Transient Little’s Law for the First and Second Moments of G/M/1/N Queue Measures

A customer in a service system and an outside observer (manager or designer of the system) estimate the system performance differently. Unlike the outside observer, the customer can never find himself in an empty system. Therefore, the sets of scenarios, relevant for the two at a given time, differ. So differ the meanings and values of the performance measures of the queue: expected queue length and expected remaining waiting time (workload). The difference between the two viewpoints can be even more significant when steady-state values of the queue measures are reached slowly, or even are never reached. In this paper, we obtain the relations between the means and variances of the measures in transient time and in steady state for a capacitated FCFS queue with exponentially distributed service time. In particular, a formula similar to Little’s law is derived for the means of the queue measures. Several examples support the validity and significance of the results.