Quantitative impact analysis of priority policy applied to BIM-based design validation
应用于基于 BIM 的设计验证的优先政策定量影响分析

Priority queueing model  优先排队模式
BIM RFIs  BIM RFI
BIM staffing  BIM 人员配置
Waiting avoidance cost  避免等待的成本
BIM ROI  BIM 投资回报率

Unexpected events can occur in construction projects. Among them, design errors are a major risk factor for reworks and schedule delays [1]. Failure to eliminate or manage risk factors during the early stages of a construction project can significantly affect the success of the project [2].
建筑项目中可能会出现意外事件。其中，设计错误是造成返工和工期延误的主要风险因素[1]。如果不能在建筑项目的早期阶段消除或管理风险因素，就会严重影响项目的成功[2]。

Building Information Modeling(BIM) is considered one of the means to reduce economic losses due to design errors [3]. The benefits of using BIM include fast and accurate design visualization and accuracy at every stage of the design phase [4]. Owing to these benefits, the use of BIM has increased [5]. Moreover, several studies have been conducted to reduce the waste of resources while using BIM [6-8].
建筑信息模型（BIM）被认为是减少因设计错误造成经济损失的手段之一[3]。使用 BIM 的好处包括快速、准确的设计可视化以及设计阶段每个阶段的准确性[4]。由于这些优点，BIM 的使用量不断增加[5]。此外，还开展了多项研究，以减少使用 BIM 时的资源浪费[6-8]。

However, despite numerous benefits of using BIM, its adoption has been slow due to the difficulties experienced while operating and maintaining it [9]. The adoption of BIM can provide information as needed. However, this incurs additional costs [10] and requires BIM experts [11]. To implement ideal BIM, BIM staff should be assigned to handle the required information and manage issues raised by various project participants [6,10,11]. Therefore, human resource allocation is important to address uncertainty in construction projects [12]. Human resource allocation is a key component of project success. However, there are not enough quantitative tools to efficiently allocate the
然而，尽管使用 BIM 有诸多好处，但由于在操作和维护方面遇到困难，其采用速度一直很慢[9]。采用 BIM 可以根据需要提供信息。然而，这会产生额外的成本[10]，并且需要 BIM 专家[11]。要实现理想的 BIM，应指派 BIM 工作人员处理所需的信息并管理各项目参与者提出的问题 [6,10,11]。因此，人力资源分配对于解决施工项目中的不确定性非常重要[12]。人力资源配置是项目成功的关键因素。然而，目前还没有足够的量化工具来有效地分配人力资源[13]。
resources [13]. Following the emphasis placed on the importance of BIM staff, human resource allocation can influence project performance or effectiveness can become a major research topic.
资源[13]。在强调 BIM 人员的重要性之后，人力资源配置对项目绩效或效果的影响可能会成为一个重要的研究课题。

Few studies have analyzed BIM ROI, including the effect of BIM staff. Neelamkavil et al. [14] included new variables in addition to the 6 performance indicators of BIM ROI analysis. LEE et al. [8] argued that BIM ROI analysis using BIM RFI should include a detailed analysis of the number of workers, quality of management, etc. Accordingly, Ham et al. [6] quantified BIM staff by tracking the BIM RFI of the actual project and analyzing the effect of BIM staff on the waiting time of project engineers. The their study applied the first come first served (FCFS) rule to the order in which the BIM staff process request for information(FRI). However, design errors have different effects on projects depending on the type, location, and time of discovery [1]. Paik et al. [58], Chahrour et al. [59] analyze BIM project data to prioritize coordination issues. The result of focusing on the priorities of BIM design coordination is improved productivity and economics of the design process. However, Chahrour et al. [59] cited the need for detailed value analysis and management of benefit realization in addition to the cost savings that can be achieved with BIM process coordination and conflict detection. Paik et al. [58] identified priorities for BIM design coordination issues, but no economic analysis was performed on them.
很少有研究分析 BIM 投资回报率，包括 BIM 人员的影响。Neelamkavil 等人[14]在 BIM 投资回报率分析的 6 个性能指标之外加入了新的变量。LEE 等人[8]认为，使用 BIM RFI 进行 BIM 投资回报率分析应包括对工人数量、管理质量等的详细分析。因此，Ham 等人[6]通过跟踪实际项目的 BIM RFI 来量化 BIM 人员，并分析 BIM 人员对项目工程师等待时间的影响。他们的研究对 BIM 人员处理信息请求（FRI）的顺序采用了先来后到（FCFS）规则。然而，设计错误会因类型、地点和发现时间的不同而对项目产生不同的影响[1]。Paik 等人[58]、Chahrour 等人[59]通过分析 BIM 项目数据来确定协调问题的优先次序。关注 BIM 设计协调优先级的结果是提高了设计过程的生产率和经济性。然而，Chahrour 等人[59] 指出，除了通过 BIM 流程协调和冲突检测可以节约成本外，还需要进行详细的价值分析和效益实现管理。Paik 等人[58]确定了 BIM 设计协调问题的优先次序，但没有对其进行经济分析。

Previous studies have revealed that there are additional items to
以往的研究表明，还需要

https://doi.org/10.1016/j.autcon.2023.105031
Received 29 August 2022; Received in revised form 19 June 2023; Accepted 15 July 2023
2022 年 8 月 29 日收到；2023 年 6 月 19 日收到修订稿；2023 年 7 月 15 日接受
Available online 26 July 2023
可于 2023 年 7 月 26 日在线查阅
0926-5805/⑳23 Elsevier B.V. All rights reserved.
0926-5805/⑳23 Elsevier B.V. 保留所有权利。保留所有权利。
consider in evaluating the benefits of BIM coordination in various ways. However, economic feasibility evaluation is not progressing because specific items cannot be presented. For example, the introduction of priorities in BIM process coordination can make a difference in the wait times of engineers and the performance of BIM staff. An additional item to consider in previous studies are the introduction of priorities for BIM process coordination. The specific item is the performance change that can change due to additional items. If so, it is necessary to analyze the change in the number of BIM staff and the waiting cost of engineers according to the application of the priority rule to BIM coordination.
在评估 BIM 协调的效益时，需要考虑各种方式。然而，由于无法提出具体项目，经济可行性评估没有取得进展。例如，在 BIM 流程协调中引入优先级，可使工程师的等待时间和 BIM 人员的工作表现有所不同。在以往的研究中，引入 BIM 流程协调的优先级是需要考虑的额外项目。具体项目是绩效变化可能因附加项目而改变。如果是这样，就有必要分析根据 BIM 协调优先规则的应用，BIM 人员数量和工程师等待成本的变化。

Accordingly, this study aims to improve the performance of BIMbased design validation by applying the priority queuing model from the field of management science. It aims to improve BIM ROI by analyzing the queue model performance indicators according to the RFI processing order of BIM staff, and quantifying the change in the number of BIM staff and the waiting cost of project engineers.
因此，本研究旨在通过应用管理科学领域的优先排队模型来提高基于 BIM 的设计验证绩效。其目的是根据 BIM 人员的 RFI 处理顺序分析排队模型性能指标，量化 BIM 人员数量的变化和项目工程师的等待成本，从而提高 BIM 投资回报率。

