Skip to main content Accessibility help


  • Hai-Bo Yu (a1), Qi-Ming He (a2) and Hanqin Zhang (a3)


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.



Hide All


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: 749750.
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: 10191022.
Gross, D. & Miller, D.R. (1984). The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operations Research 32: 343361.
He, Q.M. (1999). Partial orders and the matrix R in matrix analytic methods. SIAM Journal of Matrix Analysis and Its Applications 20: 871885.
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: 619629.
Hwang, G.U. & Sohraby, K. (2003). On the exact analysis of a discrete-time queueing system with autoregressive inputs. Queueing Systems 43: 2941.
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: 10201043.
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: 293299.
Shanthikumar, J.G. & Yao, D.D. (1992). Strong stochastic convexity: Closure properties and applications. Journal of Applied Probability 28: 131145.
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: 882893.
Weber, R.R. (1980). On the marginal benefit of adding servers to G/GI/m queues. Management Sciences 26: 946951.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed