OPS.10.1:4.4 - Construct a continuing-regime mean when that is the question

First establish the modeled regime: arrival process, service order, number of servers, availability, initial transients, return behavior and relevant dependence. Long-run parameters do not describe an arbitrary finite overload merely because their units fit.

For one continuously available first-come server, consider independent, identically distributed interarrival intervals and independent, identically distributed service times, with the two sequences independent. Let lambda be the arrival rate, E[S] the mean service duration and rho = lambda * E[S]. With finite second moments and rho < 1, a useful two-moment approximation is:

mean queue wait ≈ ((ca² + cs²) / 2) * rho / (1-rho) * E[S]
mean residence  ≈ mean queue wait + E[S]

Here ca² is interarrival variance divided by squared mean interarrival time; cs² is service-time variance divided by squared mean service time. The factor rho/(1-rho) retains the sharp rise near saturation. Both arrival and service variation matter.

Use this as a qualified mean calculation under those premises. In the Poisson-arrival case, the displayed mean wait equals the established single-server result for a general independent service distribution with finite second moment. Section 5.1 uses that special case. Neither use supplies a wait percentile.

Temporal dependence can defeat a description consisting of two moments. Count arrivals in windows at time scales relevant to the queue; examine whether their variance relative to mean changes with window length. Retain that dependence in a richer queue model or in simulated arrival paths when it can change the choice. Dispersion-based queueing analysis supplies one such mean-performance alternative; it retains its own stationarity and service assumptions.

Do not substitute the one-server formula after pooling servers, adding priorities, introducing blocking or changing the dependence. Reconstruct the model of service. A deterministic arrival every hour and a one-hour uninterrupted service can operate continuously without a queue; its feasible timing is not invalidated by a stochastic formula singular at rho = 1.