A Note on Convexity of the Expected Delay Cost in Single-Server Queues

In this note we show with an analytical counterexample that the expected delay cost for a G/G/1 queue is not necessarily convex in the arrival rate as sometimes claimed in the literature. We can prove, however, that the total expected delay cost rate is convex in the arrival rate. This cost rate is often of interest when designing queueing systems. Mathematics: Convexity. This note shows that the expected delay cost in single server queues is not necessarily convex in the arrival rate as often assumed. However, the expected delay cost rate is shown to be convex in the arrival rate. Queues: Optimization. This note explores convexity properties of expected delay cost in single server queueing systems, which are important when optimizing their design.