Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T08:55:58.800Z Has data issue: false hasContentIssue false
  • English
  • Français

ANALYTICALLY EXPLICIT RESULTS FOR THE GI/C-MSP/1/∞ QUEUEING SYSTEM USING ROOTS

Published online by Cambridge University Press:  27 April 2012

M. L. Chaudhry ,
S. K. Samanta  and
A. Pacheco
M. L. Chaudhry
Affiliation:
Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston, Ont., CanadaK7K 7B4 E-mail: [email protected]
S. K. Samanta
Affiliation:
Department of Mathematics and CEMAT, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal E-mail: [email protected]; [email protected]
A. Pacheco
Affiliation:
Department of Mathematics and CEMAT, Instituto Superior Técnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal E-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we present (in terms of roots) a simple closed-form analysis for evaluating system-length distribution at prearrival epochs of the GI/C-MSP/1 queue. The proposed analysis is based on roots of the associated characteristic equation of the vector-generating function of system-length distribution. We also provide the steady-state system-length distribution at an arbitrary epoch by using the classical argument based on Markov renewal theory. The sojourn-time distribution has also been investigated. The prearrival epoch probabilities have been obtained using the method of roots which is an alternative approach to the matrix-geometric method and the spectral method. Numerical aspects have been tested for a variety of arrival- and service-time distributions and a sample of numerical outputs is presented. The proposed method not only gives an alternative solution to the existing methods, but it is also analytically simple, easy to implement, and computationally efficient. It is hoped that the results obtained will prove beneficial to both theoreticians and practitioners.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 26 , Issue 2 , April 2012 , pp. 221 - 244
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1.Adan, I. & Zhao, Y. (1996). Analyzing GI/E r/1 queues. Operations Research Letters 19(4): 183–190.CrossRefGoogle Scholar
2.Albores-Velasco, F.J. & Tajonar-Sanabria, F.S. (2004). Analysis of the GI/MSP/c/r queueing system. Information Processes 4: 46–57.Google Scholar
3.Alfa, A.S., Xue, J., & Ye, Q. (2000). Perturbation theory for the asymptotic decay rates in the queues with Markovian arrival process and/or Markovian service process. Queueing Systems 36(4): 287–301.CrossRefGoogle Scholar
4.Bocharov, P.P., D'Apice, C., Pechinkin, A., & Salerno, S. (2003). The stationary characteristics of the G/MSP/1/r queueing system. Automation and Remote Control 64(2): 288–301.CrossRefGoogle Scholar
5.Chaudhry, M.L. (1965). Correlated queueing. Canadian Operational Research Society 3: 142–151.Google Scholar
6.Chaudhry, M.L. (1991). QPACK software package. Ontario, Canada: A&A Publications.Google Scholar
7.Chaudhry, M.L., Harris, C.M., & Marchal, W.G. (1990). Robustness of rootfinding in single-server queueing models. INFORMS Journal on Computing 2: 273–286.CrossRefGoogle Scholar
8.Chaudhry, M.L., Singh, G., & Gupta, U.C. (2012). A simple and complete computational analysis of MAP/R/1 queue using roots. Methodology and Computing in Applied Probability. DOI: 10.1007/s11009-011-9266-3Google Scholar
9.Chaudhry, M.L. & Templeton, J.G.C. (1983). A first course in bulk queues. New York: John Wiley & Sons.Google Scholar
10.Çinlar, E. (1975). Introduction to stochastic process. NJ: Prentice Hall.Google Scholar
11.Daigle, J.N. (2005). Queueing theory with applications to packet telecommunication. Berlin: Springer-Verlag.CrossRefGoogle Scholar
12.Dudin, A.N. & Klimenok, V.I. (1996). Queueing system with passive servers. Journal of Applied Mathematics and Stochastic Analysis 9(2): 185–204.CrossRefGoogle Scholar
13.Ferreira, F. & Pacheco, A. (2008). Analysis of GI X/M(n)//N systems with stochastic customers acceptance policy. Queueing Systems 58: 29–55.CrossRefGoogle Scholar
14.Gail, H.R., Hantler, S.L., & Taylor, B.A. (1996). Spectral analysis of M/G/1 and G/M/1 type Markov chains. Advances in Applied Probability 28: 114–165.CrossRefGoogle Scholar
15.Gupta, U.C. & Banik, A.D. (2007). Complete analysis of finite and infinite buffer GI/MSP/1 queue—A computational approach. Operations Research Letters 35: 273–280.CrossRefGoogle Scholar
16.Janssen, A. & Leeuwaarden, J. (2005). Analytic computation schemes for the discrete time bulk service queue. Queueing Systems 50: 141–163.CrossRefGoogle Scholar
17.Lucantoni, D.M. & Neuts, M.F. (1994). Some steady-state distributions for the MAP/SM/1 queue. Stochastic Models 10: 575–598.CrossRefGoogle Scholar
18.Neuts, M.F. (1981). Matrix-geometric solutions in stochastic models: an algorithmic approach. Baltimore: Johns Hopkins University Press, Marcel Dekker.Google Scholar
19.Ozawa, T. (2006). Sojourn time distributions in the queue defined by a general QBD process. Queueing Systems 53(4): 203–211.CrossRefGoogle Scholar
20.Pacheco, A., Tang, L.C., & Prabhu, N.U. (2009). Markov-modulated processes and semiregenerative phenomena. Singapore: World Scientific.Google Scholar
21.Ramaswami, V. & Latouche, G. (1989). An experimental evaluation of the matrix-geometric method for the GI/PH/1 queue. Stochastic Models 5: 629–667.CrossRefGoogle Scholar
22.Shioda, S. (2003). Departure process of the MAP/SM/1 queue. Queueing Systems 44(1): 31–50.CrossRefGoogle Scholar
23.Tijms, H.C. (2003). A first course in stochastic models. New York: John Wiley and Sons.CrossRefGoogle Scholar