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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. References: Gross, D., & Harris, C. M. (1998). Fundamentals of Queueing Theory. John Wiley & Sons. Kleinrock, L. (1976). Queueing Systems: Volume I - Theory. Wiley. Cooper, B. F., & Mitrani, I. (1985). Queueing Networks: A Fundamental Approach. John Wiley & Sons
134:. 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. 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. 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.
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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
3218:
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. In the more general case where jobs can
204:
An analogy often used is that of the cashier at a supermarket. (There are other models, but this one is 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
3304:
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
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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
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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.
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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.
740:
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2422:
2796:, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory. He modeled the number of telephone calls arriving at an exchange by a
490:
1169:{\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}}
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2016:
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2022:. 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 (
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1378:{\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}}}}
3169:(which allows average metrics such as throughput and sojourn times) can be computed. If the total number of customers in the network remains constant, the network is called a
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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:
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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.
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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 more
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343:
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Kendall, D.G.:Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain, Ann. Math. Stat. 1953
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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
116:
The spelling "queueing" over "queuing" is typically encountered in the academic research field. In fact, one of the flagship journals of the field is
3181:, where a network with very general service time, regimes, and customer routing is shown to also exhibit a product–form stationary distribution. The
84:, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. These ideas were seminal to the field of
643:
247:
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 increases
3277:. The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negative
73:. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of
4045:
2974:(FCFS), this principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.
2727:
1845:: the reciprocal of the mean service time (the expected number of consecutive service completions per the same unit time, e.g. per 30 seconds)
205:
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 no
4444:
3737:
2344:
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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
2905:
in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in the
419:
4233:
Dimitriou, I. (2019). "A Multiclass
Retrial System With Coupled Orbits And Service Interruptions: Verification of Stability Conditions".
166:, depending on the field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue.
4283:
4259:
2252:
5176:
4586:
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Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing.
3196:, where customers of different classes experience different priority levels at different service nodes. Another type of network are
2934:
where (material) products have a spatiotemporal existence, in the sense that products have a certain volume and a certain duration.
3500:
Using queuing theory to analyse completion times in accident and emergency departments in the light of the
Government 4-hour target
1953:
2177:
1540:{\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}
878:
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93:
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499:
A birth–death process. The values in the circles represent the state of the system, which evolves based on arrival rates
2621:
243:, which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. If
3041:
3587:"Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain"
1821:: the arrival rate (the reciprocal of the expected time between each customer arriving, e.g. 10 customers per second)
582:
5382:
4739:
Bobbio, A.; Gribaudo, M.; Telek, M. S. (2008). "Analysis of Large Scale
Interacting Systems by Mean Field Method".
2957:
77:
because the results are often used when making business decisions about the resources needed to provide a service.
336:. For a queue, these rates are generally considered not to vary with the number of jobs in the queue, so a single
5241:
3445:
3107:. When a customer is serviced at one node, it can join another node and queue for service, or leave the network.
2849:
340:
rate of arrivals/departures per unit time is assumed. Under this assumption, this process has an arrival rate of
17:
3270:
4780:
Chen, H.; Whitt, W. (1993). "Diffusion approximations for open queueing networks with service interruptions".
3062:
Several parallel servers (several queues): there are many counters and customers can decide for which to queue
2489:
1666:{\displaystyle P_{0}={\frac {1}{1+\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}}}}
5453:
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4011:
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3948:
Ramaswami, V. (1988). "A stable recursion for the steady state vector in markov chains of m/g/1 type".
3204:
in 1993: these networks do not assume exponential time distributions like the classic
Jackson network.
3018:(where a job in service can be interrupted by a higher-priority job). No work is lost in either model.
5501:
5339:
5306:
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3174:
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2819:
D stands for "deterministic", and means jobs arriving at the queue require a fixed amount of service
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5647:
5628:
5524:
5311:
5162:
4819:"Diffusion Approximation for Open State-Dependent Queueing Networks in the Heavy Traffic Situation"
3385:
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When the system arrives at a steady state, the arrival rate should be equal to the departure rate.
1785:
46:
31:
4955:
4949:
3975:
Morozov, E. (2017). "Stability analysis of a multiclass retrial system withcoupled orbit queues".
265:
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After the 1940s, queueing theory became an area of research interest to mathematicians. In 1953,
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85:
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Jockeying: customers switch between queues if they think they will get served faster by doing so
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M stands for "Markov" or "memoryless", and means arrivals occur according to a
Poisson process
1803:
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is currently busy and will take some time before it can complete service of its job. Server
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in the early 1970s. His initial contribution to this field was his doctoral thesis at the
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8:
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81:
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Several parallel servers (single queue): customers line up and there are several servers
5672:
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5587:
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5448:
5219:
5086:
Quantitative System
Performance: Computer System Analysis Using Queueing Network Models
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This principle also serves customers one at a time, but the customer with the shortest
2845:
319:
131:
101:
89:
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3882:
Pollaczek, F., Problèmes
Stochastiques posés par le phénomène de formation d'une queue
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Single queueing nodes are usually described using
Kendall's notation in the form A/S/
230:
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has just completed service of a job and thus will be next to receive an arriving job.
