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Queueing theory

From Wikipedia, the free encyclopedia
Mathematical study of waiting lines, or queues
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 theequilibrium distribution of the network, although in many applications the study of thetransient state is fundamental.

Queueing theory is the mathematical study ofwaiting lines, orqueues.[1] A queueing model is constructed so that queue lengths and waiting time can be predicted.[1] Queueing theory is generally considered a branch ofoperations 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 byAgner Krarup Erlang, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company.[1] These ideas were seminal to the field ofteletraffic engineering and have since seen applications intelecommunications,traffic engineering,computing,[2]project management, and particularlyindustrial engineering, where they are applied in the design of factories, shops, offices, and hospitals.[3][4]

Description

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Queueing theory is one of the major areas of study in the discipline ofmanagement science. Through management science, businesses are able to solve a variety of problems using different scientific and mathematical approaches. Queueing analysis is the probabilistic analysis of waiting lines, and thus the results, also referred to as the operating characteristics, are probabilistic rather than deterministic.[5] The probability that n customers are in the queueing system, the average number of customers in the queueing system, the average number of customers in the waiting line, the average time spent by a customer in the total queuing system, the average time spent by a customer in the waiting line, and finally the probability that the server is busy or idle are all of the different operating characteristics that these queueing models compute.[5] The overall goal of queueing analysis is to compute these characteristics for the current system and then test several alternatives that could lead to improvement. Computing the operating characteristics for the current system and comparing the values to the characteristics of the alternative systems allows managers to see the pros and cons of each potential option. These systems help in the final decision making process by showing ways to increase savings, reduce waiting time, improve efficiency, etc. The main queueing models that can be used are the single-server waiting line system and the multiple-server waiting line system, which are discussed further below. These models can be further differentiated depending on whether service times are constant or undefined, the queue length is finite, the calling population is finite, etc.[5]

Single queueing nodes

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Aqueue orqueueing node can be thought of as nearly ablack box.Jobs (also calledcustomers orrequests, depending on the field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue.

A black box. Jobs arrive to, and depart from, the queue.

However, the queueing node is not quite a pure black box since some information is needed about the inside of the queueing node. The queue has one or moreservers which can each be paired with an arriving job. When the job is completed and departs, that server will again be free to be paired with another arriving job.

A queueing node with 3 servers. Servera is idle, and thus an arrival is given to it to process. Serverb is currently busy and will take some time before it can complete service of its job. Serverc 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. 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 if the cashier is busy when the customer arrives, is referred to as a queue with nobuffer (or nowaiting area). A setting with a waiting zone for up ton customers is called a queue with a buffer of sizen.

Birth-death process

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See also:Survival analysis

The behaviour of a single queue (also called aqueueing node) can be described by abirth–death process, which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. Ifk denotes the number of jobs in the system (either being serviced or waiting if the queue has a buffer of waiting jobs), then an arrival increasesk by 1 and a departure decreasesk by 1.

The system transitions between values ofk bybirths anddeaths, which occur at the arrival ratesλi{\displaystyle \lambda _{i}} and the departure ratesμi{\displaystyle \mu _{i}} for each jobi{\displaystyle i}. For a queue, these rates are generally considered not to vary with the number of jobs in the queue, so a singleaverage rate of arrivals/departures per unit time is assumed. Under this assumption, this process has an arrival rate ofλ=avg(λ1,λ2,…,λk){\displaystyle \lambda ={\text{avg}}(\lambda _{1},\lambda _{2},\dots ,\lambda _{k})} and a departure rate ofμ=avg(μ1,μ2,…,μk){\displaystyle \mu ={\text{avg}}(\mu _{1},\mu _{2},\dots ,\mu _{k})}.

A birth–death process. The values in the circles represent the state of the system, which evolves based on arrival ratesλi and departure ratesμi.
A queue with 1 server, arrival rateλ and departure rateμ

Balance equations

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Thesteady state equations for the birth-and-death process, known as thebalance equations, are as follows. HerePn{\displaystyle P_{n}} denotes the steady state probability to be in staten.

μ1P1=λ0P0{\displaystyle \mu _{1}P_{1}=\lambda _{0}P_{0}}
λ0P0+μ2P2=(λ1+μ1)P1{\displaystyle \lambda _{0}P_{0}+\mu _{2}P_{2}=(\lambda _{1}+\mu _{1})P_{1}}
λn−1Pn−1+μn+1Pn+1=(λn+μn)Pn{\displaystyle \lambda _{n-1}P_{n-1}+\mu _{n+1}P_{n+1}=(\lambda _{n}+\mu _{n})P_{n}}

The first two equations imply

P1=λ0μ1P0{\displaystyle P_{1}={\frac {\lambda _{0}}{\mu _{1}}}P_{0}}

and

P2=λ1μ2P1+1μ2(μ1P1−λ0P0)=λ1μ2P1=λ1λ0μ2μ1P0{\displaystyle P_{2}={\frac {\lambda _{1}}{\mu _{2}}}P_{1}+{\frac {1}{\mu _{2}}}(\mu _{1}P_{1}-\lambda _{0}P_{0})={\frac {\lambda _{1}}{\mu _{2}}}P_{1}={\frac {\lambda _{1}\lambda _{0}}{\mu _{2}\mu _{1}}}P_{0}}.

