Calling everyone at the beginning can make a service session finish earlier while making its customers wait much longer. Asking everyone to arrive early can also increase measured waiting without changing a single service start. To understand an appointment system, keep the appointment clock, actual arrival and service start separate.
The examples below describe a fictional single-server booking desk, with all times in minutes. They are exact finite ledgers, not measured performance or a recommended clinical schedule. The purpose is to trace where time goes when an arrival rule changes.
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Choose a route below or read in order. All teaching chapters remain open; the index groups let you jump directly to a worked example or check.
Route 1 · Define the three clocks
Chapters 1–2: Define the three clocks.
Route 2 · Work the scheduled ledger
Chapters 3–4: Work the scheduled ledger.
Route 3 · Compare batching and early reporting
Chapters 5–6: Compare batching and early reporting.
Route 4 · Change the rule and test the evidence
Chapters 7–8: Change the rule and test the evidence.
Route 5 · Practise and choose the next guide
Chapters 9–12: Practise and choose the next guide.
Full chapter index · Try the transfer checks · Queueing Theory subject root
Chapter contents
Define the three clocks · Chapters 1–2
Work the scheduled ledger · Chapters 3–4
Compare batching and early reporting · Chapters 5–6
Change the rule and test the evidence · Chapters 7–8
Practise and choose the next guide · Chapters 9–12
CHAPTER 1 OF 12 · Define the three clocks
1. Separate the appointment from actual arrival
Let ti be customer i’s appointment time, ai their actual arrival, bi their service start and ci their completion. On-site waiting is bi − ai. Time after the appointment before service is a different measure: its nonnegative version is max(0, bi − ti).
If someone arrives at minute seven for a minute-ten appointment and starts at ten, on-site waiting is three minutes while appointment lateness is zero. Both statements can be accurate. A report becomes misleading when it labels one as the other or compares services using incompatible definitions.
There can be useful preparation before the booked service: forms, equipment checks or orientation. If those activities are required, define their start and completion separately. Time spent performing a necessary task is not automatically idle waiting, although it remains part of the customer’s full journey.
The existing Little’s Law boundary guide develops the population-and-time distinction. Here we keep every individual event visible so a change in a headline average can be checked against the actual schedule.
Assume one server opens at minute zero, handles one job at a time without interruption, and keeps the stated customer order. There are no missing customers, priority insertions, setup times or travel between jobs. Service durations in the first example are 3, 7, 3 and 7 minutes.
When early service is allowed, the next job starts at the latest of opening time, the previous completion and its actual arrival: bi = max(0, ci−1, ai). Completion is ci = bi + si, where si is its service duration. Take c0 = 0.
If the policy prohibits starting before the appointment, add ti to the maximum. That is a different operating rule, not an algebraic correction to the same rule. In either case the calculation is deterministic once the event times and durations have been specified.
For arrival-ordered first-come service after opening, the associated waiting recursion is Wi+1 = max(0, Wi + si − (ai+1 − ai)). This carries unfinished work to the next arrival. A stationary arrival-rate formula is unnecessary for this four-job ledger.
CHAPTER 3 OF 12 · Work the scheduled ledger
3. Work through evenly spaced appointments
Book appointments at minutes 0, 5, 10 and 15, and let everyone arrive exactly on time. Customer A starts at zero, receives three minutes of service and leaves at three. The next customer is not present until minute five, so the server is idle for two minutes.
Customer B starts at five and finishes at twelve. Customer C arrives at ten and therefore waits two minutes before starting at twelve; three minutes of service take C to fifteen. Customer D starts at fifteen and finishes at twenty-two.
The waits are 0, 0, 2 and 0. Their sum is two customer-minutes and their mean is half a minute. Total active service is twenty minutes, total idle time during the opening-to-final-completion window is two minutes, and the session ends at twenty-two.
Average service duration is five minutes, exactly the spacing between appointments. That equality does not make every wait zero. The realised long job at B carries work across C’s appointment even though the total service content is modest.
| Customer | Arrival/appointment | Service duration | Start | Finish | Wait |
|---|---|---|---|---|---|
| A | 0 | 3 | 0 | 3 | 0 |
| B | 5 | 7 | 5 | 12 | 0 |
| C | 10 | 3 | 12 | 15 | 2 |
| D | 15 | 7 | 15 | 22 | 0 |
CHAPTER 4 OF 12 · Work the scheduled ledger
4. Check waiting, work and the observation window
Customer-minutes and server-minutes have different meanings. Two people waiting simultaneously for one minute contribute two customer-minutes. One idle server during the same minute contributes one server-minute. Adding them requires a stated comparison or weighting, not just a shared unit label.
In our ledger, the area under the waiting-count curve is two customer-minutes. Adding twenty minutes of service produces twenty-two customer-minutes of total time in the system. The four individual sojourns are 3, 7, 5 and 7 minutes, which sum to the same total.
