Step semi-Markov models and application to manpower management
Barbu, Vlad Stefan1 ; D’Amico, Guglielmo2 ; Manca, Raimondo3 ; Petroni, Filippo4
1 Universitéde Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085, Avenue de l’Université, BP.12, 76801 Saint-Étienne-du-Rouvray, France.
2 Dipartimento di Farmacia, Università “G. d’Annunzio” di Chieti-Pescara, via dei Vestini 31, 66100 Chieti, Italy.
3 Dipartimento MEMOTEF, Università di Roma “La Sapienza”, via del Castro Laurenziano, 9, 00161 Rome, Italy.
4 Dipartimento di Scienze Economiche ed Aziendali, Universitá di Cagliari, via Sant’Ignazio, 17, 09123 Cagliari, Italy.
ESAIM: Probability and Statistics, Tome 20 (2016), pp. 555-571.

The purpose of this paper is to introduce a class of stochastic processes that we call step semi-Markov processes and to illustrate the modelling capacity of such processes in practical applications. The name of this process comes from the fact that we have a semi-Markov process and the transition between two states is done through several steps. We first introduce these models and the main quantities that characterize them. Then, we derive the recursive evolution equations for two-step semi-Markov processes. The interest of using this type of model is illustrated by means of an application in manpower planning.

DOI : 10.1051/ps/2016025
Classification : 60K15, 60K20, 90B25, 91B28
Mots-clés : Semi-Markov processes, manpower management, stochastic modelling
Barbu, Vlad Stefan 1 ; D’Amico, Guglielmo 2 ; Manca, Raimondo 3 ; Petroni, Filippo 4

1 Universitéde Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085, Avenue de l’Université, BP.12, 76801 Saint-Étienne-du-Rouvray, France.
2 Dipartimento di Farmacia, Università “G. d’Annunzio” di Chieti-Pescara, via dei Vestini 31, 66100 Chieti, Italy.
3 Dipartimento MEMOTEF, Università di Roma “La Sapienza”, via del Castro Laurenziano, 9, 00161 Rome, Italy.
4 Dipartimento di Scienze Economiche ed Aziendali, Universitá di Cagliari, via Sant’Ignazio, 17, 09123 Cagliari, Italy. 
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Barbu, Vlad Stefan; D’Amico, Guglielmo; Manca, Raimondo; Petroni, Filippo. Step semi-Markov models and application to manpower management. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 555-571. doi : 10.1051/ps/2016025. http://www.numdam.org/articles/10.1051/ps/2016025/

V.S. Barbu and N. Limnios, Semi-Markov chains and hidden semi-Markov models toward applications – their use in reliability and DNA analysis. Vol. 191 of Lect. Notes Stat. Springer, New York (2008). | MR | Zbl

D.J. Bartholomew, Stochastic models for social processes. Wiley, London, 3rd edition (1982). | MR | Zbl

D.J. Bartholomew, A.F. Forbes and S.I. McClean, Statistical Techniques For Manpower Planning. Wiley, London, 2nd edition (1991).

A. Charnes, W.W. Cooper, R.J. Niehaus and D. Sholtz, Multilevel models for career management and resource planning. Proc. of NATO Conference Manpower Planning Models. Cambridge, September (1971).

G. D’Amico, J. Janssen and R. Manca, Duration dependent semi-Markov models. Appl. Math. Sci. 5 (2011) 2097–2108. | MR | Zbl

G. D’Amico, G. Di Biase, F. Gismondi and R. Manca, The evaluation of generalized Bernoulli processes for salary lines construction by means of continuous time generalized non-homogeneous semi-Markov processes. Commun. Statis. – Theory and Methods 42 (2013) 2889–2901. | DOI | MR | Zbl

T. De Feyter, Modelling heterogeneity in manpower planning: dividing the personnel system into more homogeneous subgroups. Appl. Stoch. Models Bus. Ind. 22 (2006) 321–334. | DOI | MR | Zbl

T. De Feyter and M. Guerry, Evaluating recruitment strategies using fuzzy set theory in stochastic manpower planning. Stoch. Anal. Appl. 27 (2009) 1148–1162. | DOI | MR | Zbl

