This paper presents the analysis of a discrete-time renewal input multiple vacations queue with state dependent service and changeover times under policy. The service times, vacation times, and changeover times are geometrically distributed. The server begins service if there are at least units in the queue and the services are performed in batches of minimum size and maximum size . At service completion instant, if the queue size is less than but not less than a secondary limit , the server continues to serve and takes vacation if the queue size is less than . The server is in changeover period whenever the queue size is at service completion instant and at vacation completion instant. Employing the supplementary variable and recursive techniques, we have derived the steady state queue length distributions at prearrival and arbitrary epochs. Based on the queue length distributions, some performance measures of the system have been discussed. A cost model has been formulated and optimum values of the service and vacation rates have been evaluated using genetic algorithm. Numerical results showing the effect of model parameters on the key performance measures are presented. 1. Introduction The interest in discrete-time queues in which time is slotted (divided into fixed-length of contiguous intervals) has got a spectacular growth with the arrival of the digital technologies. A fundamental motive to study discrete-time queues is that they are more appropriate than their continuous-time counterparts for analyzing computer and telecommunication systems, since nowadays these systems are more digital (such as a machine cycle time, bits, and packets) than analogical. In view of this they have become increasingly important due to their applications in the study of many computer and communication systems such as asynchronous transfer mode (ATM) multiplexers, broadband integrated services digital network (B-ISDN), circuit-switched time-division multiple access (TDMA) systems, slotted carrier-sense multiple access (CSMA) protocols, and traffic concentrators in which the time axis is divided into slots. Further, the advantage of analyzing a discrete-time queue is that one can obtain the continuous-time results from it as a limiting case but the converse is not true. However, from an applied and a theoretical point of view both discrete and continuous-time queueing models have importance. One can see Bruneel and Kim [1], Takagi [2], Tran-Gia et al. [3], and Robertazzi [4] for extensive treatments of various types of discrete-time queues and their applications. Batch
References
[1]
H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, Kluwer Academic Publishers, Boston, Mass, USA, 1993.
[2]
H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation. Vol. 3. Discrete-Time Systems, North-Holland, Amsterdam, The Netherlands, 1993.
[3]
P. Tran-Gia, C. Blondia, and D. Towsley, “Discrete-time models and analysis methods,” Performance Evaluation, vol. 21, no. 1-2, pp. 1–2, 1994.
[4]
T. G. Robertazzi, Computer Networks and Systems: Queueing Theory and Performance Evaluation, Telecommunication Networks and Computer Systems, Springer, New York, NY, USA, 1990.
[5]
J. Medhi, Stochastic Models in Queueing Theory, Academic Press, Boston, Mass, USA, 1991.
[6]
M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, John Wiley & Sons, New York, NY, USA, 1983.
[7]
U. C. Gupta and V. Goswami, “Performance analysis of finite buffer discrete-time queue with bulk service,” Computers & Operations Research, vol. 29, no. 10, pp. 1331–1341, 2002.
[8]
M. L. Chaudhry and S. H. Chang, “Analysis ofthe discrete-time bulk-service queue ,” Operations Research Letters, vol. 32, no. 4, pp. 355–363, 2004.
[9]
V. Goswami, J. R. Mohanty, and S. K. Samanta, “Discrete-time bulk-service queues with accessible and non-accessible batches,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 898–906, 2006.
[10]
N. Tian and Z. G. Zhang, “The discrete-time queue with multiple vacations,” Queueing Systems, vol. 40, no. 3, pp. 283–294, 2002.
[11]
V. Goswami and P. Vijaya Laxmi, “Discrete-time renewal input multiple vacation queue with accessible and non-accessible batches,” Opsearch, vol. 48, no. 4, pp. 335–354, 2011.
[12]
L. Tadj and C. Abid, “Optimal management policy for a single and bulk service queue under Bernoulli vacation schedules,” International Journal of Applied Decision Sciences, vol. 2, no. 3, pp. 262–274, 2009.
[13]
C. Baburaj, “A discrete time policy bulk service queue,” International Journal of Information and Management Sciences, vol. 21, no. 4, pp. 469–480, 2010.
[14]
X. Chao and A. Rahman, “Analysis and computational algorithm for queues with state-dependent vacations. I. ,” Journal of Systems Science & Complexity, vol. 19, no. 1, pp. 36–53, 2006.
[15]
P. R. Parthasarathy and R. Sudhesh, “Time-dependent analysis of a single-server retrial queue with state-dependent rates,” Operations Research Letters, vol. 35, no. 5, pp. 601–611, 2007.
[16]
J. H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, Mich, USA, 1975.
[17]
R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, John Wiley & Sons, Hoboken, NJ, USA, 2nd edition, 2004.
[18]
S. S. Rao, Engineering Optimization: Theory and Practice, John Wiley & Sons, New York, NY, USA, 20009.
[19]
N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, International Series in Operations Research & Management Science, 93, Springer, New York, NY, USA, 2006.
