Utilization and L - Queueing I: Birth-Death Chains

The stationary distribution lets us compute mean queueing quantities exactly. Utilization measures how often the server is busy, while L and Lq measure average counts in the system and in the waiting line. These values come from the same exact birth-death model, so the metrics are consistent with the probabilities already computed.

highlighted = computed this step

Utilization

The utilization is 1/2. Why: rho is the long-run busy fraction for the server in this idealized queue. Interpretation: it is also the steady-state probability that the system is non-empty, because a single server is busy exactly when at least one job is present.

\rho=1/2

Mean number in system

The mean number in the system is 1. This includes the job in service, if there is one, plus any waiting jobs. Why: the geometric tail lets the expected count be summed exactly.

L=1

Mean number waiting

The mean number waiting in queue is 1/2. This excludes the job currently in service. Why: subtracting the busy server contribution leaves only the queue part, which is the congestion customers see before service starts.

L_q=1/2\quad L=L_q+\rho

Diagram note

The table shows utilization, mean system count, and mean queue count from the same recomputed queue. These are steady-state averages for the idealized model, not observations from measured operational data. Pixel positions are rounded for layout; every number shown is exact.

\text{mean counts are exact steady-state values}