On the Settling Time of the Congested Gi/g/1 Queue

We analyze a stable GI/G/1 queue that starts operating at time t = 0 with No 0 customers. First, we analyze the time TvN required for this queue to empty for the first time. Under the assumption that both the interarrival and the service time distributions are of the exponential type, we prove that limN0 .. TTN 0/No . J( ), where A and p are the arrival and the service rates. Furthermore, assuming in addition that the interarrival time distribution is of the non-lattice type, we show that the settling time of the queue is essentially equal to No/(p A); that is, we prove that lim dti( A°. c= NO --;L 0, for O 1, ''"~""" rl.1 3 ·. :'···.::·I~·::..·.-. II.·· . ·.. i where dNo(t) is the total variation distance between the distribution of the number of customers in the system at time t and its steady-state distribution. Finally, we show that there is a similarity between the queue we analyze and a simple fluid jnodel. CONGESTED QUEUE; SETTLING TIME; TRANSIENT ANALYSIS