How to use the single server queue calculator
- Enter the average number of arrivals per hour.
- Enter how many customers one server can finish per hour, which is 60 divided by the average minutes each one takes.
- Enter the variability of the service time. Use 1 for random times and 0 for a fixed time.
- Read the average wait in line, the time in the system and the number waiting.
Formula
Utilization = arrival rate / service rate
Wait in line = utilization x (1 + variability) / 2 x (average service time) / (1 - utilization)
Time in the system = wait in line + average service time
Customers waiting = arrival rate x wait in line
This is the Pollaczek and Khinchine formula for one server with random arrivals. It needs the arrival rate to be below the service rate.
8 arrivals an hour and 10 served an hour
The server is busy 80% of the time. With random service times, the average wait in line is 24.00 minutes, the time in the system is 30.00 minutes and 3.200 customers are waiting on average.
If every service takes exactly 6 minutes (variability 0) the wait in line is cut in half to 12.00 minutes and 1.600 customers are waiting, even though the average service time is the same. Highly uneven service does the opposite and lengthens the line.
Why the line gets long so fast
Waiting does not grow in a straight line with the load. As utilization rises toward 100%, the factor 1 / (1 - utilization) grows without bound, so a server at 90% busy has a line nine times longer than one at 50%, with the other inputs unchanged. A small rise in arrivals near capacity makes a large difference in the wait.
Variability matters as much as the average. Two counters with the same average service time can have very different lines if one has a mix of quick and very long jobs. Reducing the spread, by standardizing the work or handling long jobs separately, shortens the line without adding staff.
Limits of the model
The formula assumes arrivals come at random with a steady average, one line, and a server that never takes breaks. Real days have rushes and lulls, so run it for each busy period separately. For more than one server, the Erlang C Staffing Calculator is the matching model. The Little's Law Calculator links the numbers shown here, the Erlang B Blocking Calculator covers systems with no waiting room, and the Bartender Staffing Calculator applies the idea to a bar.
Assumptions and limits
- Arrivals are random and independent with a steady average rate.
- One server, one line, first come first served, and an unlimited waiting room.
- The service rate is higher than the arrival rate. Otherwise the line grows without limit.
- The figures shown are examples. Use measured rates for your own counter.
Frequently asked questions
What number do I use for the variability?
It is the variance of the service time divided by the square of its mean. Use 1 if you do not know, 0 for a fixed time, and above 1 for very uneven jobs.
Why does the tool refuse some inputs?
When arrivals are at or above the service rate the line grows forever, so there is no steady average to report.
Is the time in the system the same as the wait?
No. It adds the time being served to the time spent waiting in line.
Can I model a call center with it?
Only for a single agent. For a group of agents use a multi-server model such as Erlang C.