Abstract
In this paper we study the convexity of the waiting time, workload and the number of jobs in single stage queueing systems with respect to the bulk size of the arrival process. In particular we show that the number of jobs in a single server queueing systemG [x]/GI/1 and in a multiple server queueing systemG [x]/M/c with bulk sizesx=(x 1 ,x 2 ,x 3 ,...) is componentwise convex inx. This is in the sense of the sample path convexity introduced in Shaked and Shanthikumar [11]. These results have applications in the stochastic comparison of bulk arrival queueing systems.
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Research supported in part by NSF grant DDM-9113008.
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Shanthikumar, J.G. Convexity of single stage queueing systems with bulk arrival. Queueing Syst 16, 287–299 (1994). https://doi.org/10.1007/BF01158958
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DOI: https://doi.org/10.1007/BF01158958