The Birth-Death Chain - Queueing I: Birth-Death Chains

A birth-death chain is a state diagram for counts. In a queue, arrivals move the count upward and service completions move it downward, so adjacent states carry the whole transition story. The displayed chain is a finite window into an infinite queueing state space, and the tail remains part of the exact steady-state formulas.

highlighted = computed this step

Displayed state window

The diagram shows states 0 through 4. Why: the true queue has an infinite tail, so the graph is a finite display window. The omitted tail is not ignored by the formulas; the picture simply shows enough adjacent states to read the transition pattern.

\text{shown states }0\ldots4

Arrival moves forward

An arrival moves the chain up by one state at rate 1. This is the birth transition in the chain. Why: one more job enters the system, so the count moves to the next higher state whenever an arrival occurs.

\text{birth rate } \lambda=1

Service moves backward

A service completion moves the chain down by one state at rate 2. This is the death transition in the chain. Why: one job leaves the system, so the count moves to the next lower state whenever service completes.

\text{death rate } \mu=2

Diagram note

Forward edges are arrival transitions; backward edges are service transitions. The roles are part of the recomputed model, and the finite drawing is only a window into an infinite state space. Pixel positions are rounded for layout; every number shown is exact.

\text{birth-death roles are shown on the edges}