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Introduction
In this page, we will analyse the system with queue theory. The idea of using queue theory is to understand how a system work in order to have time efficient improvement to existing system.
Types of queue
Queue theory can be separated into 2 parts; the system of queuing and the mathematical analysis. There are 4 types of queue system. They are single channel single phase, single channel multi phase, multi channel single phase and multi channel multi phase as shown in figure 9,10,11 and 12. Single channel has only 1 server while multi channel has many servers for the queue. If the system has a series of servers that the population has to go through to complete the service, it is multi phase. If the system require only one server to get the job done, it is single phase.

Figure 9: Single Channel Single Phase Figure 11: Multi channel single phase | Figure 10: Single Channel Multi phase FIgure 12: multi channel multi phase

Common mistake
There is one very common mistake that people make in queue theory is the confusion between many single channels as shown in figure 13 and the multi channel. One way to clearly to clearly distinguish between multi channels and many single channels is looking at the queue. Multi channel has only one queue that go thorough different servers (channels) while many single channels has separate queue that go through separate servers to get the service done.
Figure 13: Many single channels single phase
Mathematical analysis
There are 2 main variable that are important in queue theory namely expected arrival per period (λ) and the expected number of service (μ). If we know both (λ) and (μ), we are able to calculate the probability distribution of n units in the system, mean number of units in the system, average length of queue, mean time waiting for service, mean time in the system. The formulas required for everything about queue theory is summarized in table 5. As I mentioned, we only need to know (λ) and (μ), then we can do all the require mathematical analysis for queue theory. There are 2 columns of formulas; one for single channel and one for multi channle. The only extra thing that we need to know for mutli channel is c, the number of channels in the system.
Table 5: Queue Formulae
APPLICATION
Figures 14 and 15 represent the number of people at the library for a certain time period. Based on this data, the expected arrival per period (λ) and the expected number of service (μ) can be obtained. Figure 14 and 15 below shows the arrival of students at libraries during normal period and exam period respectively. Based on the data below, during normal period, the expected arrival for the period of 8:00-8:30 am will be 2, the expected arrival for the period of 8:30-9:00 will be 10 and so on. Figure 16 and 17 exhibits the amount of time spent by students studying at library which is our service done per hour. For example, in figure 8, 28 students has spent 1 hour in library to study. So, 28 services has done within an hour.
Let's apply the queue theory to the library system. During normal period, library has highest population of arrival between 10:00-10:30 am and 12:00-12:30 pm. Let's take the time interval of 12:00-12:30 pm for queue theory analysis for normal period. The expected arrival for this period (λ) s 11/30 minutes. Thus, 0.367 per minutes of expected arrival. Based on figure 16, Let's assume that students do not study longer than 1 hour during normal period as most of the students only spend about an hour to study at library during normal according to the survey, there will be 28 services done within an hour based on figure 16. Thus, 0.467 services has done for a minute (μ). For our library system, I will look at it as one big server which accommodate the students and thus it will be a single channel single phase.
Figure 14: Arrival of students at library during normal period
Figure 15: Arrival of students at library during exam period.

Figure 16: The different amount of time spent by students to study at library during normal period. | Figure 17: The different amount of time spent by students to study at library during exam period.

Figure 18: Probability distribution number of students at library between 12:00-12:30 pm | If we plug in (λ) and (μ) into the first formula, we will get the probability distribution of n unit in the system will be as as shown in table 6. This result will give us the probability distribution graph as show in figure 18. From the graph, we can have an idea about the probability of the number of students will present in the library. Table 6: The probability of n number of students at library between 12:00 and 12:30 pm

Similarly, the other factors such as mean time in the system, average length of queue can be calculated easily. The result of the calculation is shown in table 7.
Table 7: All the units are in a minute
Based on the calculation, we can see that there are 0.786 people in a minute in the system (library), 2.88 minutes as average length of queue, the mean waiting time for service is 7.86 minutes and average time spent in the system is 10 minutes. This calculation is based on 0.367 per minute and 0.467 service done per minute.
Therefore, for the period of 12:00-12:30 pm during normal period, there will be about 24 persons in the library. If 0.467 service done per minute, only 14 service will be done for that period. That mean 10 students will not get their service done for that period. Each student needs to queue for 2.88 minutes and spent average time of 7.86 minutes for service waiting and total of 10 minutes needed to be spent in that queue system. This mean that every students need to wait average total of 10 minutes to study.

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Computers at Chiefly library
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