Methodology and Computing in Applied Probability

, Volume 14, Issue 3, pp 863–882 | Cite as

On the Distributions of the State Sizes of the Continuous Time Homogeneous Markov System with Finite State Capacities

  • George VasiliadisEmail author


In the present paper we study the evolution of a continuous time homogeneous Markov system whose states have finite capacities. We assume that the members of the system who overflow, due to the finite state capacities, leave the system. In order to investigate the variability of the state sizes we provide the evaluation of the intensity matrix for any time point and then we derive a formula concerning the derivate of the moments of the state sizes. As a consequence the distributions of the state sizes can be evaluated. Moreover, an alternative method of calculating the distributions of the state sizes by means of the interval transition probabilities is given. Finally we examine the distribution of the time needed for the leavers to leave the system. The theoretical results are illustrated by a numerical example.


Stochastic population systems Continuous time homogeneous Markov models Markov systems 

AMS 2000 Subject Classification

Primary 90B70 62E99; Secondary 91D35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bartholomew DJ (1982) Stochastic models for social processes, 3rd edn. Wiley, New YorkzbMATHGoogle Scholar
  2. Gani J (1963) Formulae for projecting enrolments and degrees awarded in universities. J R Stat Soc A 126:400–409CrossRefGoogle Scholar
  3. Isaacson DL, Madsen RW (1976) Markov chains: theory and applications. Wiley, New YorkzbMATHGoogle Scholar
  4. McClean SI, Millard P (2006) Using Markov models to manage high occupancy hospital care. In: 3rd international IEEE conference on intelligent systems, pp 256–260Google Scholar
  5. McClean SI, McAlea B, Millard P (1998) Using a Markov reward model to estimate spend-down costs for a geriatric department. J Oper Res Soc 10:1021–1025Google Scholar
  6. McClean SI, Papadopoulou A, Tsaklidis G (2004) Discrete time reward models for homogeneous semi Markov systems. Commun Stat, Theory Methods 33(3):623–638MathSciNetzbMATHCrossRefGoogle Scholar
  7. Patoucheas PD, Stamou G (1993) Non-homogeneous Markovian models in ecological modelling: a study of the zoobenthos dynamics in Thermaikos Gulf, Greece. Ecol Model 66:197–215CrossRefGoogle Scholar
  8. Rogoff N (1953) Recent trends in occupational mobility. Free Press, GlencoeGoogle Scholar
  9. Taylor GJ, McClean SI, Millard P (2000) Stochastic model of geriatric patient bed occupancy behaviour. J R Stat Soc 163(1):39–48CrossRefGoogle Scholar
  10. Vasiliadis G, Tsaklidis G (2007) On the moments of the state sizes of the discrete time homogeneous Markov system with a finite state capacity. In: Recent adv. in stoch. modelling and data anal. World Scientific, Singapore, pp 190–197CrossRefGoogle Scholar
  11. Vasiliadis G, Tsaklidis G (2009) On the distributions of the state sizes of closed continuous time homogeneous Markov systems. Methodol Comput Appl Probab 11:561–582MathSciNetzbMATHCrossRefGoogle Scholar
  12. Vasiliadis G, Tsaklidis G (2011) On the distributions of the state sizes of the closed discrete-time homogeneous Markov system with finite state capacities (HMS/c). Markov Processes Relat Fields 17:91–118MathSciNetzbMATHGoogle Scholar
  13. Vassiliou P-CG (1997) The evolution of the theory of non-homogeneous Markov systems. Appl Stoch Models Data Anal 13(3–4):159–176MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

Personalised recommendations