Extend fixed-joined-customer analysis to queues with reneging

Extend the probability-generating-function and computational analysis for Markovian queues with balking conditioned on exactly K joined customers to queueing systems in which customers may renege before completing service, thereby addressing settings where conditioning on the number of served customers no longer coincides with conditioning on the number of entries.

Background

The paper assumes that customers who join the system remain in service until completion, so the total number served equals the total number who entered. Under reneging, customers can depart before service completion, breaking this equivalence and requiring an extension of the proposed PGF-based and backward-in-time computational framework. The authors explicitly identify this extension as unresolved future work.

References

In this paper, we assumed that once customers join the system, they remain until service completion. Under this assumption, the total number of customers served equals the number of joined customers. However, in many real-world systems, customers may renege before completing service. In such settings, conditioning on the number of served customers no longer coincides with conditioning on the number of entries. Extending our approach to such cases, where early departures occur, remains an important direction for future work.

— Computing transient probabilities in Markovian queues with balking conditional on a fixed number of joined customers  (2608.19560 - Hayashi et al., 20 Aug 2026) in Section 6, Conclusion