COMPARATIVE ANALYSIS OF TWO QUEUING SYSTEMS M/HE2/1 WITH ORDINARY AND WITH THE SHIFTED INPUT DISTRIBUTIONS
Keywords:Hypererlangian and exponential distribution laws, shifted distributions, Lindley integral equation, spectral decomposition method, Laplace transform.
AbstractContext. In the queueing theory, studies of particular systems of the M/G/1 type are relevant in that they are still actively used in
the modern theory of teletraffic. The problem of finding a solution for the mean waiting time in a queue in the closed form of two
systems with ordinary and shifted exponential and hypererlangian input distributions is considered.
Objective. Obtaining a solution for the main system characteristic – for the average waiting time in a queue for two queuing systems
of type M/G/1 and G/G/1 with conventional and offset exponential and hypererlangian input distributions.
Method. To solve this problem, we use the classical method of spectral decomposition of the solution of the Lindley integral
equation. This method allows to obtain a solution for the average waiting time for the systems under consideration in 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. Spectral decompositions of the solution of an integral equation of Lindley for couple of systems by means of which
formulas for the average time of waiting in queue in the closed form are received. The shifted exponential distribution transforms the
system M/G/1 into the system G/G/1.
Conclusions. The spectral decompositions 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.
These expressions expand and complement the known queuing theory formulas for the average waiting time for M/G/1 and
G/G/1 systems with arbitrary laws of input flow and service time distributions. 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 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.
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.
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 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.
Tarasov V.N. Analysis of queues with hyperexponential arrival distributions. Problems of Information Transmission,
, Vol. 52, No. 1, pp. 14–23. DOI:10.1134/S0032946016010038.
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
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 queuing-networks using the two-dimensional diffusionapproximation,
Automation and Remote Control, 1983, Vol. 44, No. 8, pp. 1026–1034.
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