A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system's state. We examine two versions of the problem: "nonpreemptive," where service is uninterruptible, and "preemptive," where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to V , the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value V . The construction of these policies is based on a Hamilton-Jacobi-Bellman equation associated with V .

We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavy-traffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: n ≈ r + β √ r for some scalar β. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.)We first consider a formal weak limit, through which our queueing scheduling problem gives rise to a diffusion control problem. We show that the latter has an optimal Markov control policy, and that the corresponding Hamilton-Jacobi-Bellman (HJB) equation has a unique classical solution. The Markov control policy and the HJB equation are then used to define scheduling control policies which we prove are asymptotically optimal for our original queueing system. The analysis yields both qualitative and quantitative insights,

We consider a multi-class queueing system with multiple homogeneous servers and customer abandonment. For each customer class i, the holding cost per unit time, the service rate and the abandonment rate are denoted by c i , µ i and θ i , respectively. We prove that under a many-server fluid scaling and overload conditions, a routing policy that assigns priority to classes according to their index c i µ i /θ i , is asymptotically optimal for minimizing the overall long run average holding cost. An additional penalty on customer abandonment is easily incorporated into this model and leads to a similar index rule.

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The dependence of the optimal nonlinear filter on it's initial conditions is considered for continuous time linear filtering and for finite state space nonlinear filtering. Partial results are obtained in the high signal to noise ration case, together with a characterization of the Lyapunov exponent in the (easier) low SNR case.