A queueing system with on-demand servers: local stability of ?uid limits

Abstract We study a system where a random ?ow of customers is served by servers (called agents) invited on-demand. Each invited agent arrives into the system after a random time; after each service completion, an agent returns to the system or leaves it with some ?xed probabilities. Customers and/or agents may be impatient, that is, while waiting in queue, they leave the system at a certain rate (which may be zero). We consider the queue-length-based feedback scheme, which controls the number of pending agent invitations, depending on the customer and agent queue lengths and their changes. The basic objective is to minimize both customer and agent waiting times. We establish the system process ?uid limits in the asymptotic regime where the customer arrival rate goes to in?nity. We use the machinery of switched linear systems and common quadratic Lyapunov functions to approach the stability of ?uid limits at the desired equilibrium point and derive a variety of suf?cient local stability conditions. For our model, we conjecture that local stability is in fact suf?cient for globalstabilityof?uidlimits;thevalidityofthisconjectureissupportedbynumerical andsimulationexperiments.Whenlocalstabilityconditionsdohold,simulationsshow good overall performance of the scheme