Restricted admissibility of batches into an $M$/$G$/1 type bulk queue with modified Bernoulli schedule server vacations
ESAIM: Probability and Statistics, Tome 6 (2002) , pp. 113-125.

We investigate the steady state behavior of an $M$/$G$/1 queue with modified Bernoulli schedule server vacations. Batches of variable size arrive at the system according to a compound Poisson process. However, all arriving batches are not allowed into the system. The restriction policy differs when the server is available in the system and when he is on vacation. We obtain in closed form, the steady state probability generating functions for the number of customers in the queue for various states of the server, the average number of customers as well as their average waiting time in the queue and the system. Many special cases of interest including complete admissibility, partial admissibility and no server vacations have been discusssed. Some known results are derived as particular cases of our model.

DOI : https://doi.org/10.1051/ps:2002006
Classification : 60k25
Mots clés : steady state, compound Poisson process, Bernoulli schedule server vacations, exponential vacation periods, restricted admissibility of batches
@article{PS_2002__6__113_0,
author = {Madan, Kailash C. and Abu-Dayyeh, Walid},
title = {Restricted admissibility of batches into an $M$/$G$/1 type bulk queue with modified Bernoulli schedule server vacations},
journal = {ESAIM: Probability and Statistics},
pages = {113--125},
publisher = {EDP-Sciences},
volume = {6},
year = {2002},
doi = {10.1051/ps:2002006},
zbl = {1003.60083},
language = {en},
url = {http://www.numdam.org/item/PS_2002__6__113_0/}
}
Madan, Kailash C.; Abu-Dayyeh, Walid. Restricted admissibility of batches into an $M$/$G$/1 type bulk queue with modified Bernoulli schedule server vacations. ESAIM: Probability and Statistics, Tome 6 (2002) , pp. 113-125. doi : 10.1051/ps:2002006. http://www.numdam.org/item/PS_2002__6__113_0/

[1] N.T.J. Bailey, On queueing processes with bulk service. J. Roy. Statist. Soc. Ser. B 16 (1954) 80-87. | MR 63595 | Zbl 0055.36906

[2] U.N. Bhat, Imbedded Markov Chain analysis of single server bulk queues. J. Austral. Math. Soc. 4 (1964) 244-263. | MR 181016 | Zbl 0124.34203

[3] A. Borthakur, A Poisson queue with a general bulk service rule. J. Assam Sci. Soc. XIV (1971) 162-167.

[4] M.L Chaudhry and J.G.C. Templeton, A First Course in Bulk Queues. Wiley Inter Science, UK (1983). | MR 700827 | Zbl 0559.60073

[5] B.D. Choi and K.K. Park, The M/G/1 queue with Bernoulli schedule. Queueing Systems 7 (1990) 219-228. | MR 1079717 | Zbl 0706.60089

[6] J.W. Cohen, The Single Server Queue. North-Holland (1969). | MR 668697 | Zbl 0183.49204

[7] B.W. Conolly, Queueing at a single point with arrivals. J. Roy. Statist. Soc. Ser. B 22 (1960) 285-298. | MR 117798 | Zbl 0093.14704

[8] M. Cramer, Stationary distributions in queueing system with vacation times and limited service. Queueing Systems 4 (1989) 57-78. | Zbl 0664.60095

[9] B.T. Doshi, A note on stochastic decomposition in a GI/G/1 queue with vacations or set-up times. J. Appl. Probab. 22 (1985) 419-428. | MR 789364 | Zbl 0566.60090

[10] B.T. Doshi, Queueing systems with vacations-a survey. Queueing Systems 1 (1986) 29-66. | MR 896237 | Zbl 0655.60089

[11] S.W. Fuhrman, A note on the M/G/1 queue with server vacations. Oper. Res. 32 (1984). | MR 775265 | Zbl 0553.90046

[12] D. Gross and C.M. Harris, The Fundamentals of Queueing Theory, Second Edition. John Wiley & Sons, New York (1985). | MR 910687 | Zbl 0658.60122

[13] C.M. Harris, Some results of bulk arrival queues with state dependent service times. Management Sci. 16 (1970) 313-326. | Zbl 0191.50501

[14] A. Huang and D. Mcdonald, Connection admission control for constant bit rate traffic at a multi-buffer multiplexer using the oldest-cell-first discipline. Queueing Systems 29 (1998) 1-16. | MR 1643586 | Zbl 0913.90106

[15] N.K. Jaiswal, Time-dependent solution of the bulk service queueing problem. Oper. Res. 8 (1960) 773-781. | MR 125654 | Zbl 0105.11701

[16] B.R.K. Kashyap and M.L. Chaudhry, An Introduction to Queueing Theory. A&A Publications, Ontario, Canada (1988).

