Queueing Theory Calculator
Analyze waiting lines and service systems using queueing theory
Average number of customers arriving per unit time (e.g., per hour)
Average number of customers served per unit time per server
Number of parallel servers (1 for single-server queue)
How to Use This Calculator
Enter Arrival Rate
Enter the average number of customers arriving per unit time (e.g., 10 customers per hour).
Enter Service Rate
Enter the average number of customers one server can serve per unit time (e.g., 12 customers per hour per server).
Enter Number of Servers
Enter the number of parallel servers (e.g., 1 for a single checkout lane, 3 for three tellers).
Review Results
The calculator shows utilization, average queue length, wait times, and other performance metrics.
Formulas (M/M/1 Queue)
Utilization: ρ = λ / (s × μ)
where λ = arrival rate, μ = service rate, s = number of servers
Average Queue Length (Lq):
Lq = ρ² / (1 - ρ)
For M/M/1 queue (single server)
Average Wait Time (Wq):
Wq = Lq / λ = ρ / (μ × (1 - ρ))
Average System Time (W):
W = Wq + 1/μ = 1 / (μ - λ)
Probability of No Customers (P0):
P0 = 1 - ρ
About Queueing Theory Calculator
Queueing theory is a branch of mathematics that studies waiting lines (queues) and is widely used in operations research, telecommunications, computer science, and service industries. This calculator implements the M/M/1 and M/M/s queueing models, which are fundamental models in queueing theory.
What is Queueing Theory?
Queueing theory analyzes systems where customers arrive, wait for service, get served, and leave. It helps answer questions like: How long will customers wait? How many customers will be in the system? What's the optimal number of servers?
M/M/1 Queue Model
The M/M/1 queue is the simplest queueing model:
- M (first): Markovian/Exponential inter-arrival times (Poisson arrivals)
- M (second): Markovian/Exponential service times
- 1: Single server
Applications
- Service Industries: Banks, call centers, restaurants, healthcare
- Manufacturing: Production lines, assembly systems
- Telecommunications: Network traffic, packet routing
- Computer Systems: CPU scheduling, web server queues
- Transportation: Traffic flow, airport security lines
Key Metrics
- Utilization (ρ): Percentage of time servers are busy (must be < 1 for stability)
- Queue Length: Average number of customers waiting
- Wait Time: Average time spent waiting before service
- System Time: Total time in system (waiting + service)
Frequently Asked Questions
What does M/M/1 mean?
The notation describes the queue: M = Markovian (exponential/Poisson), 1 = one server. So M/M/1 means Poisson arrivals, exponential service times, single server. M/M/s means s servers.
Why does the system need utilization < 1?
If arrival rate exceeds service capacity (utilization ≥ 1), the queue will grow indefinitely. The system becomes unstable - customers arrive faster than they can be served, so the queue never empties.
How accurate are these calculations?
These formulas are exact for M/M/1 queues with Poisson arrivals and exponential service times. Real systems may differ, but M/M/1 provides good approximations for many practical situations.
Can I use different time units?
Yes, as long as you use the same units for arrival and service rates. For example, if arrivals are per hour, service rate must also be per hour. The results will be in the same time units.
How do I improve queue performance?
You can: increase service rate (faster servers), add more servers, reduce arrival rate (discourage arrivals during peak), or implement priority queues. Lower utilization reduces wait times but may increase server idle time.