Queuing System with Processor Sharing and Limited Resources

Consideration was given to a processor-sharing system with heterogeneous customers serviced by system resources of two types: the first, discrete, type necessarily consists of an integer number N of units (servers), and the second type (memory) may be discrete or continuous. The customer type is defined by the number of units of the first-type resources required to service it. In addition to the need for the first-type resource, each customer is characterized by a certain volume, that is, the amount of the second-type resource required to service it. The total amount of customers (total busy resource of the second type) in the system is limited by a certain positive value (memory space) V. The customer volume and its length (amount of work required to service it) are generally dependent. Their joint distribution also depends on the customer type. For this system, the stationary distribution of the number of customers sojourning in the system and the probabilities of losing customers of each type were determined.