## A Survey on Queueing Systems with Mathematical Models and Applications

Read this paper on waiting line analysis and queues. It provides a good survey of the theory and uses of this type of analysis. Pay particular attention to sections 1 through 3. How might these models be used to balance firm costs with different levels of customer satisfaction?

### 5. Performance Measures in Queueing System

#### 5.2. Little’s Law

Little's law states that the average number of customers in the system is equal to the average arrival rate of customer to the system multiplied by the average system time per customer. This can be expressed as

$L=\lambda W$

where W denotes mean response time, the mean time spent in the queue and at the server, not just simply as the mean time spent waiting to be served; L refers to the average number of customers in the system and $\lambda$ stands for mean arrival rate as usual. Little's law can be applied when we relate $L$ to the average number of customers waiting to receive service denoted by $L_{q}$ and $W$ to the mean time spent waiting for service denoted by $W_{q} .$ In this sense, the other well-known form of Little's law is

$L_{q}=\lambda W_{q}$

It may be applied to separate parts of much larger queueing systems, such as subsystems in a queueing network. In such a case, $L$ should be defined with respect to the number of customers in a subsystem and W with respect to the total time in that subsystem. Little's law may also refer to a specific class of customer in a queueing system or to subgroups of customers, and so on. Its range of applicability is very wide indeed.

Little's law seems to be independent of

• Specific assumptions regarding the arrival distribution A(t)

• Specific assumptions regarding the service time distribution B(t)

• Number of servers

• Particular queueing discipline

Little's law is important for three reasons

• It is widely applicable (it requires only very weak assumptions). It will be valuable to us in checking the consistency of measurement of data.

• It is the main task in the algorithms for evaluating several queueing network models.

• In studying computer system, we frequently find two of the quantities related by Little's law (the average number of requests in a system and the throughput of that system) and desire to know the third (the average system residence time, in this case).

Applications of Little's Law

• On rainy days, streets and highways are more crowded.

• Fast food restaurants need a smaller dining room than regular restaurants with the same customer arrival rate.

• Large buffering together with large arrival rate cause large delays.

Theorem 2: In a closed Gordon-Newell network with $m$ queues, write $\mathrm{N}=\left(\mathrm{N}_{1}, \mathrm{~N}_{2}, \ldots, \mathrm{N}_{\mathrm{m}}\right)$ for the state of network. For a customer in transit to state $\mathrm{i}$, let $\alpha_{i}\left(N-e_{i}\right)$ denotes the probability that immediately before arrival the customer sees the state of the system is $\left(\mathrm{N}-\mathrm{e}_{\mathrm{i}}\right)=\left(\mathrm{N}_{1}, \mathrm{~N}_{2}, \ldots, \mathrm{N}_{\mathrm{m}}\right)$ Then the probability $\alpha_{\mathrm{i}}\left(\mathrm{N}-\mathrm{e}_{\mathrm{i}}\right)$ is same as the steady state probability for state $\left(\mathrm{N}-\mathrm{e}_{\mathrm{i}}\right)$ for a network of the same type with one customer less.

In any of the queue, the customers want them to be served as quickly as possible. But this may not happen in all the situations. One feels quite relaxed whenever her/his turn comes for the service. To describe the nature and feeling of customers, there are some popular facts about queue. They are called Murphy’s Laws and are described as follows:

• If a customer changes queue, the one s/he has left will start to move faster than the one s/he is in.

• Customer feels that her/his queue always goes the slowest.

• Whatever queue a customer joins, no matter how short it looks, will always take the longest for her/him to get served.