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All teaching chapters stay open. Choose a direct route to the relevant explanation, or follow the full sequence from first principles.
Understand the reader problem
Chapters 1–2. Start this reading route.
Define the model and numbers
Chapters 3–5. Start this reading route.
Work the essential calculation
Chapters 6–7. Start this reading route.
Change the setting and examine evidence
Chapters 8–10. Start this reading route.
Practise and read further
Chapters 11–12. Start this reading route.
With eight customers arriving per hour and two equally fast cashiers, separate M/M/1 queues give an average wait of 20 minutes; a shared M/M/2 queue gives eight minutes under the stated assumptions. Staffing and total workload have not changed. The routing of waiting customers has.
Follow the calculation from service capacity through Erlang C, compare the results with Little’s Law and discover exactly when these familiar queue formulas stop being trustworthy.
Chapter index
Understand the reader problem · Chapters 1–2
1. Why a longer-looking shared line can move faster
2. Declare the two models and what counts as a customer
Define the model and numbers · Chapters 3–5
3. Calculate capacity before calculating the delay
4. Calculate a separate line as an M/M/1 queue
5. The common line uses the M/M/2 model
Work the essential calculation · Chapters 6–7
6. Work out the empty-system probability
7. Erlang C gives the probability of delay and its mean
Change the setting and examine evidence · Chapters 8–10
8. Compare both arrangements and check Little’s Law
9. Test the effect as demand approaches capacity
10. Know when the calculation should not be reused
Practise and read further · Chapters 11–12
11. Practice and answer the most useful FAQs
12. Continue into the connected reading library
CHAPTER 1 OF 12 · Understand the reader problem
1. Why a longer-looking shared line can move faster
Picture two supermarket checkouts. Customers can either choose a separate line and stay there, or join a single line that sends the next customer to whichever cashier becomes free. The visible shared line might look longer even while customers move through faster.
The reason is not that anyone works harder. With separate lines one cashier may have no customer while another has three waiting. A shared line can put the next person in front of whichever server is available.
We need to quantify the effect, not simply declare all pooling arrangements superior. The answer depends on how arrivals and service behave.
CHAPTER 2 OF 12 · Understand the reader problem
2. Declare the two models and what counts as a customer
Assume customers arrive according to a stationary Poisson process at rate λ=8 customers per hour. There are two identical cashiers, each with independent exponential service times averaging ten minutes. One cashier can therefore complete μ=6 customers per hour on average.
Assume unlimited queue space, no abandonment, no interruptions, first-come-first-served order and a long enough operating period for steady-state averages to apply. Every arrival is one customer, not a batch.
In the separate arrangement each customer independently chooses either cashier with probability one-half and is unable to switch. In the pooled arrangement there is one queue and two cashiers. These two disciplines are not the same queueing system.
CHAPTER 3 OF 12 · Define the model and numbers
3. Calculate capacity before calculating the delay
With two cashiers total nominal capacity is cμ=2×6=12 customers per hour. The offered load per server is ρ=λ/(cμ)=8/12=2/3.
Randomly splitting a Poisson stream in half makes two Poisson arrival streams, each at rate λi=4 customers per hour. Each separate cashier therefore operates at utilisation 4/6=2/3 as well.
Both systems are stable under their stated stationary assumptions because utilisation is below one. Equal mean utilisation is not enough to determine waiting time, however; the question of who can use an idle cashier remains.
CHAPTER 4 OF 12 · Define the model and numbers
4. Calculate a separate line as an M/M/1 queue
For one M/M/1 queue, the mean waiting time before service is Wq=λi/[μ(μ−λi)], provided λi<μ. Keep its time unit as hours until the final conversion.
Substitute λi=4 and μ=6: Wq=4/[6(6−4)]=1/3 hour=20 minutes. Because the two separate lines are symmetric, this is the per-customer mean queue wait across both lines.
An individual service lasts ten minutes on average. Thus mean arrival-to-completion time is 20+10=30 minutes. Waiting time and time in the entire system must not be interchanged.
| Line | Arrival rate | Mean queue wait | Mean total time |
|---|---|---|---|
| Counter A | 4/hour | 20 min | 30 min |
| Counter B | 4/hour | 20 min | 30 min |
| Combined experience | 8/hour total | 20 min per customer | 30 min per customer |
CHAPTER 5 OF 12 · Define the model and numbers
5. The common line uses the M/M/2 model
The pooled system keeps total λ=8 arrivals per hour and two individual service rates of μ=6 per hour. The offered load is a=λ/μ=4/3 and utilisation is ρ=a/2=2/3.
A customer waits only if both cashiers are already busy. The classical Erlang C expression supplies the stationary probability of that event for the M/M/2 model.
