Find the route that matches your question
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.
Five jobs, one server and exactly the same total work. A new “urgent jobs first” rule can cut the high-priority class’s mean wait from 12 minutes to two minutes while leaving the overall mean wait at eight. The headline statistic looks unchanged. The experience of different people does not.
We will build a five-job queue that anyone can audit with a clock, contrast first-come-first-served and nonpreemptive priority, and then change the task sizes and arrival times. The question is not whether priority is good or bad in the abstract; it is which performance measure and which group a particular rule is actually serving.
Chapter index
Understand the reader problem · Chapters 1–2
1. Why an unchanged overall mean can conceal a large change
2. Define nonpreemptive priority precisely
Define the model and numbers · Chapters 3–5
3. Build a five-job queue with transparent timing
4. Calculate the FCFS baseline job by job
5. Put high-priority jobs first, without changing work
Work the essential calculation · Chapters 6–7
6. Explain the class trade-off with weighted totals
7. An on-time fraction can tell a different story
Change the setting and examine evidence · Chapters 8–10
8. Use the correct table and describe the result honestly
9. Change the arrival time to expose preemption
10. Change service durations and the overall mean can move
Practise and read further · Chapters 11–12
11. Practise, and inspect the fairness you cannot see
12. Read from queueing mathematics into a defensible service design
CHAPTER 1 OF 12 · Understand the reader problem
1. Why an unchanged overall mean can conceal a large change
Imagine a reception desk handling ordinary and time-critical requests. Both types must be served, but a time-critical request may reasonably receive preferential access. Changing the sequence of work does not add workers or reduce the total number of minutes needed.
If someone reports only the mean wait across everyone, the result can conceal large changes by class. A better report will specify the number of jobs in each class, their individual waits, and which rules decide the next job.
Our first exercise fixes all arrival times and every job duration, so we can isolate the sequencing decision from random arrivals and service-time variability. The model is intentionally finite; it is not a stationary M/M/1 formula.
CHAPTER 2 OF 12 · Understand the reader problem
2. Define nonpreemptive priority precisely
Under first-come-first-served (FCFS), the server takes waiting jobs in arrival order. Under strict nonpreemptive priority, a free server chooses a waiting high-priority job before a low-priority one; if it has already started a job, it completes that job even when a higher-priority request arrives.
Within each priority class we use FCFS, and we stipulate a fixed tie order for jobs arriving at exactly the same instant. Thus both policies are fully specified rather than relying on a vague claim that urgent work goes first.
Preemptive priority is a different rule. It can interrupt a job already in service; we will test that distinction later. An organisation should not call its rule nonpreemptive if staff actually interrupt ongoing work.
CHAPTER 3 OF 12 · Define the model and numbers
3. Build a five-job queue with transparent timing
At time t=0, five jobs are ready. There are two high-priority jobs H1 and H2 and three low-priority jobs L1, L2 and L3. All require exactly four minutes of uninterrupted server time, and the single server is initially idle.
For the FCFS baseline, assume the original tie order is L1, L2, H1, L3, H2. Because everyone arrived together, this is a declared tie-breaking choice, not a random sample of a Poisson queue.
Queue wait is service-start time minus arrival time. As all arrival times are zero, we can read each waiting time directly from its service-start clock value. Total time to finish all work is five times four, or 20 minutes.
CHAPTER 4 OF 12 · Define the model and numbers
4. Calculate the FCFS baseline job by job
The FCFS order begins with L1 at t=0, followed by L2 at t=4, H1 at t=8, L3 at t=12 and H2 at t=16. Completion times are 4, 8, 12, 16 and 20 minutes respectively.
H1 waits eight minutes and H2 waits 16, so mean waiting for high-priority jobs is (8+16)/2=12 minutes. Low-priority waits are 0, 4 and 12, producing a mean of 16/3 minutes, about 5.33.
The sum of all five waits is 0+4+8+12+16=40 minutes. Dividing by all five jobs gives an overall mean of eight minutes. Keep that denominator visible when the policy changes.
| FCFS job order | Class | Service start (min) | Queue wait (min) |
|---|---|---|---|
| L1 | Low | 0 | 0 |
| L2 | Low | 4 | 4 |
| H1 | High | 8 | 8 |
| L3 | Low | 12 | 12 |
| H2 | High | 16 | 16 |
CHAPTER 5 OF 12 · Define the model and numbers
5. Put high-priority jobs first, without changing work
The strict nonpreemptive priority rule uses H1, H2, L1, L2, L3 when all five are waiting at t=0. The service-start times are still 0, 4, 8, 12 and 16 minutes because durations and staffing are unchanged.
High-priority waits are now zero and four minutes, giving a mean of two. The low-priority jobs wait eight, 12 and 16 minutes, giving a mean of 12. The overall mean remains (0+4+8+12+16)/5=8 minutes.
Priority therefore changed who waited, not how much aggregate waiting occurred in this special all-at-once, equal-size example. This invariance should not be claimed for arbitrary arrival patterns or service-time distributions.
CHAPTER 6 OF 12 · Work the essential calculation
6. Explain the class trade-off with weighted totals
Compared with the FCFS baseline, each of the two high-priority jobs benefits enough to reduce the class mean by ten minutes: from 12 to two. Across two people that is 20 waiting minutes removed.