[17] J. Keilson and L.D. Servi, Oscillating random walk models for G1/G/1 vacation systems with Bernoulli schedules. J. Appl. Probab. 23 (1986) 790-802. | MR 855384 | Zbl 0612.60087

[18] L. Kleinrock, Queueing Systems, Vol. 1. Wiley, New York (1975). | Zbl 0334.60045

[19] T.T. Lee, M/G/1/N queue with vacation and exhaustive service discipline. Oper. Res. 32 (1984). | MR 865578 | Zbl 0559.90032

[20] Y. Levy and U. Yechiali, Utilization of idle time in an M/G/1 queueing system. Management Sci. 22 (1975) 202-211. | Zbl 0313.60067

[21] K.C. Madan, An M/G/1 Queue with optional deterministic server vacations. Metron LVII (1999) 83-95. | MR 1796302 | Zbl 0997.60509

[22] K.C. Madan, An M/G/1 queue with second optional service. Queueing Systems 34 (2000) 37-46. | MR 1769764 | Zbl 0942.90008

[23] K.C. Madan, On a single server queue with two-stage heteregeneous service and deterministic server vacations. Int. J. Systems Sci. 32 (2001) 837-844. | MR 1959614 | Zbl 1006.90021

[24] J. Medhi and A. Borthakur, On a two server bulk Markovian queue with a general bulk service rule. Cahiers Centre Études Rech. Opér. 14 (1972) 151-158. | MR 339361 | Zbl 0245.60068

[25] J. Medhi, Waiting time distribution in a Poisson queue with a general bulk service rule. Management Sci. 21 (1975) 777-782. | Zbl 0307.60076

[26] J. Medhi, Further results in a Poison queue under a general bulk service rule. Cahiers Centre Études Rech. Opér. 21 (1979) 183-189. | MR 544040 | Zbl 0411.60095

[27] J. Medhi, Recent Developments in Bulk Queueing Models. Wiley Eastern, New Delhi (1984).

[28] R. Nadarajan and G. Sankranarayanan, A bulk service queueing system with Erlang input. J. Indian Statist. Assoc. 18 (1980) 109-116. | MR 652010

[29] M.F. Neuts, A general class of bulk queues with Poisson input. Ann. Math. Statist. 38 (1967) 759-770. | MR 211495 | Zbl 0157.25204

[30] M.F. Neuts, An algorithmic solution to the GI/M/C queue with group arrivals. Cahiers Centre Études Rech. Opér. 21 (1979) 109-119. | MR 544033 | Zbl 0423.60077

[31] M.F. Neuts, The M/G/1 queue with limited number of admissions or a limited admission period during each service time, Technical Report No. 978, University of Delaware (1984). | Zbl 0605.60081

[32] R.C. Rue and M. Rosenshine, Some properties of optimal control policies for enteries to an M/M/1 queue. Naval Res. Logist. Quart. 28 (1981) 525-532. | MR 626465 | Zbl 0534.60086

[33] S. Stidham Jr., Optimal control of arrivals to queues and networks of queues1982).

[34] M. Scholl and L. Kleinrock, On the M/G/1 queue with rest periods and certain service independent queueing disciplines. Oper. Res. 31 (1983) 705-719. | Zbl 0523.60088

[35] L.D. Servi, D/G/1 queue with vacation. Oper. Res. (1986).

[36] L.D. Servi, Average delay approximation of M/G/1 cyclic service queue with Bernoulli schedules. IEEE J. Sel. Areas Comm. (1986)

[37] J.G. Shanthikumar, On stochastic decomposition in the M/G/1 type queues with generalized vacations. Oper. Res. 36 (1988) 566-569. | MR 960257 | Zbl 0653.90023

[38] J.G. Shanthikumar and U. Sumita, Modified Lindley process with replacement: Dynamic behavior, asymptotic decomposition and applications. J. Appl. Probab. 26 (1989) 552-565. | MR 1010943 | Zbl 0698.60076

[39] H. Takagi, Queueing Analysis, Vol. 1: Vacation and Priority Systems. North- Holland, Amsterdam (1991). | MR 1149382 | Zbl 0744.60114

[40] M.H. Van Hoorn, Algorithms for the state probabilities in a general class of single server queueing systems with group arrivals. Management Sci. 27 (1981) 1178-1187. | Zbl 0465.90035