HesapHanem

M/G/1 Queue Calculator

Enter arrival rate, service rate and the service-time coefficient of variation.

Result

2.0000 jobs
0.8000
0.2500
0.3500
2.8000 jobs

How it is calculated

The tool applies the Pollaczek–Khinchine formula for one server with random arrivals and any service-time distribution. ρ = λ ÷ μ, Lq = ρ² × (1 + Cs²) ÷ (2 × (1 − ρ)), Wq = Lq ÷ λ, W = Wq + 1 ÷ μ and L = λ × W. The arrival rate λ and service rate μ must be in the same time unit, such as jobs per hour, and times come out in that unit. Cs is the coefficient of variation of the service time, its standard deviation divided by its mean, entered as Cs and not squared. Cs = 0 means perfectly steady service and Cs = 1 means exponential service, which gives the same answers as M/M/1. Results show four decimals. If λ is not below μ, or λ or μ is zero or less, nothing is shown.

Worked example

Jobs arrive at 8 per hour, one server completes 10 per hour, and service time has Cs = 0.5. Utilization: 8 ÷ 10 = 0.8000. Queue length: 0.64 × 1.25 ÷ (2 × 0.2) = 2.0000 jobs. Wait in queue: 2 ÷ 8 = 0.2500 hours. Time in system: 0.25 + 1 ÷ 10 = 0.3500 hours. Jobs in system: 8 × 0.35 = 2.8000 jobs. With Cs = 1 the queue would be 3.2 jobs.

Try this example →

Common mistakes

  • Entering the variance or Cs² in the Cs field. The tool squares the value itself, so give the standard deviation divided by the mean.
  • Mixing time units, such as arrivals per hour and service per minute. Both rates must use the same unit.
  • Entering the average service time as μ. If a job takes 6 minutes on average, μ is 10 per hour.
  • Using Cs = 1 by habit. That assumes exponential service times; if your service times are fairly steady, a smaller Cs gives a shorter and more accurate queue.
  • Expecting a result when utilization reaches 100%. The queue is unstable and the tool shows nothing.

About this calculator

The M/G/1 queue calculator models a single-server queue with Poisson (memoryless, “M”) arrivals but a general (“G”) service-time distribution — meaning service times don't have to be exponential, unlike the simpler M/M/1 model. It applies the Pollaczek–Khinchine formula, Wq = ρ(1 + Cs²) / (2μ(1 − ρ)), where ρ = λ/μ is utilization, λ is the arrival rate, μ is the service rate, and Cs² is the squared coefficient of variation of service time, to compute expected waiting time, queue length, and utilization.

The key insight the P–K formula captures is that variability in service time — not just its average — drives congestion: for a fixed utilization, a highly variable service process (Cs² > 1) produces much longer queues than a consistent one (Cs² close to 0), which is why call centers and computer networks with unpredictable job sizes need this model rather than the simpler M/M/1. Operations researchers, network engineers, and call-center capacity planners use this calculator to predict wait times from real, non-exponential service-time data.

Was this helpful?

You can pick up to 3 reactions.

Comments (1)

  • Editorial team

    Two things are easy to see by trying values. With a service rate of 10, moving the arrival rate from 8 to 9 jobs pushes the queue at Cs of 1 from 3.2 to 8.1 jobs, so delays grow sharply near full load. And making service times perfectly steady (Cs of 0) halves the queue at any load. Times come out in the same unit as your rates. What are you modelling, a help desk, a printer or a checkout?

Popular calculators

Loan Calculator

Monthly payment, total interest and full amortization schedule.

Percentage Calculator

Percent change, percent of, and ratio calculations.

Discount Calculator

Calculate final price, savings, and discount amount.

BMI Calculator

Body mass index from height and weight (metric or imperial).

Age Calculator

Compute your exact age in years, months and days from a birth date.

Compound Interest Calculator

Calculate the final amount and interest earned with compounding.

All Calculators