Queuing Rule of Thumb

People lining up for a live event

The Queuing Rule of Thumb is a simple formula used to compute a rough approximation of the proper number of servers in a queue. The formula be written as an inequality relating the number of servers s, total number of customers N, service time r, and the maximum time to finish the queue T: $s>{\frac {Nr}{T}}$ ^[1]^[2]

Compared to standard queuing formulas, the Queuing Rule of Thumb is simple enough to compute the necessary number of servers without needing to understand probability. It aims not to gain precision or optimization, but to give the layman the opportunity to understand and creatively solve queuing situations better. ^[2]

Introduction

Queuing is a process where people, materials, or information need to wait at a certain time to be given a service. It involves customers (those who are seeking service) and servers (anyone or anything providing service).^[1]^[3]

Queuing problems are usually due to too many servers or too little servers. Although having too many servers cause rapid service, workers become idle and equipment becomes under-utilized. On the other hand, having too little servers creates long queues, frustrates customers, and congests areas. Thus, correct and balanced queue management is necessary. ^[3]

To improve queuing, Queuing theory has been developed since the beginning of the 20th century. However, despite the advancement of the queuing century, many people still lack the proper understanding of proper queue management. ^[1]

Motivation

The lack of understanding on how queuing problems requires the Queuing Theory to be of public knowledge. However, Queuing Theory cannot be taught if the required background knowledge of the mathematics is fairly complicated. Hence, a rule of thumb for queuing performance formula is needed so that it can be used in more practical purposes. ^[1]

Derivation

$\lambda ={\frac {N}{T}}$
This is the arrival rate. It is the ratio of the total number of customers N and the maximum time to finish the queue T.
$\mu ={\frac {1}{r}}$
This is the service rate. It is the ratio of 1 and the service time.
$\rho ={\frac {\lambda }{\mu }}$
This is the ratio of arrival rate and service rate.
$U={\frac {\rho }{s}}<1$
This equation is a constraint within queuing theory that states that the utilization of the queuing systems must not be larger than 1.

Combining the first three equations, we arrive at $\rho ={\frac {\lambda }{\mu }}={\frac {Nr}{T}}$ . Combining this and the fourth equation, $U={\frac {\rho }{s}}={\frac {Nr}{Ts}}<1$ .

Simplifying, we get the Queuing Rule of Thumb $s>{\frac {Nr}{T}}$ .

Discussion

Eliminating and Reducing Queues

To analyze queues, one must first decide if the queue is necessary. Traditionally, the transfer of information, materials, people, or things between customers and servers always involves waiting lines. However, information transfer can now be done through the Internet. Thus, queues for information transfer (such as those for registration and requests for documents) can be eliminated. Physical transfer cannot be eliminated, but could be reduced through the Queuing Rule of Thumb. ^[3]

Applying the Queuing Rule of Thumb

The Queuing Rule of Thumb enables Queue management to focus on creating proper solutions for queue problems by simply getting the number of servers, the total number of customers, the service time, and the maximum time to finish the queue. To make a queuing system more efficient, these values must be adjusted with regards to the rule of thumb.^[3]

Examples

Conference lunch: People usually get their own food during a conference lunch. A serving table has 2 sides where people can get their food. If each person needs about 45 seconds to take his/her food and there are 1000 participants, how many serving tables must be provided so that the lunch break can be finished in an hour?

^[2]

Solution: Given r=45, N=1000, T=3600, we use the rule of thumb to get s:

s>{\frac {Nr}{T}}\Longrightarrow s>{\frac {1000\times 45}{3600}}\Longrightarrow s>12.5

. There are 2 sides of the table that can be used. So the number of tables needed is

\left\lceil {\frac {12.5}{2}}\right\rceil =6.25

. We round this up to a whole number since the number of servers must be discrete. Thus, 7 serving tables must be provided.^[2]

Student registration: A school of 10,000 students must set certain days for student registration. One working day is from 8:00 to 17:00, and there is a 1 hour lunch break. Each parent needs about 36 seconds to be served by all the serial servers. How many days are needed to register all students? ^[2]

Solution: Given s=1, N=10,000, r=36, we use the rule of thumb to get T:

s>{\frac {Nr}{T}}\Longrightarrow T>{\frac {Nr}{s}}\Longrightarrow T>{\frac {10,000\times 36}{1}}\Longrightarrow T>360,000

. Given the work hours for a day is 8 hours or 28,800 seconds, the number of registration days needed is

\left\lceil {\frac {360,000}{28,800}}\right\rceil =13

days.^[2]

Drop off: During the peak hour of the morning (6:00-7:00), about 4500 cars enter an elementary school to drop off their children. Each drop off occurs for about 60 seconds. Each car requires about 6 meters to stop and maneuver. How long should be the minimum drop off line?^[2]

Solution:Given N=4500, T=60, r=1, we use the rule of thumb to get s:

s>{\frac {Nr}{T}}\Longrightarrow s>{\frac {4500\times 1}{60}}\Longrightarrow s>75

. Given the server space for each car is 6 meters, the dropping line should be at least

75\times 6=450

meters.^[2]

Applications

The Queuing Rule of Thumb can solve various real-world problems such as those of the following field: ^[4]

References

1 2 3 4 Teknomo, Kardi. "Queuing Rule of Thumb based on M/M/s Queuing Theory with Applications in Construction Management".
1 2 3 4 5 6 7 8 Teknomo, Kardi. "Queuing Rule of Thumb".
1 2 3 4 Teknomo, Kardi (April 2016). Queuing Rule of Thumb. MathCon.
↑ "Queueing theory". Wikipedia. Retrieved 4 August 2017.

External Links

Queuing Rule of Thumb Calculator

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[paperq-1] 1 2 3 4 Teknomo, Kardi. "Queuing Rule of Thumb based on M/M/s Queuing Theory with Applications in Construction Management".

[webq-2] 1 2 3 4 5 6 7 8 Teknomo, Kardi. "Queuing Rule of Thumb".

[confq-3] 1 2 3 4 Teknomo, Kardi (April 2016). Queuing Rule of Thumb. MathCon.

[4] "Queueing theory". Wikipedia. Retrieved 4 August 2017.