This study first presents a theoretical review of BIM and management issues. Research issues are identified by reviewing previous studies on human resource allocation in construction, BIM design validation, and BIM ROI, and by critically reviewing existing approaches, including the discussion of the queuing model. We further present a detailed framework for comparing performance indicators according to the BIM RFI processing sequence using a queuing model for quantitative analysis when priority rules are applied. We discuss, including detailed descriptions of two real-world projects, where BIM was applied in the construction phase, to use the model.
本研究首先对 BIM 和管理问题进行了理论回顾。通过回顾以往关于建筑业人力资源分配、BIM 设计验证和 BIM 投资回报率的研究，以及对现有方法的批判性回顾，包括对排队模型的讨论，确定了研究问题。我们进一步提出了一个详细的框架，用于根据 BIM RFI 处理顺序比较性能指标，在应用优先规则时使用排队模型进行定量分析。我们讨论了该模型的使用，包括对两个在施工阶段应用了 BIM 的实际项目的详细描述。

In a construction project, risks and uncertainties are the highest at the beginning of the project [15]. Recognizing risks and uncertainties at the design stage does not significantly affect the cost, while recognizing them at the construction stage increases the risk significantly [2]. The use of BIM in the design phase can improve the overall process [16]. Consequently, the use of BIM at the beginning of the project can significantly influence on the project and minimize costs.
在建筑项目中，项目开始阶段的风险和不确定性最高[15]。在设计阶段认识到风险和不确定性不会对成本产生重大影响，而在施工阶段认识到风险和不确定性则会大大增加风险[2]。在设计阶段使用 BIM 可以改善整个流程[16]。因此，在项目开始阶段使用 BIM 可以对项目产生重大影响，并最大限度地降低成本。

BIM-based coordination is an important step prior to construction, as the building is equipped with many mechanical, electrical and plumbing (MEP) systems [17]. The general concept of coordinating construction design involves defining the location and dimensions of building components to avoid clash between two or more disciplines including architecture, structural, mechanical, electrical, plumbing, and fire protection [18]. This allows for the identification of potential conflicts early and the avoidance of conflicts that affect project costs and schedules.
基于 BIM 的协调是施工前的一个重要步骤，因为建筑物配备了许多机械、电气和管道系统 (MEP)[17]。协调施工设计的一般概念包括定义建筑构件的位置和尺寸，以避免两个或多个专业（包括建筑、结构、机械、电气、管道和消防）之间的冲突[18]。这样可以及早发现潜在冲突，避免冲突影响项目成本和进度。

In the construction sector, the Design-Bid-Build (DBB) approach is the most common industry standard for project delivery [19]. The DBB approach can separate architects from contractors to build projects, which are used for most public and private projects [20], but can result in risks such as fragmentation and discontinuity, which are well-known in the AEC industry [21]. This DBB method problem can be improved through the introduction of BIM [21].
在建筑领域，设计-招标-建造（DBB）方法是最常见的项目交付行业标准[19]。DBB 方法可以将建筑师与承包商分开来建设项目，大多数公共和私人项目都采用这种方法[20]，但可能会导致分散和不连续性等风险，这在 AEC 行业是众所周知的[21]。这种 DBB 方法的问题可以通过引入 BIM 得到改善[21]。

However, BIM does not fit well with the current widespread procurement system (ex:DBB) [21]. Although the Design-Build (DB) method allows for overlap between design and construction phases and allows for easy adjustment of design errors [22], the DBB method is difficult to use to maximize the benefits of BIM-based coordination [23]. The DBB approach results in a certain level of separation between the design phase and the construction phase [16]. Consequently, the DB method can easily adjust design errors. However, there is not enough time for BIM-based coordination when using DBB method. BIM-based design validation is performed in a short period before construction begins [6]. When suing BIM, construction modelers inevitably discover errors and inconsistencies as they create architectural information
然而，BIM 与当前广泛使用的采购系统（例如：DBB）并不十分匹配[21]。虽然设计-建造（DB）方法允许设计阶段和施工阶段重叠，并允许轻松调整设计错误[22]，但 DBB 方法很难最大限度地发挥基于 BIM 的协调优势[23]。DBB 方法导致设计阶段与施工阶段之间存在一定程度的分离[16]。因此，DB 方法很容易调整设计错误。然而，在使用 DBB 方法时，没有足够的时间进行基于 BIM 的协调。基于 BIM 的设计验证是在施工开始前的短时间内进行的[6]。在使用 BIM 时，施工建模人员在创建建筑信息时不可避免地会发现错误和不一致之处。
models [24]. Resolving all design errors in a short period from the detailed design to the pre-construction stage requires a lot of additional cost. However, the use of BIM must be done within BIM contract costs.
模型[24]。要在短时间内解决从详细设计到施工前阶段的所有设计错误，需要大量的额外成本。然而，BIM 的使用必须在 BIM 合同成本范围内进行。

In many studies, the number of BIM RFIs was tracked for the economic analysis of the adoption of BIM [6-8,10]. Barlish et al. [7] design errors were analyzed by dividing them into error identification potential, effect on schedule, and effect on quality. Consequently, there were differences in schedule, quality, possibility of error identification, direct cost, and indirect cost depending on design errors. However, the effect of each design error was not considered in the BIM ROI analysis. Ham et al. [6] improved BIM ROI by quantifying the BIM workforce through the number of RFIs in three projects. However, there was a precondition that the effects of design errors were not considered and were managed in the order in which they were discovered.
许多研究跟踪了 BIM RFI 的数量，用于采用 BIM 的经济分析[6-8,10]。Barlish 等人[7]将设计错误分为错误识别可能性、对进度的影响和对质量的影响进行分析。结果表明，设计错误在进度、质量、错误识别可能性、直接成本和间接成本方面都存在差异。但是，在 BIM 投资回报率分析中没有考虑每个设计错误的影响。Ham 等人[6]通过三个项目中的 RFI 数量来量化 BIM 劳动力，从而提高了 BIM 投资回报率。然而，前提条件是不考虑设计错误的影响，并按照发现错误的顺序进行管理。

Ajibade et al. [61] studied the frequency of occurrence of RFI and the time of occurrence of RFI according to the characteristics of the project. It was revealed that the larger and more complex the project, the more RFIs and the longer the processing time. Importantly, they found research data that the processing of RFIs is related to the number and level of staffing. The question of how to estimate and prepare the staff needed to handle RFIs was raised as a future study. Vishal et al. [60] presented a framework for classifying and specifying functional and technical requirements for a BIM server to serve as a platform. Based on the contingency theory, it is argued that the collaboration requirements would vary from project to project, and, hence, support technical requirements should be an integral part of the BIM-server development rather than an afterthought. Therefore, in order to maximize the technological benefits of using BIM, it is essential to identify the potential impact of design errors on a project and address them in order of priority.
Ajibade 等人[61]根据项目的特点研究了 RFI 发生的频率和 RFI 发生的时间。结果显示，项目越大、越复杂，RFI 就越多，处理时间就越长。重要的是，他们发现研究数据表明 RFI 的处理与人员配置的数量和水平有关。他们提出了如何估算和准备处理 RFI 所需的人员这一问题，并将其作为今后的研究课题。Vishal 等人[60]提出了一个框架，用于分类和指定 BIM 服务器作为平台的功能和技术要求。根据权变理论，他们认为不同项目的协作要求会有所不同，因此，支持技术要求应该是 BIM 服务器开发的一个组成部分，而不是事后才想到的。因此，为了最大限度地发挥使用 BIM 的技术优势，必须识别设计错误对项目的潜在影响，并按优先顺序解决这些问题。

