Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T12:04:31.477Z Has data issue: false hasContentIssue false
  • English
  • Français

Transient analysis of state-dependent queueing networks via cumulant functions

Part of: Special processes Operations research and management science Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Timothy I. Matis  and
Richard M. Feldman
Timothy I. Matis*
Affiliation:
New Mexico State University
Richard M. Feldman*
Affiliation:
Texas A & M University
*
∗ Postal address: Department of Industrial Engineering, New Mexico State University, Las Cruces, NM 88003-8001, USA.
∗∗ Postal address: Department of Industrial Engineering, Texas A & M University, College Station, TX 77843-3131, USA. Email address: richf@tamu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A new procedure that generates the transient solution of the first moment of the state of a Markovian queueing network with state-dependent arrivals, services, and routeing is developed. The procedure involves defining a partial differential equation that relates an approximate multivariate cumulant generating function to the intensity functions of the network. The partial differential equation then yields a set of ordinary differential equations which are numerically solved to obtain the first moment.

Keywords

Queuesqueueing networkscumulant generating functionstransient solutions

MSC classification

Primary: 90B22: Queues and service 60K25: Queueing theory 90B15: Network models, stochastic 65C99: None of the above, but in this section 90B99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 841 - 859
Copyright
Copyright © by the Applied Probability Trust 2001 

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

Bailey, N. T. J. (1964). Elements of Stochastic Processes. John Wiley, New York.Google Scholar
Bartlett, M. S. (1955). An Introduction to Stochastic Processes, with Special Reference to Methods and Applications. Cambridge University Press.Google Scholar
Dshalalow, J. H. (1997). Queueing systems with state-dependent parameters. In Frontiers in Queueing: Models and Applications in Science and Engineering, CRC Press, New York, pp. 61–116.Google Scholar
Gupta, U. C., and Rao, T. S. S. (1998). On the analysis of single server finite queue with state dependent arrival and service processes: M(n)/G(n)/i/K . OR Spektrum 20, 83–89.Google Scholar
Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions. John Wiley, New York.Google Scholar
Kendall, M. G. (1940). The derivation of multivariate sampling formulae from univariate formulae by symbolic operation. Ann. Eugenics 10, 92–402.Google Scholar
Kendall, M. G. (1994). Advanced Theory of Statistics. Halsted Press, New York.Google Scholar
Mandelbaum, A., and Pats, G. (1995). State-dependent queues: approximations and applications. In Stochastic Networks (IMA Vols Math. Appl. 71), eds Kelly, F. P. and Williams, R. J., Springer, New York, pp. 239–282.Google Scholar
Mandelbaum, A., and Pats, G. (1998). State-dependent stochastic networks. Part I: Approximations and applications with continuous diffusion limits. Ann. Appl. Prob. 8, 569–646.Google Scholar
Mandelbaum, A., Massey, W. A., and Reiman, M. I. (1998). Strong approximations for Markovian service networks. Queueing Systems 30, 149–201.Google Scholar
Matis, J. H., and Kiffe, T. R. (2000). Stochastic Population Models—A Compartmental Perspective. Springer, New York.Google Scholar
Matis, T. I., and Feldman, R. F. (2001). Using cumulant functions in queueing theory. To appear in Queueing Systems.Google Scholar
Miyazawa, M. (1997). Structure-reversibility and departure functions of queueing networks with batch movements and state dependent routing. Queueing Systems 25, 45–75.Google Scholar
Parthasarathy, P. R., and Lenin, R. B. (1997). On the exact transient solution of finite birth and death processes with specific quadratic rates. Math. Scientist 22, 92–105.Google Scholar
Parthasarathy, P. R., and Vijajayalaksmi, V. (1995). Transient solution of a state dependent many server poisson queue with balking. J. Appl. Statist. Sci. 2, 207–220.Google Scholar
Renshaw, E. (1998). Saddlepoint approximations for stochastic processes with truncated cumulant generating functions. IMA J. Math. Appl. Med. Biol. 15, 1–12.CrossRefGoogle Scholar
Rhee, Y., and Perros, H. G. (1996). Analysis of an open tandem queueing network with population constraints and constant service times. Europ. J. Operat. Res. 92, 99–111.Google Scholar
Serfozo, R. (1999). Introduction to Stochastic Networks. Springer, New York.Google Scholar
Smith, P. J. (1995). A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. Amer. Statistician 49, 217–218.Google Scholar
Towsley, D. (1980). Queueing network models with state-dependent routing. J. Assoc. Comput. Mach. 27, 323–337.Google Scholar
Whitt, W. (1999). Decomposition approximations for time-dependent Markovian queueing networks. Operat. Res. Lett. 24, 97–103.Google Scholar
Whittle, P. (1957). On the use of the normal approximation in the treatment of stochastic processes. J. R. Statist. Soc. B 19, 268–281.Google Scholar
Zheng, Q. (1998). Computing relations between statistical moments and cumulants. Tech. Rep., National Center for Toxicological Research, Jefferson, AR.Google Scholar
Zheng, Q. (1998). MomCumConvert. MATHEMATICA 4.0 Package. Available at http://www.mathsource.com/.Google Scholar