On two product form modifications for finite overflow systems

Overflow mechanisms can be found in a variety of queueing models. This paper studies a simple and generic overflow system that allows the service times to be both job type and station dependent. This system does not exhibit a product form. To justify simple product form computations, two product form modifications are given, as by a so-called call packing principle and by a stop protocol. The provided proofs are self-contained and straightforward for the exponential case and of merit by itself. Next, it is numerically studied whether and when, or under which conditions, the modifications lead to a reasonable approximation of the blocking probability, if not an ordering. The numerical results indicate that call packing provides a rather accurate approximation when the overflow station is not heavily utilized. Moreover, when overflowed jobs have an equal or faster service rate, the approximation is consistently found to be pessimistic, which can be useful for practical purposes. The stop protocol, in contrast, appears to be less accurate for most natural situations. Nevertheless, for an extreme situation the order might change. In addition, for the stop protocol the product form is proven to be insensitive (i.e. to also apply for arbitrary non-exponential service times). For call packing, this numerically appears not to be the case, as of interest by itself. However, from a practical viewpoint the sensitivity seems light. The results are intriguing for both theoretical and practical further research.