Arrival processes and model assumptions

An average of 100 requests per second can mean regular spacing or a synchronized burst every second. Queueing depends on that temporal structure. Poisson arrivals provide an analyzable baseline, not an automatic consequence of having many requests.

What Poisson arrivals assume

A homogeneous Poisson process has independent increments and a constant intensity $\lambda$. In an interval of length $\Delta$:

$$P\{A(\Delta )=n\}=e^{-\lambda \Delta }\frac{(\lambda \Delta {)}^n}{n!},E[A(\Delta )]=\mathrm{Var}(A(\Delta ))=\lambda \Delta .$$

Successive gaps $I$ are independent exponential random variables: $P(I>t)=e^{-\lambda t}$ and $E[I]=1/\lambda$. At $\lambda =2$/s, a one-second window has about a 13.5% chance of no arrivals. A mean gap of 0.5 s does not imply an arrival every 0.5 s.

Exponential memorylessness means:

$$P(I>s+t\mid I>s)=e^{-\lambda t}.$$

Having waited longer does not make the next arrival closer in this model. A deterministic interval behaves differently: elapsed time reveals how much remains.

Hold $\lambda$ fixed and widen the counting window. Count mean and variance increase, but the interarrival survival curve stays unchanged. Counts are discrete and gaps continuous; the charts represent different random variables.

Variability beyond the mean

$$C_a^2=\frac{\mathrm{Var}(I)}{E[I{]}^2},C_s^2=\frac{\mathrm{Var}(S)}{E[S{]}^2}.$$

Deterministic gaps have $C_a^2=0$ and exponential gaps have 1. Larger values indicate greater dispersion relative to the mean. These moments describe marginal distributions, not autocorrelation, periodic synchronization or sustained bursts. Reordering the same short and long gaps can preserve their mean and variance while changing waiting behavior.

Inspect rates by time segment, count variance at multiple time scales, gap distributions and dependence. Scheduled batches, daily cycles, cache expiration and synchronized retries can violate homogeneity or independence. Superposing independent sparse sources can sometimes approximate Poisson traffic; shared triggers undermine that rationale.

Kendall notation labels assumptions

$A/S/c/K$ specifies interarrival type, service type, server count and total system capacity. $M$ denotes memoryless exponential gaps or service, $D$ deterministic and $G$ general distributions. $GI$ explicitly denotes independent, identically distributed general intervals. Here an omitted $K$ means infinite capacity; scheduling and abandonment assumptions are stated separately.

PASTA says exogenous Poisson arrivals see the time-stationary state distribution just before arriving, under appropriate nonanticipation conditions. It lets a busy-time fraction become an arrival's probability of waiting. General arrivals lack this guarantee. PASTA also does not make successive waiting times independent.

Check your understanding

1. Exactly 100 requests arrive together at each integer second. Is this Poisson traffic at 100/s?

2. Does measuring $C_a^2\approx 1$ establish independent exponential gaps?

Further reading

The CMU modeling route emphasizes translating systems into models. The MIT lecture supplies queueing notation. Subsequent calculations retain these assumption labels.