V.A. Dimitriou and N. Tsantas, Evolution of a time dependent Markov Model for training and recruitment decisions in manpower planning. Linear Algebra and its Applications 433 (2010) 1950–1972. | DOI | MR | Zbl

V.A. Dimitriou and N. Tsantas, The augmented semi-Markov system in continuous time, Commun. Stat.: Theory Methods 41 (2012) 88–107. | DOI | MR | Zbl

A.T. Ernst, H. Jiang, M. Krishnamoorthy and D. Sier, Staff scheduling and rostering: a review of applications, methods and models. Eur. J. Oper. Res. 153 (2004) 3–27. | DOI | MR | Zbl

A.C. Georgiou and N. Tsantas, Modelling recruitment training in mathematical human resource planning. Appl. Stoch. Models Bus. Ind. 18 (2002) 53–74. | DOI | MR | Zbl

F. Gismondi, R. Manca and A.V. Swishchuk, Salary lines forecasting by means of generalized binomial processes, Int. J. Manag. Sci. Eng. Manag. 4 (2010) 309–320.

M.A. Guerry, On the evolution of stock vectors in a deterministic integer-valued Markov system. Linear Algebra Appl. 429 (2008) 1944–1953. | DOI | MR | Zbl

J. Janssen and R. Manca, Salary cost evaluation by means of non-homogeneous semi-Markov processes. Stoch. Models 18 (2002) 7–23. | DOI | MR | Zbl

J. Janssen and R. Manca, Applied Semi-Markov Processes. Springer, New York (2006). | MR | Zbl

J. Janssen and R. Manca, Semi-Markov risk models for finance, insurance and reliability. Springer, New York (2007). | MR | Zbl

J. Janssen, R. Manca and E. Volpe di Prignano, Semi-Markov modelization for salary line evolution. Proc. of the VIII Applied Stochastic Models and Data Analysis, edited by J. Janssen and N.C. Lauro. Anacapri (Napoli), Italy, Napoli (1997).

M. Karaliopoulou, On the number of word occurrences in a semi-Markov sequence of letters. ESAIM: PS 13 (2009) 328–342. | DOI | Numdam | MR | Zbl

G.L. Lilien and A. Rao, A model for man-power management. Manage. Sci. 21 (1975) 1447–1457. | DOI

N. Limnios and G. Oprişan, Semi-Markov processes and reliability. Birkhäuser, Boston (2001). | MR | Zbl

S.I. Mcclean, A semi-Markov model for a multigrade population with Poisson recruitment. J. Appl. Probab. 17 (1980) 846–852. | DOI | MR | Zbl

S.I. McClean, Semi-Markov models for manpower planning, In Semi-Markov models: theory and applications, edited by J. Janssen. Plenum (1986). | MR

S.I. Mcclean, Semi-Markov models for human resource modelling. IMA J. Manage. Math. 4 (1992) 307–315. | DOI | Zbl

S.I. Mcclean and O. Gribbin, A non-parametric competing risk model for man power planning. Appl. Stoch. Models Data Anal. 7 (1991) 327–341. | DOI

S.I. McClean and E. Montgomery, Estimation for semi-Markov manpower models in a stochastic environment. In Semi-Markov models and applications, edited by J. Janssen and N. Limnios. Kluwer (2000). | MR | Zbl

S.I. Mcclean, E. Montgomery and F. Ugwuowo, Non-homogeneous continuous-time Markov and semi-Markov manpower models. Appl. Stoch. Models Data Anal. 13 (1998) 191–198. | DOI | MR | Zbl

A.A. Papadopolou and P.-C.G. Vassiliou, Non-homogeneous semi-Markov systems and maintainability of the state sizes. J. Appl. Probab. 29 (1992) 519–534. | DOI | MR | Zbl

A.A. Papadopoulou and P.-C.G. Vassiliou, Asymptotic behaviour of non-homogeneous semi-Markov systems. Linear Algebra Appl. 210 (1994) 153–198. | DOI | MR | Zbl

A.A. Papadopoulou and P.-C.G. Vassiliou, Continuous time non-homogeneous semi-Markov systems. In Semi-Markov Models and Applications, edited by J. Janssen and N. Limnios. Kluver Academic Publishing (1999) 241–251. | MR | Zbl

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