Equilibrium Sets of Some GI/M/1 Queues (with more examples)

We view queue as a service facility and consider the situation when users offer arrival rate at stationarity that depends on the Quality of Service (QoS) they experience. We study the equilibrium points and the equilibrium sets associated with this interaction and their interpretations in terms of business cycles. The queue implements the optimal admission control (either discounted or ergodic) policy in the presence of holding cost and admission charge for admitted customers; threshold policies are known to be optimal for such queues. We consider two QoS measures: the long run fraction of customers lost, L, and the long run rate of customers lost, L1. Our first result is that both the QoS measures for GI/M/1 queues are locally continuous with respect to the arrival rate. ForM/M/1 queues we show that control limit is finite and that (Assumption A1) the QoS measures increase with arrival rate. We also show that multiple optimal policies lead to equilibrium sets. We observe that various combinations of cost criteria and QoS measures lead to differing equilibrium behaviour. Under A1, similar results hold for D/M/1 queue. We study a GI/M/1 queue with discrete arrival rate supports whose arrival rate is locally continuous. We illustrate that A1 need not hold, but a relaxed assumption may hold. Nonetheless, change of support is another cause for emergence of equilibrium sets and equilibrium behaviour is interesting as there may be multiple equilibrium points/sets, etc. Generalized equilibrium sets may also exist; these are usually due to the non-contraction nature of the QoS measure, the rate of customers lost, L1. We present details of some computational examples that were are not presented in (8). keywords: admission control of queues; quality of service; multiple optimal policies; change of support; invariant sets; generalized equilibrium sets; non-contraction maps; fixed points; parametrized MDPs