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Choose a route or begin at Chapter 1. Teaching chapters stay open; the chapter index expands by section.
Define the objects
Chapters 1–2. Name the capacity and the admission rule
Build the argument
Chapters 3–5. Work through ten arrivals
Compare and interpret
Chapters 6–8. Calculate from a free-space distribution
Check and practise
Chapters 9–10. Audit a real measurement
Questions and next routes
Chapters 11–12. Questions readers ask
Full chapter index · Worked practice · Queueing Theory subject root
A service system can reject one quarter of arriving groups while rejecting more than two fifths of arriving jobs. Both statements can be correct. The denominator changes because larger groups carry more jobs and may be harder to fit into the remaining space.
This guide builds the distinction from a finite observation log, then derives a probability formula. It studies all-or-nothing admission into a finite-capacity queue. Numerical cases are original teaching examples; the example occupancy frequencies are stipulated observations, not a solved stationary distribution.
Chapter index
Define the objects · Chapters 1–2
1. Name the capacity and the admission rule
2. Keep an event ledger and a job ledger
Build the argument · Chapters 3–5
3. Work through ten arrivals
4. Derive the general probability relationship
5. Make the independence assumption visible
Compare and interpret · Chapters 6–8
6. Calculate from a free-space distribution
7. Separate fractions from rates
8. Change the policy to partial admission
Check and practise · Chapters 9–10
9. Audit a real measurement
10. Try a changed case and check the answer
Questions and next routes · Chapters 11–12
11. Questions readers ask
12. Sources and connected learning routes
Let K be the maximum number of jobs allowed in the system, including any job being served. Immediately before event i, let Nᵢ be the number present. An arriving group contains Bᵢ positive-integer jobs. Its free space is Fᵢ = K − Nᵢ.
Under whole-batch rejection, the entire batch enters when Bᵢ ≤ Fᵢ. Otherwise none enters. Define Rᵢ = 1 for rejection and zero for admission. The lost job count is then BᵢRᵢ, not just Rᵢ.
Write this rule before calculating. A capacity that counts only waiting positions requires a different occupancy definition. A policy that admits part of a batch changes the loss variable. Neither change is a small correction to the same calculation.
For m arrival events, the rejected-event fraction is ΣRᵢ/m. The lost-job fraction is ΣBᵢRᵢ/ΣBᵢ. These are exact finite-log identities. No Poisson process, exponential service time or independence assumption is needed to count them.
The first asks what fraction of arriving groups fail admission. The second asks what fraction of offered jobs are lost. A random event and a random offered job are different sampling procedures. A size-three batch appears once in the first procedure and contributes three opportunities to the second.
Keep offered, admitted and rejected jobs in separate columns. Their counts must reconcile: offered = admitted + rejected. A rejection counter alone cannot reconstruct lost jobs unless rejected batch sizes are also known.
Take K = 4 and the ten-event log in the table. The pre-arrival occupancies are observations. Between arrivals, enough service completions occur to make the displayed sequence possible; service timing is deliberately not modelled.
Events 3, 6 and 8 are rejected. Thus the event loss is 3/10 = 30%. They contain 2 + 4 + 3 = 9 jobs. All offered batches contain 22 jobs, so job loss is 9/22, approximately 40.91%.
Thirteen jobs enter. Checking 13 + 9 = 22 catches a common mistake: counting one lost unit for each rejected group. This is a group admission rule with job-level consequences.
| Event | N before arrival | B offered | Jobs admitted | Jobs lost |
|---|---|---|---|---|
| 1 | 0 | 2 | 2 | 0 |
| 2 | 1 | 1 | 1 | 0 |
| 3 | 3 | 2 | 0 | 2 |
| 4 | 2 | 2 | 2 | 0 |
| 5 | 1 | 3 | 3 | 0 |
| 6 | 2 | 4 | 0 | 4 |
| 7 | 0 | 3 | 3 | 0 |
| 8 | 2 | 3 | 0 | 3 |
| 9 | 1 | 1 | 1 | 0 |
| 10 | 0 | 1 | 1 | 0 |
CHAPTER 4 OF 12 · Build the argument
4. Derive the general probability relationship
For a stationary arrival-event distribution with finite positive E[B], let R be its rejection indicator. The corresponding event loss probability is P(R = 1). The job-weighted loss fraction is E[BR]/E[B]. Interpreting sample ratios as these expectations requires an appropriate law of large numbers, such as stationarity with ergodicity.
Expand E[BR] as E[B]E[R] + Cov(B,R). Therefore job loss minus event loss equals Cov(B,R)/E[B]. This equation identifies exactly what an event percentage leaves out.
If rejected events tend to be larger, the covariance is positive. The ten-event log illustrates this. It does not prove that every admission system must have positive covariance: batch size could also be related to how much free space is available.
Suppose the arriving batch size B is independent of the pre-arrival free space F. Then, for a given batch size b, rejection probability is P(F < b), a nondecreasing function of b. Larger batches cannot be easier to fit under this assumption.
