Cost and Profit Analysis of Markovian Queuing System with Two Priority Classes: A Computational Approach

Authors: S. S. Mishra, D. K. Yadav

Abstract:

This paper focuses on cost and profit analysis of single-server Markovian queuing system with two priority classes. In this paper, functions of total expected cost, revenue and profit of the system are constructed and subjected to optimization with respect to its service rates of lower and higher priority classes. A computing algorithm has been developed on the basis of fast converging numerical method to solve the system of non linear equations formed out of the mathematical analysis. A novel performance measure of cost and profit analysis in view of its economic interpretation for the system with priority classes is attempted to discuss in this paper. On the basis of computed tables observations are also drawn to enlighten the variational-effect of the model on the parameters involved therein.

Keywords: Cost and Profit, Computing, Expected Revenue, Priority classes

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1072387

References:

[1] A. M. K. Tarabia, "Analysis of M/M/1 Queuing system with two priority classes", Opsearch, vol.44, No.4, p.346-365, 2007.
[2] Afeche, Philipp, Mendelson and Hairm, "Pricing and priority auctions in queuing system with a generalized delay cost structure", Management Sciences, Article, 2004.
[3] C. Heathcofe, "The time-dependent problem for a queue with preemptive priorities", Operations Research, vol.7, p.670-680, 1959.
[4] Chapra, S. C. and Canale, R. P, Numerical Methods for Engineers, WBC/Mc Graw-Hill, USA, 1998.
[5] F.F. Stephan, "Two queues under preemptive priority with Poisson arrival and service rates", Operations Research, vol.6, p.399-418, 1958.
[6] G. Bitran and R. Caldentey, "Two-class priority system with statedependent arrivals", Queueing System, vol.40, 4, 2002.
[7] Gross, D. and Harris, C. M., Fundamental for queuing theory, John Wiley New- York; 1974.
[8] H. A. Taha, Operations Research, An Introduction, Prentice-Hall of India, 1997.
[9] Harrison, P. G. Zhang, Y., "Delay analysis of modulated traffic", 18th IEEE International Symposium, Sept.2005.
[10] J. Walraevens, S. Wittevrangel and H. Bruneel, "A discrete-time priority queue with train arrivals", Stochastic models vol.23, Issu.3, p.489-512, 2007.
[11] J. Y. Barry, "A priority queueing problem", Opre.Res.vol.4, p.385-386, 1956.
[12] Jeffery, J. Leader, Numerical Analysis and Scientific Computation; Pearson International addition New-York, 2004.
[13] Ke, J. C. and Wang, K. H., "Cost analysis of the M/M/R machine repair problem with balking reneging and server breakdowns", Journal of the Operational Research Society, vol.50, p.275-282,1999.
[14] Koole, G., "Assigning a single server to inhomogeneous queues with switching costs", Theoretical Computer Science, 182, p.203-216, 1997.
[15] Larson, R.C. and C. Schaak, "An N-server cut-off priority queue", Oper.Res.vol.34, p. 257-266, 1986.
[16] Miller, R.G., "Priority queues", Ann.Math.Statist.Vol.31, p.86-103, 1960.
[17] M.M. Ali and X. Song, "A performance analysis of a discrete-time priority queuing system with correlated arrivals", Performance Evaluation,vol.57, Issu.3, p.307-339, 2004.
[18] M. F .Neuts, Matrix-geometric solutions in stochastic models; The Johs Hopkins University Press, Baltimore, 1981.
[19] Morse, P. M; Queues, Inventories and Maintenance. Wiley, New York,1958.
[20] Miller, R.D., "Computation of steady-state probabilities for M/M/1 priority queues", Operations Research, vol.29, p.945-958, 1981.
[21] Mishra, S. S. and Pal, S. "A computational approach to the M/M/1/N interdependent queuing model with controllable arrival rates", Journal of Indian Statistical Association, vol.41, 25-35, 2003.
[22] Mishra S.S. and Yadav D. K, "Cost and Profit analysis of M/Ek /1 queuing system with removable service station", Bulgarian Journal of Applied Mathematical Sciences, vol. 2, No. 56, 2777-2784 , 2008.
[23] Mishra, S. S. and Pandey, N. K., "Cost analysis of bulk queuing model M/M (a, b)/2 for non-identical servers with vacations", International Journal of Management and Systems, vol.20, p.291-300,2004.
[24] Mishra S.S. and Mishra V., "The Cost Analysis of Machine Interference Model with Balking, Reneging and Spares", OPSEARCH, 42, 3, 35-46 (2004).
[25] Mishra S.S. and Yadav D. K, "Computational approach to profit optimization of a loss-queueing system", Communicated to International Journal of Contemporary Engineering Sciences, 2008.
[26] Mishra S S and Shukla D C, "A computational approach to the cost analysis of machine interference model", American Journal of Mathematical and Management Sciences, 2008, to be published.
[27] S. S. Franti," Algorithms for a dynamic priority queue with two types of customers", Ph.D. Thesis, Drexel University Philandelphia, 1985.
[28] S. Drekic and G. D. Woolford, "A preemptive priority queues with balking", Eur.J.Oper.Res.vol.164, p.387-401, 2005.
[29] T. Nishida, "Approximate analysis for heterogeneous multiserver systems with priority jobs", Performance Evaluation, vol.15, p.77-88, 1992.
[30] T. Katayama and K. Kobayashi, "Analysis of a no preemptive priority queue with exponential time and server vacations", Performance evaluation; vol.64, Issu.6, p.495-506, Jul.2007.
[31] Wang, K. H. and Wu, J. D., "Cost analysis of the M/M/R machine repair problem with spares and two modes of failures", Journal of the Operational Research society, vol.46, p.783-790, 1995.
[32] Wang, K. H. and Ming, Y., "Profit Analysis of M/Ek/1 machine repair problem with a non-reliable service station", Computers and Industrial Engineering, vol.32, p. 587-594, 1997.
[33] Yue, D., Yue, W., and Sun, Y.," Performance analysis of an M/M/C/N queueing system with balking, reneging, and synchronous vacation of partial servers". International Symposium on OR and its Applications, Chaina, 2006.