Approximation analysis of multi-class closed queueing maintenance networks with a parts inventory system and two-phase Coxian time distributions

Abstract

We consider a maintenance network where a set of bases is supported by a replacement parts inventory system and a centrally located repair depot. The ordering policy for the parts is the (S, Q) inventory policy. We extended the previous results to the network, where processing times at each node follow a two-phase Coxian distribution. The proposed network was modeled as a multi-class closed queueing network with a synchronization station. To make the analysis of the network computationally tractable, we developed a two-phase approximation method. In the first phase of the method, the proposed network was analyzed with the previous algorithm based on a product-form approximation. In the second phase, a sub-network was again analyzed with the procedure of a product-form approximation method such that the state space of the sub-network was reduced. In the analysis of a sub-network, a recursive method was also used to solve balance equations by exploiting the special structure of the Markov chain. The new algorithm provided a good estimation of the performance measures of interest. In addition to being accurate, the new algorithm is simple and converges rapidly.

Introduction

For industries and services with technically advanced equipments, such as expensive vehicles, electronic devices and airplane engines, high degree of availability is essential. One way to achieve high degree of availability is to supply sufficient spare parts to replace failed components. However, as critically important components of these advanced systems are expensive, it is often profitable to repair and reuse these components when they fail. For this reason, manufacturing and service industries resort to repair facilities to support their field service operations. Because the cost of investment on repairable items and repair facilities is known to be considerable, a proper interaction between the spare inventories and the capacity of repair facilities should be considered for efficient and effective system operation.

In the literature, considerable attention is paid to spare inventory models designed to address this question. Muckstadt [1], Graves [2] and Schultz [3] are only a few among the researchers who have investigated the spare inventories for repairable items. The majority of the research on spare inventories is based on a method called the multi-echelon technique for recoverable item control (METRIC), which was designed for the US Air Force by Sherbrooke [4]. This method considers a two-echelon repairable item inventory for the determination of the base and depot stock levels to minimize the expected backorders. After the presentation of this model, several extensions have been made (see [5], [6] for surveys). There are four main streams of research regarding the spare provisioning problem for repairable items, as shown in Table 1.

Because the METRIC model adopted the assumption of the infinite repair capacity and the infinite item sources, most research on the first stream is based on the assumption that repair capacity is ample, and an infinite number of items is operating at bases. Lau et al. [7] considered a multi-echelon repairable item inventory system with infinite repair capacity and failure rates that depend on time. Kranenburg and Van Houtum [8] studied a multi-item, single-stage spare parts inventory model with multiple customer classes. Andersson and Melchiors [9] studied a single-item two-echelon inventory with lost sales and a batch ordering policy. Al-Rifaia and Rossettib [10] and Topan et al. [11] considered a two-echelon inventory system in which a batch ordering policy was used for the central warehouses and a base-stock policy was used for the local warehouses. Under the assumption that a failed item is immediately replaced by a spare item, De Smidt-Destombes et al. [12] presented a spare inventory model where only the fewest items needed are operational while other items are switched when required. Due to substantial computing time required for an exact analysis, most of the research mentioned above [7], [9], [10], [12] is based on approximation methods using the queueing theory or Markov models.

The second stream of research in this area adopts the finite repair capacity and infinite number of items operating at bases. As Kim et al. [13] have noted, the models in this stream are more realistic than the comparable METRIC models, but certainly more difficult to solve due to the huge multidimensional state spaces involved. Most research in this stream assumes that operating items fail according to a Poisson process, and repair times have general distributions. Sleptchenko et al. [14] analyzed the trade-off between inventory levels and repair capacity in a multi-echelon multi-indentured model under the assumption of infinite items operating at each base. Díaz and Fu [15], Spanjers et al. [16], Jung et al. [17] and Kim et al. [13] developed an approximation method for a multi-item multi-echelon model with finite repair capacity and infinite items. Rappold and Van Roo [18] presented an approach to model and solve the joint problem of facility location, inventory allocation, and capacity investment when demands are stochastic, and an infinite number of items are operating at bases.

The third stream of research in this area assumes the finite repair capacity and the finite number of items operating at bases. As the number of items is finite, researchers in this type of stream generally model their networks as closed systems, assuming that holding times at all nodes follow independent exponential distributions. Gross et al. [19] and Madu [20] proposed mathematical models using a closed queueing network that consists of one base and a two-stage repair depot. Albright and Gupta [21] analyzed a multi-echelon multi-indentured model under the assumption that the finite number of items is operating at the bases, and the finite number of identical repairmen exists. Under block replacement sharing limited spare items and repair capacity, De Smidt-Destombes et al. [22], [23] considered an installed base consisting of N identical components, of which at least k components are required for the operation of the system.

