On the input and output processes for a general birth-and-death queueing model

Abstract

The steady-state input and output processes are considered for a birth-and-death queueing model with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and an arbitrary queueing discipline. Let an index n indicate that the quantity in question depends on the system state but not on time t. The instantaneous arrival rate is λ, the probability of balking (i.e., not trying to obtain service) being ξn. The instantaneous departure rate, μn, of customers having joined the system is the sum of the rate of service completions and the rate of defections before service completion. Three cases are considered. We start by ignoring balking customers; in the first case treating a lost customer neither as an input nor as an output, then secondly as both. Finally, balking and lost customers are considered both as inputs and outputs.