An Ergodic Control Problem for Many-Server Multiclass Queueing Systems with Cross-Trained Servers

Abstract

Abstract.A Markovian queueing network is considered with d independent customer classes and d server pools in Halfin-Whitt regime. Class i customers has priority for service in pool i for i 1, . . . , d, and may access some other pool if the pool has an idle server and all the servers in pool i are busy. We formulate an ergodic control problem where the running cost is given by a non-negative convex function with polynomial growth. We show that the limiting controlled diffusion is modelled by an action space which depends on the state variable. We provide a complete analysis for the limiting ergodic control problem and establish asymptotic convergence of the value functions for the queueing model.

On Viscosity Solution of HJB Equations with State Constraints and Reflection Control

“…Such queueing systems might be referred to as queueing networks with help. Similar type queueing network is studied in [4] for a M/M/N+M queueing network with ergodic type costs. The service time distributions of the customers at different servers are assumed to be exponential.…”

“…The main goal of this article is to characterize the value function (2.4) in terms of the solution of HJB (2.5). However, in the many server settings above questions are addressed in [4]. The limiting HJB in [4] has a classical solution which is used to obtain an optimal control for the limiting problem.…”

“…However, in the many server settings above questions are addressed in [4]. The limiting HJB in [4] has a classical solution which is used to obtain an optimal control for the limiting problem. This optimal control is then used to obtain convergence result of the value functions for the queuing model and asymptotic optimality.…”

Infinite horizon asymptotic average optimality for large-scale parallel server networks

“…For multiclass multi-pool networks, Ward and Armony [30] have studied blind fair routing policies, and used simulations to validate their performance, and compared them with non-blind policies derived from the limiting diffusion control problem. Biswas [31] recently studied a specific multiclass multi-pool network with "help" where each server pool has a dedicated stream of a customer class, and can help with other customer classes only when it has idle servers. For this network model, the control policies may not be work-conserving, and the associated controlled diffusion has a uniform stability property, which is not satisfied for general multiclass multi-pool networks.…”

Infinite-Horizon Average Optimality of the N-Network in the Halfin–Whitt Regime

“…For the same model, Armony and Ward [5] showed that a threshold policy is asymptotically optimal for minimizing the steady-state expected queue length and waiting time subject to a "fairness" constraint on the workload division. Biswas [11] has recently studied a multiclass multi-pool network with "help" under an ergodic cost criterion, where each server pool has a dedicated stream of a customer class, and can help with other customer classes only when it has idle servers. The N-network does not belong to the class of models considered in Biswas [11].…”