Management skills and workforce planning can significantly affect productivity in construction management [25]. The workflow management(WfM) system is the most successful type of systems that support collaborative work [26]. The WfM system is evaluated in terms of lead time, service time, latency and resource utilization, and the above adjustments can improve efficiency [27]. The WfM system has its roots in computer science and management science [28], and queue networks are used for the quantitative analysis of efficiency [28-30].
管理技能和劳动力规划会极大地影响施工管理的生产率[25]。工作流管理系统（WfM）是支持协同工作的最成功的系统类型[26]。WfM 系统从准备时间、服务时间、延迟和资源利用率等方面进行评估，上述调整可以提高效率[27]。WfM 系统起源于计算机科学和管理科学[28]，队列网络被用于效率的定量分析[28-30]。

Queueing theory is a mathematical theory focused on modeling and analysis serving random demand; queuing model is an abstract description of these systems [31]. It can be considered statistically through several performance measures, including the average latency of a queue or system, and the number of expected waits or incoming services [30].
队列理论是一种数学理论，侧重于为随机需求提供建模和分析服务；队列模型是对这些系统的抽象描述[31]。它可以通过几种性能指标进行统计，包括队列或系统的平均延迟、预期等待或传入服务的数量 [30]。

The queuing theory describes the types of waits as shown in Fig. 1 [32], and expresses the types of queuing systems that actually appear [32]. Customers arrive at the queue system randomly for service. Typically, one or more servers provide the service. Each customer leaves the queue after being individually served by the server.
排队理论描述了如图 1 所示的等待类型[32]，并表达了实际出现的排队系统类型[32]。顾客随机到达队列系统寻求服务。通常，一个或多个服务器提供服务。每个顾客在接受服务器的单独服务后离开队列。

Queue systems are subject to time-dependent changes in system
队列系统会随着时间的推移而发生变化。

Fig. 1. Basic queueing system ( $C$$C$CC represents a customer, and $S$$S$SS a server).
图 1.基本排队系统（ $C$$C$CC 代表顾客， $S$$S$SS 代表服务器）。
parameters such as arrival rate or number of servers [33]. Unusual parameters such as arrival rate and number of servers include patients and medical staff performing care, port overload and port size. Changing time-dependent parameters can significantly affect the performance of a queuing system $[6,33]$$[6,33]$[6,33][6,33]. Table 1 presents research areas that use timedependent queuing systems.
到达率或服务器数量等参数[33]。到达率和服务器数量等非正常参数包括进行护理的病人和医务人员、端口过载和端口大小。改变与时间相关的参数会显著影响队列系统的性能 $[6,33]$$[6,33]$[6,33][6,33] 。表 1 列出了使用随时间变化的排队系统的研究领域。

Bailey et al. [34] minimized the waiting time of customers to increase customer satisfaction by adjusting the appointment time of outpatients. Cayirli et al. [35] minimizes the waiting and idle time between the patient being served at the hospital and the medical staff performing the treatment through the queueing model. Buzacott et al. [36] presented a solution to a design optimization problem that occurs in a manufacturing system using a queuing model. Zhang et al. [37] performed quantitative analysis through the queuing model to solve the port overload problem with the port size expansion and minimum input. Kim et al. [38] proposed a 4D digital twin framework through the performance index analysis of the queuing model. Wee et al. [39] quantitatively analyzed the effect of the construction team allocation on the performance of frame construction using the queuing theory. Kim et al. [40] quantitatively analyzed the apartment frame construction period and applied the queuing theory to determine the validity of the estimated period.
Bailey 等人[34] 通过调整门诊病人的预约时间，最大限度地减少了顾客的等待时间，从而提高了顾客满意度。Cayirli 等人[35]通过排队模型最大限度地减少了在医院接受服务的病人与进行治疗的医务人员之间的等待和空闲时间。Buzacott 等人[36] 利用排队模型提出了制造系统中出现的设计优化问题的解决方案。Zhang 等人[37]通过排队模型进行了定量分析，解决了端口过载问题，并扩大了端口尺寸和最小输入量。Kim 等人[38]通过排队模型的性能指标分析提出了 4D 数字孪生框架。Wee 等[39]利用排队理论定量分析了施工队伍分配对框架施工性能的影响。Kim等人[40]定量分析了公寓框架施工工期，并应用排队理论确定了估计工期的正确性。

Lin et al. [41] improved the error detection process and error correction process of the software system by using a priority queuing model. Kim et al. [42] analyzes the queue network in the call center to prioritize high-value customers to avoid client loss. Ingolfsson et al. [43] examined the structure of patrol car shifts and personnel scheduling methods that model stochastic and time-varying demand processes. Nan et al. [44] examined the resource optimization problem for multimedia cloud. By comparing priorities and multi-service scenarios, they achieved the minimum response time for cloud resources and derived the minimum resource cost. De Souza et al. [45] integrates users of different priority classes into the system queue to support different patients according to the medical severity of the Emergency Medical Service (EMS). Through this, they evaluated the performance of the average travel time and average waiting time of various user classes.
Lin 等人[41]利用优先队列模型改进了软件系统的错误检测过程和纠错过程。Kim 等人[42]分析了呼叫中心的队列网络，以优先处理高价值客户，避免客户流失。Ingolfsson 等人[43]研究了巡逻车轮班结构以及模拟随机和时变需求过程的人员调度方法。Nan 等人[44]研究了多媒体云的资源优化问题。通过比较优先级和多服务场景，他们实现了云资源的最小响应时间，并得出了最小资源成本。De Souza 等人[45]根据紧急医疗服务（EMS）的医疗严重程度，将不同优先级的用户整合到系统队列中，以支持不同的病人。为此，他们评估了不同用户等级的平均旅行时间和平均等待时间的性能。

As shown in Table 1, the queueing model is useful for quantitatively analyzing systems in various fields. However, it is important to note that studies on better resource optimization and performance improvement were conducted through comparative analysis when the priority rule
如表 1 所示，排队模型对于定量分析各领域的系统非常有用。但需要注意的是，在优先权规则为 "0 "或 "0 "时，通过比较分析，可以更好地优化资源和提高性能。

Table 1  表 1
Research areas that use time-dependent queuing systems.
使用随时间变化的排队系统的研究领域。

rather than the FCFS rule, which is the basic assumption of the queueing model.
而不是 FCFS 规则，后者是队列模型的基本假设。

Recently, studies applying the queuing model have also begun to emerge in the construction field. However, there is no study that quantitatively analyzes resource optimization and performance improvement in the construction field by applying the priority rule. Schwarz et al. [33] provide an overview of approaches for the performance evaluation of time-dependent queueing systems. According to their study, all areas of application, a systematic test of other evaluation approaches may represent a worthwhile investigation. They argue that the opportunity for the development of other evaluation approaches lies in the combination of existing ideas concerning approximation. The development of new approaches can be a combination of priority policy items and BIM ROI items. The optimal BIM staffing and the variation in engineer waiting costs for high priority RFIs can be used for systematic testing.
最近，建筑领域也开始出现应用排队模型的研究。然而，目前还没有应用优先权规则对建筑领域的资源优化和性能提升进行定量分析的研究。Schwarz 等人[33]概述了与时间相关的排队系统性能评估方法。根据他们的研究，在所有应用领域，对其他评估方法进行系统测试可能是一项值得研究的工作。他们认为，开发其他评估方法的机会在于结合现有的近似思想。新方法的开发可以是优先政策项目和 BIM 投资回报率项目的结合。最佳 BIM 人员配置和高优先级 RFI 的工程师等待成本变化可用于系统测试。

