Jian-Qiang Hu

Traffic grooming, routing, and wavelength assignment in optical WDM mesh networks

In this paper, we consider the traffic grooming, routing, and wavelength assignment (GRWA) problem for optical mesh networks. In most previous studies on optical mesh networks, traffic demands are usually assumed to be wavelength demands, in which case no traffic grooming is needed. In practice, optical networks are typically required to carry a large number of lower rate (sub-wavelength) traffic demands. Hence, the issue of traffic grooming becomes very important since it can significantly impact the overall network cost. In our study, we consider traffic grooming in combination with traffic routing and wavelength assignment. Our objective is to minimize the total number of transponders required in the network. We first formulate the GRWA problem as an integer linear programming (ILP) problem. Unfortunately, for large networks it is computationally infeasible to solve the ILP problem. Therefore, we propose a decomposition method that divides the GRWA problem into two smaller problems: the traffic grooming and routing problem and the wavelength assignment problem, which can then be solved much more efficiently. In general, the decomposition method only produces an approximate solution for the GRWA problem. However, we also provide some sufficient condition under which the decomposition method gives an optimal solution. Finally, some numerical results are provided to demonstrate the efficiency of our method.

Optimality of hedging point policies in the production control of failure prone manufacturing systems

We study the necessary and sufficient conditions for the optimality of the hedging point policy for production systems in which the failure rate of machines depends on the rate of production. We focus on a one machine one part-type and infinite horizon discounted cost problem. It is shown that when the failure rate is independent of the rate of production and a constant, the hedging point policy is provably optimal. The main result of this paper is to show that the linearity of the failure rate function is both necessary and sufficient for the optimality of the hedging point policy. >

Sensitivity Analysis for Monte Carlo Simulation of Option Pricing

Monte Carlo simulation is one alternative for analyzing options markets when the assumptions of simpler analytical models are violated. We introduce techniques for the sensitivity analysis of option pricing which can be efficiently carried out in the simulation. In particular, using these techniques, a single run of the simulation would often provide not only an estimate of the option value but also estimates of the sensitivities of the option value to various parameters of the model. Both European and American options are considered, starting with simple analytically tractable models to present the idea and proceeding to more complicated examples. We then propose an approach for the pricing of options with early exercise features by incorporating the gradient estimates in an iterative stochastic approximation algorithm. The procedure is illustrated in a simple example estimating the option value of an American call. Numerical results indicate that the additional computational effort required over that required to estimate a European option is relatively small.

Addendum to "Extensions and generalizations of smoothed perturbation analysis in a generalized semi-Markov process framework"

Under a very general framework, both in terms of finite-time performance measures and system structure, the authors derive smoothed perturbation analysis (SPA) estimators and prove their unbiasedness. The commuting condition, which has been key in previous work, is not required a priori, and thus the framework includes such systems as the GI/G/1/K queue and multiclass queueing networks, which do not satisfy the commuting condition. The generality achieved is traded off against the fact that the estimator is not always easily implementable on a single sample path. The use of the commuting condition in a local sense is proposed to help simplify the estimators derived: queueing and multiclass queueing networks are used as illustrative examples. For a simple multiclass closed queueing network, some simulation results are provided. When the commuting condition is satisfied globally, the framework allows the recovery of previous results on IPA and SPA estimators as corollaries of the main theorems. >

Production rate control for failure-prone production systems with no backlog permitted

In this paper, we consider systems in which backlog is not allowed. We show that the hedging point policy is still optimal. For systems with backlog, it is usually quite straightforward to show that their optimal cost-to-go functions are convex-a key property that is needed for the hedging point policy to be optimal. With no backlog permitted, it becomes much more difficult to establish the convexity property, and the explicit formulas for the optimal hedging point and the optimal cost-to-go functions have to be obtained, based on which the convexity property can then be verified. The method we use in this paper to derive these explicit formulas is mainly based on an interesting relationship between the inventory process of the system under the hedging point policy and some stochastic processes which are well studied in queueing theory. >

Simulation Allocation for Determining the Best Design in the Presence of Correlated Sampling

We consider the problem of efficiently allocating simulation replications in order to maximize the probability of selecting the best design under the scenario in which system performances are sampled in the presence of correlation. In the case of two designs, we are able to derive the optimal allocation exactly, and find that in the presence of positive correlation, unless the variance of one design is significantly larger than that of the other, the number of simulation replications should be identical. In extending to a general number of competing designs, an approximation for the asymptotically optimal allocation is obtained. The approximation coincides with the independent case derived previously in the limit as the correlation vanishes and also agrees with the two-design exact solution. Furthermore, the allocations prescribed by the results seem to match intuition, in terms of the relationship to correlations and relative variances between designs, again suggesting that equal allocation is optimal for sufficiently high positive correlation. An allocation algorithm based on the approximation is proposed and tested on several numerical examples.

THE MACLAURIN SERIES FOR THE GI/G/1 QUEUE

We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G /1 queue. The coefficients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light traffic derivatives can be obtained from these series. For the M/G /1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek–Khinchin mean-value formula.

Convexity of sample path performance and strong consistency of infinitesimal perturbation analysis estimates

A strong consistency of infinitesimal perturbation analysis (IPA) estimates for a class of stochastic processes satisfying certain convex properties is established. The approach is based on convex analysis of sample paths. Applications of the results to tandem and cyclic queueing systems are discussed. Possible extensions to more general systems are also illustrated. >

Formulation of the traffic engineering problems in MPLS based IP networks

The growth of the Internet has fuelled the development of new technologies that enable IP backbone networks to be engineered efficiently. One such prominent technology, multiprotocol label switching (MPLS) enables IP networks with quality of service to be traffic engineered well. We mathematically formulate the traffic engineering problems in MPLS based IP networks including constraint based routing, connection admission control, rerouting and capacity planning problems. Unfortunately, obtaining the optimal solution of the traffic engineering problems has undesirable computational complexity since they can be shown to be NP-complete. It is intended that this work will articulate the details and provide insights into the inherent structure of the problems as well as motivate the development of efficient solution techniques.

The queueing equivalence to a manufacturing system with failures

The authors consider optimal production rate control in a failure prone manufacturing system. It is well known that the hedging point policy is the optimum controller for such a system. They show that under the hedging point policy the system can be treated as an M/M/1 queue. Therefore, existing results in queuing theory can be readily applied to obtaining the steady-state probability density function of the production surplus, based on which the optimal hedging point policy can be computed. To a large extent, the approach is based on sample path analysis. It not only provides an alternative way to solve the problem but also reveals some interesting insights. >