Over the complete window from zero to twenty-two, mean number present is 22/22 = 1. The completed-cohort rate is 4/22 per minute and mean sojourn is 22/4 = 5.5 minutes, reproducing that finite-window accounting identity. No unfinished customer was discarded at the end.
If observation stopped at minute fifteen, D’s future completion would not yet be available. A completed-only average would describe a different cohort. Preserve unfinished cases and window boundaries before treating a shorter report as evidence of faster service.
CHAPTER 5 OF 12 · Compare batching and early reporting
5. Bring the same four jobs in as one batch
Now make all four actual arrivals minute zero, retaining their service order and durations. Starts become 0, 3, 10 and 13; completions become 3, 10, 13 and 20. The server has no idle gap before the final completion.
Customer waits are 0, 3, 10 and 13 minutes. Total waiting is twenty-six customer-minutes, or a mean of 6.5 minutes. The session ends two minutes earlier than before, but customers collectively wait twenty-four additional minutes.
No job became longer and no service capacity was added. The earlier finish comes from filling the original two-minute idle gap. The larger customer-time burden comes from moving their arrival times earlier than the server could usefully serve them.
This is arrival batching, not batch service. The server still completes one job at a time, with total service duration twenty. Use batch arrivals and event versus job rates for the stochastic counting distinction; a simultaneous booking block is not a new fast multi-customer service mechanism.
CHAPTER 6 OF 12 · Compare batching and early reporting
6. Ask everyone to report three minutes early
Restore the original appointment times 0, 5, 10 and 15, but set actual arrivals to −3, 2, 7 and 12. A negative time simply means before the opening reference. First impose the rule that service cannot start before the booked appointment.
Service starts and completions remain exactly those in the original scheduled ledger: 0, 5, 12, 15 and 3, 12, 15, 22. The server still has two idle minutes. Actual waits become 3, 3, 5 and 3, totalling fourteen customer-minutes and averaging 3.5 minutes.
Every customer’s on-site wait rose by three minutes, so the total rose by 4×3 = twelve. Positive appointment lateness remains 0, 0, 2 and 0. An unchanged appointment-lateness report therefore conceals the additional time customers now spend present.
This does not prove that early reporting is always unnecessary. Preparation, accessibility and coordination can make it useful. It establishes what the reporting instruction does in this explicitly gated example. Any real justification should identify the activity or improvement that uses the extra time.
CHAPTER 7 OF 12 · Change the rule and test the evidence
7. Allow early service and recalculate
Keep the three-minute-early arrivals but remove the appointment gate. The server opens at zero, finishes A at three and can start B immediately, because B arrived at two. B finishes at ten; C has already arrived at seven and finishes at thirteen; D finishes at twenty.
Starts are 0, 3, 10 and 13, giving actual waits of 3, 1, 3 and 1 minutes. Their total is eight customer-minutes, mean two minutes. Completion is again at twenty, with no server idle gap.
The early-service rule improves on the gated early-arrival case in this path. It still produces more customer waiting than the original on-time arrivals, whose total was two. A result can improve one comparison and worsen another; name the baseline.
| Scenario | Total customer wait | Mean wait | Server idle time | Final completion |
|---|---|---|---|---|
| On-time arrivals | 2 | 0.5 | 2 | 22 |
| All arrive at opening | 26 | 6.5 | 0 | 20 |
| Three minutes early; appointment gate | 14 | 3.5 | 2 | 22 |
| Three minutes early; early service allowed | 8 | 2 | 0 | 20 |
The calculation assumes the customer is ready and eligible for early service, and that order is retained. If eligibility, preparation or priority changes, the start-time rule must change too. Do not apply a work-conserving maximum to a process intentionally waiting for a prerequisite.
CHAPTER 8 OF 12 · Change the rule and test the evidence
8. Change job order without changing the average work
For a new path, retain on-time arrivals at 0, 5, 10 and 15 but change service durations to 7, 3, 7 and 3. The four starts are 0, 7, 10 and 17; waits are 0, 2, 0 and 2. Total waiting is four customer-minutes and the session ends at twenty.
The same twenty minutes of service and the same average duration now give a different waiting total and finish time. Order matters because a long early job can pass work into later appointments, while a short job can let the system catch up.
If all original durations were known in advance, appointments at 0, 3, 10 and 13 would produce zero waiting and zero idle time for that path. Those are hindsight-perfect appointments. Replacing uncertain future durations with their observed values makes planning look easier than it is.
Compare equal mean service times and different waits and priority rules and class trade-offs. The new ledger concerns timing and sequence, while those guides examine further reasons that an aggregate average is insufficient.
