• Published : 2009.07.31


We analyze $MAP_1,\;MAP_2$/G/1 finite queues with service scheduling function dependent upon queue lengths. The customers are classified into two types. The arrivals of customers are assumed to be the Markovian Arrival Processes (MAPs). The service order of customers in each buffer is determined by a service scheduling function dependent upon queue lengths. Methods of embedded Markov chain and supplementary variable give us information for queue length of two buffers. Finally, the performance measures such as loss probability and mean waiting time are derived. Some numerical examples also are given with applications in telecommunication networks.


$MAP_1$;$MAP_2$/G/1;service scheduling function;loss;mean delay


  1. S. Asmussen and G. Koole, Marked point processes as limits of Markovian arrival streams, J. Appl. Probab. 30 (1993), no. 2, 365–372
  2. D. I. Choi, B. D. Choi, and D. K. Sung, Performance analysis of priority leaky bucket scheme with queue-length-threshold scheduling policy, IEE Proc. Comm. 145 (1998), no. 6, 395–401
  3. B. D. Choi, Y. C. Kim, D. I. Choi, and D. K. Sung, An analysis of M, MMPP/G/1 queue with QLT scheduling policy and Bernoulli schedule, IEICE Tran. on Comm. E81-B (1998), no. 1, 13–22
  4. D. I. Choi and Y. Lee, Performance analysis of a dynamic priority queue for traffic control of bursty traffics in ATM networks, IEE Proc. Commun. 148 (2001), no. 6, 395–401
  5. E. Cinlar, Markov renewal theory, Advances in Appl. Probability 1 (1969), 123–187
  6. S. S. Fratini, Analysis of a dynamic priority queue, Comm. Statist. Stochastic Models 6 (1990), no. 3, 415–444
  7. A. Graham, Kronecker Products and Matrix Calculus: with applications, Ellis Horwood Series in Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1981
  8. C. Knessl, D. I. Choi, and C. Tier, A dynamic priority queue model for simultaneous service of two traffic types, SIAM J. Applied Mathematics 63 (2002), no. 2, 398–422
  9. D. M. Lucantoni, K. S. Meier-Hellstern, and M. F. Neuts, A single-server queue with server vacations and a class of nonrenewal arrival processes, Adv. in Appl. Probab. 22 (1990), no. 3, 676–705
  10. H. Nassar and H. Al. Mahdi, Queueing analysis of an ATM multimedia multiplexer with nonpreemptive priority, IEE Proc. Commun. 150 (2003), no. 3, 189–196
  11. D. I. Choi, T. S. Kim, and S. M. Lee, Analysis of a queueing system with a general service scheduling function, with applications to telecommunication network traffic control, European J. Oper. Res. 178 (2007), no. 2, 463–471
  12. H. Heffes and D. M. Lucantoni, A Markov-modulated characterization of packetized voice and data traffic and related statistical multiplexer performance, IEEE J. Select. Areas Commun. 6 (1986), 856–868
  13. A. Sugahara, T. Takine, Y. Takahashi, and T. Hasegawa, Analysis of a nonpreemptive priority queue with SPP arrivals of high class, Performance Evaluation 21 (1995), no. 3, 215–238