The Stationary Distribution

Balance Equations

Stationary probabilities are not guessed from a simulation. They come from exact flow balance between neighboring states: in steady state, the upward probability flow and downward probability flow match across every adjacent cut. For this queue, that balance says each next probability is a fixed fraction of the current one.

highlighted = computed this step

Balance across a cut

At steady state, arrival flow across a cut equals service flow back. Think of the cut between adjacent states: probability mass moves upward by arrivals and downward by service completions. Why: if those flows did not balance, the stationary probability on one side would drift instead of staying steady.

λPn=μPnext\lambda P_n=\mu P_{\text{next}}
Balance relationExact rates imply the probability ratio.lambda mu rho P0 P1lambdamurhoP0P1121/21/21/4

Probability ratio

The traffic ratio is 1/2. This ratio is also the utilization: the long-run busy fraction for the server and the probability the system is non-empty in this idealized model. Why: dividing the arrival rate by the service rate gives the multiplier from one probability to the next.

ρ=λ/μ=1/2\rho=\lambda/\mu=1/2
Balance relationExact rates imply the probability ratio.lambda mu rho P0 P1lambdamurhoP0P1121/21/21/4

First balance step

The first two probabilities shown are 1/2 and 1/4. Why: the next probability equals rho times the current probability. Interpretation: each step farther into the tail is less likely here because service outpaces arrivals.

Pnext=ρPnP_{\text{next}}=\rho P_n
Balance relationExact rates imply the probability ratio.lambda mu rho P0 P1lambdamurhoP0P1121/21/21/4

Diagram note

The table entries are the recomputed rates, ratio, and first two stationary probabilities. They describe the steady-state distribution, not a simulated trajectory or a forecast of the next customer event. Pixel positions are rounded for layout; every number shown is exact.

balance produces a geometric ratio\text{balance produces a geometric ratio}
Balance relationExact rates imply the probability ratio.lambda mu rho P0 P1lambdamurhoP0P1121/21/21/4