Queues, variability and resource utilization

Why it matters

A queue can form in front of a resource even though its average load is below its capacity. If jobs do not arrive at regular intervals, some pile up at the same time. If processing times vary, one long job holds up those behind it. A later slack may make the average capacity look sufficient, but it does not make the earlier waiting not have happened. For this reason a fully loaded schedule prepared only with average times can produce long lead times in the real flow. A queue study does not aim to keep the worker busy every minute. It helps to understand how arrival, service, variability and resource room together show up in the flow; it stops local utilization from being the only success measure.

The approximation given in this lesson is for a flow with a single service resource, balanced over a sufficiently long period, with an arrival rate below the service rate. Jobs are processed in order of arrival; the resource does not move to another job before finishing one. The variability of the arrival intervals and of the service times is summarized. Multiple parallel resources, priority changes, batch holding, blocking by the next station or frequent interruptions are outside this simple scope. In such cases the approach does not directly give an exact result. First ask which flow the model represents. Having a formula at hand does not mean every production or administrative process has the same queue structure. The arrival intervals are taken as independent and identically distributed among themselves, and the service times likewise independent and identically distributed among themselves; the two series are taken as independent of each other. The means and variances must be finite. The queue space is assumed unlimited and no one leaves while waiting. This approximate expression is especially useful at high utilization; for general distributions it is not an exact equality or a delivery guarantee.

Utilization is the ratio of the average arrival rate to the average service rate. When arrivals increase while the service rate is held constant, the recovery room in front of the resource shrinks. Waiting does not rise linearly with utilization alone; in the given approximation, the ratio of utilization to the unused share grows. For this reason the effect of going from 80% to 95% cannot be brushed aside as a change of only 15 percentage points. Even if variability stays the same, waiting can grow several times over. Bringing the same average work in more evenly, or reducing unnecessary fluctuation in service time, can also be effective. Adding capacity is not the only option; the conditions of workable arrangements must be verified separately.

Little's law links work in process, output rate and time in the system under a common boundary and stable long-run averages. This relationship is not an equation that explains the cause of a queue; it makes three measures of the same flow consistent. Substituting waiting in the queue for total time in the system, or treating today's last stock as a long-run average, breaks the scope. In a temporary pile-up, first record the inflow, outflow and change in stock. Because jobs that started on different days may be inside the period, the average time of those completed may not on its own represent the time in the system of the whole system. The right equation does not by itself correct a wrong time and entity boundary.

Formula