Do not replace two exponential-service cashiers with a fictional single exponential server that operates at 12/hour. That model has a different service process despite matching the headline capacity.
CHAPTER 6 OF 12 · Work the essential calculation
6. Work out the empty-system probability
For a pooled M/M/2 queue the probability that the entire system is empty is P0=1/[1+a+a²/(2(1−ρ))]. The final term accounts for the queueing tail.
Since a=4/3 and ρ=2/3, a²/[2(1−ρ)]=(16/9)/(2/3)=8/3. The denominator is 1+4/3+8/3=5.
Therefore P0=1/5=0.20. Do not confuse an empty system with an immediately available cashier: one cashier may already be working while the other is free.
CHAPTER 7 OF 12 · Work the essential calculation
7. Erlang C gives the probability of delay and its mean
The probability that an arriving customer has to wait is C=[a²/(2(1−ρ))]P0. Therefore C=(8/3)(1/5)=8/15, about 53.3%.
The mean waiting time before service is Wq=C/(cμ−λ). For our numbers this is (8/15)/(12−8)=2/15 hour=8 minutes.
Add ten minutes of expected service to obtain 18 minutes of mean time in the pooled system. Two staff members are still doing the work; the efficiency improvement comes from how they share waiting customers.
CHAPTER 8 OF 12 · Change the setting and examine evidence
8. Compare both arrangements and check Little’s Law
The two separate lines require 20 minutes mean queueing and 30 minutes mean total time. The shared queue requires eight minutes mean queueing and 18 minutes mean total time, all for a customer arriving under the stationary model.
Little’s Law checks the average waiting population. With total arrival rate eight per hour, the separate-line total is Lq=8×20/60=8/3 customers; the pooled queue is Lq=8×8/60=16/15 customers.
Neither figure is a guaranteed headcount at a particular moment. They are expected values calculated using the same underlying arrival population.
| Quantity | Two separate lines | One pooled M/M/2 line |
|---|---|---|
| Total arrivals | 8/hour | 8/hour |
| Total service capacity | 12/hour | 12/hour |
| Utilisation | 2/3 per server | 2/3 per server |
| Probability of waiting | 2/3 | 8/15 |
| Mean queue wait | 20 min | 8 min |
| Mean total time | 30 min | 18 min |
| Mean number waiting | 8/3 | 16/15 |
CHAPTER 9 OF 12 · Change the setting and examine evidence
9. Test the effect as demand approaches capacity
At utilisation 2/3, either system has enough capacity on average, yet stochastic arrivals and services create queues. As demand approaches capacity, small changes in arrival rate or service interruptions can multiply delays.
A shared line prevents the particular problem of one cashier sitting idle while another cashier has customers waiting, provided both cashiers can serve the same kind of customer.
If skill requirements, priority rules or switching time make the cashiers non-interchangeable, a single pooled model may no longer represent the decision. Record what the system actually does.
CHAPTER 10 OF 12 · Change the setting and examine evidence
10. Know when the calculation should not be reused
Real arrivals may come in groups, service times may be non-exponential, and customers may switch lines, abandon waiting or receive priority service. Each of these changes can alter the answer.
In a separate system where everyone joins whichever line is shortest, the independent 50:50 Poisson-splitting assumption does not hold. It would be incorrect to quote the same 20-minute result for that policy without recomputation.
Use timestamped arrivals, real service-start and finish times, staff availability, line choice and abandonment data. A simulation or a different stochastic model may be more credible than forcing a formula to fit.
CHAPTER 11 OF 12 · Practise and read further
11. Practice and answer the most useful FAQs
Practice: reproduce a=4/3, ρ=2/3, P0=1/5 and C=8/15. Then use C/(12−8) to obtain eight minutes and confirm that adding ten minutes of service yields 18 minutes.
FAQ: Is eight minutes a maximum wait? No. Does pooling add a cashier? No. Does the visible shared line have to be shorter? No. Does equal utilisation mean equal delays? No: dispatch and workload sharing matter.
The usual mistakes are forgetting that μ belongs to one server, counting queue wait as total time, and applying the steady-state formula to a short bursty shift with unavailable staff.
CHAPTER 12 OF 12 · Practise and read further
12. Continue into the connected reading library
The Queueing Theory root introduces arrival rates, utilisation, queues and measurement. Compare this to grouped arrivals and service-time variability, where other model ingredients change.
For independently readable calculations, use the M/M/1 and M/M/2 waiting-time formulas and Erlang C and resource-pooling notes. These sources do not replace a local service study.
The transferable habit is to distinguish arrival flow, capacity, routing discipline and customer experience before deciding whether the queue has been improved.
Continue through Queueing Theory, The Gold Standard of Education and How X Works. Original worked examples use their stated assumptions; source references checked 22 October 2026.