The Stationary Distribution - Queueing I: Birth-Death Chains

The balance recurrence gives a geometric tail. Normalization fixes the first probability, and the rest follow by repeated multiplication by the traffic ratio. The table shows a finite prefix of an infinite stationary distribution, making the exact arithmetic visible without pretending the queue has only the displayed states.

highlighted = computed this step

Normalize the tail

The first probability is 1/2. Why: total probability is one, so the geometric tail starts at one minus rho. Interpretation: this is the empty-system probability, the chance that no job is present in steady state.

P_{\text{zero}}=1/2

Geometric probabilities

The displayed probabilities are 1/2, 1/4, 1/8, and 1/16. Why: each step multiplies by rho. The distribution is geometric, so the same traffic ratio carries the probability from one state to the next.

P_n=\rho^n(1-\rho)

Finite display window

The table stops after the fourth shown probability. Why: the model has an infinite tail, but the lesson displays only an exact finite prefix. The unshown probabilities continue by the same recurrence and are still part of the normalized steady state.

\text{shown probabilities are a prefix of the tail}

Diagram note

Every probability in the table is recomputed from rho and normalized exactly. The table is a finite prefix of an infinite geometric distribution, so the stopping point is a display choice rather than a mathematical cutoff. Pixel positions are rounded for layout; every number shown is exact.

\text{stationary probabilities are exact fractions}