【正文】
hte queues had a product form [18]. In the proposed approach, the whole chain is deposed into multiple singlestage multiclass inventory queues (an inventoryqueue is a queueing model that incorporates certain inventory replenishment policies such as base stock). The inputs (raw materials or ponents arrival processes) of each singlestage multiclass inventory queue are used to capture the characteristics of input flows of the original chain. The rest of the paper is anized as follows. Section 2 provides a review of the relevant literature. In Sect. 3, the operations and the principal characteristics of the developed model are described. A deposition method that divides the whole logistics chain into multiple singlestage queuing works is proposed and the performance measures by analyzing the singlestage queueing work are obtained in Sect. 4. Section 5 presents some numerical results. Section 6 summarizes this research and gives some future research directions. 2 Literature review Significant literature exists on inventory management in logistics chains. In the following, some prior studies devoted to the issues which are similar to the above described problems are reviewed. Some important work on singleproduct multistage systems is reviewed. Lee and Zipkin [11, 12] and Duri et al. [9] used the deposition method to analyze the tandem queues and processing works. They transformed the production system into a multiechelon model with limited production capacities. Azaron et al. [3] developed an open queueing work for multistage assemblies in which each service station represents a manufacturing or assembly operation. In the proposed model, not only the manufacturing and assembly processing times are considered as the functions of the arrival and service rates of the various stages of the manufacturing process, but also the role of transport times between the service stations in the manufacturing lead time is considered. An assumption is that the arrival processes of the individual parts of the product are independent Poisson processes with equal rates. In each service station, there is a server with exponential distribution of processing time. The transport times between the service stations are assumed to be independent random variables with exponential distributions. By applying the longest path analysis in queueing works, the distribution function of time spent by a product in the system or the manufacturing lead time can be obtained. The study in this paper is more similar to Liu et al. [13]. They developed a multistage inventory queue model and a jobqueue deposition approach that evaluates the performance of serial manufacturing and supply chain systems with inventory control at every stage. In this paper, the proposed method deposes a queue at each stage into two ponents, a backlog queue and a material queue. Instead of the single type product queues in their model [13], the queues contain a multiclass of items