Single Server Systems -- II. Busy Periods

This is the second of two papers dealing with single server systems. In the first paper, 1 two subjects were discussed, (i) a method which led to the average values of several quantities of interest and (ii) the statistical behavior of a single server loss system. The present paper is concerned with the busy periods (the periods during which the server is continuously busy) in a single server system when no restrictions are placed on the queue length. The distribution of the busy period lengths has been studied by a number of investigators, among them E. Borel," D. G. Kendall/ F. Pollaczek, 4 ' 5 L. Takacs, 6,7 and B. W. Connolly. 8 ' 9 A closely related problem is that of the storage of water behind a dam. An interesting survey of this subject has been given by J. Gani.10 The moving server problem treated by McMillan and Riordan 11 and by Karlin, Miller and Prabhu 12 is also related to the busy period problem. Again, for Poisson input, the distribution of the busy period lengths is related to the distribution of the delays in "last come, first served" type of service (see Riordan' 0 and the references to earlier work given there). The most general results are those due to Pollaczek. His work leads to the joint distribution function of n,S,y where n is the number of services 279