By mathematical induction,

Pn=λn−1λn−2⋯λ0μnμn−1⋯μ1P0=P0∏i=0n−1λiμi+1{\displaystyle P_{n}={\frac {\lambda _{n-1}\lambda _{n-2}\cdots \lambda _{0}}{\mu _{n}\mu _{n-1}\cdots \mu _{1}}}P_{0}=P_{0}\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}}.

The condition∑n=0∞Pn=P0+P0∑n=1∞∏i=0n−1λiμi+1=1{\displaystyle \sum _{n=0}^{\infty }P_{n}=P_{0}+P_{0}\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}=1} leads to

P0=11+∑n=1∞∏i=0n−1λiμi+1{\displaystyle P_{0}={\frac {1}{1+\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}}}}

which, together with the equation forPn{\displaystyle P_{n}}(n≥1){\displaystyle (n\geq 1)}, fully describes the required steady state probabilities.

Kendall's notation

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Main article:Kendall's notation

Single queueing nodes are usually described using Kendall's notation in the form A/S/c whereA describes the distribution of durations between each arrival to the queue,S the distribution of service times for jobs, andc the number of servers at the node.[6][7] For an example of the notation, theM/M/1 queue is a simple model where a single server serves jobs that arrive according to aPoisson process (where inter-arrival durations areexponentially distributed) and have exponentially distributed service times (the M denotes aMarkov process). In anM/G/1 queue, the G stands forgeneral and indicates an arbitraryprobability distribution for service times.

Example analysis of an M/M/1 queue

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Consider a queue with one server and the following characteristics:

  • λ{\displaystyle \lambda }: the arrival rate (the reciprocal of the expected time between each customer arriving, e.g. 10 customers per second)
  • μ{\displaystyle \mu }: the reciprocal of the mean service time (the expected number of consecutive service completions per the same unit time, e.g. per 30 seconds)
  • n: the parameter characterizing the number of customers in the system
  • Pn{\displaystyle P_{n}}: the probability of there beingn customers in the system in steady state

Further, letEn{\displaystyle E_{n}} represent the number of times the system enters staten, andLn{\displaystyle L_{n}} represent the number of times the system leaves staten. Then|En−Ln|∈{0,1}{\displaystyle \left\vert E_{n}-L_{n}\right\vert \in \{0,1\}} for alln. 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{\displaystyle E_{n}=L_{n}}) or not (|En−Ln|=1{\displaystyle \left\vert E_{n}-L_{n}\right\vert =1}).

When the system arrives at a steady state, the arrival rate should be equal to the departure rate.

Thus the balance equations

μP1=λP0{\displaystyle \mu P_{1}=\lambda P_{0}}
λP0+μP2=(λ+μ)P1{\displaystyle \lambda P_{0}+\mu P_{2}=(\lambda +\mu )P_{1}}
λPn−1+μPn+1=(λ+μ)Pn{\displaystyle \lambda P_{n-1}+\mu P_{n+1}=(\lambda +\mu )P_{n}}

imply

Pn=λμPn−1, n=1,2,…{\displaystyle P_{n}={\frac {\lambda }{\mu }}P_{n-1},\ n=1,2,\ldots }

The fact thatP0+P1+⋯=1{\displaystyle P_{0}+P_{1}+\cdots =1} leads to thegeometric distribution formula

Pn=(1−ρ)ρn{\displaystyle P_{n}=(1-\rho )\rho ^{n}}

whereρ=λμ<1{\displaystyle \rho ={\frac {\lambda }{\mu }}<1}.

Simple two-equation queue

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A common basic queueing system is attributed toErlang and is a modification ofLittle's Law. Given an arrival rateλ, a dropout rateσ, and a departure rateμ, length of the queueL is defined as:

L=λ−σμ{\displaystyle L={\frac {\lambda -\sigma }{\mu }}}.

Assuming an exponential distribution for the rates, the waiting timeW 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:

μλ=e−Wμ{\displaystyle {\frac {\mu }{\lambda }}=e^{-W{\mu }}}

The second equation is commonly rewritten as:

W=1μlnλμ{\displaystyle W={\frac {1}{\mu }}\mathrm {ln} {\frac {\lambda }{\mu }}}

The two-stage one-box model is common inepidemiology.[8]

History

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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.[9][10][11] He modeled the number of telephone calls arriving at an exchange by aPoisson process and solved theM/D/1 queue in 1917 andM/D/k queueing model in 1920.[12] In Kendall's notation:

  • M stands forMarkov ormemoryless, and means arrivals occur according to a Poisson process
  • D stands fordeterministic, and means jobs arriving at the queue require a fixed amount of service
  • k describes the number of servers at the queueing node (k = 1, 2, 3, ...)

