The M/M/1 Queue - Queueing I: Birth-Death Chains

A single-server memoryless queue tracks the number of jobs in the system. Arrivals increase that count, and service completions decrease it, so the whole model can be drawn as a birth-death chain. The lesson starts with the rates because every later probability and mean is bookkeeping built from those two event roles.

highlighted = computed this step

Single-server model

The arrival rate is 1 and the service rate is 2. These are the two clocks that move the model. An arrival is a birth because it increases the count, and a service completion is a death because it decreases the count. Why: the queue state changes only when one of those events happens.

\lambda=1\quad \mu=2

State is the count

The state is the number of jobs in the system. It includes any job in service and any jobs waiting. Why: for this idealized memoryless queue, that count is enough to determine the next birth or death rate, so no extra history is needed.

\text{state}=\text{number in system}

Service is faster

The service rate is larger than the arrival rate. That inequality is the stability gate: service must outpace incoming work on average. Why: otherwise the count has no steady-state balance and the queue drifts upward instead of settling.

\lambda<\mu

Diagram note

The graph labels the forward arrival role and the backward service role; rates are recomputed from the queue parameters. It is a steady-state model diagram, not a transient sample path of individual customers moving through time. Pixel positions are rounded for layout; every number shown is exact.

\text{chain edges are exact roles from the model}