How many servers does the queue actually need?
The multi-server queueing formula needs a sum of factorials, which is exactly why most people give up and copy numbers out of the table at the back of the book. A recursive helper column produces the same terms without a factorial function, so the model is live and editable rather than looked up.
Operations Advanced Data Table free
After you install, this is the model to open.
How Long Will the Line Get?
- In your spreadsheet, click the Sortia icon in the strip of icons down the right-hand edge. No strip? Click the arrow at the bottom-right to open it. You can also use Extensions, then Sortia, then Open Sortia.
- Click Start from a template and put that name in the search box.
- Pick the card with that name and click Load this template. It arrives on a new tab with real numbers already in it.
The answer
Six rows showing the wait collapsing and the cost curve turning back up:
- 3 servers
- 7.02 min wait, $236.01 an hour
- 4 servers
- 1.07 min wait, $127.99 an hour
- 5 servers
- 0.26 min wait, $135.87 an hour
- Cheapest
- 4 servers
Three servers run hot at 83.3% utilization, with 3.51 people in line and a 7.02 minute wait costing $236.01 an hour once waiting time is priced. Adding a fourth server cuts the wait by about 85%, to 1.07 minutes, and nearly halves the cost to $127.99. Adding a fifth cuts the wait again, to 0.26 minutes, but total cost turns back up to $135.87. So the cost curve has a genuine bottom at four rather than always wanting more staff, and where that bottom sits depends entirely on what you think a customer’s hour is worth, which is an editable cell rather than something buried in a formula. Two limits the sheet states plainly: it runs to fifteen servers, and below three there is no steady state at all, so the model reports that in words instead of returning a confident negative queue length.
The model
Arrivals and service rate set the offered load. A helper block builds each term of the queueing series from the one above it, which is what makes the multi-server case computable here at all. From that the sheet derives the probability the system is empty, the average queue length, the average wait, and the hourly cost of staffing plus customer waiting time.
| Arrivals | 30 an hour |
| Service rate | 12 an hour for each server |
| Offered load | 2.5, so one server cannot cope |
| Staff cost | $26 an hour a server, editable |
| Cost of waiting | $45 an hour a customer, editable |
| Servers on duty | swept from 3 to 8 |
| Ceiling | fifteen servers, set by the helper block |
Once it is in your sheet
- The model arrives with real numbers in it and runs as it stands, so you can press the button first and understand it second.
- Change the numbers to yours. The sheet marks which cells are inputs and which hold formulas, and most labels carry a note explaining the row.
- Press the run button at the bottom of the panel. It is labeled for the tool you are in, and the result lands on its own tab, with a written reading of it beside the figures.
Never used Google Sheets? Start here goes the whole way, in seven steps, and assumes nothing.
Next question
- Can you actually keep your wait-time promise?What Shape Are Your Wait Times?
- Does the night shift really produce less?Shift or Line: What Moves Output?
- Is one line underfilling, or is that just spread?Are Two Fill Lines Filling the Same?
- Two machines, same average. Which one wanders?Is the New Machine More Consistent?
- What is next month's demand, give or take?Smooth the Noise, See the Trend
- Is overselling worth the bumps at the gate?How Many Tickets Should You Oversell?
Every model like this one, and the method behind them: What-if analysis in Google Sheets.