If the node has more jobs than servers, then jobs will queue and wait for service.

TheM/G/1 queue was solved byFelix Pollaczek in 1930,[13] a solution later recast in probabilistic terms byAleksandr Khinchin and now known as thePollaczek–Khinchine formula.[12][14]

After the 1940s, queueing theory became an area of research interest to mathematicians.[14] In 1953,David George Kendall solved the GI/M/k queue[15] and introduced the modern notation for queues, now known asKendall's notation. In 1957, Pollaczek studied the GI/G/1 using anintegral equation.[16]John Kingman gave a formula for themean waiting time in aG/G/1 queue, now known asKingman's formula.[17]

Leonard Kleinrock worked on the application of queueing theory tomessage switching in the early 1960s andpacket switching in the early 1970s. His initial contribution to this field was his doctoral thesis at theMassachusetts Institute of Technology in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in theARPANET, a forerunner to the Internet.

Thematrix geometric method andmatrix analytic methods have allowed queues withphase-type distributed inter-arrival and service time distributions to be considered.[18]

Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing.[19]

Modern day application of queueing theory concerns among other thingsproduct development where (material) products have a spatiotemporal existence, in the sense that products have a certain volume and a certain duration.[20]

Problems such as performance metrics for theM/G/k queue remain an open problem.[12][14]

Service disciplines

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"First come, first served" redirects here. For the Kool Keith album, seeFirst Come, First Served.

Various scheduling policies can be used at queueing nodes:

First in, first out
First in first out (FIFO) queue example
Also calledfirst-come, first-served (FCFS),[21] this principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.[22]
Last in, first out
This principle also serves customers one at a time, but the customer with the shortestwaiting time will be served first.[22] Also known as astack.
Processor sharing
Service capacity is shared equally between customers.[22]
Priority
Customers with high priority are served first.[22] Priority queues can be of two types:non-preemptive (where a job in service cannot be interrupted) andpreemptive (where a job in service can be interrupted by a higher-priority job). No work is lost in either model.[23]
Shortest job first
The next job to be served is the one with the smallest size.[24]
Preemptive shortest job first
The next job to be served is the one with the smallest original size.[25]
Shortest remaining processing time
The next job to serve is the one with the smallest remaining processing requirement.[26]
Service facility
  • Single server: customers line up and there is only one server
  • Several parallel servers (single queue): customers line up and there are several servers
  • Several parallel servers (several queues): there are many counters and customers can decide for which to queue
Unreliable server

Server failures occur according to a stochastic (random) process (usually Poisson) and are followed by setup periods during which the server is unavailable. The interrupted customer remains in the service area until server is fixed.[27]

Customer waiting behavior
  • Balking: customers decide not to join the queue if it is too long
  • Jockeying: customers switch between queues if they think they will get served faster by doing so
  • Reneging: customers leave the queue if they have waited too long for service

Arriving customers not served (either due to the queue having no buffer, or due to balking or reneging by the customer) are also known asdropouts. The average rate of dropouts is a significant parameter describing a queue.

Queueing networks

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Queue networks are systems in which multiple queues are connected bycustomer routing. When a customer is serviced at one node, it can join another node and queue for service, or leave the network.

For networks ofm nodes, the state of the system can be described by anm–dimensional vector (x1,x2, ...,xm) wherexi represents the number of customers at each node.

The simplest non-trivial networks of queues are calledtandem queues.[28] The first significant results in this area wereJackson networks,[29][30] for which an efficientproduct-form stationary distribution exists and themean value analysis[31] (which allows average metrics such as throughput and sojourn times) can be computed.[32] If the total number of customers in the network remains constant, the network is called aclosed network and has been shown to also have a product–form stationary distribution by theGordon–Newell theorem.[33] This result was extended to theBCMP network,[34] where a network with very general service time, regimes, and customer routing is shown to also exhibit a product–form stationary distribution. Thenormalizing constant can be calculated with theBuzen's algorithm, proposed in 1973.[35]

Networks of customers have also been investigated, such asKelly networks, where customers of different classes experience different priority levels at different service nodes.[36] Another type of network areG-networks, first proposed byErol Gelenbe in 1993:[37] these networks do not assume exponential time distributions like the classic Jackson network.