The three low-priority jobs move from an average of 16/3 to 12 minutes, a rise of 20/3 minutes each. Across three jobs that adds exactly 20 waiting minutes. The changes cancel in the overall sum.
This arithmetic explains why the same overall mean does not imply unchanged service fairness. It also reminds us that the class averages must be weighted by the number of jobs before they are combined; averaging two class means equally would be wrong here.
CHAPTER 7 OF 12 · Work the essential calculation
7. An on-time fraction can tell a different story
Suppose the service promise is that a job begins within eight minutes, inclusive. In the FCFS order, three jobs meet that threshold: L1, L2 and H1. In the priority order three also qualify: H1, H2 and L1.
The overall on-time share is therefore 3/5=60% in both cases. But the high-priority class moves from 1/2=50% to 2/2=100%, while the low-priority class moves from 2/3 to 1/3.
A report that provides the overall 60% alone conceals the policy decision. The service threshold, the category definitions, the count in each category and the full wait distribution should be visible together.
CHAPTER 8 OF 12 · Change the setting and examine evidence
8. Use the correct table and describe the result honestly
Compare class waiting means and on-time shares instead of publishing one pooled average. The baseline and priority schedules do not differ in number of jobs or staff, so the observed differences come solely from the ordering rule.
The example establishes a trade-off under an explicitly specified finite batch, not a claim that urgent work is unimportant. Real priority rules may be justified by safety, legal deadlines or disproportionate consequences of delay. Those priorities still deserve transparent measurement.
If you have more than one server, mixed task sizes or changing arrival times, rerun the calculation or simulation. The table below is not a ready-made performance estimate for a clinic or call centre.
| Measure | FCFS order | High-priority-first |
|---|---|---|
| High-priority mean wait | 12 min | 2 min |
| Low-priority mean wait | 16/3 min ≈ 5.33 | 12 min |
| All-job mean wait | 8 min | 8 min |
| High-priority within 8 min | 1/2 | 2/2 |
| Low-priority within 8 min | 2/3 | 1/3 |
CHAPTER 9 OF 12 · Change the setting and examine evidence
9. Change the arrival time to expose preemption
Consider a new two-job situation. A low-priority job L starts at t=0 and requires ten minutes of work. A four-minute high-priority job H arrives at t=2. Under nonpreemptive priority, L continues to t=10; H waits from t=2 to t=10, then completes at t=14. H has eight minutes of queue waiting.
Under preemptive-resume priority, H interrupts at t=2, serves from t=2 to t=6 and finishes. L resumes its eight remaining minutes from t=6 to t=14. H now has zero initial queue waiting, while L completes four minutes later than under the nonpreemptive rule.
Both schedules finish the same 14 minutes of total work by t=14, but the interruption changes the time each job is complete. We have assumed cost-free interruption and perfect service resumption—two assumptions that can be seriously unrealistic in human work.
CHAPTER 10 OF 12 · Change the setting and examine evidence
10. Change service durations and the overall mean can move
Now restore five jobs all arriving at t=0, but set L1 to ten minutes, L2 and L3 to two minutes each, and H1 and H2 to four minutes each. Under the original FCFS order L1, L2, H1, L3, H2, start times are 0, 10, 12, 16 and 18 minutes; the total of waits is 56 minutes.
Under strict high-priority-first order H1, H2, L1, L2, L3, start times become 0, 4, 8, 18 and 20. The total of waits is 50 minutes. Different job sizes permit the overall mean to change with the service order even though total processing work remains 22 minutes.
This example is a reason not to export the equal-job-size invariant beyond its stated setting. To evaluate priority under a continuing stochastic arrival stream, use class-specific arrival rates, service-time distributions and an appropriate queueing model.
CHAPTER 11 OF 12 · Practise and read further
11. Practise, and inspect the fairness you cannot see
Exercise: repeat the five equal four-minute jobs but change the policy to low-priority first: L1, L2, L3, H1, H2. The wait values are still 0, 4, 8, 12 and 16 minutes. Compute the two class means and show that the grand mean remains eight minutes.
FAQ: Is nonpreemptive priority the same as interrupting a low-priority job? No. Does priority increase raw service capacity? No. Is the overall mean sufficient to establish equitable outcomes? No. Are priorities always arbitrary? No; some correspond to real urgency or contractual obligations that should be stated openly.
In an actual operation record class, arrival time, time service begins, completion time, work duration, interruptions, switching cost, deadline and eventual outcome. Report distributions and tail percentiles by class and consider how long a low-priority item might wait during sustained high-priority demand.
CHAPTER 12 OF 12 · Practise and read further
12. Read from queueing mathematics into a defensible service design
Return to The Gold Standard of Queueing Theory for arrival rates, stability and Little’s Law. The pooled-versus-separate-line example shows how sharing capacity matters even without priority, while batch arrivals isolates clustering.
For the formal distinction between preemptive and nonpreemptive classes, consult the Encyclopedia of Mathematics account of queues with priorities and Carnegie Mellon University’s nonpreemptive priority model. For an applied scheduling analogue, see the Kubernetes documentation on non-preempting priority classes. These sources inform the general mechanism; the numerical schedules here are original.
The transferable decision rule is to define the service classes and their stakes, calculate each class’s outcomes under a reproducible policy, then decide whether the trade-off matches the service’s real obligations. No one headline mean can make that decision for you.
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.