Calculator Inputs
Enter arrival rate, service rate, and total finite system capacity.
Capacity K includes the customer in service and all waiting spaces.
Example Data Table
This table shows sample queue scenarios. Use it to compare light traffic,
balanced traffic, and overloaded finite capacity behavior.
| Scenario |
λ |
μ |
K |
ρ |
Expected Meaning |
| Small help desk |
8 |
10 |
12 |
0.80 |
Moderate waiting with low blocking. |
| Busy service window |
14 |
12 |
10 |
1.17 |
High blocking limits system growth. |
| Balanced kiosk |
9 |
9 |
8 |
1.00 |
Equal rates create uniform state probabilities. |
| Fast counter |
6 |
15 |
9 |
0.40 |
Short waits and high idle chance. |
How to Use This Calculator
- Enter the arrival rate λ for the chosen time unit.
- Enter the service rate μ using the same time unit.
- Enter K as the maximum number of customers allowed inside the system.
- Select the unit and decimal precision.
- Press the calculate button to view L, Lq, PK, W, and Wq.
- Use CSV or PDF export buttons to save the result.
Understanding Average Customers in an M/M/1/K Queue
An M/M/1/K queue is a finite waiting line model. It describes one service
channel with random arrivals and random service times. The letter K means
the system has a fixed customer limit. This limit includes the person being
served and all people waiting. When the system is full, new arrivals cannot
enter. That rejected arrival is called a blocked customer.
Why This Model Matters
Many real systems have limited space. A small clinic may have one counter
and only a few chairs. A call center route may have one active agent and
limited queue slots. A machine repair bay may accept only a fixed number
of jobs. In each case, the average number of customers helps planners
measure congestion before service quality falls.
Important Inputs
The arrival rate shows how often customers enter the system. The service
rate shows how fast the server completes work. Both rates must use the
same time unit. The capacity value K controls how many customers can exist
inside the system at once. A larger K may reduce blocked arrivals, but it
may increase waiting and crowding.
Reading the Result
The main result is L, the average number of customers in the system. This
includes customers waiting and the customer in service. Lq only counts
customers waiting in line. Blocking probability shows the chance that a new
arrival finds the system full. Effective arrival rate adjusts the original
arrival rate after blocked customers are removed.
Planning With the Output
A high blocking value suggests that the system rejects many customers. A
high waiting value suggests that accepted customers spend too long inside
the system. A high busy probability means the server is heavily loaded.
Use these outputs together. They help compare capacity changes, staffing
choices, and service speed improvements in a simple way.
Frequently Asked Questions
1. What does M/M/1/K mean?
M/M/1/K means Poisson arrivals, exponential service times, one server, and a finite system capacity of K customers.
2. Does K include the customer being served?
Yes. K includes everyone in the system. It counts the customer in service plus all customers waiting in line.
3. What is L in this calculator?
L is the average number of customers in the whole system. It includes both waiting customers and the customer receiving service.
4. What is Lq?
Lq is the average number of customers waiting in the queue. It excludes the customer currently being served.
5. What happens when ρ equals 1?
When ρ equals 1, all system states have equal probability. The average number in the system becomes K divided by 2.
6. Why is blocking probability important?
Blocking probability shows how often new arrivals are rejected because the system is full. It helps measure lost demand.
7. Can ρ be greater than 1?
Yes. In a finite queue, ρ can exceed 1. The system remains bounded because excess arrivals are blocked.
8. Why must rates use the same time unit?
Arrival and service rates must share one time unit. Mixing hours and minutes creates wrong traffic intensity and waiting values.