Stochastic Modeling of Server Capacity Utilization by Geometric Sums

In this paper we consider a region which is divided into serveral subregions. Each region is served by a station with a finite capacity. This means that the station can only serve customers within the limits of its capacity (maximum number of connections simultaneously). In connection with this the following question arises: What is the first occurrence time for this event? More exactly, what is the distribution of the first occurrence time of congestion. In the present paper we will show that this problem can be handled with geometric sums (Kalashnikov [3]), and we will present a general representation of the distribution function for the first occurrence time of capacity congestion.