The $M^\theta/G/1/m$ queues with the time of service, depending on the length of the queue (in Ukrainian)

Author K. Yu. Zhernovyi
e-mail
Ivan Franko National University of Lviv

Abstract We study the $M^\theta/G/1/m$ queue with the time of service, depending on the length of the queue at the service initiation. By using Korolyuk's potential method, we derive the average duration of the busy time and the stationary distribution of the number of customers in such a system. Similar results for the $M^\theta/G/1/m$ queue with one threshold of switching of service modes are obtained.
Keywords M^theta/G/1/m queue; queue length dependent service time; stationary distribution of the number of customers
Reference 1. M. Finneran, Problems of the high-quality voice over IP-based networks: compression, delay and echo. Part 1, Elektronnyie Komponenty, 11 (2008), 83–85. (in Russian)

 2. K. Sriram, D.M. Lucantoni, Traffic smoothing effects of bit dropping in a packet voice multiplexer, IEEE Trans. Comm., 37 (1989), ¹7, 703–712.

 3. J.H. Dshalalow, Queueing systems with state dependent parameters, In: Frontiers in Queueing: Models and Appl. in Science and Eng., CRC Press, Boca Raton, FL, (1997), 61–116.

 4. B.D. Choi, Y.Ch. Kim, Y. Shin, Ch.E.M. Pearce, The M^X/G/1 queue with length dependent service times, J. of Appl. Math. and Stoch. Anal., 14 (2001), ¹4, 399–419.

 5. V.S. Korolyuk, Boundary Problems for Compound Poisson Processes, Naukowa Dumka, Kyiv, 1975. (in Russian)

 6. V.S. Korolyuk, M.S. Bratiychuk, B. Pirdzhanov, Boundary Problems for Random Walks, Ylym, Ashgabat, 1987. (in Russian)

 7. M.S. Bratiychuk, B. Borowska, Explicit formulae and convergence rate for the system M^alpha/G/1/N as N to infty, Stochastic Models, 18 (2002), ¹1, 71–84.

 8. A.M. Bratiychuk, The Study of Systems with Limited Queue, PhD thesis, Kyiv, 2008. (in Ukrainian)

 9. M. Bratiychuk, Yu. Zhernovyi, Study of M/G/1/m and M/G/1 queues with group arrivals and threshold blocking of an input flow, Visnyk Lviv. Univ., Ser.Mech-Math., 71 (2010), 26–39. (in Ukrainian)

 10. K.Yu. Zhernovyi, Investigation of the M^theta/G/1/m system with service regime switchings and threshold blocking of the input flow, J. of Communicat. Technology and Electronics, 56 (2011), ¹12, 1570–1584.

 11. K.Yu. Zhernovyi, Stationary characteristics of the M^theta/G/1/m system with the threshold functioning strategy, J. of Communicat. Technology and Electronics, 56 (2011), ¹12, 1585–1596.

 12. Yu.V. Zhernovyi, Simulation of Queueing Systems, Lviv. Nats. Univ., 2007. (in Ukrainian)
Pages 93-105
Volume 38
Issue 1
Year 2012
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML