Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T16:19:52.288Z Has data issue: false hasContentIssue false
  • English
  • Français

MONOTONICITY AND CONVEXITY OF SOME FUNCTIONS ASSOCIATED WITH DENUMERABLE MARKOV CHAINS AND THEIR APPLICATIONS TO QUEUING SYSTEMS

Published online by Cambridge University Press:  12 December 2005

Hai-Bo Yu ,
Qi-Ming He  and
Hanqin Zhang
Hai-Bo Yu
Affiliation:
The College of Economics and Management, Beijing University of Technology, Beijing 100022, People's Republic of China, E-mail: [email protected]
Qi-Ming He
Affiliation:
Department of Industrial Engineering, Dalhousie University, Halifax, Nova Scotia, Canada B3J 2X4, E-mail: [email protected]
Hanqin Zhang
Affiliation:
Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, People's Republic of China, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Motivated by various applications in queuing theory, this article is devoted to the monotonicity and convexity of some functions associated with discrete-time or continuous-time denumerable Markov chains. For the discrete-time case, conditions for the monotonicity and convexity of the functions are obtained by using the properties of stochastic dominance and monotone matrix. For the continuous-time case, by using the uniformization technique, similar results are obtained. As an application, the results are applied to analyze the monotonicity and convexity of functions associated with the queue length of some queuing systems.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 20 , Issue 1 , January 2006 , pp. 67 - 86
Copyright
© 2006 Cambridge University Press

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

Çinlar, E. (1975). Introduction to stochastic processes. Englewood Cliffs, NJ: Prentice Hall.
Cohen, J.W. (1982). The single server queue, rev. ed. Amsterdam: North-Holland.
Dijk, N.M.V. (1990). On a simple proof of uniformization for continuous and discrete-state continuous-time Markov chains. Advances in Applied Probability 22: 749–750.Google Scholar
Doorn, E.V. (1980). Stochastic monotonicity and queueing applications of birth–death processes. New York: Springer-Verlag.
Dyer, M.E. & Proll, L.G. (1977). On the validity of marginal analysis for allocating servers in M/M/c queues. Management Sciences 23: 1019–1022.Google Scholar
Gross, D. & Miller, D.R. (1984). The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operations Research 32: 343–361.Google Scholar
He, Q.M. (1999). Partial orders and the matrix R in matrix analytic methods. SIAM Journal of Matrix Analysis and Its Applications 20: 871–885.Google Scholar
Hsu, G.H. (1985). Stochastic service systems. Beijing: Science Press.
Hwang, G.U., Choi, B.D., & Kim, J.K. (2002). The waiting time analysis of a discrete-time queue with arrivals as a discrete autoregreessive process of order 1. Journal of Applied Probability 39: 619–629.Google Scholar
Hwang, G.U. & Sohraby, K. (2003). On the exact analysis of a discrete-time queueing system with autoregressive inputs. Queueing Systems 43: 29–41.Google Scholar
Kemeny, J.G., Snell, J.L., & Knapp, A.W. (1976). Denumerable Markov chains. New York: Springer-Verlag.
Li, H.J. & Xu, S.H. (2000). On the dependence structure and bounds of correlated parallel queues and their applications to synchronized stochastic systems. Journal of Applied Probability 37: 1020–1043.Google Scholar
Lindvall, T. (1992). Lecture on the coupling method. New York: Wiley.
Medhi, J. (1991). Stochastic models in queueing theory. London: Academic Press.
Meyn, S.P. & Tweedie, R.L. (1993). Markov chains and stochastic stability. New York: Springer-Verlag.
Neuts, M.F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Baltimore: The Johns Hopkins University Press.
Ross, S.M. (1983). Stochastic processes. New York: Wiley.
Shaked, M. & Shanthikumar, J.G. (1994). Stochastic orders and their applications. New York: Academic Press.
Shanthikumar, J.G. & Yao, D.D. (1991). Multiclass queueing systems: Polymatroidal structure and optimal scheduling control. Operations Research 40: 293–299.Google Scholar
Shanthikumar, J.G. & Yao, D.D. (1992). Strong stochastic convexity: Closure properties and applications. Journal of Applied Probability 28: 131–145.Google Scholar
Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: Wiley.
Wang, C.L. (1999). On the transient delays of M/G/1 queues. Journal of Applied Probability 36: 882–893.Google Scholar
Weber, R.R. (1980). On the marginal benefit of adding servers to G/GI/m queues. Management Sciences 26: 946–951.Google Scholar