Moments in Tandem Queues

We settle a conjecture concerning necessary conditions for finite mean steady-state customer delay at the second node of a tandem queue, using as an example a stable tandem queue with mutually independent i.i.d. interarrival and service times. We assume that the service times have infinite variance at the first node, finite variance at the second node, and smaller mean at the first node than at the second node. We show that this causes infinite mean stationary delay at the second node. Thus, in general, when the mean service time is smaller at the first node than at the second, finite variance of service times at node 1 is necessary for finite mean delay at node 2. This confirms a conjecture made by Wolff. Our result complements sufficiency conditions previously published by Wolfson; together these necessary and sufficient conditions are presented as a theorem at the conclusion of the paper. Our proof uses a known duality between risk processes and queues.