Queuing Theory Calculator
How to use Queuing Theory Calulator?

First, select units of time you want to analyze.

It can be any time units depends on the specific context and requirements.

Example: the time units is 1 hour and average customers per hour is 10 customers per hour, then the arrival rate (λ) is 10 customers/hour.

Select a queueing model base on the data and context.

M/M/c If you know average number of customers that be served per server.

Example: A cashier can serve 20 customers per hour on average, then the service rate (μ) is 20 customers/hour.

M/M/c/K Similar to M/M/c but if the system have limit maximum number of customers.

Example: A cafe only have 10 tables for customers, then K is 10.

M/D/1 If the service times are deterministic (constant). It can be estimate by Little’s law.

Example: A cashier will take 3 minues per service on average, then in 1 hour the service rate (μ) is 60/3 = 20 customers/hour.

c mean number of servers, if the number is dynamic recommend to use 1 or use simulation techniques instead.

What is Queuing Theory?

Queuing theory is a mathematical modeling tool used to analyze waiting lines, or queues. It is used to predict the length of queues, the amount of time customers spend waiting, and the number of servers needed to provide a given level of service.

Queuing Theory Use Cases

Queuing theory is a powerful tool that can be used to improve the efficiency and effectiveness of a wide variety of systems. By understanding how queues work, businesses can make better decisions about staffing, scheduling, and inventory management. This can lead to reduced waiting times for customers, improved customer satisfaction, and increased profits.

Examples of queuing theory:

  • Waiting in line at the grocery store. The grocery store is a classic example of a queuing system. Customers arrive at the store and wait in line to be checked out. The number of customers in line, the length of time they wait, and the number of checkout lanes open all affect the efficiency of the system.
  • Calling a customer service line. When you call a customer service line, you are joining a queue of other callers. The length of time you wait on hold depends on the number of callers in the queue, the number of customer service representatives available, and the average time it takes to handle a call.
Queuing Theory Models

Kendall’s notation is a standardized way to describe and classify queuing models in the field of queuing theory.

The notation is written in the form A/S/c/K/N/D, where:

  • A denotes the time distribution between arrivals to the queue. Common symbols include:

    • M: Markov or memoryless, i.e., exponential distribution or exponential time
    • D: Degenerate distribution, i.e., A deterministic or constant time
    • G: General distribution i.e., arbitrary distribution
  • S denotes the time distribution of the service rate. It uses the same symbols as A (M, D, G).

  • c denotes the number of servers in the system.

  • K (optional) denotes the capacity of the system, i.e., the maximum number of customers that can be in the system at the same time (including both in queue and in service). If not specified, it’s assumed to be infinite.

  • N (optional) denotes the population size, i.e., the total number of potential customers. If not specified, it’s assumed to be infinite.

  • Z (optional) denotes the queue discipline, the order in which customers are served. FIFO is the most common one.