# Queue model: primary sources and interpretation

Reviewed 30 October 2026. Scope: the fixed-capacity queue explanation in Volume IV, `theories/congestion-is-variability.html`, and the proposed independent queue calculation. No traffic data or experiments were run for this source check.

**Finding:** publishable as a limited mathematical/computational check. Distinguish larger temporary queues from instability. The proposed model can challenge the claim that variance alone changes its stability threshold; it cannot settle urban traffic breakdown or policy cost effectiveness.

## Three primary references

1. **R. M. Loynes, “The stability of a queue with non-independent inter-arrival and service times.” July 1962**, *Mathematical Proceedings of the Cambridge Philosophical Society* 58(3), 497–520. [Original publisher introduction and metadata](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/stability-of-a-queue-with-nonindependent-interarrival-and-service-times/7BDE17CC3B08D8DF0E1641E85ED2FA6F).

   The introduction explicitly classifies its single-server queues as subcritical/stable and supercritical/unstable. Its critical class has multiple possible behaviors. Tiny quote: “Subcritical: stable (Theorem 3).” The original structural assumptions are referred to §2.1, outside the accessible introduction. Therefore cite it for this classification, and use reference 2's explicit independent-input assumptions for our model. The publisher's 2008 online date is digitization, not the paper's publication year.

2. **Peter W. Glynn and Assaf Zeevi, “Bounding Stationary Expectations of Markov Processes.” 2008**, *IMS Collections* 4, 195–214, DOI 10.1214/074921708000000381. [Author-hosted paper, §5.1, printed pp.208–209](https://web.stanford.edu/~glynn/papers/2008/GZeevi08.pdf).

   This primary paper explicitly uses the reflected recursion `X[n+1] = max(0, X[n] + Z[n+1])` with independent, identically distributed increments. Strict negative mean increment gives a unique stationary distribution. With a finite second moment it derives the Kingman upper bound `E_stationary[X] <= E[Z²] / (2 |E[Z]|)`. For our bounded independent batch arrivals, `Z=A−1`, so these conditions hold when `E[A]<1`. The bound shows why variability can increase delay while mean spare capacity remains relevant. It is an upper bound, not an exact formula or proof of monotonicity for every distributional comparison. Use the explicit strict-negative-drift statement; an earlier line has an apparent `<=`/`<` inconsistency.

3. **J. F. C. Kingman, “On Queues in Heavy Traffic.” July 1962**, *Journal of the Royal Statistical Society: Series B* 24(2), 383–392. [Original publisher summary](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1962.tb00465.x).

   This is a heavy-traffic limiting result for equilibrium waiting times with load below, but approaching, one, under additional regularity conditions. It supports discussing substantial delay in a stable queue near capacity. It does not establish a variance-only instability threshold. Do not apply a continuous-time approximation as an exact numerical prediction for our discrete batch model.

## Exact example: independent mathematical check

Specify an infinite-buffer, work-conserving queue, one unit of service per slot, initially empty, with post-service measurement:

`Q[t+1] = max(0, Q[t] + A[t] − 1)`.

Constant fluid input `A=.8` has post-service queue zero. Periodic input `[4,0,0,0,0]` has the same mean `.8`, variance `2.56` across the five-slot period, and repeating post-service queues `[3,2,1,0,0]`. Its peak before service is four; its peak after service is three. It drains within each period and remains bounded. These follow directly from the recurrence, without a stochastic theorem or simulation. Fractional input is an amount of work, not a literal fraction of a vehicle.

## Limits and wording for publication

- Define stability as a proper stationary workload distribution for the random-input model; do not equate it with every sample path being bounded, finite mean delay under arbitrary heavy-tailed/dependent input, or clearing every slot. The periodic example is separately pathwise bounded.
- Bounded independent batch arrivals have finite moments. For load above one, cumulative arrivals exceed cumulative potential service asymptotically. Finite simulations illustrate this; they do not prove stationarity below one.
- For Bernoulli batches of size `b`, probability `p=rho/b` must be at most one. Size one at load `1.05` is infeasible and must be omitted or explicitly marked.
- Hold service capacity fixed. Do not infer results for spillback, capacity drop, finite road storage, interacting intersections, nonstationary demand, breakdown probabilities, or equivalent-cost policy comparisons.
- A stable bursty example refutes a universal “bursts must destabilize an underloaded queue” explanation. It does not refute the existential possibility of real roads with matched means and different breakdown risks.

Suggested conclusion: **“In this fixed-service queue, burstier arrivals created temporary backlog without changing the load-based stability condition. This narrows the proposed mechanism; road-network predictions and the relative value of smoothing versus added capacity remain untested.”**