Queuing Rule of Thumb

A derivation of the QROT formula follows. The arrival rate is the ratio of the total number of customers N and the maximum time needed to finish the queue T.

${\displaystyle \lambda ={\frac {N}{T}}}$

The service rate is the reciprocal of service time r.

${\displaystyle \mu ={\frac {1}{r}}}$

It is convenient to consider the ratio of the arrival rate and the service rate.

${\displaystyle \rho ={\frac {\lambda }{\mu }}}$

Assuming s servers, the utilization of the queuing system must not be larger than 1.

${\displaystyle U={\frac {\rho }{s}}<1}$

Combining the first three equations gives ${\displaystyle \rho ={\frac {\lambda }{\mu }}={\frac {Nr}{T}}}$. Combining this and the fourth equation yields ${\displaystyle U={\frac {\rho }{s}}={\frac {Nr}{Ts}}<1}$.

Simplifying, the formula for the Queuing Rule of Thumb is ${\displaystyle s>{\frac {Nr}{T}}}$.

The Queuing Rule of Thumb assists queue management to resolve queue problems by relating the number of servers, the total number of customers, the service time, and the maximum time needed to finish the queue. To make a queuing system more efficient, these values can be adjusted with regards to the rule of thumb.[3]

The following examples illustrate how the rule may be used:

1. Conference lunch
   • Problem: Conference lunches are usually self-service. Each serving table has 2 sides where people can pick up their food. If each of 1000 attendees needs 45 seconds to do so, how many serving tables must be provided so that lunch can be served in an hour?[2]
   • Solution: Given r = 45, N = 1000, T = 3600, we use the rule of thumb to get s: ${\displaystyle s>{\frac {Nr}{T}}\Longrightarrow s>{\frac {1000\times 45}{3600}}\Longrightarrow s>12.5}$. There are two sides of the table that can be used. So the number of tables needed is ${\displaystyle \left\lceil {\frac {12.5}{2}}\right\rceil =6.25}$. We round this up to a whole number since the number of servers must be discrete. Thus, 7 serving tables must be provided.[2]
2. Student registration
   • Problem: A school of 10,000 students must set certain days for student registration. One working day is 8 hours. Each student needs about 36 seconds to be registered. How many days are needed to register all students?[2]
   • Solution: Given s = 1, N = 10,000, r = 36, the rule of thumb yields T: ${\displaystyle s>{\frac {Nr}{T}}\Longrightarrow T>{\frac {Nr}{s}}\Longrightarrow T>{\frac {10,000\times 36}{1}}\Longrightarrow T>360,000}$. Given the work hours for a day is 8 hours (28,800 seconds), the number of registration days needed is ${\displaystyle \left\lceil {\frac {360,000}{28,800}}\right\rceil =13}$ days.[2]
3. Drop off
   • Problem: During the peak hour of the morning about 4500 cars drop off their children at an elementary school. Each drop-off requires about 60 seconds. Each car requires about 6 meters to stop and maneuver. How much space is needed for the minimum drop off line?[2]
   • Solution: Given N = 4500, T = 60, r = 1, the rule of thumb yields s: ${\displaystyle s>{\frac {Nr}{T}}\Longrightarrow s>{\frac {4500\times 1}{60}}\Longrightarrow s>75}$. Given the space for each car is 6 meters, the line should be at least ${\displaystyle 75\times 6=450}$ meters.[2]

• Little's law

• Murugan, Arumugaselvi; Aminu, Halima; Subramanian, Ganesan. "Mathematical Analysis on Quality of Service in Cloud servers". International Journal of Enhanced Research in Science, Technology & Engineering. 4 (10).
• Mouhaffel, Adib Guardiola; Dominguez, Carlos Martinez; Martin, Ricardo Diaz; Seck, Assane; Ahmadou, Wague; Fall, Melissa; Sall, Ddjibril. "Assessment of Carbon Footprint and Economic Evaluation Schedule of Agriculture Workland by Theoretical Queuing". Indian Journal of Science and Technology. 10 (19).
• Stintzing, Josefin; Norrman, Frederik. Prediction of queuing behaviour through the use of artificial neural networks (Thesis). Kth Royal Institute of Technology.
• Sheikh, Afshan; Lakshmipathy, M.; Prakash, Arokia. "Application of Queuing Theory for Effective Equipment Utilization and Maximization of Productivity in Construction Management". International Journal of Applied Engineering Research. 11 (8).
• Burkul, Vinod Bandu; Oh, Joon-Yeoul; Peel, Larry; Tang, Hee Joong. "Reducing Customer Waiting Time with New Layout Design". Cite journal requires |journal= (help)
• Ikwunne, Tochukwu Arinze; Orji, Rita. Persuasive Technology for Reducing Waiting and Service Cost: A Case Study of Nigeria Federal Medical Centers. Proceedings of the First African Conference on Human Computer Interaction. Nairobi, Kenya. pp. 24–35.

• Queueing Rule of Thumb Calculator