Investigation of Probability Generating Function in an Interdependent M/M/1:(∞; GD) Queueing Model with Controllable Arrival Rates Using Rouche’s Theorem

Vishwa Nath Maurya

Shekhawati Engineering College, Rajasthan Technical University, Kota, India.

DOI: 10.4236/ojop.2012.12006                

Abstract

Present paper deals a M/M/1:(∞; GD) queueing model with interdependent controllable arrival and service rates where- in customers arrive in the system according to poisson distribution with two different arrivals rates-slower and faster as per controllable arrival policy. Keeping in view the general trend of interdependent arrival and service processes, it is presumed that random variables of arrival and service processes follow a bivariate poisson distribution and the server provides his services under general discipline of service rule in an infinitely large waiting space. In this paper, our central attention is to explore the probability generating functions using Rouche’s theorem in both cases of slower and faster arrival rates of the queueing model taken into consideration; which may be helpful for mathematicians and researchers for establishing significant performance measures of the model. Moreover, for the purpose of high-lighting the application aspect of our investigated result, very recently Maurya [1] has derived successfully the expected busy periods of the server in both cases of slower and faster arrival rates, which have also been presented by the end of this paper.

 

Keywords

Interdependent Queueing Model; Bivariate Poisson Process; Controllable Arrival Rates; Probability Generating Function; Laplace Transform; Rouche’s Theorem; Performance Measures

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	V. N. Maurya, “Determination of Expected Busy Periods in Faster and Slower Arrival Rates of an Interdependent Queueing Model with Controllable Arrival Rates,” International Journal of Engineering Research & Technology, Vol. 1, No. 5, 2012, pp. 1-5.
[2]	I. J. B. F. Adan and J. Resing, “A Note on a Fluid Queue Driven by an Queue,” The Annals of Mathematical Statistics, Vol. 27, 1956, pp. 768-778.
[3]	P. Purdue, “The Queue in a Markovian Environment,” Operations Research, Vol. 22, No. 3, 1974, pp. 562-569. doi:10.1287/opre.22.3.562
[4]	S. Pal, “The Cost Analysis of Interdependent Queueing Model with Controllable Arrival Rates,” Ganita Sandesh, Vol. 17, No. 2, 2003, pp. 13-20.
[5]	V. N. Maurya, “A Study of Use of Stochastic Processes in Some Queueing Models,” Ph. D. Thesis, Department of Mathematics & Statistics, Dr. Ram Manohar Lohia Avadh University, Faizabad, 2000.
[6]	V. N. Maurya, “Performance Analysis of an Queueing Model with Second Phase Optional Service and Bernoulli Vacation Schedule,” International Journal of Emerging Research in Management and Technology, Vol. 1, No. 2, 2012.
[7]	V. N. Maurya, “Sensitivity Analysis on Significant Performance Measures of Bulk Arrival Retrial Queueing Model with Second Phase Optional Service and Bernoulli Vacation Schedule,” International Journal of Engineering Research and Technology, Vol. 1, No. 10, 2012.
[8]	K. Srinivasa Rao, T. Shobha and P. Srinivas Rao, “The Interdependent Queueing Model with Controllable Arrival Rates,” Opsearch, Vol. 37, No. 1, 2000.
[9]	M. Thiagarajan and A. Srinivasan, “The Interdependent Queueing Model with Controllable Arrival Rates,” Journal of Decision and Mathematical Sciences, Vol. 11, No. 1-3, 2006, pp. 7-24.
[10]	V. N. Maurya, “On the Expected Busy Periods of an Interdependent Queueing Model Using Bivariate Poisson Process and Controllable Arrival Rates,” Proceedings of 2nd International Conference on Computer Research and Development, IEEE Computer Society, Kuala Lumpur, 7-10 May 2010, pp. 558-564. doi:10.1109/ICCRD.2010.119
[11]	L. Norman and S. K. Johnson, “Discrete Distributions,” John Wiley & Sons, New York, 1969.
[12]	D. Gross and C. M. Harris, “Fundamentals of Queueing Theory,” John Wiley, New York, 1974.