информационные технологии. – 2014. – № 3(57). – С. 92–96.
8. Анализ входящего трафика на уровне трех моментов распреде-
лений временных интервалов / [В. Н. Тарасов, Н.Ф. Бахарева,
Г. А. Горелов, С. В. Малахов] // Информационные техноло-
гии. – 2014. – № 9. – С. 54–59.
9. HTTPS://tools.ietf.org/html/rfc3393. RFC 3393 IP Packet Delay
Variation Metric for IP Performance Metrics (IPPM) (дата обра-
щения: 26.02.2016).
10. Myskja A. An improved heuristic approximation for the GI/GI/1
queue with bursty arrivals / A. Myskja // Teletraffic and datatraffic
in a Period of Change, ITC-13. Elsevier Science Publishers. –
1991. – P. 683–688.
11. Алиев Т. И. Основы моделирования дискретных систем /
Т. И. Алиев. – СПб. : СПбГУ ИТМО, 2009. – 363 с.
12. Алиев Т. И. Аппроксимация вероятностных распределений в
моделях массового обслуживания / Т. И. Алиев // Научно-
технический вестник информационных технологий, механики и
оптики. – 2013. – № 2(84). – С. 88–93.
13. Aras A. K. Many-server Gaussian limits for overloaded non-
Markovian queues with customer abandonment / A. K. Aras,
X. Chen & Y. Liu // Queueing Systems. – 2018. – Vol. 89,
No. 1. – P. 81–125. DOI: https://doi.org/10.1007/s11134-018-
9575-0
14. Jennings O. B. Comparisons of ticket and standard queues. /
O. B. Jennings & J. Pender // Queueing Systems. – 2016. –
Vol. 84, No. 1. – P. 145–202. DOI:
https://doi.org/10.1007/s11134-016-9493-y







Copyright (c) 2019 V.N. Tarasov, N. F. Bakhareva

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Address of the journal editorial office:
Editorial office of the journal «Radio Electronics, Computer Science, Control»,
National University "Zaporizhzhia Polytechnic", 
Zhukovskogo street, 64, Zaporizhzhia, 69063, Ukraine. 
Telephone: +38-061-769-82-96 – the Editing and Publishing Department.
E-mail: rvv@zntu.edu.ua

The reference to the journal is obligatory in the cases of complete or partial use of its materials.