What it means
Every queue has three basic parts: things arriving, a place where they are served, and the time it takes to serve them. Queuing theory describes these with two numbers, the arrival rate (how many arrive per hour) and the service rate (how many can be handled per hour).
From them it estimates average waiting times and the length of the line. The most useful lesson is that waiting time grows sharply as the system gets busier.
A team running at 70% of its capacity may have short waits, but at 95% the line can become very long even though the average workload seems fine. This is why running staff at full capacity often backfires.
In finance and operations, the ideas are used for call centres, payment processing, loan approvals, invoice handling and warehouse loading. A manager can compare the cost of adding another person with the cost of customers or invoices waiting.
For example, late approvals can hold up supplier payments and risk missing early-payment discounts. The simplest model is called M/M/1, which assumes random arrivals, random service times and a single server.
More complex models handle several servers, priority customers and limited waiting space. The formulas become more involved, so many businesses use simulation software for detailed cases.
The nuance is that real queues are rarely as tidy as the maths. Arrivals often come in bursts, such as invoices at month end, and service times vary with the complexity of each item.
The models give a useful guide, but managers should check them against actual data. Queuing ideas also help with fairness and customer experience, not only cost.
People tolerate a wait more easily if it is predictable and if a single shared line feeds several servers. Many service businesses therefore use one common queue and tell customers their expected wait.
In practice
Real-world examples.
Example
A bank branch manager sees long lines at lunchtime and uses arrival data to decide how many tellers to schedule. Adding one teller during the peak cuts the wait sharply, while a second would add cost with little extra benefit. The analysis helps justify a split shift.
Example
A manufacturer's loading dock receives delivery trucks at random times throughout the day. Using queuing models, the logistics team learns that trucks wait for over an hour when only one bay is open. They add a second bay at peak times, saving on driver waiting charges.
Example
A customer support team handles billing queries for a software company. Management sets a target that calls are answered within 60 seconds and uses the model to work out how many agents are needed. The result is a staffing plan by hour of the day.
Formula
Calculation
Utilisation = arrival rate / service rate; average number in the system = utilisation / (1 - utilisation); average time in the system = 1 / (service rate - arrival rate)
Suppose an accounts payable clerk receives 12 invoices an hour and can process 15 an hour.
Step 1: utilisation = 12 / 15 = 0.80, or 80%.
Step 2: average number of invoices in the system = 0.80 / (1 - 0.80) = 0.80 / 0.20 = 4 invoices.
Step 3: average time in the system = 1 / (15 - 12) = 1 / 3 hour = 20 minutes.
If each invoice waiting costs the business $30 per hour in delayed approvals, the average cost is 4 x $30 = $120 per hour, which can be compared with the cost of hiring help.Case study
Seen in the real world.
Pinecrest Insurance is a fictional claims handler used for illustration. Its single approvals team processed claims at an average of 30 a day but received 28 a day, so utilisation was about 93%. Claimants were complaining that decisions took more than a week.
An analyst applied queuing theory and showed that adding just one part-time reviewer would cut average waiting time dramatically, because the system was close to its limit. In this illustrative story, the extra salary was far smaller than the cost of complaints and customers leaving. The manager adopted the principle of keeping utilisation below about 85% for time-sensitive work.
Six months later average decision time had dropped from eight days to three. The manager reviewed utilisation each month, and moved a spare reviewer between teams when claims surged after bad weather. The simple utilisation target became part of the department's standard reporting.
Watch out
Common mistakes.
- Believing a team is fine because its average workload is below capacity. Waiting time rises sharply as utilisation approaches 100%.
- Ignoring variability in arrivals. Bursts of work create queues even when the average looks manageable.
- Treating the formula as exact in real life. It is a guide that should be checked against observed waiting times.
Questions
People also ask.
What does M/M/1 mean?
It describes a queue with random arrivals, random service times and one server.
Where is queuing theory used in finance?
It is used in payment processing, call centres, loan approval, invoice handling and trading systems.
Why not run staff at 100% utilisation?
At that level any small surge creates a growing backlog, so waiting times become very long.
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