Therefore, this study quantitatively analyzes the effect of BIM-based design verification to which the priority rule is applied. This includes a comparison of basic performance analyses such as $\mathrm{L},\mathrm{Lq},\mathrm{W},\mathrm{Wq}$$\mathrm{L},\mathrm{Lq},\mathrm{W},\mathrm{Wq}$L,Lq,W,Wq\mathrm{L}, \mathrm{Lq}, \mathrm{W}, \mathrm{Wq}, and Wq , based on the study of Ham et al. [6] with FCFS rules applied, which is the basic assumption of the queue model. Additionally, the quantitative analysis of optimal server input (ex: BIM staff) and customer waiting costs(ex: project engineers) under priority rules quantifies BIM ROI that can be improved.
因此，本研究定量分析了应用优先级规则的基于 BIM 的设计验证的效果。这包括根据 Ham 等人的研究[6]，在应用 FCFS 规则（这是队列模型的基本假设）的情况下，对 $\mathrm{L},\mathrm{Lq},\mathrm{W},\mathrm{Wq}$$\mathrm{L},\mathrm{Lq},\mathrm{W},\mathrm{Wq}$L,Lq,W,Wq\mathrm{L}, \mathrm{Lq}, \mathrm{W}, \mathrm{Wq} 和 Wq 等基本性能分析进行比较。此外，对优先级规则下的最佳服务器输入（例如：BIM 人员）和客户等待成本（例如：项目工程师）进行量化分析，以量化可提高的 BIM 投资回报率。

The adoption of BIM incurs additional costs. Therefore, owners and managers in the Architecture, Engineering, and Construction (AEC) industry are primarily concerned with ROI [10]. ROI analysis is used to evaluate a proposed investment [24]. Various studies have been conducted to calculate and improve BIM ROI. Cost savings from BIM adoption have generally been estimated based on RFI, design changes, and construction duration or delays [7,8,46]. Lee et al. [8] analyzed BIM ROI based on the cost of avoiding rework owing to design errors. Lu et al. [47] calculated cost savings by comparing BIM and non-BIM projects. Giel et al. [26] calculated BIM ROI through tracking RFI. Ahn et al. [11] show that BIM ROI can be improved through the importance of BIM experts and process improvement.
采用 BIM 会产生额外成本。因此，建筑、工程和施工（AEC）行业的业主和管理人员主要关注投资回报率[10]。投资回报率分析用于评估建议的投资[24]。为了计算和提高 BIM 投资回报率，已经开展了多项研究。一般根据 RFI、设计变更和施工工期或延误来估算采用 BIM 所节省的成本[7,8,46]。Lee 等人[8]根据避免因设计错误造成返工的成本分析了 BIM 投资回报率。Lu 等人[47]通过比较 BIM 项目和非 BIM 项目，计算了节省的成本。Giel 等人[26]通过跟踪 RFI 计算了 BIM 投资回报率。Ahn 等人[11]的研究表明，BIM 投资回报率可以通过 BIM 专家的重要性和流程改进来提高。

In the studies mentioned above, only traceable data, such as labor costs and capital costs, were considered BIM investment costs. However, Jin et al. [48] confirmed through a survey that BIM practitioners’ experience of using BIM and collaboration based on priorities increases BIM ROI. Ham et al. [6] analyzed the RFI response latency of project participants and improved BIM ROI by quantifying the number of BIM staff. $[6,48]$$[6,48]$[6,48][6,48] revealed that the analysis of latency and cost according to priority policies and workforce quantification should be considered for the increase of ROI.
在上述研究中，只有人工成本和资本成本等可追溯数据被视为 BIM 投资成本。然而，Jin 等人[48]通过调查证实，BIM 从业人员的 BIM 使用经验和基于优先级的协作提高了 BIM 投资回报率。Ham 等人[6]分析了项目参与者的 RFI 响应延迟，并通过量化 BIM 人员数量提高了 BIM 投资回报率。 $[6,48]$$[6,48]$[6,48][6,48] 发现，为提高投资回报率，应考虑根据优先级政策和劳动力量化分析延迟和成本。

In fact, the optimal scenario was derived in many studies by considering the problem of determining how and when to execute each activity to minimize the total cost [49-51]. Time-cost trade-offs have long been studied because they are the two most important quantifiable goals in construction projects [51]. Klerides et al. [49] proposed a programming approach to the time-cost trade-off problem of examining various scenarios and accommodating variations by applying priority rules. El-Rayes et al. [50] developed an optimal model between time, cost and quality by optimizing activity priority information and activity cost to improve construction quality. Hu et al. [51] presented a complex time-cost-quality optimization model based on correlations between construction activities.
事实上，在许多研究中，最佳方案都是通过考虑确定如何以及何时执行每项活动以最小化总成本而得出的[49-51]。长期以来，人们一直在研究时间成本权衡问题，因为这是建筑项目中两个最重要的可量化目标[51]。Klerides 等人[49]针对时间成本权衡问题提出了一种编程方法，即研究各种方案，并通过应用优先权规则来适应各种变化。El-Rayes 等人[50]通过优化活动优先级信息和活动成本，建立了时间、成本和质量之间的最优模型，以提高施工质量。Hu 等人[51]根据施工活动之间的相关性提出了一个复杂的时间-成本-质量优化模型。

The best-case scenario for improving BIM ROI is to find a trade-off between time and cost. It is difficult to solve many design errors discovered through the verification of BIM design in a short time. The degree of rework caused by design errors depends on how long the errors remain undetected [52]. Peansupap et al. [53] classified the rank of the effect of design errors from negligible levels (1) to severe levels (5) in the
提高 BIM 投资回报率的最佳方案是在时间和成本之间找到平衡点。通过 BIM 设计验证发现的许多设计错误很难在短时间内解决。设计错误造成的返工程度取决于错误未被发现的时间长短[52]。Peansupap 等人[53]将设计错误的影响等级划分为可忽略等级（1）到严重等级（5）。
study on the effect levels owing to design errors. Therefore, among the many design errors discovered through the verification of BIM design, there are design errors that must be managed first. Resolving high-level design errors first and delaying low-level design errors can be the best way through which to minimize the total cost of the project.
设计错误造成的影响程度的研究。因此，在通过 BIM 设计验证发现的众多设计错误中，有一些设计错误必须先行管理。先解决高层次的设计错误，延迟解决低层次的设计错误，是将项目总成本降至最低的最佳途径。

Therefore, this study quantitatively analyzes the effectiveness of the verification of the BIM design with priority rules. To find a time-cost trade-off when the priority rule is applied, the performance is compared with the performance applied by the FCFS rule in Ham et al. (6), and the optimal number of BIM staff is sought according to the performance. Additionally, we analyzed the avoidance cost of engineers’ waiting for high-priority design errors. Changing the optimal number of BIM staff under the priority rule and reducing the waiting cost of engineers for higher priorities can reduce the total cost of the project. This is the best-case scenario where BIM ROI is improved.
因此，本研究定量分析了采用优先规则验证 BIM 设计的有效性。为了找到应用优先规则时的时间成本权衡，我们将其性能与 Ham 等人（6）中的 FCFS 规则所应用的性能进行了比较，并根据性能寻求最佳的 BIM 人员数量。此外，我们还分析了工程师等待高优先级设计错误的避免成本。改变优先级规则下的最佳 BIM 人员数量，减少工程师对高优先级的等待成本，可以降低项目的总成本。这是提高 BIM 投资回报率的最佳情况。

This section identifies the problem through preliminary research on two real-world projects. For both projects, BIM design validation was conducted in the pre-construction stage after detailed design. Through this, 1228 design errors were found in project A and 908 design errors were found in project B (Table 2). The research was conducted under the premise of solving all the design errors found through BIM design validation in the pre-construction stage.
本节通过对两个实际项目的初步研究来确定问题所在。这两个项目都在详细设计后的施工前阶段进行了 BIM 设计验证。通过这项工作，在项目 A 中发现了 1228 个设计错误，在项目 B 中发现了 908 个设计错误（表 2）。研究的前提是解决施工前阶段通过 BIM 设计验证发现的所有设计错误。

