# Queuing Rule of Thumb

The Queuing Rule of Thumb is a mathematical formula, known as the queuing constraint equation when it is used to find an approximation of servers required to service a queue. The formula is written as an inequality relating the number of servers (s), total number of service requestors (N), service time (r), and the maximum time to finish the queue (T): $s>{\frac {Nr}{T}}$ Compared to standard queuing formulas, QROT is simple enough to compute the necessary number of servers without involving probability. It serves as a rough heuristic to address queue problems. 

## Formula

1. $\rho ={\frac {\lambda }{\mu }}$ This is the ratio of the arrival rate and the service rate.

2. $U={\frac {\rho }{s}}<1$ This equation states that the utilization of the queuing system must not be larger than 1.

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

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

## Usage

The Queuing Rule of Thumb assists queue management to resolve queue problems by providing 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 must be adjusted with regards to the rule of thumb.

The mathematics underlying Queuing Theory is too complicated for use in most settings.The rule of thumb is simpler and therefore more practical.

• Conference lunch: 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?


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.
• Student registration: A school of 10,000 students must set certain days for student registration. One working day is 8 hours. Each parent needs about 36 seconds to be served. How many days are needed to register all students? 
Solution: Given s=1, N=10,000, r=36, the rule of thumb yields 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 (28,800 seconds), the number of registration days needed is $\left\lceil {\frac {360,000}{28,800}}\right\rceil =13$ days.
• Drop off: 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?
Solution: Given N=4500, T=60, r=1, the rule of thumb yields s: $s>{\frac {Nr}{T}}\Longrightarrow s>{\frac {4500\times 1}{60}}\Longrightarrow s>75$ . Given the space for each car is 6 meters, the line should be at least $75\times 6=450$ meters.