The queue is arithmetic

Waiting is not a service attitude, it is a ratio. On the sample desk 12,000 contacts a month arrive at 25.0 an hour against an 11.31-minute handling time, which needs six seats on average, ten in the busiest hour, and leaves one paid seat-hour in five with nothing being handled.

Open the partner account Start with the ceiling

Desk stub

Arrivals: 25.0 a staffed hour, 12,000 a month
Handling time: 11.31 minutes, weighted across channels
Offered load: 4.71 erlangs, held at an 80% target
Seats: 6 for the average hour, 10 for the peak
Idle: 618.0 paid seat-hours, 21.5% of the roster

the turnOne numbered serving of a contact, with its clock and what it cost the desk

the ceilingWhat a level may give away without asking, set by the evidence it can see

the transcriptThe only record that a contact happened, kept 24 months on this sample

Direct answerA queue is described by two numbers and one ratio: how many contacts arrive an hour, how long one takes, and how much of each seat’s time is being used. On the sample desk that is 25.0 arrivals an hour at 11.31 minutes each - 4.71 erlangs of work - which needs six seats to hold utilisation at 80%, ten in the evening peak, and leaves 618.0 paid seat-hours a month doing nothing.

Subject why a wait is a ratio rather than an attitudeUnit the seat - one staffed position for one hourStaffed hours 16 hours a day across 30 days = 480 hoursTarget used 80% utilisation, a staffing decision and not a natural constantBoundary the standards the desk publishes are on /the-standards

From contacts to seats

The whole derivation, so a reader can move any one number and see the queue change contacts in the month 12,000 staffed hours 480 16 h x 30 days arrivals a staffed hour 25.0 12,000 / 480 handling time per contact 11.31 min work offered = 25.0 x 11.31 / 60 4.71 erlangs seats to hold 80% utilisation 5.89 -> 6 seats roster bought: 6 x 480 2,880 seat-hours work actually handled 2,262.0 hours paid seat-hours with nothing in them 618.0 21.5% of the roster what those idle hours cost 12,978.00 618.0 x 21.00 utilisation achieved 78.5% 2,262.0 / 2,880 the peak: 40% of contacts in a 4-hour window 4,800 contacts over 120 hours 40.0/h work offered 40.0 x 11.31 / 60 7.54 erlangs seats at 80% 9.43 -> 10 seats the same 7.54 erlangs on 8 seats 94.3% and waiting grows

ratio 01

The average hour never happens

Six seats is what the month averages to. The busiest four hours of a day carry 40% of the month’s contacts on this sample, and that window needs ten seats at the same utilisation target. A roster is bought for the peak, which is why the average utilisation ends up below the target: the idle hours are the price of being staffed when everyone arrives at once.

That is also why a queue that looks over-staffed on paper can be under-staffed in the evening.

ratio 02

Waiting does not grow in a straight line

As utilisation approaches the capacity of the seats that exist, the time a contact waits stops rising gently and starts rising sharply, because every seat is occupied and the arrival pattern is random. Eight seats instead of ten for the same 7.54 erlangs is a utilisation of 94.3% and a materially different wait, which is why small staffing differences do not produce small service differences.

A desk’s published response target is really a statement about how much it is willing to pay for the shape of that curve.

ratio 03

What the player sees of all this

A queue position. “You are fourth in the queue” is computed from the seats free and the contacts ahead, and it moves backwards when several contacts arrive at once - which is the one part of the queue a reader experiences directly, and the part no published standard describes.

The record of the wait is in the transcript, which is why the arrival time is worth keeping.

What the numbers do not say

A ratio is not a service. Nothing here says a wait is acceptable, only where it comes from. The 80% target is a staffing decision, the handling time is a property of the work, and the arrival pattern is a property of the product’s schedule - three numbers, one wait.

Ask for the arrival time of the contact if a wait has to be documented.
Count the contacts rather than the minutes when comparing two waits, because arrivals are what the queue is made of.
Remember that a peak hour is a different queue from the average hour, even at the same desk.
Treat a published response target as a statement about staffing, and check whether it is a target or an obligation.

6 seats, 10 at peak 2,262.0 worked hours 12,978.00 of idle time when utilisation passes the curve arithmetic, not advice

What the desk promises

Targets, obligations and the tail.

What happens at the end

Abandoned and auto-closed contacts.

Which door is fastest

Four queues, four handling times.

The month in one picture

Every figure above, drawn once.