CHAPTER 9 OF 12 · Practise and choose the next guide
9. Separate an exact path from a scheduling prediction
A real schedule is chosen before all durations, arrivals and interruptions are known. Predicting its performance requires a suitable model or data representing those uncertainties. Our arithmetic can evaluate one completed path exactly without establishing its expected future cost.
Service duration may depend on job type or conditions. Early and late arrivals may be related to travel, preparation instructions or previous experiences. Treating every input as independent merely because a simulation offers that option can misrepresent the process.
No-shows are a separate event. In the current ledgers everybody attends, so no overbooking conclusion follows. A lower realised workload caused by absence also does not mean that every intended customer received the service.
Use time-varying arrivals and capacity arriving too late to connect appointment blocks with peaks, and server vacations and residual delays when availability is interrupted. Adding complexity should answer an observed question rather than decorate a model.
CHAPTER 10 OF 12 · Practise and choose the next guide
10. Build the evidence record from individual events
Record appointment, requested reporting time, actual arrival, readiness for service, start and finish. If readiness is unobserved, state that limitation. A person who arrived but is completing a required task may not yet belong to the ready-to-serve queue.
Report on-site waiting and appointment delay separately, with their denominators and observation windows. Include late arrivals, unattended bookings and unfinished cases under explicit categories. Otherwise a rule that deters attendance can appear to improve the completed-customer mean.
Inspect the distributions and the experience of different groups. A mean of half a minute in our first example comes from three zero waits and one two-minute wait. The same mean could be distributed differently, with different implications for a time-sensitive task.
Check whether an intervention changed service quality or moved work elsewhere. A shorter desk interaction that leaves unresolved preparation downstream is not necessarily a shorter full journey. The broader queueing theory and waiting-times guide connects the ledger with system-level interpretation.
CHAPTER 11 OF 12 · Practise and choose the next guide
11. Practise a changed opening and a changed policy
Check one: keep the original scheduled arrivals and durations but open the server at minute two. Starts become 2, 5, 12 and 15, with waits 2, 0, 2 and 0. Total waiting is four and final completion remains twenty-two. The original idle gap absorbs this particular delayed opening.
Check two: with original starts held fixed and no service before appointment, ask everyone to arrive five minutes early instead of three. The waits become 5, 5, 7 and 5, summing to twenty-two. Positive appointment lateness is still only two in total.
Check three: someone reports zero overtime after batching. If planned closing is twenty minutes, that statement is correct for our batch ledger. It says nothing about the twenty-six customer-minutes of waiting. Check the outcome omitted from the report.
For transfer, use the same event definitions at a repair desk or equipment-booking session. Rebuild the readiness and start-time rules before using the maximum equation. The useful skill is following the work and the clocks, not declaring one arrival pattern universally best.
CHAPTER 12 OF 12 · Practise and choose the next guide
12. Questions, reading routes and sources
Is arriving early the cause of every long wait?
No. Long service, interruptions, prerequisites and schedule design can all contribute. The early-arrival examples isolate one change so its accounting effect is visible.
Does a faster finish prove a better schedule?
It establishes a faster finish under the specified conditions. Customer time, access, reliability and service quality remain separate outcomes. The objective must name the trade-offs.
Can the maximum recursion handle a priority change?
The start-time ledger can be rebuilt for a declared new sequence and eligibility rule. The fixed arrival-ordered recursion should not be applied unchanged to an order it no longer describes.
Choose the next question
Define arrivals, waits and service decisions
- Queueing Theory, Arrival Rates and Waiting Times
- Queues, Waiting Times and Service Decisions
- Little’s Law and the Waiting-Room Boundary That Changes the Answer
- Batch Arrivals, Event Rates and Job Counts
- Priority Rules, Class Waits and Hidden Trade-Offs
- Equal Mean Service Times, Different Mean Waits
Add practical complications carefully
- Time-Varying Arrivals and Capacity That Arrives Too Late
- Server Vacations and the Residual Delay Seen by an Arrival
- Abandonment and the Waits Hidden by Answered-Only Statistics
- Retries, Attempt Rates and Unique Jobs
- Finite Buffers, Blocking and Admitted Throughput
- Two Servers, One Pooled Queue or Separate Lines?
- When Batch Size and Job Duration Move Together
- Whole-Batch Rejection and the Difference Between Event and Job Loss Rates
- Finite-Source Repair Queues and State-Dependent Arrivals
Sources and scope
Karl Sigman’s Columbia single-server notes explain the waiting-work recursion. Mandelbaum and colleagues’ appointment-scheduling research develops scheduling under uncertain durations and punctuality. De Vuyst and colleagues’ university publication record documents finite-session waiting and idle-time evaluation. The four-job ledgers and practice variations here are original calculations.
Sources checked 22 October 2026. Return through The Gold Standard of Queueing Theory, The Gold Standard of Education and How X Works to practise matching an improved number with the experience it measures.