Using the results on the M|G|? queue busy period, it is presented an application of this queue system in the unemployment periods parameters and distribution fubction study.info:eu-repo/semantics/publishedVersio

Coelho, M.

Ferreira, M. A. M.

Filipe, J. A.

English

Repositório Institucional do ISCTE-IUL

Int.	J	Latest	Trends	Fin.	Eco.	Sc.	 	 	 	 	 	 	 	 														Vol‐4	No.	1	March,	2014		677 Unemployment Modelled Through M|G|∞ Systems Manuel Alberto M. Ferreira*1, José António Filipe*2, Manuel Coelho# * Lisbon University Institute ISCTE-IUL, BRU-IUL  Portugal 1manuel.ferreira@iscte.pt 2jose.filipe@iscte.pt #SOCIUS, ISEG-University of Lisbon  Portugal coelho@iseg.utl.pt   Abstract - Using results on the M|G|∞ queue busy period, it is presented an application of this queue system in the unemployment periods parameters and distribution function study. Keywords ‐  | |M G  , busy period, unemployment.    1. The Model In the M|G| queue system ‐ The customers arrive according to a Poisson process at rate   ‐ Receive a service which time length is a positive random variable with distribution function  .G and mean   ‐ When they arrive, each one finds immediately an available server ‐ Each customer service is independent from the other customers’ services and from the arrivals process ‐ The traffic intensity is   .  It is easy to understand how the | |M G   queue can be applied to the unemployment study. Then  ‐  is the rate at which occur the firings, supposed to occur according to a Poisson process ‐ The service time is the time between the worker firing and the moment he/she finds a new job. In a queue system a busy period is a period that begins when a costumer arrives at the system finding it empty, ends when a costumer abandons the system letting it empty and in it there is always at least one customer present. So in a queuing system there is a sequence of idle and busy periods. In the | |M G  queue system the idle periods have an exponential length with mean 1 . But the busy period’s distribution is much more complicated. In spite of it, it is possible to present some results as it will be seen. For what interests in this work ‐ A busy period is a period of unemployment ‐ An idle period is a period of full employment. The results that will be presented are on unemployment periods length and their number in a certain time interval. 2. Unemployment Periods Length Call D the random variable unemployment period length. According to the results known for the | |M G   queue busy period length distribution -   1         (2.1)eE D   ____________________________________________________________________________________International	Journal	of	Latest	Trends	in	Finance	&	Economic	Sciences	IJLTFES,	E‐ISSN:	2047‐0916	Copyright	©	ExcelingTech,	Pub,	UK	(http://excelingtech.co.uk/)	Int.	J	Latest	Trends	Fin.	Eco.	Sc.	 	 	 	 	 	 	 	 														Vol‐4	No.	1	March,	2014		678 whichever is a worker unemployment time length distribution, see (Takács, 1962) - As for  Var D  depends on the whole unemployment time length distribution probabilistic structure. But Sathe, see (Sathe, 1985), demonstrated that  2 2 2 2 2max 2 1;0se e e Var D                 222 1 1 1         (2.2),se e e             where s  the unemployment time length coefficient of variation - If a worker unemployment time length distribution function is     , 0,   (2.3)1 teG t te e e         the D distribution function is      1 1 , 0       2.4 ,e tD t e e t       see (Ferreira, 1991) - If the unemployment time length of a worker is such that  111 , 0        (2.5)1teG t te e        the D distribution function is      111 , 0       2.6 ,e tD t e t      see (Ferreira, 1995) - For   and   great enough (very intense unemployment conditions) since  .G  is such that for   great enough   0, 0G t t  ,    1 , 0       2.7 ,e tD t e t     see (Ramalhoto and Ferreira, 1994).        Note: - As for this last result, begin noting that many probability distributions fulfill the condition   0, 0G t t   for   great enough. The exponential distribution is one example. - As for the meaning of   and   great enough, computations presented in (Ramalhoto and Ferreira, 1994) show that for 1  , after 10   it is reasonable to admit (2.7) for many distributions. 3. Mean Number of Unemployment Periods in a Time Interval After the renewal processes theory, see (Çinlar, 1975), calling ( )R t  the mean number of unemployment periods that begin in  0, t , being 0t   the beginning instant of an unemployment period, it is possible to obtain, see (Ferreira, 1995),      0 01 10      (3.1)t utG v dv G v dvR t e e du              and , consequently,    1 1        (3.2),e t R t t        see Ferreira (2004). Also, A)     , 01 teG t te e e          1       (3.3)R t e t    B)  111 , 0 1teG t te e               2 11 1    3.4teR t e e e t e e e                 Int.	J	Latest	Trends	Fin.	Eco.	Sc.	 	 	 	 	 	 	 	 														Vol‐4	No.	1	March,	2014		679 C)   0, 1, tG tt       1, 1 , tR te t t      (3.5) D) If the unemployment time length is exponentially distributed  1 1  (3.6)t te ee e t R t e t                      4. Concluding Remarks So that this model can be applied it is necessary that the firings occur according to a Poisson process at constant rate. It is an hypothesis that must be tested. Among the results presented, (2.1), (2.2), (2.7) and (3.2) are remarkable for its simplicity and also for requiring only the knowledge of the firings rate  , the mean unemployment time  , and the unemployment time variance.      The other results are more complex and demand the goodness of fit test for the distributions indicated to the unemployment times. References [1] E. Çinlar (1975). Introduction to Stochastic Processes. New Jersey, Prenctice-Hall, Inc. [2] L.Takács (1962). An introduction to queueing theory. Oxford University Press. [3] M. A. M. Ferreira (1991). Um sistema M|G| com período de ocupação exponencial. Actas das XV Jornadas Luso-Espanholas de Matemática, Vol. IV, D63-D66, Universidade de Évora, Évora. Accession Number: WOS: A1991BV70Y00123 . [4] M. A. M. Ferreira (1995). Comportamento transeunte e período de ocupação de sistemas de filas de espera sem espera. PhD thesis. ISCTE.  [5] M. A. M. Ferreira (2004). | |M G   queue busy cycle renewal function for some particular service time distributions. Proceedings of the International Conference Quantitaive Methods in Economics (Multiple Criteria Decision Making XII), 42-47, Virt, Slovakia. Accession Number: WOS:000288041600006 .  [6] M. F. Ramalhoto (1996), M. A. M. Ferreira. Some further properties of the busy period of an  | |M G  queue. Central European Journal for Operations Research and Economics, 4(4), 251-278. [7] Y. S. Sathe (1985). Improved bounds for the variance of the busy period of the | |M G   queue. Advances in Applied Probability, 17(4), 913-914. http://dx.doi.org/10.2307/1427096   

Unemployment Modelled Through M|G|? Systems

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Unemployment Modelled Through M|G|? Systems

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