سیستم صف بندی موجودی با دو دسته از مشتریان

We consider a queueing-inventory system with two classes of customers. Customers arrive at a service facility according to Poisson processes. Service times follow exponential distributions. Each service uses one item in the attached inventory supplied by an outside supplier with exponentially distributed lead time. We find a priority service rule to minimize the long-run expected waiting cost by dynamic programming method and obtain the necessary and sufficient condition for the priority queueing-inventory system being stable. Formulating the model as a level-dependent quasi-birth-and-death (QBD) process, we can compute the steady state probability distribution by Bright–Taylor algorithm. Useful analytical properties for the cost function are identified and extensive computations are conducted to examine the impact of different parameters to the system performance measures.

Research on queueing systems with inventory control has captured much attention of researchers over the last decades. In this system, customers arrive at the service facility one by one and require service. In order to complete the customer service, an item from the inventory is needed. A served customer departs immediately from the system and the on-hand inventory decreases by one at the moment of service completion. The inventory is supplied by an outside supplier. This system is called a queueing-inventory system (Schwarz et al., 2006). The queueing-inventory system is different from the traditional queueing system because the attached inventory influences the service. If there is no inventory on hand, the service will be interrupted. Also, it is different from the traditional inventory management because the inventory is consumed at the serving rate rather than the customers’ arrival rate when there are customers queued up for service. Berman and Kim (1999) analyzed a queueing-inventory system with Poisson arrivals, exponential service times and zero lead times. The authors proved that the optimal policy is “never to order when the system is empty”. Berman and Sapna (2000) studied queueing-inventory systems with Poisson arrivals, arbitrary distribution service times and zero lead times. The optimal value of the maximum allowable inventory which minimizes the long-run expected cost rate has been obtained. Berman and Sapna (2001) discussed a finite capacity system with Poisson arrivals, exponential distributed lead times and service times. The existence of a stationary optimal service policy has been proved. Berman and Kim (2004) addressed an infinite capacity queuing-inventory system with Poisson arrivals, exponential service times and exponential lead times. The authors identified a replenishment policy which maximized the system profit. Berman and Kim (2001) studied internet-based supply chains with Poisson arrivals, exponential service times and the Erlang lead times and found that the optimal ordering policy has a monotonic threshold structure. Schwarz et al. (2006) derived stationary distributions of joint queue length and inventory processes in explicit product form for M /M /1 queuing-inventory system with lost sales under various inventory management policies such as (r ,Q ) policy and (r ,S ) policy. The M /M /1 queueing-inventory system with backordering was investigated by Schwarz and Daduna (2006). The authors derived the system steady state behavior under Π(1)Π(1) reorder policy which is (0, Q) policy with an additional threshold 1 for the queue length as a decision variable. Krishnamoorthy et al. (2006a) discussed an (s,S) inventory system with service time where the server keeps processing the items even in the absence of customers. Krishnamoorthy et al. (2006b) introduced an additional control policy (N-policy) into (s,S) inventory system with positive service time. In Manuel et al., 2007 and Manuel et al., 2008 the perishable queueing-inventory systems with Markovian arrival process (MAP) were studied. The joint probability distributions of the number of customers in the system and the inventory level were obtained for the steady state case. The stationary system performance measures and the total expected cost rate were both calculated. Some related works in the production industry are He and Jewkes (2000) and He et al., 2002a and He et al., 2002b. He and Jewkes (2000) developed two algorithms for computing the average total cost per product and other performance measures for a make-to-order inventory-production system with Poisson arrivals, exponential production times and zero lead times. He et al. (2002b) studied the inventory replenishment policy of an M/M/1 make-to-order inventory-production system with zero lead times. They explored the structure of the optimal replenishment policy which minimizes the average total cost per product. For the M/PH/1 make-to-order inventory-production system with Erlang distributed lead times, He et al. (2002a) quantified the value of information used in inventory control. All the above studies about queueing-inventory systems are limited to one class of customers. To our knowledge, we have not found any literature studying a queueing-inventory system with two or more classes customers with different priority. In fact such system is popular in reality. For example, in an assembly manufactory buyers with long-term supply contracts have higher priority than buyers who do not. In a hospital, accident victims who are seriously injured will get treatment with priority. The real-life problems stimulate us to study the queueing-inventory system with two classes of customers. An important issue in the queueing-inventory system with two classes of customers is the priority assignment problem. If two classes of customers are both on the queue, the server needs to make a choice between the two classes of customers whenever it would begin a service. In this paper we figure out the optimal service rule to minimize the long-run expected waiting cost. This is different with the previous study about traditional inventory systems with multiple demand classes in which the optimization problem is based on the inventory costs (see Melchiors, 2003, Nahmias and Demmy, 1981, Teng, 2009 and Teunter and Klein Haneveld, 2008). In this paper we consider a queueing system with inventory management in which two classes of customers arrive at a service facility according to Poisson processes, service times follow exponential distributions, each service uses one item in the attached inventory supplied by an outside supplier with exponentially distributed lead times. The rest of the paper is organized as follows. We define the basic queueing-inventory system with two classes of customers in Section 2 and prove a priority service rule to minimize the long-run expected waiting cost in Section 3. In Section 4 we show the stability condition for the priority queueing-inventory system. In Section 5 we compute the joint steady state distributions. In Section 6, we provide some numerical examples. The paper is concluded in Section 7.

In this paper, we consider a queueing-inventory system with Poisson inputs, exponential service times and (r ,Q ) replenishment policy. We show that it is optimal to serve the customer with the larger μbμb to minimize the long-run expected waiting cost. Formulating the system process into a single-dimensional level-dependent quasi-birth-and-death (LDQBD) process, we compute the steady state probability of the system. Numerical investigations exhibit the monotonic behaviors of the inventory replenishment policy and system performance measures which lead to managerial insights into controlling such queueing systems with inventory. The current work can be generalized in two directions. The first one includes investigating the dynamic replenishment policy depending on the queue length information. Some results about queueing-inventory system with one class of customers have been reported in Berman and Kim, 2001 and Berman and Kim, 2004. The second research direction is to consider the general arrival process, service time distribution and lead time distribution which might cause non-Markovian process. The problem becomes much more challenging than the current one.