The Haeundae L project is a high-rise construction project with a maximum height of 412 m , in which Company A participated as a general contractor. Landmark tower ( 5 stories below ground to 101 above, 412 m above the ground) comprising a tourist hotel, a residence hotel, an observatory, etc., residential tower A ( 5 below the ground - 85 above, 339 m ) for apartment houses and ancillary facilities, residential tower B (underground), It comprises a podium(1st floor - 7th floor), comprising 5 floors - 85 floors above ground, 333 m ), and commercial facilities.
Haeundae L 项目是 A 公司作为总承包商参与的最高高度为 412 米的高层建筑项目。地标塔楼（地下 5 层至地上 101 层，地上 412 米）包括旅游酒店、住宅酒店、天文台等；住宅楼 A（地下 5 层至地上 85 层，339 米）包括公寓和附属设施；住宅楼 B（地下）包括裙楼（1 层至 7 层）（地上 5 层至 85 层，333 米）和商业设施。

Project A was conducted in a fast-track method. Furthermore, a CoreWall, which divides the plane into a core part and an outer peripheral part and divides it into construction, was adopted. For high-rise buildings, it is important to shorten the construction period and manage the process [54]. It is important to manage design errors in the early stages of the project owing to the nature of the high-rise building and the fasttrack project. However, there was a risk that the construction company could bear additional costs other than the contract amount during the quality review process of design documents in the pre-construction stage after detailed design. Accordingly, Company A signed a contract with
项目 A 采用快速施工法。此外，还采用了将平面分为核心部分和外围部分并分段施工的 CoreWall。对于高层建筑来说，缩短工期和进行过程管理非常重要[54]。由于高层建筑的性质和快速通道项目的特点，在项目初期对设计错误进行管理非常重要。然而，在详细设计之后的施工前阶段，在设计文件的质量审查过程中，建筑公司有可能承担合同金额之外的额外费用。因此，A 公司与

Table 2  表 2
Description of real-world projects.
实际项目描述。
(\left.$$\text{Missing end{array}}$$$$\text{Missing end{array}}$$"Missing \end{array}"\begin{array}{lll}\hline \text { Project } & \text { A } & \text { B } \\ \hline \text { Project name } & \text { Haeundae L Project } & \begin{array}{l}\text { Gwangmyeong Ikea } \\ \text { Project }\end{array} \\ \begin{array}{c}\text { Project delivery } \\ \text { method }\end{array} & \text { DBB(design-bid-build) } & \text { DBB } \\ \begin{array}{c}\text { Total construction } \\ \text { cost(KRW) }\end{array} & 1,400,000,000,000 & 101,441,003,035 \\ \begin{array}{l}\text { Floor area }\left(m^{2}\right) \\ \text { Construction } \\ \text { duration } \\ \text { (months) }\end{array} & 661,134 & 135,000 \\ \begin{array}{c}\text { BIM-based } \\ \text { construction } \\ \text { service period } \\ \text { (months) }\end{array} & 50 & 16 \\ \begin{array}{ll}\text { Total BIM service } & 34\end{array} \begin{array}{l}\text { BIM-based design validation, } \\ \text { BIM-based design information } \\ \text { management, BIM-based }\end{array} & Master Model & Management, MEP
(左起：模型与管理大师，MEP

Company C for BIM-based construction services worth KRW $3,200,000,000$$3,200,000,000$3,200,000,0003,200,000,000. The contract period was from September 2015 to June 2018 (34 months). 6 BIM staffs were deployed to address the low design quality. During this process, 1228 design errors were found. Table 3 shows the classification of the discovered design errors by construction type. Design errors can result in serious losses if not addressed in advance [6]. Therefore, BIM staff must resolve 1228 design errors found through design validation before construction.
C 公司提供基于 BIM 的建筑服务，价值 $3,200,000,000$$3,200,000,000$3,200,000,0003,200,000,000 韩元。合同期为 2015 年 9 月至 2018 年 6 月（34 个月）。为解决设计质量低的问题，部署了 6 名 BIM 工作人员。在此过程中，发现了 1228 个设计错误。表 3 按建筑类型对发现的设计错误进行了分类。如果不提前解决设计错误，可能会造成严重损失[6]。因此，BIM 工作人员必须在施工前解决通过设计验证发现的 1228 项设计错误。

The Gwangmyeong Ikea project is the first Ikea store project in Korea. The main structures consist of pre-cast concrete structures and steel structures. The building with a total floor area of $135,000{\mathrm{m}}^2$$135,000{\mathrm{m}}^2$135,000m^(2)135,000 \mathrm{~m}^{2} has $53,739{\mathrm{m}}^2$$53,739{\mathrm{m}}^2$53,739m^(2)53,739 \mathrm{~m}^{2} store floors, excluding parking lots, and consists of a show room, a warehouse space, a restaurant and a children’s play area.
光明宜家项目是韩国第一个宜家商店项目。主要结构由预制混凝土结构和钢结构组成。建筑总面积为 $135,000{\mathrm{m}}^2$$135,000{\mathrm{m}}^2$135,000m^(2)135,000 \mathrm{~m}^{2} ，有 $53,739{\mathrm{m}}^2$$53,739{\mathrm{m}}^2$53,739m^(2)53,739 \mathrm{~m}^{2} 个楼层，不包括停车场，由一个展厅、一个仓库空间、一个餐厅和一个儿童游乐区组成。

This project discovered many risks of MEP clash during the design document review process in the pre-construction stage after detailed design. MEP coordination is critical to project success as it requires significant time and human resources [55,56]. Accordingly, Company W signed a BIM-based construction support service with Company F. Before contracting the BIM-based construction support service, the LOD (Level of Detail) of the project stayed at 300. A more detailed BIM model was required to present the results applicable for immediate construction and detailed design review required in the construction phase. Therefore, in the pre-construction stage, the process of increasing the detail level of the BIM model to 400 or higher was conducted.
该项目在详细设计后的施工前阶段的设计文件审查过程中发现了许多 MEP 冲突的风险。MEP 协调对项目成功至关重要，因为它需要大量的时间和人力资源[55,56]。因此，W 公司与 F 公司签订了基于 BIM 的施工支持服务合同。在签订基于 BIM 的施工支持服务合同之前，项目的 LOD（详细程度）保持在 300。需要更详细的 BIM 模型来展示适用于即时施工和施工阶段所需的详细设计审查的结果。因此，在施工前阶段，将 BIM 模型的详细程度提高到 400 或更高。

The contract period was from July 2013 to September 2014 (15 months). This BIM-based construction support service contract includes Master Model Management, MEP Conflict check, and Solution. One BIM staff was assigned to solve the low design quality. During this process, 904 MEP clash errors were found. Table 4 shows the MEP clash errors of Project B discovered through the BIM design validation.
合同期为 2013 年 7 月至 2014 年 9 月（15 个月）。这份基于 BIM 的施工支持服务合同包括主模型管理、MEP 冲突检查和解决方案。一名 BIM 工作人员负责解决设计质量低的问题。在此过程中，发现了 904 个 MEP 冲突错误。表 4 显示了通过 BIM 设计验证发现的项目 B 的 MEP 冲突错误。

