Load balancing in parallel infinite-server queues with action delay via phase representation

2026-07-30

AI Summary

This work addresses load balancing in parallel infinite-server queues under action delays by explicitly incorporating delay into the state representation for the first time. The authors model the delay process using an Erlang phase-type structure, thereby constructing a finite-dimensional Markov jump system and employing ordinary differential equations to explicitly track tasks in transit. Exploiting symmetry between two servers, the system dynamics are reduced to a single mode capturing load imbalance. Theoretical analysis establishes equivalence between this formulation and the delayed-information model in their linearized dynamics, overcoming limitations of traditional delay-differential-equation approaches. The study derives the characteristic equation governing imbalance dynamics for an arbitrary number of phases, and numerical experiments confirm the accuracy of the fluid approximation while quantifying the effects of phase count, routing sensitivity, and mean delay on transient response.

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.

Problem

• load balancing
• action delay
• infinite-server queues
• state-dependent routing
• spatially distributed service systems

Innovation

• action delay
• phase-type distribution
• load balancing
• Markov jump process
• fluid approximation