×

Analysis of a discrete-time queue with load dependent service under discrete-time Markovian arrival process. (English) Zbl 1304.60105

Summary: In the study of normal queueing systems, the server’s average service times are generally assumed to be constant. However, in numerous applications this assumption may not be valid. To prevent congestion in overload control telecommunication networks, the transmission rates vary depending on the number of packets waiting in the queue. As traffics in telecommunication networks are of bursty nature and correlated, we assume that arrivals follow the discrete-time Markovian arrival process. This paper analyzes a queueing model in which the server changes its service times (rates) only at the beginning of service depending on the number of customers waiting in the queue. We obtain the steady-state probabilities at various epochs and some performance measures. In addition, varieties of numerical results are discussed to display the effect of the system parameters on the performance measures.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Al-Seedy, R. O., Analytical solution of the state-dependent Erlangian queue: \(M / E_j / 1 / N\) with balking, Microelectronics Reliability, 36, 2, 203-206 (1996)
[2] Altman, E.; Avrachenkov, K.; Barakat, C.; Núñez-Queija, R., State-dependent \(M / G / 1\) type queueing analysis for congestion control in data networks, Computer Networks, 39, 789-808 (2002)
[3] Blondia, c., A discrete time batch Markovian arrival process as B-ISDN traffic model, Belgian Journal of Operations Research, Statistics and Computer Science, 32, 3-23 (1993) · Zbl 0781.60097
[4] Bruneel, H.; Kim, B. G., Discrete-time models for communication systems including ATM (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Boston
[5] Chaudhry, M. L.; Gupta, U. C., On the analysis of the discrete-time \(G e o m(n) / G(n) / 1 / N\) queue, Probability in the Engineering and Informational Sciences, 10, 3, 415-428 (1996) · Zbl 1095.60511
[6] Chaudhry, M. L.; Gupta, U. C., Queue length distributions at various epochs in discrete-time \(D-M A P / G / 1 / N\) queue and their numerical evaluations, International Journal of Information and Management Sciences, 14, 3, 67-83 (2003) · Zbl 1046.90018
[7] Chaudhry, M. L.; Gupta, U. C.; Templeton, J. G.C., On the relations among the distributions at different epochs for discrete-time \(G I / G e o m / 1\) queues, Operations Research Letters, 18, 247-255 (1996) · Zbl 0855.90057
[8] Choi, B. D.; Choi, D. I., Queueing system with queue length dependent service times and its application to cell discarding scheme in ATM networks, IEE Proceedings of Communications, 143, 1, 5-11 (1996)
[9] Choi, D. I.; Kim, T. S.; Lee, S., Analysis of an \(M M P P / G / 1 / K\) queue with queue length dependent arrival rates, and its application to preventive congestion control in telecommunication networks, European Journal of Operational Research, 187, 2, 652-659 (2008) · Zbl 1149.90038
[10] Choi, D. I.; Knessl, C.; Tier, C., A queueing system with queue length dependent service times, with applications to cell discarding in ATM networks, Journal of Applied Mathematics and Stochastic Analysis, 12, 1, 35-62 (1999) · Zbl 0936.60086
[11] Grassmann, W. K.; Taksar, M. I.; Heyman, D. P., Regenerative analysis and steady state distributions for Markov chains, Operations Research, 33, 5, 1107-1116 (1985) · Zbl 0576.60083
[12] Gravey, A.; Hébuterne, G., Simultaneity in discrete time single server queues with Bernoulli inputs, Performance Evaluation, 14, 123-131 (1992) · Zbl 0752.60079
[13] Gray, W. J.; Wang, P.; Scott, M., An \(M / G / 1\)-type queuing model with servic times depending on queue length, Applied Mathematical Modelling, 16, 12, 652-658 (1992) · Zbl 0764.60097
[14] Gupta, U. C.; Samanta, S. K.; Sharma, R. K.; Chaudhry, M. L., Discrete-time single-server finite-buffer queues under discrete Markovian arrival process with vacations, Performance Evaluation, 64, 1-19 (2007)
[15] Gwiggner, C.; Nagaoka, S., Data and queueing analysis of a Japanese air-traffic flow, European Journal of Operational Research, 235, 1, 265-275 (2014) · Zbl 1305.90277
[16] Hunter, J. J., (Discrete-time models: techniques and applications. Discrete-time models: techniques and applications, Mathematical techniques of applied probability, Vol. II. (1983), Academic Press: Academic Press New York) · Zbl 0539.60065
[17] Jain, R., Congestion control and traffic management in ATM networks: recent advances and a survey, Computer Networks and ISDN System, 28, 13, 1723-1738 (1996)
[18] Johnson, M. A.; Narayana, S., Descriptors of arrival process burstiness with application to the discrete time Markovian arrival process, Queueing Systems, 23, 107-130 (1996) · Zbl 0879.60105
[19] Kim, N. K.; Chang, S. H.; Chae, K. C., On the relationships among queue lengths at arrival, departure, and random epochs in the discrete-time queue with D-BMAP arrivals, Operations Research Letters, 30, 25-32 (2002) · Zbl 1002.60085
[20] Liu, D.; Neuts, M. F., A queueing model for an ATM rate control scheme, Telecommunication Systems, 2, 321-348 (1994)
[21] Rosenshine, M., Queues with state-dependent service times, Transportation Research, 1, 2, 97-104 (1967)
[22] Sriram, K.; Lucantoni, D. M., Traffic smoothing effects of bit dropping in a packet voice multiplexer, IEEE Transactions on Communications, 37, 7, 703-712 (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.