From the studies mentioned above, repairs for failed items are not assumed to require parts. Therefore, when an item fails, the item is sent directly to a repair depot. While this assumption is realistic in some cases, parts are required to repair failed items in many practical cases. Abboud and Daigle [24] presented a spares provisioning problem incorporating a parts inventory system. They assumed that the system consists of a base, a parts inventory system and a repair depot. When an item in the base fails, the failed item is supplied with a part at the parts inventory system and sent to the repair depot. These researchers assumed that an (S−1,S) inventory policy is used at the parts inventory system. By definition, the (S−1,S) inventory policy sets maximum stock level as S and places an order whenever the stock balance (on hand plus on order) falls to S−1. Therefore, the (S−1,S) inventory policy places an order whenever a demand occurs (see e.g. [25] for further details of the (S−1,S) inventory policy). For special cases when the value of S is zero or infinite, Abboud and Daigle presented an algorithm that searches for the optimal number of items and the repair channels. To determine the initial feasible solution, the algorithm uses Little's law, which means that the average number of customers in a system is equal to the average effective arrival rate multiplied by the average time a customer spends in the system (see e.g. [26] for further details of Little's law).

Recently, Park and Lee [27] extended the model of Abboud and Daigle by relaxing the following assumptions: first, while there is only one base in the model of Abboud and Daigle, their model assumed that there are multiple bases, and the mean time before failure for the items owned by one base differs from the mean time before failure of the items owned by other bases. Second, a continuous review (S, Q) inventory policy is used in the model of Park and Lee [27]. In the (S, Q) inventory policy, each time the accumulated demand for parts reaches level Q, a batch order of size Q is placed such that the sum of on-hand inventory and on-order amount becomes a target number, S. After order lead times, the ordered parts with batch size Q arrive at the parts inventory system to replenish the inventory. Therefore, the (S−1,S) policy is a special case of the (S, Q) inventory policy with Q=1. The proposed system was modeled as a multi-class closed queueing network with a synchronized station and analyzed by a decomposition approximation method based on a product-form approximation method [28].

In this study, we generalize the spares provisioning problem with a parts inventory system given in Abboud and Daigle [24] and Park and Lee [27]. These authors assume that operational times of an item before failure, the order lead times and the repair times follow an exponential distribution. The generalization is performed in this study by allowing a two-phase Coxian (C₂) distribution. Because a variety of data sets or distributions can be presented adequately by a C₂ distribution, more accurate modeling of real maintenance networks is possible. In detail, we assume that the operational times of an item before failure, the order lead times and the repair times follow C₂ distributions. The sojourn times at each phase in C₂ distributions are exponentially distributed by definition. Thus, due to the imbedded memoryless property, the C₂ distribution is computationally tractable and extensively used [29]. In addition, our model also assumes the (S, Q) inventory policy and multiple bases, as the model in Park and Lee [27] did. Although these assumptions are more adequate to reflect the actual circumstances in the industries, analysis of the proposed system becomes much more complicated.

In detail, to analyze the sub-network in the decomposition approximation method, Park and Lee [27] proposed a recursive method to exploit the special structure of the Markov chain with an exponential distribution. However, because dimension of the Markov chain with a C₂ distribution is double compared to the one in Park and Lee [27], the structural configuration of the Markov chain becomes considerably more complicated and the number of balance equations increases rapidly. Furthermore, as the (S, Q) policy at the parts inventory system is still maintained in this study, it is considerably more difficult to exploit the special structure of the Markov chain with expanded dimensions. Therefore, an efficient procedure should be devised to overcome this difficulty.

The purpose of this paper is to develop a new approximate algorithm for the analysis of the proposed system. The new algorithm developed in this paper is based on the previous algorithm proposed by Park and Lee [27]. Therefore, the new algorithm yields the same results as the algorithm of Park and Lee [27] if processing times at each node follow an exponential distribution. However, in this study, to solve the case of C₂ distributions, a two-phase approximation method is devised to make the system computationally tractable by reducing the dimension of the state spaces involved. In the first phase of the approximation method, the proposed network is analyzed with the algorithm of Park and Lee [27]. In the second phase, the procedure of a product-form approximation method is applied to the analysis of a sub-network to allow for efficient computation. The new algorithm provides a fairly good estimation of the performance measures of interest. In addition to being accurate, the new algorithm is simple and converges rapidly.

As for the rest of the paper, the network under study is described and modeled as a multi-class closed queueing network in Section 2. In Section 3, the network is analyzed with a product-form approximation method. In 4 Analysis of the sub-network, 5 Analysis in isolation of the base, an approximation method is proposed to analyze sub-networks. In Section 6, performance measures are defined, and the results obtained from the algorithm are compared with those obtained by simulation. Finally, we present the conclusion of this paper in Section 7.