Load balancing in parallel infinite-server queues with action delay via phase representation
arXiv:2607.27976
The paper analyzes load balancing in parallel infinite‑server queues when routing decisions are subject to an action delay, modeling the delay with an Erlang phase structure to obtain a finite‑dimensional Markov jump process and deriving fluid approximations for the server imbalance dynamics.
Abstract
Spatially distributed service systems rely on state-dependent routing to allocate users, tasks, or requests to less-loaded service nodes. In practice, a routing decision does not take effect immediately: the assigned job reaches the selected node only after a lag caused by travel time, communication latency, or actuation. We call this lag the action delay. Whereas delayed-information models treat such a lag as stale information and analyze the model via a delay differential equation, the decision uses the current state and only its execution is deferred, so the job experiencing an action delay must be tracked as part of the state. Representing the action delay by an Erlang phase structure, we obtain a finite-dimensional Markov jump process and build ordinary differential equations that explicitly track the jobs in the delay phase. Exploiting the two-server symmetry, we reduce the dynamics to a difference mode for the server imbalance and derive its characteristic equation for an arbitrary number of phases. This equation coincides with that of the delayed-information model, showing that the two different delays share the same linearized imbalance dynamics. Numerical experiments confirm the fluid approximation and illustrate how the number of phases, the routing sensitivity, and the mean delay govern the transient response of the server imbalance.