Convexity of single stage queueing systems with bulk arrival

In this paper the convexity of the waiting time, workload and the number of jobs in single stage queueing systems is studied with respect to the bulk size of the arrival process. In particular it is shown that the number of jobs in a single server queueing system and in a multiple server queueing system with bulk sizes is componentwise convex in x. This is in the sense of the sample path convexity introduced in Shaked and Shanthikumar. These results have applications in the stochastic comparison of bulk arrival queueing systems