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 server's 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.

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.

This paper studies stability of network models that capture macroscopic features of data communication networks, including the Internet. The network model consists of a set of links and a set of possible routes that are fixed subsets of links. A connection is dynamically established along one of the routes to transmit data as requested and is terminated after the transmission is over. The transmission bandwidth of a link is dynamically allocated, according to specific bandwidth allocation policy, to ongoing connections that traverse the link. A network model is said to be stable under a given bandwidth allocation policy if, roughly, the number of ongoing connections in the network will not blow up over time. We consider a stationary and a bursty network model; the former assumes stochastically stationary arrival processes of connections as did many theoretical studies, while the latter allows more realistic bursty and correlated arrival processes. For both models under a necessary stability condition (i.e., the average offered transmission load on each link is within its bandwidth capacity), we show that the proportionally fair, the minimum potential delay, the max-min fair, and a class of utility-maximizing bandwidth allocation policies ensure network model stability, while some priority-oriented and maximum throughput policies do not. Interestingly, the bandwidth allocation policy that maximizes the arctan(·) utility ensures the stability of the stationary model but not the bursty model. This raises a serious concern about the current practice in the Internet protocol design, since such a policy is thought of as a good approximation of one of the most widely used TCP in the Internet.

It is known that a data network may not be stable at the connection level under some unfair bandwidth allocation policies, even when the normal offered load condition is satisfied, i.e., the average traffic load at each link is less than its capacity. In this note, we show that, under the normal offered load condition, a data network is stable when the bandwidth of the network is allocated so as to maximize a class of general utility functions. Using the microscopic model proposed by Kelly (1997, 2001) for a transmission control protocol (TCP) congestion control algorithm, we argue that the bandwidth allocation in the network dominated by this algorithm can be modeled as our bandwidth allocation model, and hence that the network is stable under the normal offered load condition. This result may shed light on the stability issue of the Internet since the majority of its data traffic is dominated by the TCP.

This paper addresses two practical problems from a liner shipping company, i.e. the container flow management problem and the ship deployment problem, at the tactical planning level. A sequential model and a joint optimisation model are formulated to solve the problems. Our results show that the company should implement the joint optimisation model at the tactical planning level to improve the shipping capacity utilisation rather than the sequential model used in the current practice. Repositioning empty containers also need to be considered jointly with the nonempty container flow at the tactical planning level. Some important managerial insights into the operational and business processes are gained.

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.