Routing algorithms

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See also:Stochastic scheduling

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 service node.[21] In the more general case where jobs can visit more than one node,backpressure routing gives optimal throughput. Anetwork scheduler must choose aqueueing algorithm, which affects the characteristics of the larger network.[38]

Mean-field limits

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Mean-field models consider the limiting behaviour of theempirical measure (proportion of queues in different states) as the number of queuesm approaches 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.[39]

Heavy traffic/diffusion approximations

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Main article:Heavy traffic approximation

In a system with high occupancy rates (utilisation near 1), a heavy traffic approximation can be used to approximate the queueing length process by areflected Brownian motion,[40]Ornstein–Uhlenbeck process, or more generaldiffusion process.[41] The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negativeorthant.

Fluid limits

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Main article:Fluid limit

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.[42]

Queueing Applications

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Queueing theory finds widespread application in computer science and information technology. In networking, for instance, queues are integral to routers and switches, where packets queue up for transmission. By applying queueing theory principles, designers can optimize these systems, ensuring responsive performance and efficient resource utilization.

Beyond the technological realm, queueing theory is relevant to everyday experiences. Whether waiting in line at a supermarket or for public transportation, understanding the principles of queueing theory provides valuable insights into optimizing these systems for enhanced user satisfaction. At some point, everyone will be involved in an aspect of queuing. What some may view to be an inconvenience could possibly be the most effective method.Queueing theory, a discipline rooted in applied mathematics and computer science, is a field dedicated to the study and analysis of queues, or waiting lines, and their implications across a diverse range of applications. This theoretical framework has proven instrumental in understanding and optimizing the efficiency of systems characterized by the presence of queues. The study of queues is essential in contexts such as traffic systems, computer networks, telecommunications, and service operations.

Queueing theory delves into various foundational concepts, with the arrival process and service process being central. The arrival process describes the manner in which entities join the queue over time, often modeled using stochastic processes like Poisson processes. The efficiency of queueing systems is gauged through key performance metrics. These include the average queue length, average wait time, and system throughput. These metrics provide insights into the system's functionality, guiding decisions aimed at enhancing performance and reducing wait times.[43][44][45]

See also

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References

[edit]
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  2. ^Lawrence W. Dowdy, Virgilio A.F. Almeida, Daniel A. Menasce."Performance by Design: Computer Capacity Planning by Example".Archived from the original on 2016-05-06. Retrieved2009-07-08.
  3. ^Schlechter, Kira (March 2, 2009)."Hershey Medical Center to open redesigned emergency room".The Patriot-News.Archived from the original on June 29, 2016. RetrievedMarch 12, 2009.
  4. ^Mayhew, Les; Smith, David (December 2006).Using queuing theory to analyse completion times in accident and emergency departments in the light of the Government 4-hour target.Cass Business School.ISBN 978-1-905752-06-5. Archived fromthe original on September 7, 2021. Retrieved2008-05-20.
  5. ^abcTaylor, Bernard W. (2019).Introduction to management science (13th ed.). New York: Pearson.ISBN 978-0-13-473066-0.
  6. ^Tijms, H.C,Algorithmic Analysis of Queues, Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003
  7. ^Kendall, D. G. (1953)."Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain".The Annals of Mathematical Statistics.24 (3):338–354.doi:10.1214/aoms/1177728975.JSTOR 2236285.
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  19. ^Morozov, E. (2017). "Stability analysis of a multiclass retrial system withcoupled orbit queues".Proceedings of 14th European Workshop. Lecture Notes in Computer Science. Vol. 17. pp. 85–98.doi:10.1007/978-3-319-66583-2_6.ISBN 978-3-319-66582-5.
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  26. ^Harchol-Balter, M. (2012). "Scheduling: SRPT and Fairness".Performance Modeling and Design of Computer Systems. pp. 518–530.doi:10.1017/CBO9781139226424.041.ISBN 978-1-139-22642-4.
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  35. ^Buzen, J. P. (1973)."Computational algorithms for closed queueing networks with exponential servers"(PDF).Communications of the ACM.16 (9):527–531.doi:10.1145/362342.362345.S2CID 10702.Archived(PDF) from the original on 2016-05-13. Retrieved2015-09-01.
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  37. ^Gelenbe, Erol (Sep 1993). "G-Networks with Triggered Customer Movement".Journal of Applied Probability.30 (3):742–748.doi:10.2307/3214781.JSTOR 3214781.S2CID 121673725.
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  42. ^Bramson, M. (1999)."A stable queueing network with unstable fluid model".The Annals of Applied Probability.9 (3):818–853.doi:10.1214/aoap/1029962815.JSTOR 2667284.
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  45. ^Cooper, B. F., & Mitrani, I. (1985).Queueing Networks: A Fundamental Approach. John Wiley & Sons.

Further reading

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External links

[edit]
Look upqueueing orqueuing in Wiktionary, the free dictionary.
Single queueing nodes
Arrival processes
Queueing networks
Service policies
Key concepts
Limit theorems
Extensions
Information systems
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