Insensitivity for coupled infinite-server queues with general service times

Establish whether the insensitivity property holds for potentially coupled state- and time-dependent M_{Q,t}/G_t/∞ queues and whether such queues have the target distribution π as their limiting distribution.

Background

The paper constructs a temporal point-process sampler that can be viewed as a system of potentially coupled M_{Q,t}/D/∞ queues with deterministic service times. By introducing service-time redraws and taking a limiting regime, the authors show that the queue-length process converges to a time-inhomogeneous birth-death process, equivalently related to M_{Q,t}/M_t/∞ queues, while retaining the target distribution π as its limiting distribution.

The authors conjecture that this limiting-distribution property is insensitive to the service-time distribution and therefore extends to more general potentially coupled M_{Q,t}/G_t/∞ queues. Proving this would generalize the deterministic-service and exponential-service cases treated in the paper.

References

With the point process sampler taking the form of potentially coupled M_{Q, t}/D/\infty queues and the birth-death sampler taking the form of potentially coupled M_{Q, t}/M_t/\infty queues, we conjecture that the insensitivity property holds and more general potentially coupled M_{Q, t}/G_t/\infty queues exist which have \pi as their limiting distribution.

— Sampling on Discrete Spaces with Temporal Point Processes  (2603.09089 - Stewart et al., 10 Mar 2026) in Section 4, subsection “Connection to sampling with birth-death processes”