In discrete-time queueing theory, the service process is traditionally modeled using the notion of service time, the time it takes the server to completely process one customer. In our research, we take a different approach and model the service process by means of two more basic quantities: service demands and service capacities. The service demands are independent and identically distributed (i.i.d.) random variables that describe the number of work units that each customer requires from the system, whereas the service capacities are i.i.d. random variables that describe the amount of work units that the server can process per timeslot. If a customer requires more work units than the server can provide in a slot, the service continues in the next slot. Conversely, if the service capacity in a slot is higher than the customer in service still requires, the remaining capacity is used for the next customer in line, and more than one customer might be served in that slot.
This type of model has been studied in previous work, but with either the restriction that the service capacities follow a geometric distribution or that they are deterministically equal to a given constant. In our research, we analyze the system with the restriction that the service capacity distribution must have finite support. The numbers of customers arriving per slot and the service demands of the customers can be general i.i.d. random variables.
The analysis is performed using probability generating functions (pgfs), and as a result we obtain expressions for the pgfs of the delay of a random customer, the amount of unfinished work and the number of customers in the system in a random slot. These pgfs are then used to derive expressions for the moments of these quantities, and to approximate the probability mass function of these quantities with much higher precision than simulations could provide