Both case projects had a common problem. As both projects used the DBB delivery method, risks had to be managed in the pre-construction stage according to the design quality. To review all design errors within the short period of the pre-construction stage after the design, high BIM staff labor costs are incurred. It is important to keep the BIM investments within the set contract amount budget limits.
这两个案例项目都有一个共同的问题。由于两个项目都采用了 DBB 交付方式，因此必须在施工前阶段根据设计质量管理风险。在设计完成后的施工前阶段的短时间内审查所有设计错误，会产生高昂的 BIM 人员人工成本。将 BIM 投资控制在既定的合同金额预算范围内非常重要。

To quantify the effect of BIM on labor productivity in [57], issues such as too much activity or biased budgets of the BIM staff must be addressed in the data collection stage to obtain the right level of detail. However, Ham et al. [6] is the only study that uses the FCFS rule that can quantify the number of BIM staff.
要量化[57]中 BIM 对劳动生产率的影响，必须在数据收集阶段解决 BIM 人员活动过多或预算有偏差等问题，以获得正确的详细程度。然而，Ham 等人的研究[6] 是唯一使用 FCFS 规则对 BIM 人员数量进行量化的研究。

In fact, construction companies do not process design errors discovered through the validation of the BIM design in order. Therefore, from a practical point of view, the BIM design validation performance should be analyzed when the priority order of design error handling
事实上，建筑公司并没有按顺序处理通过 BIM 设计验证发现的设计错误。因此，从实际角度出发，在分析 BIM 设计验证性能时，应按设计错误处理的优先顺序排列

Table 3  表 3
Project A Design Errors by Work Type.
按工种划分的项目 A 设计错误。

Table 4  表 4
Project B MEP clash errors by Work Type.
项目 B 按工程类型划分的 MEP 冲突错误。

rather than the FCFS rule is applied. The validation performance of the BIM design when priorities are applied shows different results than when FCFS rules are applied. The research method is described in the following section.
而不是 FCFS 规则。在应用优先级时，BIM 设计的验证性能显示出与应用 FCFS 规则时不同的结果。下一节将介绍研究方法。

In this section, we present the research methods used to address the research problems found in the Literature review and preliminary investigation(Fig. 2). We tracked all RFIs recorded from A and B projects. In addition, according to the progress and characteristics of the project, priority classification criteria were set for handling design errors (priority classification criteria are listed in the case study).
在本节中，我们将介绍用于解决文献综述和初步调查中发现的研究问题的研究方法（图 2）。我们对 A 和 B 项目中记录的所有 RFI 进行了跟踪。此外，我们还根据项目的进度和特点，制定了处理设计错误的优先级分类标准（优先级分类标准已在案例研究中列出）。

First, all RFIs were classified according to classification criteria, and characteristics were identified of the queueing model(M/M/s model, priority model) to analysis performance when priority rules are applied to the order of RFI responses. Through performance analysis of the two selected models, we analyze the effect of BIM RFI processing order on the optimal number of BIM staffs.
首先，根据分类标准对所有 RFI 进行分类，并确定排队模型（M/M/s 模型、优先模型）的特征，以分析优先规则应用于 RFI 响应顺序时的性能。通过对所选两个模型的性能分析，我们分析了 BIM RFI 处理顺序对最佳 BIM 人员数量的影响。

Second, analyze changes in the engineers’ waiting cos as highpriority design errors are addressed first. The reduction in waiting costs can be considered waiting avoidance cost of engineers through the use of BIM. Therefore, we quantitatively analyze the input(BIM staff) for using BIM and the output(waiting avoidance cost of engineer) through the use of BIM.
其次，分析工程师在优先处理高优先级设计错误时等待成本的变化。等待成本的减少可视为工程师通过使用 BIM 而避免的等待成本。因此，我们对使用 BIM 的投入（BIM 人员）和使用 BIM 的产出（避免工程师等待成本）进行了定量分析。

In a queuing system, the time between customer arrivals is referred to as inter-arrival time. It is almost impossible to predict when the next customer will arrive in a typical queuing system. However, there is enough data about the customers arriving at the queuing system, you can estimate the average number of customers arriving per unit time. This is referred to as the ‘average arrival rate’ and is denoted by the Greek letter ’ $\lambda$$\lambda$lambda\lambda '. Because $\lambda$$\lambda$lambda\lambda is the average rate of the arrival of customers arriving at the waiting system, the mean of the probability distribution
在排队系统中，顾客到达之间的时间称为到达间隔时间。在典型的排队系统中，几乎不可能预测下一位顾客何时到达。但是，如果有足够的关于到达排队系统的顾客的数据，就可以估算出单位时间内到达的顾客的平均数量。这就是 "平均到达率"，用希腊字母" $\lambda$$\lambda$lambda\lambda "表示。由于 $\lambda$$\lambda$lambda\lambda 是到达等待系统的顾客的平均到达率，因此概率分布的平均值为

Performance analysis of queueing model according to the design error processing sequence
根据设计误差处理顺序对排队模型进行性能分析
BIM staffs’ Performance $$$$quad\quad Engineers’ waiting time
BIM 人员的表现 $$$$quad\quad 工程师的等待时间
BIM Design Validation Performance Analysis under Priority Rules
优先规则下的 BIM 设计验证性能分析

Fig. 2. Research method.  图 2.研究方法。
over the time between arrivals is $1/\lambda$$1/\lambda$1//lambda1 / \lambda. Customers arrive randomly. The exponential distribution is a probability distribution that adequately describes the random arrival property and is referred to as the ‘Markovian property’.
$1/\lambda$$1/\lambda$1//lambda1 / \lambda 。客户随机到达。指数分布是一种能充分描述随机到达特性的概率分布，被称为 "马尔可夫特性"。

A system with only one server is referred to as a single server system, and a system with two or more servers is referred to as a multi-server system. In a basic queuing system, each customer is individually serviced by one server. The time from the start of the customer’s service to the moment it is completed is referred to as service time. The queuing model assumes that service times have a uniform probability distribution across all servers. Table 5 shows the server and queue model assumptions.
只有一台服务器的系统称为单服务器系统，有两台或更多服务器的系统称为多服务器系统。在基本的排队系统中，每个客户都由一台服务器单独提供服务。从客户服务开始到服务结束的时间称为服务时间。队列模型假定所有服务器的服务时间具有均匀的概率分布。表 5 显示了服务器和队列模型假设。

The symbol for the mean of the service time probability distribution is $1/\mu .\mu$$1/\mu .\mu$1//mu.mu1 / \mu . \mu represents the average number of customers that one server can serve per unit time. This is referred to as the average service rate. For example, if $1/\mu$$1/\mu$1//mu1 / \mu is 30 min , then the average service rate $(\mu )$$(\mu )$(mu)(\mu) is 2 customers/h. In the queuing model, $M=$$M=$M=M= exponential distribution (Markovian) as a symbol representing the probability distribution for arrival-to-arrival times and service times.
服务时间概率分布均值的符号为 $1/\mu .\mu$$1/\mu .\mu$1//mu.mu1 / \mu . \mu 代表一名服务器在单位时间内能服务的客户平均数量。这被称为平均服务率。例如，如果 $1/\mu$$1/\mu$1//mu1 / \mu 为 30 分钟，则平均服务速率 $(\mu )$$(\mu )$(mu)(\mu) 为 2 个客户/小时。在排队模型中， $M=$$M=$M=M= 指数分布（马尔可夫分布）作为一个符号，代表了到达到到达时间和服务时间的概率分布。

