In this paper we consider a queueing system having n exponential servers, each with its own queue and service rate. Customers arrive according to a Poisson process with rate λ, and upon arrival each customer must be routed to some server's queue. No jockeying amongst the queues is allowed and each server serves its queue according to a first-come-first-served discipline.
Each server i, 1 ≦ i ≦ n, provides service with a state-dependent rate μ
(i)(k), k = 0, 1, …. In addition, at every queue i, there is a deterministic holding cost which occurs at rate h
(i)(k) while there are k customers at the queue.
An admissible routing policy is a policy that assigns each arriving customer to one of the queues. A decision at time t may be randomized and dependent on the queue lengths and decisions till time t
. An optimal routing policy is an admissible policy that minimizes the long-run average holding cost.
In this study, we bound the optimal cost from below, by considering an ideal system, where each server optimally selects a given proportion of customers, irrespective of other servers' selections. From this ideal system we construct a class of admissible routing policies, the overflow routing class, that approximates the ideal situation for each server. Finally, we evaluate the policies and compare them to the lower bound.