The job loss fraction is Σ b pᵦ P(F < b) / E[B], while event loss is Σ pᵦ P(F < b). Here pᵦ is the probability of an event bringing b jobs. The weighting by b is the difference.
Independence is useful but must be justified. Appointments, advance reservations and adaptive dispatch can couple large groups to quiet periods. In such cases use the joint arrival distribution of B and F rather than multiplying separate marginals.
CHAPTER 6 OF 12 · Compare and interpret
6. Calculate from a free-space distribution
Consider a second teaching case: batches have size one or three, each with probability one half. At arrival events, F equals 0, 1 or 3 with probabilities 0.1, 0.3 and 0.6. Assume B and F are independent.
Size-one batches are rejected with probability 0.1. Size-three batches are rejected with probability 0.4. Event loss is 0.5×0.1 + 0.5×0.4 = 0.25. Expected lost jobs per event are 0.5×1×0.1 + 0.5×3×0.4 = 0.65.
Since E[B] = 2, job loss is 0.65/2 = 0.325. An event loss of 25% accompanies a job loss of 32.5%. These figures use the stated arrival distribution; they are not a prediction obtained from K and service rate alone.
| Batch size | Event probability | Rejection probability | Lost jobs per overall event |
|---|---|---|---|
| 1 | 0.5 | 0.1 | 0.05 |
| 3 | 0.5 | 0.4 | 0.60 |
| Total | 1 | Event loss 0.25 | 0.65 |
If the second case receives a long-run rate of 12 events per hour, it offers 12×2 = 24 jobs per hour. It loses 12×0.25 = 3 events per hour and 12×0.65 = 7.8 jobs per hour. The admitted job rate is 16.2 per hour.
Rates carry units. Fractions do not. Multiplying the job arrival rate by the event loss fraction would give 6 lost jobs per hour, which is incorrect here because it uses incompatible weighting.
The rate arithmetic does not require Poisson arrivals. A Poisson assumption can support additional sampling results, but an arrival-event occupancy distribution is not automatically the same as a time-sampled occupancy distribution for an arbitrary arrival process.
If a system fills remaining positions and rejects only the excess, lost jobs become max(B − F, 0). In the second example, size-one batches lose an expected 0.1 job per event of that size. Size-three batches lose 3×0.1 + 2×0.3 = 0.9.
The unconditional expected loss is 0.5×0.1 + 0.5×0.9 = 0.5 job per event. Dividing by E[B] gives 25% job loss when evaluated on the same stipulated B,F distribution.
This is a conditional comparison at the same arrival states. In a running queue, changing admission also changes subsequent occupancy. Do not use this calculation as the complete stationary performance comparison between two policies.
Record event time, batch identifier, offered size, pre-arrival occupancy, capacity definition, policy and admitted size. Retain events that are rejected before they enter the visible queue; otherwise the denominator quietly becomes accepted demand.
Check retries. If a rejected group returns, decide whether the report concerns attempts, unique groups or unique jobs. A retry can add another event without adding another person to the population being served.
Check observation boundaries and duplicate records. A partially observed batch must not be counted as complete. State whether priority classes, reservations or variable job sizes change the meaning of a capacity position.
Four events bring batches of sizes 1, 1, 4 and 4. Only the final size-four batch is rejected. Before reading further, calculate both loss fractions and explain which denominator each uses.
One of four events is lost, so event loss is 25%. Four of ten jobs are lost, so job loss is 40%. Six jobs enter. If these events occur over two hours, offered job rate is five per hour and lost job rate is two per hour.
Now reject the first size-one event instead. Event loss stays 25%, but job loss becomes 10%. This changed log shows why the batch identities matter. It does not assert that both logs arise from the same stationary queue model.
Does a 10% rejection rate mean one in ten jobs is lost? Only if the rate is defined over jobs, or if the batch-size weighting happens to make the two fractions equal. Ask for the denominator.
Can finite capacity make every offered load manageable? It bounds occupancy by discarding some offered work. Bounded queue length alone says nothing about acceptable access or loss.
Can I obtain rejection from mean utilisation? Not in general. Admission policy, batch distribution, service dynamics and arrival-state occupancy all matter. Two systems with similar averages can have different loss behaviour.
Should all rejected groups count equally? That depends on the question. Event and job measures answer different questions; report both when group-level experience and job-level throughput matter.
CHAPTER 12 OF 12 · Questions and next routes
12. Sources and connected learning routes
The linked research paper establishes complete rejection as a recognised finite-buffer batch-arrival model. The counting identities, covariance argument and numerical examples here are derived explicitly and do not use that paper’s approximation.
Continue with the batch-arrival guide for the distinction between event and job rates, then the batch-duration guide when jobs in large groups have different service requirements. Return to the subject root for service variability, system boundaries and broader applications.
To transfer the method, ask a learner to rebuild the two ledgers for parcels in consignments or tasks in submitted bundles. Preserve the counting unit before moving a formula into the new setting.
References checked 22 October 2026
Complete rejection in finite-buffer batch queues
Continue through Queueing Theory
Batch arrivals, event rates and job counts
When batch size and job duration move together
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