## 数学代写|运筹学作业代写operational research代考|Queueing Theory

Every one of us has waited in line at a bank, a supermarket, or a bus stop. Waiting is generally annoying, and much productivity is lost because of it. Queueing theory cannot help us eliminate waiting but does give us insight into how waiting times depend on the queueing system’s design. This insight can be used to reduce the waiting times.

Queueing problems occur in many diverse real-life situations. In telephony and telecommunication problems, calls and messages wait for a free line; in seaports and airports, ships wait for loading and unloading facilities and planes for a runway to become available; in a production hall, machines wait for repair; and so on. The design of all kinds of systems raises questions such as “How many telephone lines are needed to guarantee a certain degree of service?”, “Which appointment system should a hospital use to keep a patient’s average waiting time below a given value?”, “What is the effect on the downtime of machines in a production hall if the number of repairmen is increased?”, and so on. In many cases, queueing theory can be useful in answering these types of questions.

In situations with uncertainty in the arrival pattern of customers and the lengths of the service times, it is inevitable that periods in which customers wait for service and periods in which the servers wait for customers alternate. One cannot suppress both forms of waiting at the same time. If the aim is to provide reasonable service to customers, then the system’s load must not be too close to the maximum processing capacity. The price of good service is that the service station will, from time to time, be without work. A rule of thumb in queueing theory is that in the case of a single service station, on average, this station may not be filled to its full capacity more than $80 \%$ of the time if it is to provide reasonable service. Queueing theory will teach us that in a heavily loaded system, a small increase in the load can lead to an enormous increase in a customer’s average waiting time. Nonlinear effects occur. A customer arrival rate that is twice as high does not generally lead to an average waiting time that is twice as long; the increase in the queue length is much higher. The cause of this phenomenon lies in the variability of the arrival times and the service times. The essence of waiting is beautifully expressed in the so-called waiting time paradox.

## 数学代写|运筹学作业代写operational research代考|The Waiting Time Paradox

Suppose that every 20 minutes, a bus is meant to arrive at a given bus stop. Someone who does not know the timetable goes to the bus stop hoping for the best. How long will he wait on average? Is it 10 minutes, half of 20 minutes? This is only correct if the buses arrive exactly every 20 minutes, without any variability. Otherwise, it is not correct, and the average waiting time is always longer than 10 minutes. This phenomenon can be clarified using a numerical example. Suppose that, on average, half of the time, the buses arrive after 10 minutes and half of the time, they arrive after 30 minutes. Then the average waiting time is not $\frac{1}{2} \times 5+\frac{1}{2} \times 15=10$ minutes, but $\frac{3}{4} \times 15+\frac{1}{4} \times 5=12.5$ minutes. After all, on average, a passenger will arrive during a long 30 -minute interval 3 times out of 4 and during a short 10-minute interval 1 time out of 4 . The explanation of the waiting time paradox therefore lies in the fact that the probability of arriving during a long interval is higher than that of arriving during a short interval. The same holds for checkout counters at the supermarket: the probability of arriving while the checker is helping a customer with a relatively long handling time is relatively high.

A simple mathematical formula can be given for the waiting time paradox. For this, we need the notion of the coefficient of variation of a random variable. If the random variable $T$ denotes the time between the departure of two consecutive buses, then the coefficient of variation $c_T$ of $T$ is defined as the quotient of the standard deviation of $T$ and the expected value of $T$. So
$$c_T=\frac{\sigma(T)}{\mathbb{E}[T]}$$
The coefficient of variation is dimensionless. It measures how variable the interdeparture time $T$ is. If the interdeparture time $T$ is constant, then $\sigma(T)=0$ and therefore $c_T=0$. The interdeparture time $T$ is said to be strongly variable if $\sigma(T)>\mathbb{E}[T]$, that is, $c_T>1$. If we assume that the buses’ interdeparture times are independent of one another and have the same distribution as the random variable $T$, then one can prove that the average waiting time at the bus stop is given by
$$\frac{1}{2}\left(1+c_T^2\right) \mathbb{E}[T]$$
for someone who arrives at the bus stop at a random point in time. Indeed, if the buses ride exactly on schedule $\left(c_T=0\right)$, then the average waiting time is equal to $\frac{1}{2} \mathbb{E}[T]$, as expected. Otherwise, the average waiting time is always longer than that. If the interdeparture times vary strongly $\left(c_T>1\right)$, then the average waiting time is even longer than the average interdeparture time.

# 运筹学代考

## 数学代写|运筹学作业代写operational research代考|The Waiting Time Paradox

$$c_T=\frac{\sigma(T)}{\mathbb{E}[T]}$$

$c_T>1$. 如果我们假设公共汽车的发车间隔时间相互独立并且与随机变量具有相同的分 布 $T$ ，那么可以证明在公共汽车站的平均等待时间为
$$\frac{1}{2}\left(1+c_T^2\right) \mathbb{E}[T]$$

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