- History
- Model and elements
- -Elements
- Source of entry or potential population
- The tail
- -The tail system
- - Service mechanism
- -Client
- -Queue capacity
- -Discipline of the queue
- Models
- Types of queuing systems
- Terminology
- What is the theory for
- Elements present in formulas
- Examples
- Part A
- Part b
- References
The queuing theory is the branch of mathematics that studies the phenomena and behaviors in waiting lines. They are defined when a user who demands a certain service decides to wait for the server to be processed.
Study the elements that are present in the waiting lines of any type, whether they are human elements, or data processing or operations. His conclusions are of constant application in production, registration and processing lines.
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Its values serve in the parametrization of processes prior to their implementation, serving as a key organizational element for the correct planning management.
History
The main responsible for its development was the Danish-born mathematician Agner Kramp Erlang, who worked at the Copenhagen telephone Exchange telecommunications company.
Agner noted the growing needs emerging in the company's telephone service delivery system. That is why the study of mathematical phenomena that could be quantified in the waiting line system began.
His first official publication was an article entitled Queuing Theory, which was published in 1909. His focus was mainly directed to the problem of sizing lines and telephone switching centers for call service.
Model and elements
There are different models of queues where some aspects are responsible for defining and characterizing each of them. Before defining the models, the elements that make up every queue model are presented.
-Elements
Source of entry or potential population
It is the set of possible applicants for the service. This applies to any type of variable, from human users to data packet sets. They are classified into finite and infinite depending on the nature of the set.
The tail
It refers to the set of elements that are already part of the service system. Which have already agreed to wait for the availability of the operator. They are in a state of waiting for system resolutions.
-The tail system
It is made up of the triad formed by the queue, the service mechanism and the discipline of the queue. It gives structure to the system protocol, governing the selection criteria for elements in the queue.
- Service mechanism
It is the process by which the service is provided to each user.
-Client
It is any element belonging to the potential population that demands a service. It is important to know the rate of entry of clients, as well as the probability that the source has of generating them.
-Queue capacity
It refers to the maximum capacity of items that can be waiting to be served. It can be considered finite or infinite, being in most cases infinite by criteria of practicality.
-Discipline of the queue
It is the protocol by which the order in which the customer is served is determined. It serves as a processing and order channel for users, being responsible for their disposition and movement within the queue. According to your criteria, it can be of different types.
- FIFO: From the acronym in English First in first out, also known as FCFS first come first served. Which means, respectively, First in first out and First in first to be served. Both forms denote that the first customer to arrive will be the first to be served.
- LIFO: Last in first out also known as stack or LCFS last come first served. Where the customer who has arrived last is served first.
- RSS: Random selection of service also called SIRO service in random order, where customers are selected according to random or random criteria.
Models
There are 3 aspects that govern the queuing model to consider. These are the following:
- Distribution of time between arrivals: refers to the rate at which units are added to the queue. They are functional values and are subject to different variables depending on their nature.
- Distribution of service time: time used by the server to process the service requested by the client. It varies according to the number of operations or procedures that are established.
These 2 aspects can take the following values:
M: exponential exponential distribution (Markoviana).
D: Degenerate distribution (constant times).
E k: Erlang distribution with shape parameter k.
G: General distribution (any distribution).
- Number of servers: Service gates open and available to process clients. They are essential in the structural definition of each queuing model.
In this way, the queuing models are defined, first taking the initials in capital letters of the arrival time distribution and the service time distribution. Finally, the number of servers is studied.
A fairly common example is MM 1, which refers to an exponential type of arrival and service time distribution, while working with a single server.
Other types of queue models are MM s, MG 1, ME 1, DM 1, among others.
Types of queuing systems
There are several types of queuing systems where multiple variables serve as indicators of the type of system presented. But fundamentally it is governed by the number of queues and the number of servers. The linear structure to which the user is subjected to obtain the service also applies.
- A queue and a server. It is the usual structure, where the user through the arrival system enters the queue, where after completing his wait according to the discipline of the queue, and is processed by the only server.
- One queue and multiple servers. The user, at the end of her waiting time, can go to different servers that can be executors of the same processes, as well as they can be private for different procedures.
- Multiple queues and multiple servers. The structure can be divided for different processes or serve as a wide channel to cover a high demand for common service.
- A queue with sequential servers. Users go through different stages. They enter and take a place in the queue, and when they are served by the first server, they pass to a new stage that requires previous fulfillments made in the first service.
Terminology
- λ: This symbol (Lambda) represents in queuing theory the expected value of inputs per time interval.
- 1 / λ: Corresponds to the expected value between the arrival times of each user who enters the system.
- μ: The symbol Mu corresponds to the expected number of clients that complete the service per unit of time. This applies to every server.
- 1 / μ: Service time expected by the system.
- ρ: The symbol Rho denotes the utilization factor of the server. It is used to measure how much time the server will be busy processing users.
ρ = λ / sμ
If p> 1, the system will be transitory, it will tend to grow, because the server's utility rate is below the number of users entering the system.
If p <1 the system will remain stable.
What is the theory for
It was created in order to optimize the telephone service provision processes. This demarcates a utility with respect to the phenomena of waiting lines, where it is sought to reduce time values and cancel any type of rework or redundant process that slows down the process of users and operators.
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At more complex levels, where the input and service variables take mixed values, calculations performed outside of queuing theory are almost unthinkable. The formulas provided by the theory opened up advanced calculus within this branch.
Elements present in formulas
- Pn: Value referring to the probability that “n” units are within the system.
- Lq: Length of the queue or average value of users in it.
- Ls: Average of units in the system.
- Wq: Average waiting rate in the queue.
- Ws: Average waiting rate in the system.
- _λ: Average number of clients that enter the service.
- Ws (t): Value referring to the probability that a customer remains more than “t” units in the system.
- Wq (t): Value referring to the probability that a customer remains more than “t” units in the queue.
Examples
A registry has a single server to process the passports of the users who come. An average of 35 users per hour attend the registry. The server has the capacity to serve 45 users per hour. It is previously known that users spend an average of 5 minutes in the queue.
You want to know:
- Average time each user spends on the system
- Average number of customers in the queue
We have λ = 35/45 Customers / minutes
μ = 45/60 clients / minutes
Wq = 5 minutes
Part A
Average time in the system can be calculated with Ws
Ws = Wq + 1 / μ = 5 minutes + 1.33 = 6.33 minutes
In this way, the total time that the user will be in the system is defined, where 5 minutes will be in the queue and 1.33 minutes with the server.
Part b
Lq = λ x Wq
Lq = (0.78 clients minutes) x (5 minutes) = 3.89 clients
There can be more than 3 clients in the queue simultaneously.
References
- Operations management. Editorial Vértice, Apr 16. 2007
- Theory of queues or waiting line. Germán Alberto Córdoba Barahona. Pontificia Universidad Javeriana, 2002
- Systems theory solved problems. Roberto Sanchis Llopis. Publications of the Universitat Jaume I, 2002
- Quantitative methods of industrial organization II. Joan Baptista Fonollosa Guardiet, José María Sallán Laws, Albert Suñé Torrents. Univ. Politèc. from Catalunya, 2009
- Inventory theory and its application. Editorial Pax-México, 1967