Abstract
A queue system with batch service is investigated. The batch size depends on the state both of the system and the state of some random environment. The server state changes under the action of the random environment. When the system is empty, a birth-death process acts as the random environment, whereas when the system is busy the random environment is disregarded and affects only the initial service probabilistic characteristics (in this sense, the random environment in the busy state can be regarded as arbitrary). An analytical model of the system is constructed and studied in terms Laplace (Laplace–Stieltjes) images. The generating function of the queue length is derived.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Purdue, P., The M/M/1 Queue in a Markovian Environment, Oper. Res., 1974, vol. 22, no. 3, pp. 562–569.
Neuts, M., Matrix-Geometric Solution in Stochastic Models, London: Wiley, 1981.
Halfin, S., The Backlog of Data in Buffers with Variable Input and Output Rates, Perform. Comput. Commun. Syst., 1984, pp. 307–319.
Helm, W.E. and Waldman, K.H., Optimal Control of Arrivals to Multiserver Queues in A Random Environment, J. Appl. Prob., 1984, pp. 602–615.
Dudin, A.N., A Server System with Variable Operation Mode, Avtom. Vychisl. Tekh., 1985, no. 2, pp. 27–29.
Fukuda, A., Analysis of Queuing Systems with Randomly Changing Population States, IECE Jpn., 1986, vol. J69-A, no. 8, pp. 925–932.
Bacelli, F. and Makowski, A.M., Stability and Bounds for Single-Server Queues in Random Environment, Commun. Statist. Stoch. Models 2, 1986, no. 2, pp. 281–291.
Regterschot, G.J.K. and De Smit, J.H.A., The Queue M/G/1 with Markov Modulated Arrivals and Services, Math. Oper. Res., 1986, vol. 11, no. 3, pp. 456–483.
Sotelo, W. and Fukuda, A., A Comparison of Synchronous and Asynchronous Time-Variant M/M/1 Models, Natl. Conv. Rec. IEICE, 1987, pp. 1123–1127.
Sotelo, W. and Fukuda, A., On Multiserver Queues with Synchronous Fluctuation of Traffic Intensity, IEEE Trans., 1987, vol. E70, no. 10, pp. 951–959.
Sotelo, W., Mukumoto, K., and Fukuda, A., On Multiserver Queues with m-Phase Synchronous Fluctuation of Traffic Intensity, Trans. IEICE, 1987, vol. E70, no. 12, pp. 1187–1193.
Gelenbe, E. and Rosenberg, C., Queues with Slowly Varying Arrival and Service Processes, Manag. Sci., 1990, vol. 36, no. 8, pp. 928–937.
Korotaev, I.A. and Spivak, L.R., Queueing Systems in a Semi-Markovian Random Environment, Avtom. Telemekh., 1992, no. 7, pp. 86–92.
Lucantoni, D.M., New Results of the Single Server Queue with a Batch Markov Arrival Process, Commun. Statist. Stoch. Models, 1991, vol. 7, no. 1, pp. 1–46.
Lucantoni, D.M., Choudhury, G.L., and Whitt, W., The Transient BMAP/G/1 Queue, Commun. Statist. Stoch. Models, 1994, vol. 10, no. 1, pp. 145–182.
Dudin, A.N. and Klimenok, V.I., Computation of Characteristics of a Single-Server Queueing System in a Markov Synchronous Random Environment, Avtom. Telemekh., 1997, no. 1, pp. 74–84.
Dudin, A.N. and Klimenok, V.I., A BMAP/G/1 Queue System with Alternating Operation Mode, Avtom. Telemekh., 1999, no. 10, pp. 97–107.
Gnedenko, B.V. and Kovalenko, I.N., Vvedenie v teoriyu massovogo obsluzhivaniya (Introduction to Queueing Theory), Moscow: Nauka, 1987.
Mikadze, I.S., A Queueing System with Time and Information Redundancy, Kibern. Sist. Anal., 1991, no. 5, pp. 163–176.
Kakubava, R.V. and Mikadze, I.S., Redundant Queueing System: An Analysis, Avtom. Telemekh., 1984, no. 1, pp. 160–166.
Kakubava, R.V. and Mikadze, I.S., Redundant Queueing System, Kibernetika, 1985, no. 3, pp. 92–98.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kakubava, R.V. Analysis of Queues under Batch Service in an M/G/1 System in a Random Environment. Automation and Remote Control 62, 782–788 (2001). https://doi.org/10.1023/A:1010278924923
Issue Date:
DOI: https://doi.org/10.1023/A:1010278924923