亿点知识 | FlexSim排队系统基础

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排队系统是由顾客和服务台组成的系统，它的应用非常广泛，是最基本的建模构造可以利用一些特征来描述不同的排队系统，并且根据这些特征的不同用不同的符号来表示这 些排队系统。

Queuing system is a system composed of customers and service desk. It is widely used, and it is the most basic modeling construction that can use some features to describe different queuing systems, and represent these queuing systems with different symbols according to these different features.

排队系统的特点

1. 拟到达总体

排队系统的关键元素是顾客和服务台。潜在顾客的总体称为拟到达总体，也称为顾客 源，可以是有限的，也可能是无限或近似无限的。在拥有大量潜在顾客的系统中，通常假设 拟到达总体是无限的。

2. 系统容量

系统容量指系统可以容纳的最大顾客数量，可以是有限的，也可能是无限或近似无限 的。对于一个加油站，可能系统只能容纳10辆车，当车辆到达发现系统已饱和时，不会进入 系统。对一个不限制排队人数的网上门票销售系统，可以认为系统容量是无限的。

3. 顾客到达过程

顾客到达过程一般用到达时间间隔来表征，可分为确定性到达及随机性到达。随机性 到达采用概率分布来表征到达时间间隔，如到达时间间隔服从指数分布。另外，顾客可能独 自到达，也可能成批到达，每批到达的数目可能固定，也可能随机。

最重要的顾客到达过程称为泊松到达过程，而其中平稳泊松到达过程又是最基础的一 种。对平稳泊松到达过程来说，顾客到达速率人（单位时间内到达的顾客数，如10个/分钟） 是恒定的，相应地，顾客到达时间间隔的均值也是恒定的。顾客到达时间间隔服从均值为 1/人（如0. 1分钟）的指数分布。在长度为t的时间段中到达的顾客数目N（i）服从均值为Mt 的泊松分布。而非平稳泊松到达过程顾客到达速率随时间的不同而不同。

4. 服务时间与服务机制

服务时间指服务台为顾客服务的时间，可以是确定的，也可以是随机的。服务机制指服 务台的数量及其连接形式（串联还是并联），顾客是单个还是成批接受服务。

5. 排队行为与排队规则

顾客的排队行为可以分为以下几种：

（1） 拒绝进入：顾客到达系统时发现队列过长立即离开（不进入队列）。

（2） 中途离队：顾客排队一段时间后未接受服务中途离开系统。

（3） 换队：顾客排队一段时间后换队（换到较短的队列）。

1. Plposed to reach the whole

The key elements of the queuing system are the customer and the service desk. The potential customer population is called the quasi-arrival population, also called customer sources, and can be finite or infinite or approximately infinite. In systems with large numbers of potential customers, it is often assumed that the proposed arrival population is unlimited.

2. System capacity

System capacity refers to the maximum number of customers a system can accommodate, which can be finite or approximately unlimited. For a gas station, the system may only accommodate 10 vehicles and will not enter when the vehicle arrives and finds that the system is saturated. For an online ticket sales system that does not limit the number of queues, the system capacity can be considered unlimited.

3. Customer arrival process

The customer arrival process is generally characterized by the arrival time interval, which can be divided into deterministic arrival and random arrival. Random arrival uses a probability distribution to characterize the arrival interval, such as that following an exponential distribution. In addition, customers may arrive alone or in batches, and the number of arrivals in each batch may be fixed or random.

The most important customer arrival process is called the Poisson arrival process, among which the smooth Poisson arrival process is the most basic one. For the smooth Poisson arrival process, the customer arrival rate (the number of customers arriving per unit time, such as 10 per minute) is constant, and accordingly, the average of the customer arrival time interval is also constant. The arrival interval follows an exponential distribution of 1 / person (e. g., 0. 1 min). The number of customers arriving in length t N (i) follows a Poisson distribution with mean Mt. Instead of the smooth Poisson arrival process, the customer arrival rate varies over time.

4. Service time and service mechanism

Service time refers to the time when the service desk serves customers, which can be determined or random. Service mechanism refers to the number of service desks and the form of connection (series or parallel), and whether the customer receives the service individually or in batches.

