, Volume 8, Issue 3, pp 175–189

# Un modelo de la teoria de colas con numero variable de canales

• José Romaní
Article

## Summary

In this paper the author gives the state probabilities of the following queue model.

The arrival and departure precess are at random with λ-mean arrival and μ=mean servicing time in a single channel.

The number of channels is in this model a random number depending upon the length of the queue; when the number of waiting units is M and it arrives another unit, the number of channels is increased in one, and this channel services the first unit in queue; when a channel is free and none unit being in queue, this channel es cancelled. Only in the case of the last channel this should be open. So we have the length of the queue is never greater than M.

The notation used in the paper is:r=number of units in queue.s=number of channels (of number of elements being serviced).$$\rho = \frac{\lambda }{\mu }$$ P(r,s;t)=Probability of the state (r, s) on timet.p(r s)=Probability of the state (r, s) in the steady state of the model.

With this notation, we have
$$\begin{gathered} P\left( {r,1} \right) = \frac{{\sum\limits_0^r {\rho ^{ - K} } }}{{\left( {M + 1} \right)e^\rho + \sum\limits_1^M {\left( {M - {\rm K} + 1} \right)\rho ^{ - K} } }} \hfill \\ p\left( {r,s} \right) = \frac{1}{{\left( {M + 1} \right)e^\rho + \sum\limits_1^M {\left( {M - {\rm K} + 1} \right)\rho ^{ - K} } }} \hfill \\ \end{gathered}$$
(1)

This model includes as particular cases the model of a single channel, equivale to M-→∞ and the model of infinite channels, equivalent to M=0.

## Bibliografia

1. [1]
Kendall, D. G.: «Some problems in the theory of queues».Jour. Roy. Stat. Soc. Series B, vol. 13, 1951, pp. 151–185.
2. [2]
Romaní, J.: «La teoría de las colas aplicadas a un problema de producción industria».Trabajos de estadistica. Vol. VI. Cuad. III, 1955, pp. 235–259.Google Scholar
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Feller, W.: «On the integrodifferential equations on completely discontinuos Markoff processes».Tran. Am. Math. Soc. Vol. 48, 1940, pp. 488–515.
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Feller, W.: «An introduction to Probability Theory and its Applications». Wiley, New York, 1950.