The M/M/1 model is a single server model with one server. Both arrival-to-arrival and service times follow an exponential distribution. If $s$$s$ss is the symbol for the number of servers, then the $M/M/s$$M/M/s$M//M//sM / M / s model is a model with servers of $s$$s$ss. The first symbol (M) is the probability distribution over the arrival interval, and the second symbol (M) is the probability distribution over the service time. (M) denotes an exponential distribution with Markovian properties. The performance of the standby system can be analyzed by the number of customers waiting and the waiting time of the customers. Table 6 presents the notation used in this study.
M/M/1 模型是一种只有一台服务器的单服务器模型。到达-到达时间和服务时间都服从指数分布。如果 $s$$s$ss 是服务器数量的符号，那么 $M/M/s$$M/M/s$M//M//sM / M / s 模型就是服务器数量为 $s$$s$ss 的模型。第一个符号 (M) 是到达间隔的概率分布，第二个符号 (M) 是服务时间的概率分布。(M) 表示具有马尔可夫特性的指数分布。备用系统的性能可以通过等待的客户数量和客户等待时间来分析。表 6 列出了本研究中使用的符号。

In a waiting system, customers are unproductive members of the organization while waiting. Customers care about how long they have to wait for service, and customer satisfaction can be expressed in appropriate waiting times. Therefore, it is important to control the waiting time of customers. The basic equations for deriving $L,L_q,W$$L,L_q,W$L,L_(q),WL, L_{q}, W, and $W_q$$W_q$W_(q)W_{q} from the queue model are: (1)-(5).
在等待系统中，顾客在等待期间是组织中的非生产性成员。顾客关心的是等待服务的时间，而顾客满意度可以用适当的等待时间来表示。因此，控制顾客的等待时间非常重要。从队列模型中推导出 $L,L_q,W$$L,L_q,W$L,L_(q),WL, L_{q}, W 和 $W_q$$W_q$W_(q)W_{q} 的基本方程是(1)-(5).
$\mathrm{L}=\lambda W$$\mathrm{L}=\lambda W$L=lambda W\mathrm{L}=\lambda W ( $\lambda$$\lambda$lambda\lambda mean arrival rate)
$\mathrm{L}=\lambda W$$\mathrm{L}=\lambda W$L=lambda W\mathrm{L}=\lambda W （ $\lambda$$\lambda$lambda\lambda 平均到达率）
$L_q=\frac{P_0(\lambda /\mu {)}^s\rho }{s!(1-\rho {)}^2}=\frac{P_0{\lambda }^{s+1}}{(s-1)!{\mu }^{s-1}(s\mu -\lambda {)}^2}$$L_q=\frac{P_0(\lambda /\mu {)}^s\rho }{s!(1-\rho {)}^2}=\frac{P_0{\lambda }^{s+1}}{(s-1)!{\mu }^{s-1}(s\mu -\lambda {)}^2}$L_(q)=(P_(0)(lambda//mu)^(s)rho)/(s!(1-rho)^(2))=(P_(0)lambda^(s+1))/((s-1)!mu^(s-1)(s mu-lambda)^(2))L_{q}=\frac{P_{0}(\lambda / \mu)^{s} \rho}{s!(1-\rho)^{2}}=\frac{P_{0} \lambda^{s+1}}{(s-1)!\mu^{s-1}(s \mu-\lambda)^{2}}
$W=W_q+\frac{1}{\mu }$$W=W_q+\frac{1}{\mu }$W=W_(q)+(1)/(mu)W=W_{q}+\frac{1}{\mu}
$W_q=L_q/\lambda$$W_q=L_q/\lambda$W_(q)=L_(q)//lambdaW_{q}=L_{q} / \lambda
$P_0=\frac{1}{\sum _{n=0}^{s-1}\frac{(\lambda /\mu {)}^n}{n!}+\frac{(\lambda /\mu {)}^s}{s!}\left(\frac{1}{1-\lambda /s\mu }\right)}$$P_0=\frac{1}{\sum _{n=0}^{s-1} \frac{(\lambda /\mu {)}^n}{n!}+\frac{(\lambda /\mu {)}^s}{s!}\left(\frac{1}{1-\lambda /s\mu }\right)}$P_(0)=(1)/(sum_(n=0)^(s-1)((lambda//mu)^(n))/(n!)+((lambda//mu)^(s))/(s!)((1)/(1-lambda//s mu)))P_{0}=\frac{1}{\sum_{n=0}^{s-1} \frac{(\lambda / \mu)^{n}}{n!}+\frac{(\lambda / \mu)^{s}}{s!}\left(\frac{1}{1-\lambda / s \mu}\right)}
The relational expression of (1) was first proved by D.C. Little and is referred to as Little’s formula. The above relation is important because if one performance of $L,L_q,W$$L,L_q,W$L,L_(q),WL, L_{q}, W, and $W_q$$W_q$W_(q)W_{q} is obtained from these relations, the other performance can be obtained.
(1) 的关系式由 D.C. Little 首次证明，被称为 Little 公式。上述关系式非常重要，因为如果从这些关系式中得到 $L,L_q,W$$L,L_q,W$L,L_(q),WL, L_{q}, W 和 $W_q$$W_q$W_(q)W_{q} 的一个性能，就可以得到另一个性能。

Table 5  表 5
Assumptions of the queueing model.
队列模型的假设。

Table 6  表 6
Notation.  记号

In the queuing model with priority, queuing rules are based on priority. The order in which customers are selected from the queue for service is determined by a predetermined priority. Important customers are prioritized. When one server serves one customer, the customer with the highest priority is selected and served. The distribution of arrival interval times and service times for each priority class assumes an exponential distribution. Priority queuing assumes that the average service time is the same for all classes. However, average arrival rates may vary by class.
在有优先权的排队模式中，排队规则以优先权为基础。从队列中选择客户提供服务的顺序由预先确定的优先级决定。重要客户优先。当一台服务器为一位客户提供服务时，优先级最高的客户会被选中并得到服务。每个优先级的到达间隔时间和服务时间的分布假设为指数分布。优先排队假定所有类别的平均服务时间相同。然而，平均到达率可能因等级而异。

There are two priority queue models namely non-preemptive and preemptive. Non-preemptive means that when a higher-priority customer arrives in the system, a lower-priority customer in service continues to be served until the moment the service is complete. Therefore, once a server starts serving a customer, they do not stop until completion. Preemptive means that whenever a higher-priority customer enters the system, lower-priority customers in service are evicted. The server immediately begins servicing new, high-priority customers. If both models ignore the distinction for customers at different priority levels, they are the same as the $M/M/s$$M/M/s$M//M//sM / M / s model except in the order of service.
有两种优先队列模式，即非抢占式和抢占式。非抢占式是指当一个优先级较高的客户到达系统时，系统会继续为优先级较低的客户提供服务，直到服务完成。因此，服务器一旦开始为客户提供服务，就不会停止，直到服务完成。抢占式指的是，每当有优先级较高的客户进入系统时，正在服务的优先级较低的客户就会被驱逐。服务器立即开始为新的高优先级客户提供服务。如果这两种模式都忽略了不同优先级客户的区别，那么除了服务顺序外，它们与 $M/M/s$$M/M/s$M//M//sM / M / s 模式是一样的。

