Heng-Qing Ye

A Stochastic Network Under Proportional Fair Resource Control---Diffusion Limit with Multiple Bottlenecks

We study a multiclass stochastic processing network operating under the so-called proportional fair allocation scheme, and following the head-of-the-line processor-sharing discipline. Specifically, each servers capacity is shared among the job classes that require its service, and it is allocated, in every state of the network, among the first waiting job of each class to maximize a log-utility function. We establish the limiting regime of the network under diffusion scaling, allowing multiple bottlenecks in the network, and relaxing some of the conditions required in prior studies. We also identify the class of allocation schemes among which the proportional fair allocation minimizes a quadratic cost objective function of the diffusion-scaled queue lengths, and we illustrate the limitation of this asymptotic optimality through a counterexample.

Heavy-Traffic Optimality of a Stochastic Network Under Utility-Maximizing Resource Allocation

We study a stochastic network that consists of a set of servers processing multiple classes of jobs. Each class of jobs requires a concurrent occupancy of several servers while being processed, and each server is shared among the job classes in a head-of-the-line processor-sharing mechanism. The allocation of the service capacities is a real-time control mechanism: in each network state, the resource allocation is the solution to an optimization problem that maximizes a general utility function. Whereas this resource allocation optimizes in a “greedy” fashion with respect to each state, we establish its asymptotic optimality in terms of (a) deriving the fluid and diffusion limits of the network under this allocation scheme, and (b) identifying a cost function that is minimized in the diffusion limit, along with a characterization of the so-called fixed-point state of the network.

Utility-Maximizing Resource Control: Diffusion Limit and Asymptotic Optimality for a Two-Bottleneck Model

We study a stochastic network that consists of two servers shared by two classes of jobs. Class 1 jobs require a concurrent occupancy of both servers while class 2 jobs use only one server. The traffic intensity is such that both servers are bottlenecks, meaning the service capacity is equal to the offered workload. The real-time allocation of the service capacity among the job classes takes the form of a solution to an optimization problem that maximizes a utility function. We derive the diffusion limit of the network and establish its asymptotic optimality. In particular, we identify a cost objective associated with the utility function and show that it is minimized at the diffusion limit by the utility-maximizing allocation within a broad class of “fair” allocation schemes. The model also highlights the key issues involved in multiple bottlenecks.

Asymptotic Optimality of Balanced Routing

Consider a system with K parallel servers, each with its own waiting room. Upon arrival, a job is routed to the queue of one of the servers. Finding a routing policy that minimizes the total workload in the system is a known difficult problem in general. Even if the optimal policy is identified, the policy would require the full queue length information at the arrival of each job; for example, the join-the-shortest-queue policy (which is known to be optimal for identical servers with exponentially distributed service times) would require comparing the queue lengths of all the servers. In this paper, we consider a balanced routing policy that examines only a subset of c servers, with 1 ≤ c ≤ K: specifically, upon the arrival of a job, choose a subset of c servers with a probability proportional to their service rates, and then route the job to the one with the shortest queue among the c chosen servers. Under such a balanced policy, we derive the diffusion limits of the queue length processes and the workload processes. We note that the diffusion limits are the same for these processes regardless the choice of c, as long as c ≥ 2. We further show that the proposed balanced routing policy for any fixed c ≥ 2 is asymptotically optimal in the sense that it minimizes the workload over all time in the diffusion limit. In addition, the policy helps to distribute work among all the servers evenly.

Diffusion limit of a two-class network: stationary distributions and interchange of limits

In [11], we have established the diffusion limit for a network with two servers (both being bottlenecks) and two job classes under a proportional fair resouce control. Here, we show that the usual traffic condition is necessary and sufficient for the existence and uniqueness of the stationary distributions of both the diffusion limit and the pre-limit processes. Furthermore, the interchange of limits is also justified, i.e., the sequence of stationary distributions of the pre-limit networks converges to that of the diffusion limit.

Diffusion Limit of Fair Resource Control -- Stationarity and Interchange of Limits

We study a resource-sharing network where each job requires the concurrent occupancy of a subset of links (servers/resources), and each link’s capacity is shared among job classes that require its service. The real-time allocation of the service capacity among job classes is determined by the so-called “proportional fair” scheme, which allocates the capacity among job classes taking into account the queue lengths and the shadow prices of link capacity. We show that the usual traffic condition is necessary and sufficient for the diffusion limit to have a stationary distribution. We also establish the uniform stability of the prelimit networks, and hence the existence of their stationary distributions. To justify the interchange of two limits, the limit in time and limit in diffusion scaling, we identify a bounded workload condition, and show it is a sufficient condition to justify the interchange for the stationary distributions and their moments. This last result is essential for the validity of the diffusion limit as an approximation to the stationary performance of the original network. We present a set of examples to illustrate justifying the validity of diffusion approximation in resource-sharing networks, and also discuss extensions to other multiclass networks via the well-known Kumar-Seidman/Rybko-Stolyar model.

Asymptotic optimality of threshold control in a stochastic network based on a fixed-point approximation

In Li and Yao [5], a stochastic network with simultaneous resource occupancy is studied, and a threshold control policy is proposed based on a fixed-point approximation. Here, we establish the asymptotic optimality of this control policy under fluid and diffusion scaling.

On the Optimality of Reflection Control, with Production--Inventory Applications

We study the control of a Brownian motion (BM) with a negative drift, so as to minimize a long--run average cost objective. We show the optimality of a class of reflection controls that prevent the BM from dropping below some negative level r, by cancelling out from time to time part of the negative drift; and this optimality is established for any holding cost function h(x) that is increasing in |x|. Furthermore, we show the optimal reflection level can be derived as the fixed point that equates the long--run average cost to the holding cost. We also show the asymptotic optimality of this reflection control when it is applied to production--inventory systems driven by discrete counting processes.

Interchange of limits in heavy traffic analysis under a moment condition

We develop an approach to prove the interchange of limits in heavy traffic analysis of stochastic processing networks, using a moment condition on the primitive data, the interarrival and service times. The approach complements the one in [8], where a bounded workload condition is required instead.

Asymptotic Optimality of the Max-Min Fair Allocation

Multiple classes of jobs are processed in a stochastic network that consists of a set of servers. Each class of jobs requires a concurrent occupancy of a subset of servers to be processed, and each server is shared among the job classes in a head-of-the-line processor-sharing mechanism. In each state of the network, the server capacities are allocated among the job classes according to the so-called max-min fair policy. We derive the fluid and diffusion limits of the network under this resource control policy. Furthermore, we provide a characterization of the fixed-point state associated with the fluid limit, and identify a cost function that is minimized in the diffusion regime