Queueing Fundamentals. A basic queueing system is a service system where âcustomersâ arrive to a bank of âserversâ and require some service from one of them. Itâs important to understand that a âcustomerâ is whatever entity is waiting for service and does not have to be a person. Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. 1 Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service. Probability Density Function (pdf): Cumulative Distribution Function (cdf): Mean: a Variance: a2 Coefficient of Variation = (Std Deviation)/mean = 1 Memoryless: Expected time to the next arrival is always a regardless of the time since the last arrival Remembering the past history does not help. F(x)= 1 a eâx/a F(x)=P(X.
Introduction Queuing theory is a branch of mathematics that studies and models the act of waiting in lines. This paper will take a brief look into the formulation of queuing theory along with examples of the models and applications of their use. In queueing theory these interarrival times are usually assumed to be independent and identicallydistributedrandomvariables.Theotherrandomvariableistheservicetime, sometimesitiscalledservicerequest,work.ItsdistributionfunctionisdenotedbyB(x), thatis B(x) = P( servicetime. Queuing Theory Queuing theory, the mathematical study of waiting in lines, is a branch of operations research because the results often are used when making business decisions about the resources needed to provide service.
Queuing theory is the mathematical study of queuing, or waiting in lines. Queues contain customers (or âitemsâ) such as people, objects, or information. Queues form when there are limited resources for providing a service. For example, if there are 5 cash registers in a grocery store, queues will form if more than 5 customers wish to pay for their items at the same time.
A basic queuing system consists of an arrival process (how customers arrive at the queue, how many customers are present in total), the queue itself, the service process for attending to those customers, and departures from the system.
Mathematical queuing models are often used in software and business to determine the best way of using limited resources. Queueing models can answer questions such as: What is the probability that a customer will wait 10 minutes in line? What is the average waiting time per customer?
The following situations are examples of how queueing theory can be applied:
Characterizing a Queuing System
Queuing models analyze how customers (including people, objects, and information) receive a service. A queuing system contains:
Mathematics of Queuing Theory
Kendallâs notation is a shorthand notation that specifies the parameters of a basic queuing model. Kendallâs notation is written in the form A/S/c/B/N/D, where each of the letters stand for different parameters.
Littleâs law, which was first proven by mathematician John Little, states that the average number of items in a queue can be calculated by multiplying the average rate at which the items arrive in the system by the average amount of time they spend in it.
Although Littleâs law only needs three inputs, it is quite general and can be applied to many queuing systems, regardless of the types of items in the queue or the way items are processed in the queue. Littleâs law can be useful in analyzing how a queue has performed over some time, or to quickly gauge how a queue is currently performing.
For example: a shoebox company wants to figure out the average number of shoeboxes that are stored in a warehouse. The company knows that the average arrival rate of the boxes into the warehouse is 1,000 shoeboxes/year, and that the average time they spend in the warehouse is about 3 months, or ¼ of a year. Thus, the average number of shoeboxes in the warehouse is given by (1000 shoeboxes/year) x (¼ year), or 250 shoeboxes.
Queuing Theory FormulaKey Takeaways
Sources
Queue networks are systems in which single queues are connected by a routing network. In this image, servers are represented by circles, queues by a series of rectangles and the routing network by arrows. In the study of queue networks one typically tries to obtain the equilibrium distribution of the network, although in many applications the study of the transient state is fundamental.
Queueing theory is the mathematical study of waiting lines, or queues.[1] A queueing model is constructed so that queue lengths and waiting time can be predicted.[1] Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.
Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange.[1] The ideas have since seen applications including telecommunication, traffic engineering, computing[2]and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management.[3][4]
Spelling[edit]
The spelling 'queueing' over 'queuing' is typically encountered in the academic research field. In fact, one of the flagship journals of the profession is named Queueing Systems.
'Queueing' is, incidentally, the only English word having more than four consecutive vowels.
Single queueing nodes[edit]
A queue, or 'queueing node' can be thought of as nearly a black box. Jobs or 'customers' arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue (see Fig. 1).
Fig. 1. A black box. Jobs arrive to, and depart from, the queue.
The queueing node is not quite a pure black box, however, since there is some information we need to specify about the inside of the queuing node. The queue has one or more 'servers' which can each be paired with an arriving job until it departs, after which that server will be free to be paired with another arriving job (see Fig. 2).
Fig. 2. A queueing node with 3 servers. Server a is idle, and thus an arrival is given to it to process. Server b is currently busy and will take some time before it can complete service of its job. Server c has just completed service of a job and thus will be next to receive an arriving job.
An analogy often used is that of the cashier at a supermarket. There are other models, but this is one commonly encountered in the literature. Customers arrive, are processed by the cashier, and depart. Each cashier processes one customer at a time, and hence this is a queueing node with only one server. A setting, where a customer will leave immediately, when in arriving he finds the cashier busy, is called a queue with no buffer (or no 'waiting area', or similar terms). A setting with a waiting zone for up to n customers is called a queue with a buffer of size n.
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The behaviour / state of a single queue (also called a 'queueing node') can be described by a Birth-death process, which describe the arrivals and departures from the queue, along with the number of jobs (also called 'customers' or 'requests', or any number of other things, depending on the field) currently in the system. An arrival increases the number of jobs by 1, and a departure (a job completing its service) decreases k by 1 (see Fig. 3).
Fig. 3. Birth / death process. The values in the circles represent the state of the birth-death process, k. The system transitions between values of k by 'births' and 'deaths' which occur at rates given by various values of λi and μi, respectively. For a queueing system, k is the number of jobs in the system (either being serviced or waiting if the queue has a buffer of waiting jobs). Further, for a queue, the arrival rates and departure rates are generally considered not to vary with the number of jobs in the queue so that we consider a single average rate of arrivals/departures per unit time to the queue. Hence, for a queue, this diagram has arrival rates of λ = λ1, λ2, .., λk and departure rates of μ = μ1, μ2, .., μk (see Fig. 4).