5. Queue behavior and queuing rules

Customer queuing behavior can be divided into the following categories:

(1) Refuse entry: when the customer arrives at the system, he finds the queue is too long and leaves immediately (do not enter the queue).

(2) Leave the team midway: the customer leaves the system after a period of time.

(3) Change change: customers queue for a period of time (change to a shorter queue).

排队系统的符号表示

为了区别各种排队系统，人们对不同的排队系统给出了不同的符号表示，其中最著名的 是肯道尔（Kendall, 1953）提出的一种被广泛采用的排队系统符号表示，完整的排队系统表 达方式通常用到6个符号并取固定格式：A/B/C/D/E/F.各符号的意义如下：

（1） A表示顾客到达时间间隔分布，常用下列符号。

① M：指数分布

② D：常数

③ Et: k阶爱尔朗分布

④ G或GJ：任何分布，到达过程是独立到达。

（2） B表示服务时间分布，所用符号与表示顾客到达时间间隔分布相同。

① M：指数分布

② D：常数

③ E：阶爱尔朗分布

④ G或GI：任何分布，到达过程是独立到达。

（3） 表示服务台个数：“1"表示单个服务台,-5，（5>1）表示多个服务台。

（4） D表示系统容量，分有限与无限两种。8表示系统容量无限，此时8也可省略 不写。

（5） E表示顾客源数目，分有限与无限两种。8表示顾客源无限，此时8也可省略 不写。

（6） F表示服务规则，常用下列符号。

① FCFS：表示先到先服务的排队规则

② LCFS：表示后到先服务的排队规则

③ PR：表示优先权服务的排队规则

例如：某排队系统为M/M/S/oo/co/FCFS,则表示顾客到达间隔时间为指数分布（泊 松流），服务时间为指数分布，有多个服务台，系统容量无限，顾客源无限，采用先到先服务 规则。

多数情况下排队问题仅用上述表达形式中的前3~5个符号。如不特别说明，则均理 解为系统容量无限，顾客源无限，先到先服务。例如M/M/1,代表单服务台系统，顾客到达 间隔和服务时间都服从指数分布。

In order to distinguish between various queuing systems, people give different symbol representations for different queuing systems, among which the most famous is a widely used queuing system symbol representation proposed by Kendall (1953). The complete queuing system expression usually uses six symbols and takes a fixed format: A / B / C / D / E / F. The meaning of each symbol is as follows:

(1) A indicates the distribution of customer arrival time interval, and the following symbols are commonly used.

①M: exponential distribution

②D: constant

③ Et: k-order Alran distribution

④ G or GJ: Any distribution, the arrival process is reached independently.

(2) B indicates the service time distribution, and the symbol used is the same as the distribution of the customer arrival time interval.

①M: exponential distribution

②D: constant

③ E " Order Alran distribution

④ G or GI: Any distribution, the arrival process is an independent arrival.

（3） 。The number of service desks: "1" means a single service desk, -5,, (5> 1) means multiple service desks.

(4) D represents the system capacity, either finite or infinite.8 It means that the system capacity is unlimited, and 8 can also be omitted.

(5) E represents the number of customer sources, either limited or unlimited.8 Represents that the customer source is unlimited, and 8 can also be omitted at this time.

(6) F represents a service rule, and the following symbols are commonly used.

① FCFS: first come, first service

② LCFS: A queuing rule for the back-to-forward service

③ PR: A queuing rule for a priority service

For example, a queuing system is M / M/S / oo / co / FCFS, which means that the customer arrival interval is exponential distribution (Poisson flow), the service time is exponential distribution, with multiple service desks, unlimited system capacity, unlimited customer source, using the first-come-first service rule.

In most cases, the queuing problem only uses the first 3 to 5 symbols in the above expression form. If not specified, it is understood that the system capacity, unlimited customer source, first come, first service. For example, M / M / 1, represents the single service desk system, customer arrival interval and service time obey the exponential distribution.

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参考资料：《实用系统仿真建模与分析》

翻译：金山词霸

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