Busy Period and Busy Cycle Distributions and Parameters for a Particular M / G /∞ Queue System

Solving a Riccati equation a collection of service times distributions, very general, is determined. For it the ∞ / / G M queue busy period and busy cycle probabilistic study, very comfortable, is performed. In addition the properties of that distributions collection are deduced and also presented.


Introduction
In the ∞ / / G M queue system the costumers arrive according to a Poisson process at rate λ . They receive a service which length is a positive random variable with distribution function G(.) and mean α . There are always available servers. The traffic intensity is λα ρ = . At the operation of a ∞ / / G M queue system, such as in any other queue system, there is a sequence of idle periods and busy periods. An idle period followed by a busy period is called a busy cycle.
When applying this queue system to real problems, the busy period and the busy cycle length probabilistic study is very important. Along this work it is shown that the solution of the problem may be obtained solving a Riccati equation. The solution is a collection of service distributions for which both the busy period and the busy cycle have lengths with quite simple distributions, generally related with exponential distributions and the degenerated at the origin one.
Some results for that collection of service distributions are also presented.

A Riccati Equation Important in the M/G/ Occupation Study
(2) see [1]. Note that, for

Busy Period and Busy Cycle Study
After (2) and considering ( ) for the ∞ / / G M system busy period length distribution function, being * the convolution operator.
So, if the service distribution is given by (6), the busy period distribution function is the one of the mixture of a degenerated distribution at the origin with an exponential.
that is if the service distribution is degenerated at the origin with probability 1 it happens the same with the busy period distribution.
So, if the service time distribution function is given by (6) the busy cycle distribution function is the mixture of two exponential distributions.

Busy Period and Busy Cycle Parameters
As for the busy period moments, calling B the busy period random variable, note that (3) is equivalent to, see Differentiating n times, using Leibnitz's formula and  For the busy cycle moments, if Z is the busy cycle random variable, The M|G|∞ queue busy period "peak" is the Laplace transform value at α 1 , see [6]. It is a parameter that characterizes the busy period distribution and contains information about all its moments. For the collection of service distributions (2) the "peak", called pi , is

Collection (2) Distributions Moments Computation
If, in (2), as it had to happen since the differential equation considered is a Riccati one.
And computing,     So it is evident now that this distributions collection moments computation is a complex task. This was already true for the study of[10] where the results presented are a particular situation of these ones for 0 = p . The consideration of the approximation ( ) 1 1 ,

Conclusions
The function ( ) t β , defined in 1., that leads to the equation (1), the source of the results presented, is induced by the M/G/∞ transient probabilities monotony study, see [1] and [3]. Being well known the close relations existing among the transient behaviour and the busy period distributions, it is not surprising that the service distribution functions, solutions of (1), give rise to so friendly distributions to the M/G/∞ queue busy period. This is a quite unusual situation, that allows simple computations for the probabilities and parameters of the busy period and busy cycle.
This so simple structure lies on the exponential distribution and the deterministic one. In many of the situations seen there are purely deterministic and exponential distributions, mixtures of deterministic and exponential distributions and mixtures of two exponential distributions. The great presence of the exponential distribution in this context is due certainly to the so friendly properties of the Poisson Process and to the structure of the function resulting from the inversion of the Laplace Transform given by (4).
Even the stochastic processes related with this queue become quite simple. They are very close to the Poisson Process and in certain situations are even Poisson Processes.
In conclusion, all of this results in a very simple expression for the inversion of (4), with the service distributions solutions of (1). In consequence, as it was shown in this work, the distribution functions and the parameters for the busy period and, consequently, for the busy cycle result very simple to compute and interpret.
Many of these results are true for other service distribution functions, in the case of insensibility: parameters that depend on the service distribution only through its mean. Some others may result in good approximations.
Note still that when the distribution functions for ( )