The difference from the $\mathrm{M}/\mathrm{M}/\mathrm{s}$$\mathrm{M}/\mathrm{M}/\mathrm{s}$M//M//s\mathrm{M} / \mathrm{M} / \mathrm{s} model is the distribution of waiting time derived from the first-come-first-served service rule. If you have a priority rule, the variance of the waiting time distribution is even greater. This is because the waiting time of the customer with the highest priority will be shorter than that of the first-come-first-served basis, and the waiting time of the customer with the lowest priority will be larger. In the queue model, the probability distribution of service time is determined by the properties of the queue system.
与 $\mathrm{M}/\mathrm{M}/\mathrm{s}$$\mathrm{M}/\mathrm{M}/\mathrm{s}$M//M//s\mathrm{M} / \mathrm{M} / \mathrm{s} 模型的不同之处在于先到先得服务规则得出的等待时间分布。如果有优先权规则，等待时间分布的方差会更大。这是因为优先级最高的客户的等待时间会短于先到先得基础上的等待时间，而优先级最低的客户的等待时间会更长。在队列模型中，服务时间的概率分布是由队列系统的特性决定的。

How long it takes for BIM staffs to process a project engineer’s design errors may vary from design error to design error. However, this study uses an exponential distribution as the probability distribution of service times (time taken to process design errors). A priority queuing model can be used, assuming a certain probability distribution of service times and times between arrivals of customers (ex: design errors). Both arrival-to-arrival and service times and performance indicators such as $L,L_q,W$$L,L_q,W$L,L_(q),WL, L_{q}, W, and $W_q$$W_q$W_(q)W_{q} are used.
BIM 人员处理项目工程师设计错误所需的时间可能因设计错误而异。不过，本研究采用指数分布作为服务时间（处理设计错误所需时间）的概率分布。假设服务时间和客户到达时间（例如：设计错误）之间存在一定的概率分布，则可使用优先排队模型。到达到到达时间和服务时间以及 $L,L_q,W$$L,L_q,W$L,L_(q),WL, L_{q}, W 和 $W_q$$W_q$W_(q)W_{q} 等性能指标均可使用。

The comparison of performance metrics in the queue model is as follows: First, we analyze the performance of the BIM staff according to the application of priority rules and FCFS rules. This allows for the analysis of the performance of BIM staff when priority rules are applied. Second, a quantitative comparison of waiting avoidance costs for highpriority design errors in FCFS rules and priority rules is possible.
队列模型的性能指标比较如下：首先，我们根据优先级规则和 FCFS 规则的应用情况分析 BIM 人员的绩效。这样就可以分析在应用优先规则时 BIM 工作人员的绩效。其次，可以对 FCFS 规则和优先级规则中高优先级设计错误的等待避免成本进行量化比较。

In this research, the preemptive queue model was excluded from the performance analysis. Through the BIM design validation, all BIM RFIs discovered in a short period of time from the detailed design to the preconstruction stage were tracked. There are no new RIFs to discover. Therefore, the non-preemptive queue model is suitable for performance analysis that handles design errors discovered through design verification. The priority queue model is the same as the $M/M/s$$M/M/s$M//M//sM / M / s model, except that there is a priority, and the service order and waiting time are distributed accordingly.
在本研究中，性能分析不包括抢先队列模型。通过 BIM 设计验证，跟踪了从详细设计到施工前阶段短时间内发现的所有 BIM RFI。没有新的 RIF 需要发现。因此，非优先队列模型适用于处理通过设计验证发现的设计错误的性能分析。优先队列模型与 $M/M/s$$M/M/s$M//M//sM / M / s 模型相同，只是有一个优先级，服务顺序和等待时间也相应分配。

If the average waiting time in the system of a class k customer in a steady state is $W_k$$W_k$W_(k)W_{k}, then $s=$$s=$s=s= number of servers, $\mu =$$\mu =$mu=\mu= average service rate per server, ${\lambda }_i=$${\lambda }_i=$lambda_(i)=\lambda_{i}= average arrival rate of class $i$$i$ii customers,
如果稳定状态下 k 类客户在系统中的平均等待时间为 $W_k$$W_k$W_(k)W_{k} ，则 $s=$$s=$s=s= 服务器数量， $\mu =$$\mu =$mu=\mu= 每台服务器的平均服务速率， ${\lambda }_i=$${\lambda }_i=$lambda_(i)=\lambda_{i}= $i$$i$ii 类客户的平均到达速率、

$A=s!\frac{s\mu -\lambda }{r^s}\sum _{j=0}^{s-1r^j}+s\mu ,B_0=1$$A=s!\frac{s\mu -\lambda }{r^s}\sum _{j=0}^{s-1r^j} +s\mu ,B_0=1$A=s!(s mu-lambda)/(r^(s))sum_(j=0)^(s-1r^(j))+s mu,B_(0)=1A=s!\frac{s \mu-\lambda}{r^{s}} \sum_{j=0}^{s-1 r^{j}}+s \mu, B_{0}=1,
$B_k=1-\frac{\sum _{i=1}^k{\lambda }_i}{s\mu }$$B_k=1-\frac{\sum _{i=1}^k {\lambda }_i}{s\mu }$B_(k)=1-(sum_(i=1)^(k)lambda_(i))/(s mu)B_{k}=1-\frac{\sum_{i=1}^{k} \lambda_{i}}{s \mu}, This result assumes (6)
$B_k=1-\frac{\sum _{i=1}^k{\lambda }_i}{s\mu }$$B_k=1-\frac{\sum _{i=1}^k {\lambda }_i}{s\mu }$B_(k)=1-(sum_(i=1)^(k)lambda_(i))/(s mu)B_{k}=1-\frac{\sum_{i=1}^{k} \lambda_{i}}{s \mu} ，这个结果假设 (6)
$\sum _{i=1}^k{\lambda }_i<s\mu$$\sum _{i=1}^k {\lambda }_i<s\mu$sum_(i=1)^(k)lambda_(i) < s mu\sum_{i=1}^{k} \lambda_{i}<s \mu.
Therefore, grade $k$$k$kk can reach a steady state. Because Little’s formula can be applied by rank, let $L_k$$L_k$L_(k)L_{k} be the average number of rank $k$$k$kk customers in the queued system at steady state, which is $L_k={\lambda }_k{W}_k,k=1,2,\dots ,N$$L_k={\lambda }_k{W}_k,k=1,2,\dots ,N$L_(k)=lambda_(k)W_(k),k=1,2,dots,NL_{k}=\lambda_{k} W_{k}, k=1,2, \ldots, N. The average wait time in the queue for Tier $k$$k$kk customers is $W_k$$W_k$W_(k)W_{k} minus $1/\mu$$1/\mu$1//mu1 / \mu, and the average queue length is multiplied by ${\lambda }_k$${\lambda }_k$lambda_(k)\lambda_{k}. If the performance of $L$$L$LL, $L_q,W$$L_q,W$L_(q),WL_{q}, W, and $W_q$$W_q$W_(q)W_{q} is found, the other performance can be found as well. Therefore, we can analyze the performance of the queue.
因此，等级 $k$$k$kk 可以达到稳定状态。由于利特尔公式可以按等级应用，设 $L_k$$L_k$L_(k)L_{k} 为稳定状态下排队系统中等级 $k$$k$kk 客户的平均人数，即 $L_k={\lambda }_k{W}_k,k=1,2,\dots ,N$$L_k={\lambda }_k{W}_k,k=1,2,\dots ,N$L_(k)=lambda_(k)W_(k),k=1,2,dots,NL_{k}=\lambda_{k} W_{k}, k=1,2, \ldots, N 。等级 $k$$k$kk 客户在队列中的平均等待时间为 $W_k$$W_k$W_(k)W_{k} 减去 $1/\mu$$1/\mu$1//mu1 / \mu ，平均队列长度乘以 ${\lambda }_k$${\lambda }_k$lambda_(k)\lambda_{k} 。如果找到了 $L$$L$LL 、 $L_q,W$$L_q,W$L_(q),WL_{q}, W 和 $W_q$$W_q$W_(q)W_{q}