Fig. 4. A queue with 1 server, arrival rate λ and departure rate μ.
Single queueing nodes are usually described using Kendall's notation in the form A/S/c where A describes the distribution of durations between each arrival to the queue, S the distribution of service times for jobs and c the number of servers at the node.[5][6] For an example of the notation, the M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process (inter-arrival durations are exponentially distributed) and have exponentially distributed service times. In an M/G/1 queue, the G stands for general and indicates an arbitrary probability distribution for service times.
Overview of the development of the theory[edit]
In 1909, Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory.[7][8][9] He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920.[10] In Kendall's notation:
If there are more jobs at the node than there are servers, then jobs will queue and wait for service
The M/G/1 queue was solved by Felix Pollaczek in 1930,[11] a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the PollaczekâKhinchine formula.[10][12]
After the 1940s queueing theory became an area of research interest to mathematicians.[12] In 1953 David George Kendall solved the GI/M/k queue[13] and introduced the modern notation for queues, now known as Kendall's notation. In 1957 Pollaczek studied the GI/G/1 using an integral equation.[14]John Kingman gave a formula for the mean waiting time in a G/G/1 queue: Kingman's formula.[15]
Leonard Kleinrock worked on the application of queueing theory to message switching and packet switching. His initial contribution to this field was his doctoral thesis at the Massachusetts Institute of Technology in 1962, published in book form in 1964 in the field of digital message switching. His theoretical work after 1967 underpinned the use of packing switching in the ARPANET, a forerunner to the Internet.
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The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered.[16]
Problems such as performance metrics for the M/G/k queue remain an open problem.[10][12]
Service disciplines[edit]
First in first out (FIFO) queue example.
Various scheduling policies can be used at queuing nodes:
Arriving customers not served (either due to the queue having no buffer, or due to balking or reneging by the customer) are also known as dropouts and the average rate of dropouts is a significant parameter describing a queue.
Simple two-equation queue[edit]
A common basic queuing system is attributed to Erlang, and is a modification of Little's Law. Given an arrival rate λ, a dropout rate Ï, and a departure rate μ, length of the queue L is defined as:
Assuming an exponential distribution for the rates, the waiting time W can be defined as the proportion of arrivals that are served. This is equal to the exponential survival rate of those who do not drop out over the waiting period, giving:
The second equation is commonly rewritten as:
The two-stage one-box model is common in epidemiology.[22]
Queueing networks[edit]
Networks of queues are systems in which a number of queues are connected by what's known as customer routing. When a customer is serviced at one node it can join another node and queue for service, or leave the network.
For networks of m nodes, the state of the system can be described by an mâdimensional vector (x1, x2, .., xm) where xi represents the number of customers at each node.
The simplest non-trivial network of queues is called tandem queues.[23] The first significant results in this area were Jackson networks,[24][25] for which an efficient product-form stationary distribution exists and the mean value analysis[26] which allows average metrics such as throughput and sojourn times to be computed.[27] If the total number of customers in the network remains constant the network is called a closed network and has also been shown to have a productâform stationary distribution in the GordonâNewell theorem.[28] This result was extended to the BCMP network[29] where a network with very general service time, regimes and customer routing is shown to also exhibit a product-form stationary distribution. The normalizing constant can be calculated with the Buzen's algorithm, proposed in 1973.[30]
Networks of customers have also been investigated, Kelly networks where customers of different classes experience different priority levels at different service nodes.[31] Another type of network are G-networks first proposed by Erol Gelenbe in 1993:[32] these networks do not assume exponential time distributions like the classic Jackson Network.
Example analysis of an M/M/1 queue[edit]
Consider a queue with 1 server and the following variables:
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Further, let En represent the number of times the system enters state n, and Ln represent the number of times the system leaves state n. For all n, we have |En â Ln| â {0, 1}, that is, the number of times the system leaves a state differs by at most 1 from the number of times it enters that state, since it will either return into that state at some time in the future (En = Ln) or not (|En â Ln| = 1).
When the system arrives at a steady state, the arrival rate should be equal to the departure rate.
Routing algorithms[edit]
In discrete time networks where there is a constraint on which service nodes can be active at any time, the max-weight scheduling algorithm chooses a service policy to give optimal throughput in the case that each job visits only a single person [17] service node. In the more general case where jobs can visit more than one node, backpressure routing gives optimal throughput. A network scheduler must choose a queuing algorithm, which affects the characteristics of the larger network[citation needed]. See also Stochastic scheduling for more about scheduling of queueing systems.
Mean field limits[edit]
Mean field models consider the limiting behaviour of the empirical measure (proportion of queues in different states) as the number of queues (m above) goes to infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as the original model.[33]
Fluid limits[edit]
Fluid models are continuous deterministic analogs of queueing networks obtained by taking the limit when the process is scaled in time and space, allowing heterogeneous objects. This scaled trajectory converges to a deterministic equation which allows the stability of the system to be proven. It is known that a queueing network can be stable, but have an unstable fluid limit.[34]
Heavy traffic/diffusion approximations[edit]
In a system with high occupancy rates (utilisation near 1) a heavy traffic approximation can be used to approximate the queueing length process by a reflected Brownian motion,[35]OrnsteinâUhlenbeck process or more general diffusion process.[36] The number of dimensions of the RBM is equal to the number of queueing nodes and the diffusion is restricted to the non-negative orthant.
See also[edit]
References[edit]
Further reading[edit]
External links[edit]
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