Basics of queuing theory applied to calculate average waiting time

What is a Queuing System? A flow of customers from a finite or infinite population towards your service facility forms a queue on account of lack of capability to serve them all at the same time. To manage this, you must have a queue system that consists of customers arriving for service, waiting for service if it is not immediate, and leaving the system after being served.

There is a complete explanation of queuing theory below, along with the characteristics, math and formulas that you need to calculate these factors. But if you simply want to calculate the average waiting time your customers are facing, make use of this M/M/1 queuing theory calculator below.

Elements of Queuing System

Input process

The pattern in which customers arrive in the system.

Size of the queue

The size of the input service is either finite or infinite.

Arrival distribution

The pattern in which the customers arrive at the service system or inter-arrival time. The most common stochastic models use Poisson distribution for arrival rate and/or exponential distribution for inter-arrival times.

Customer’s behaviour

Customers may wait in the queue regardless of its length, or they may leave upon observing the queue's length.

Queue Discipline

FIFO – First In First Out
FCFS – First Come First Served
LIFO – Last In First Out

Service Mechanism

Single queue and one server
Single queue and several servers
Several queues and one server
Several servers

Capacity of the Queuing System

A finite source limits the customers arriving for service, while an infinite source is forever abundant and unrestricted.

Operating Characteristics

E(n) or L

Expected number of customers in the system

E(m) or Lq

Expected number of customers in the Queue

Expected waiting line in the system

Expected waiting time in the queue

Service Utilisation Factor (P = λ/µ)

The proportion of time that a server actually spends with a customer where λ is the average number of customers arriving per unit of time and µ is the average number of customers completing service per unit time.

Probability Distributions

Models counting only arrivals are "pure birth" models. Queuing theory typically involves "birth-death" processes, as new customers increase arrivals while service completions decrease the count via departures.

Distribution of Arrivals

Let P_n(t) be the probability of n arrivals in a time interval of length t (n ≥ 0). Assuming a Poisson process with mean λt:

P_{n} (t) = \frac{{(λ t)}^{n} e^{- λ t}}{n!}, n \geq 0

Distribution of Inter-arrival Times

If the arrival process follows a Poisson distribution, the time between successive arrivals follows an exponential distribution:

f (t) = λ e^{- λ t}, t \geq 0

Distribution of Departures

For N customers at t=0, with departures at rate µ, the probability P_n(t) of n customers remaining is the Truncated Poisson distribution:

P_{n} (t) = \frac{{(μ t)}^{N - n} e^{- μ t}}{(N - n)!}, 1 \leq n \leq N

P_{0} (t) = 1 - \sum_{n = 1}^{N} P_{n} (t)

Distribution of Service Times

The probability for completing the service of n customers in time t follows the exponential distribution:

s (t) = μ e^{- μ t}, t \geq 0

Classification of Queuing Models

Models are typically described as (a / b / c) : (d / e)

Inter-arrival time distribution

Inter-service time distribution

Number of servers

System capacity

Queue discipline

Model 1: (M/M/1) : (∞/FIFO)

This model features a single server channel, Poisson input, exponential service, infinite capacity, and First-In-First-Out discipline.

Key Characteristics (ρ = λ/µ)

Avg. customers in system E(n)

\frac{λ}{μ - λ}

Avg. queue length E(m)

\frac{ρ^{2}}{1 - ρ}

Avg. waiting time E(w)

\frac{λ}{μ (μ - λ)}

Total time in system E(v)

\frac{1}{μ - λ}