Considerations on MG Queue Busy Period Variance Manuel Al berto M. Ferreira Instituto Universitário de Lisboa ISCTE IUL ISTAR - IUL

2025-04-27 0 0 248.37KB 6 页 10玖币

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Considerations on M/G/∞ Queue Busy Period Variance

Manuel Alberto M. Ferreira

Instituto Universitário de Lisboa (ISCTE – IUL), ISTAR- IUL

Lisboa, Portugal

manuel.ferreira@iscte.pt

ABSTRACT

Some important results on the variance of the M/G/∞ queue busy period are

presented. Often, this parameter depends on the whole structure of the service time

distribution. So, the importance of the bounds presented, depending only on some

parameters. Also, some bounds for service time distributions important in reliability

theory, with technological and financial applications, are presented and a particular

stochastic ordination is explored to explain those results. Finally, an interesting result

about the exponentiality of the M/G/∞ queue busy period, resulting from the coefficient

of variation study is shown.

Keywords: M/G/∞, busy period, variance.

1. BOUNDS FOR THE VARIANCE OF THE M/G/∞ QUEUE BUSY

PERIOD

The ∞busy period variance is given by, see for instance (1),















where  is the customers arrival rate,  is the traffic intensity, given by

,

being α mean service time, and G(.) is the service time distribution function. B

designates the busy period random variable. So, in general,  depends on the

whole structure of the service time distribution.

In (2) are deduced lower and upper bounds for the∞ busy period variance

that depend only on,  and - the service time coefficient of variation – that are then

improved in (3) assuming the form





Note that

- With  the bounds are coincident and are  value for the ∞

queue system – constant services with value α,

- For the ∞ queue system – exponential service times





∞

 



In (3) it is shown that (1.1) is equivalent to











 



where  and  are, respectively, the variance and the coefficient of variation of the

service time and









As 

 then follow the bounds in formula (1.3). But, for

the∞ system, the upper bound may be improved:

- 











 









 









 









.

- So,

 









 













 

.

- But

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摘要：

ConsiderationsonM/G/∞QueueBusyPeriodVarianceManuelAlbertoM.FerreiraInstitutoUniversitáriodeLisboa(ISCTE–IUL),ISTAR-IULLisboa,Portugalmanuel.ferreira@iscte.ptABSTRACTSomeimportantresultsonthevarianceoftheM/G/∞queuebusyperiodarepresented.Often,thisparameterdependsonthewholestructureoftheservicetimedis...

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Considerations on MG Queue Busy Period Variance Manuel Al berto M. Ferreira Instituto Universitário de Lisboa ISCTE IUL ISTAR - IUL

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