An ODE for an Overloaded X Model Involving a Stochastic Averaging Principle

We study an ordinary differential equation (ODE) arising as the many-server heavy-traffic fluid limit of a sequence of overloaded Markovian queueing models with two customer classes and two service pools. The system, known as the X model in the call-center literature, operates under the fixed-queue-ratio-with-thresholds (FQR-T) control, which we proposed in a recent paper as a way for one service system to help another in face of an unanticipated overload. Each pool serves only its own class until a threshold is exceeded; then one-way sharing is activated with all customer-server assignments then driving the two queues toward a fixed ratio. For large systems, that fixed ratio is achieved approximately. The ODE describes system performance during an overload. The control is driven by a queue-difference stochastic process, which operates in a faster time scale than the queueing processes themselves, thus achieving a time-dependent steady state instantaneously in the limit. As a result, for the ODE, the driving process is replaced by its long-run average behavior at each instant of time; i.e., the ODE involves a heavy-traffic averaging principle (AP).

💡 Research Summary

This paper investigates the fluid‑scale dynamics of an overloaded “X‑model” queueing system, a canonical representation of many modern call‑center architectures that feature two customer classes and two dedicated service pools. In the X‑model each pool normally serves only its own class, but when an unexpected overload occurs one pool may lend capacity to the other. The authors study this sharing mechanism under the Fixed‑Queue‑Ratio‑With‑Thresholds (FQR‑T) control, a policy they introduced previously. FQR‑T continuously monitors the difference between the two queue lengths, D(t)=Q₁(t)−r·Q₂(t), where r is a pre‑specified target ratio. When D(t) exceeds a pre‑set threshold θ (in the appropriate direction), one‑way sharing is turned on; all service capacity that becomes idle in the helping pool is immediately redirected to the overloaded pool, driving the queues toward the target ratio.

The central technical contribution is the derivation of an ordinary differential equation (ODE) that captures the system’s evolution during overload, by exploiting a stochastic averaging principle (AP). The authors observe that the queue‑difference process D(t) evolves on a much faster time‑scale than the queue‑length processes themselves. In the many‑server heavy‑traffic regime (the number of servers N→∞ while arrival rates scale proportionally), D(t) essentially reaches its stationary distribution instantaneously at each fluid‑time instant. Consequently, the fast process can be replaced by its long‑run average conditional on the current slow state (Q₁,Q₂). This separation of time‑scales yields a deterministic ODE whose drift terms involve the expected value of the sharing indicator under the stationary distribution of D(t).

Formally, the original stochastic model is a continuous‑time Markov chain with arrival rates λ₁, λ₂, service rates μ₁, μ₂, and pool capacities s₁, s₂. The state is (Q₁,Q₂,Z), where Z∈{0,1} indicates whether sharing is active. Under the scaling, the fluid variables q₁(t)=Q₁(t)/N, q₂(t)=Q₂(t)/N evolve according to

dq₁/dt = λ₁ – μ₁·min{q₁,s₁} – β·E_{μ_{(q₁,q₂)}}