Xiuli Chao

Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations

This note introduces reliability issues to the analysis of queueing systems. We consider an MIGI1 queue with Bernoulli vacations and server breakdowns. The server uptimes are assumed to be exponential, and the server repair times are arbitrarily distributed. Using a supplementary variable method we obtain a transient solution for both queueing and reliability measures of interest. These results provide insight into the effect of server breakdowns and repairs on system performance.

A queueing network model with catastrophes and product form solution

Consider a network of queues with the possibility of catastrophes at each station. When a catastrophe occurs at one service station, all the customers there are destroyed immediately, and the server is then sent for repair. The catastrophes may come either from outside the system or from another service station. Steady-state probability of the queue sizes is obtained, and is shown to have a product form solution. This model can be used, for instance, to analyze migration processes with catastrophes, and computer networks with virus infections.

Analysis of multi-server queues with station and server vacations

Abstract In this paper, we consider GI / M / c queues with two classes of vacation mechanisms: Station vacation and server vacation. In the first one, all the servers take vacation simultaneously whenever the system becomes empty, and they also return to the system at the same time, i.e., station vacation is a group vacation for all servers. This phenomenon occurs in practice, for example, when the system consists of a set of machines monitored by a single operator, or the system consists of inseparable interconnected parallel machines. In such situations the whole station has to be treated as a single entity for vacation when the system is utilized for a secondary task. For the second class of vacation mechanisms, each server takes its own vacation whenever it complexes a service and finds no customers waiting in the queue, which occurs, for instance in the post office, when each server is a relatively independent working unit, and can itself be used for other purposes. For both models, we derive steady state probabilities that have matrix geometric form, and develop computational algorithms to obtain numerical solutions. We also analyze and make comparisons of these models based on numerical observations.

Dynamic balancing of inventory in supply chains

Abstract We consider the dynamic version of the classic problem of allocation of inventories to a set of retailers to rectify the imbalance of inventories amongst them. While most research is focussed on analyzing different allocation strategies with a predetermined time of shipment (static policy), we investigate the benefit of using real time demand (inventory) information to schedule rebalancing shipments in a retail network. We model the dynamic rebalancing problem that has two decisions, the timing of the balancing shipments and determination of the new stocking levels at the retailers, as a dynamic program (DP). We obtain structural properties for the optimal allocation, rebalancing and timing strategies. We also present conditions under which a greedy heuristic to decide how much to ship from one retailer to another is optimal. The DP for determining the optimal timing and quantity of shipments has a very large state space. We present an algorithm to solve this DP efficiently. We also provide a heuristic solution procedure to the dynamic problem that performs very close to optimal. Numerical results show that dynamic allocation policies can lead to substantial benefits over the static policy especially in systems in which the starting inventories at the retailers are balanced or when high service levels are required.

Optimal Control of Inventory Systems with Multiple Types of Remanufacturable Products

Product returns have become a significant feature of many manufacturing systems. Because products are returned under different operational conditions, they usually require different remanufacturing effort/costs. Motivated by a project with a major energy company that manages its inventory through options of ordering and remanufacturing returned products (cores) in various condition, in this paper, we study a single-product, periodic-review inventory system with multiple types of cores. The serviceable products used to fulfill stochastic customer demand can be either manufactured from new parts or remanufactured from the cores, and the objective is to minimize the expected total discounted cost over a finite planning horizon. We show that the optimal manufacturing--remanufacturing--disposal policy has a simple structure and can be completely characterized by a sequence of constant control parameters when manufacturing and remanufacturing leadtimes are the same. To demonstrate the value of the optimal policy, we conduct a numerical study that compares its performance with two simple heuristics, namely, pull policy without and with sorting. The results show that the reduction in system cost by using the optimal policy can be significant. When manufacturing and remanufacturing leadtimes are different, we develop a heuristic method for computing the near-optimal control policy that performs quite well as demonstrated numerically.

Modeling and performance evaluation of a cellular mobile network

An analytic model of cellular mobile communications networks with instantaneous movement is investigated in this paper. This cellular mobile network is showed to be equivalent to a queueing network and furthermore the equilibrium distribution of this cellular mobile network is proved to have a product form. The explicit expressions for handoff rates of calls from one cell to another, the blocking probability of new calls and handoff calls are then obtained. Actual call connection time (ACCT) of a call in this cellular mobile network is characterized in detail, which is the total time a mobile user engages in communications over the network during a call connection and can be used to design appropriate charging schemes. The average ACCT for both complete call and incomplete call, as well as the probability for a call to be incomplete or complete, are derived. Our numerical results show how the above measures depend on the new call arrival process for some specific reserved channels numbers in each cell. The results presented in this paper are expected to be useful for the cost analysis for updating location and paging in cellular mobile network.

Call admission control for an adaptive heterogeneous multimedia mobile network

A novel call admission control (CAC) scheme for an adaptive heterogeneous multimedia mobile network with multiple classes of calls is investigated here. Different classes of calls may have different bandwidth requirement, different request call holding time and different cell residence time. At any time, each cell of the network has the capability to provide service to at least a given number of calls for each class of calls. Upon the arrival (or completion or hand off) of a call, a bandwidth degrade (or upgrade) algorithm is applied. An arriving call to a cell, finding insufficient bandwidth available in this cell, may either be disconnected from the network or push another call out of the cell toward a neighboring cell with enough bandwidth. We first prove that the stationary distribution of the number of calls in the network has a product form and then show how to apply this result in deriving explicit expressions of handoff rates for each class of calls, in obtaining the disconnecting probabilities for each class of new and handoff calls, and in finding the grade of service of this mobile network

On generalized networks of queues with positive and negative arrivals

Consider a generalized queueing network model that is subject to two types of arrivals. The first type represents the regular customers; the second type represents signals. A signal induces a regular customer already present at a node to leave. Gelenbe [5] showed that such a network possesses a product form solution when each node consists of a single exponential server. In this paper we study a number of issues concerning this class of networks. First, we explain why such networks have a product form solution. Second, we generalize existing results to include different service disciplines, state-dependent service rates, multiple job classes, and batch servicing. Finally, we establish the relationship between these networks and networks of quasi-reversible queues. We show that the product form solution of the generalized networks is a consequence of a property of the individual nodes viewed in isolation. This property is similar to the quasi-reversibility property of the nodes of a Jackson network: if the arrivals of the regular customers and of the signals at a node in isolation are independent Poisson, the departure processes of the regular customers and the signals are also independent Poisson, and the current state of the system is independent of the past departure processes.

TRANSIENT ANALYSIS OF IMMIGRATION BIRTH–DEATH PROCESSES WITH TOTAL CATASTROPHES

Very few stochastic systems are known to have closed-form transient solutions. In this article we consider an immigration birth and death population process with total catastrophes and study its transient as well as equilibrium behavior. We obtain closed-form solutions for the equilibrium distribution as well as the closed-form transient probability distribution at any time t ≥ 0. Our approach involves solving ordinary and partial differential equations, and the method of characteristics is used in solving partial differential equations.

Stochastic convexity for multidimensional processes and its applications

A multidimensional stochastic process is considered which is a function of a parametric process. The parametric process may be multidimensional as well. Two such processes are compared that differ only in their parametric processes. Known stochastic convexity results for one-dimensional stochastic processes are extended to multidimensional processes. These results are used to obtain comparison results for various queuing systems that are subject to different parametric processes, which may be the arrival processes, service processes, etc. Based on these comparison results it is shown how the performances of queueing systems can be affected by the variability of parametric processes. >