Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic

Abstract

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 .

State Space Collapse in Many-Server Diffusion Limits of Parallel Server Systems

“…Atar et al [6] proved that the policies obtained from this approach are asymptotically optimal. Atar [4,5] followed a similar approach to that in [6] to find asymptotically optimal policies for tree-like systems. In Mandelbaum et al [33], they used a uniform acceleration technique to obtain the fluid and diffusion limit of a Markovian service network.…”

Service-Level Differentiation in Call Centers with Fully Flexible Servers

“…Our initial investigation suggests that the insights gained from studying the V-model are useful in analyzing more complicated network structures (Gurvich 2004). Other researchers have also tackled this general skill-based routing (SBR) problem (e.g., Bassamboo et al 2006aBassamboo et al , 2006bAtar 2005;Tezcan and Dai 2007;Gurvich and Whitt 2006), but many issues remain unresolved.…”

Control of systems with flexible multi-server pools: a shadow routing approach

“…Papers [24] and [7] formulated a diffusion control problem to study a V-parallel server system with impatient customers in many-server heavy traffic; [7] proved that the policies obtained from this approach are asymptotically optimal. Works [5,6] followed a similar approach to that in [7] to find asymptotically optimal policies for tree-like systems. In [38] and [15] it is shown that a greedy policy is asymptotically optimal first for N-systems and then for general systems when the service rates are only server pool dependent.…”

“…To improve exposition, from this point on in the proof, consider a special system with I = {1, 2, 3, 4}, J = {5, 6, 7, 8}, and basic activities E b = {(1, 5), (2,5), (2,6), (2,7), (3,7), (3,8), (4,5)}, see Fig. 1.…”

“…The service rates are μ 15 = 4, μ 16 = 5, μ 25 = 2, μ 26 = 4, μ 27 = 1, μ 36 = 2, μ 37 = 1, μ 38 = 1, μ 35 = 1.5, μ 45 = 4, μ 48 = 1.5 and all other μ ij 's are zero (for non-feasible activities); β 5 = β 6 = β 7 = β 8 = 1. The input rate parameters are λ 1 = 2, λ 2 = 4.8, λ 3 = 1.6, and λ 4 = 1.2, which makes all feasible activities basic, except activities (3,5), (3,6), (1,6), and (4,8). We use the scaling parameter r = 100, so that there are 100 servers in each pool.…”