Open AccessThe object of research in this article is to obtain the analytical expressions for the probability distribution functions (PDF) of states of the closed system of mass service (SMS) like "model of the repairman" Mr|GIr|1 || Nr in a stationary mode. The SMS has a source of requests of final capacity with exponentially distributed time of residence of different class requests in the source, one queue with discipline of requests service i.e. the non-preemptive priorities, one serving device (SD) and arbitrary PDF of a residence time of each class requests.
The method includes the following steps.
The first step, while servicing the non-uniform population of requests, distinguishes a sequence of time points, so-called points of regeneration in which a process behaves as Markov's. It is shown that such points are time points to complete a service of requests in SD. Further, for these points an embedded Markov's chain is designed, and the states space of this chain is defined. Analytical expressions to calculate the elements of transition probability matrix in such chain are obtained. These expressions reflect a relationship between these probabilities and the source parameters, the residence time PDF parameters, the number of each class requests, and a discipline of their service. Further, the solution for PDF of this chain states is found.
The second step establishes connection between the PDF of SMS states in a stationary mode and the previously received PDF of states of the embedded Markov's chain. Such connection is defined by a system of the equations of global balance for the SMS states in a stationary mode and states of the embedded Markov's chain. The physical sense of such balance is that each of SMS state in a stationary mode is in equilibrium, i.e. the intensities of transitions from the state and in the state are equal. The space of states of initial SMS in a stationary mode is defined. It is shown that in intervals of request service time in SD the process behaves as a process of net reproduction when intensity of death – transition of SMS to "lower" states – is equal to zero. The paper presents the obtained analytical expressions to calculate the stationary PDF of given SMS depending on the PDF of the embedded Markov' chain, the discipline of service, and the PDF of residence times to serve different class requests in SD.
The third step provides expressions to calculate the average values of residence and waiting times of each class requests in SMS, as well as expressions to define the SD loading by requests of each class and the general loading of SD. The formulas development is based on conservatism (work preservation), fair for discipline of service with non-preemptive priorities, and also using Little's result and expressions for average number of requests of each class in SMS which, in turn, are defined by the PDF of SMS states in a stationary mode.