Qi-Ming He

Markov chains with marked transitions

Several useful point processes such as the Markovian arrival process, the input and departure processes of finite-capacity Markovian queues, and various models for counters and biological phenomena are obtained by considering Markov chains with marked transitions. This point of view yields many examples of interesting dependent point processes and provides a unified formalism for their study. This paper presents some characterizations of Markov chains with marked transitions.

Fundamentals of Matrix-Analytic Methods

Fundamentals of Matrix-Analytic Methods targets advanced-level students in mathematics, engineering and computer science. It focuses on the fundamental parts of Matrix-Analytic Methods, Phase-Type Distributions, Markovian arrival processes and Structured Markov chains and matrix geometric solutions. New materials and techniques are presented for the first time in research and engineering design. This book emphasizes stochastic modeling by offering probabilistic interpretation and constructive proofs for Matrix-Analytic Methods. Such an approach is especially useful for engineering analysis and design. Exercises and examples are provided throughout the book.

On the Relation Between the Benefits of Risk Pooling and the Variability of Demand

The benefits of pooling risks, manifested in inventory management by consolidating multiple random demands in one location, are well known. What is less well understood are the determinants of the magnitude of the savings. Recently there has been speculation about the impact of demand variabilities on the benefits of risk pooling. We provide an example where increased variability of the individual demands actually reduces the benefits of risk pooling. We prove, however, that if we restrict increased variability to a common linear transformation, the greater the demand variabilities the larger the benefits of consolidating them, in agreement with intuition. We also provide bounds on the benefits of the consolidation of demands. Our results do not require independence of the demands, apply to any number of pooled demands, and are obtained in a pure cost-driven model.

Optimal and near-optimal inventory control policies for a make-to-order inventory-production system

Abstract This paper examines several inventory replenishment policies for a make-to-order inventory–production system that consists of a production workshop and a warehouse. Demands arrive to the production workshop according to a Poisson process, and are processed in an FCFS manner. The production workshop requires that the warehouse provides, as needed, raw materials for use in the production process. The warehouse inventory is replenished according to an inventory replenishment policy. The optimal replenishment policy, which minimizes the average total cost per product, is derived using a Markov decision process approach. The structure of the optimal replenishment policy is explored. Simple “order-up-to”, “myopic”, and heuristic replenishment policies are introduced. The myopic and heuristic replenishment policies are easy to compute, and yet perform almost as well as the optimal replenishment policy.

Spectral Polynomial Algorithms for Computing Bi-Diagonal Representations for Phase Type Distributions and Matrix-Exponential Distributions

In this paper, we develop two spectral polynomial algorithms for computing bi-diagonal representations of matrix-exponential distributions and phase type (PH) distributions. The algorithms only use information about the spectrum of the original representation and, consequently, are efficient and easy to implement. For PH-representations with only real eigenvalues, some conditions are identified for the bi-diagonal representations to be ordered Coxian representations. It is shown that every PH-representation with a symmetric PH-generator has an equivalent ordered Coxian representation of the same or a smaller order. An upper bound of the PH-order of a PH-distribution with a triangular or symmetric PH-generator is obtained.

On matrix exponential distributions

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

The value of information used in inventory control of a make-to-order inventory-production system

This paper studies a make-to-order inventory-production system consisting of a warehouse and a workshop. The concept of information level as the detail available on the number of unfilled demands at the workshop is introduced. The focal point is the value of the information used in inventory control in the warehouse. Dynamic programming is used to develop an algorithm for computing the optimal replenishment policy and the average total inventory cost per product. Numerical analysis is carried out and the results show that information used in inventory control can reduce the total inventory cost significantly. It is shown that the classical (Q, R) policy may not perform well if information about the number of demands is partially or fully available.

The Versatility of MMAP[K] and the MMAP[K]/ G [K]/1 Queue

This paper studies a single server queueing system with multiple types of customers. The first part of the paper discusses some modeling issues associated with the Markov arrival processes with marked arrivals (MMAP[K], where K is an integer representing the number of types of customers). The usefulness of MMAP[K] in modeling point processes is shown by a number of interesting examples. The second part of the paper studies a single server queueing system with an MMAP[K] as its input process. The busy period, virtual waiting time, and actual waiting times are studied. The focus is on the actual waiting times of individual types of customers. Explicit formulas are obtained for the Laplace–Stieltjes transforms of these actual waiting times.

Performance measures of a make-to-order inventory-production system

This paper develops two algorithms for computing the average total cost per product and other performance measures for a make-to-order inventory-production system. The two algorithms are developed by using matrix analytic methods. The first algorithm is based on the matrix-geometric solution of the Quasi-Birth-and-Death (QBD) Markov process. The second algorithm is based on the fundamental period of the QBD Markov process. The advantages and disadvantages of the two algorithms are discussed

OPTIMAL INSPECTION ORDER WHEN PROCESS’ FAILURE RATE IS CONSTANT

Suppose that a lot has been produced by a process with a constant failure rate. So either the entire lot is good, or all units up to some point are good and from that point on are all defective. We wish to determine the order in which units in such lot should be inspected so as to minimize the expected number of inspections needed to identify all defectives. Unlike previous work in this area, we do not a priori assume that the last unit in the lot is defective, and that key difference turns out to dramatically influence the nature of the optimal inspection policy and the expected number of units inspected. After analyzing the optimal policy, we suggest a very simple and intuitive heuristic, which turns out to perform extremely well. Numerical results are provided.