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4642:
4537:
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3934:
3800:
3719:
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The next job to serve is the one with the smallest remaining processing requirement.
3010:
Customers with high priority are served first. Priority queues can be of two types:
862:{\displaystyle \lambda _{n-1}P_{n-1}+\mu _{n+1}P_{n+1}=(\lambda _{n}+\mu _{n})P_{n}}
409:{\displaystyle \lambda ={\text{avg}}(\lambda _{1},\lambda _{2},\dots ,\lambda _{k})}
108:, where they are applied in the design of factories, shops, offices, and hospitals.
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If the node has more jobs than servers, then jobs will queue and wait for service.
542:
118:
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4110:
4012:"Simulation and queueing network modeling of single-product production campaigns"
3325:
3315:
3158:
3154:
2841:
2797:
2596:
1769:
50:
5470:
4556:"Computational algorithms for closed queueing networks with exponential servers"
3985:
3851:
3832:
3518:
5529:
5083:
Lazowska, Edward D.; John Zahorjan; G. Scott Graham; Kenneth C. Sevcik (1984).
4512:"Open, closed and mixed networks of queues with different classes of customers"
4507:
3813:
Pollaczek, F., Ueber eine Aufgabe der Wahrscheinlichkeitstheorie, Math. Z. 1930
3365:
3345:
1777:
1768:
is a simple model where a single server serves jobs that arrive according to a
184:
4876:
4859:
4835:
4818:
4715:
3961:
3918:
3792:
3711:
3603:
3586:
3296:
known that a queueing network can be stable but have an unstable fluid limit.
3173:
and has been shown to also have a product–form stationary distribution by the
3095:. The average rate of dropouts is a significant parameter describing a queue.
5641:
5582:
5567:
5544:
5356:
3193:
4703:
3572:, Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003
3087:
Reneging: customers leave the queue if they have waited too long for service
735:{\displaystyle \lambda _{0}P_{0}+\mu _{2}P_{2}=(\lambda _{1}+\mu _{1})P_{1}}
5539:
5366:
4656:
4350:
3892:
3774:
3653:
3644:
3627:
3390:
3201:
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2872:
2779:
1756:
describes the distribution of durations between each arrival to the queue,
538:
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4528:
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169:
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4741:
2008 Fifth International Conference on Quantitative Evaluation of Systems
4313:
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2938:
2880:
2805:
2801:
1781:
1765:
30:"First come, first served" redirects here. For the Kool Keith album, see
2768:{\displaystyle W={\frac {1}{\mu }}\mathrm {ln} {\frac {\lambda }{\mu }}}
130:
Queueing theory is one of the major areas of study in the discipline of
5285:
4885:
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4634:
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3693:
3612:
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the number of servers at the node. For an example of the notation, the
66:
4492:
4321:
2930:
Modern day application of queueing theory concerns among other things
70:
5361:
3895:; Atiyah (October 1961). "The single server queue in heavy traffic".
3103:
Queue networks are systems in which multiple queues are connected by
3036:
The next job to be served is the one with the smallest original size.
2663:
Assuming an exponential distribution for the rates, the waiting time
2417:{\displaystyle P_{n}={\frac {\lambda }{\mu }}P_{n-1},\ n=1,2,\ldots }
151:
97:
4672:
4626:
3439:
1851:: the parameter characterizing the number of customers in the system
1796:
Consider a queue with one server and the following characteristics:
1776:) and have exponentially distributed service times (the M denotes a
485:{\displaystyle \mu ={\text{avg}}(\mu _{1},\mu _{2},\dots ,\mu _{k})}
3979:. Lecture Notes in Computer Science. Vol. 17. pp. 85–98.
3246:(proportion of queues in different states) as the number of queues
37:
5154:
4939:
3628:"An application of queuing theory to SIS and SEIS epidemic models"
2863:
queue and introduced the modern notation for queues, now known as
4927:
Introduction to Queueing Theory and Stochastic Teletraffic Models
3278:
3081:
Balking: customers decide not to join the queue if it is too long
2906:
337:
4613:(1975). "Networks of Queues with Customers of Different Types".
2328:{\displaystyle \lambda P_{n-1}+\mu P_{n+1}=(\lambda +\mu )P_{n}}
4335:
Jackson, James R. (Oct 1963). "Jobshop-like Queueing Systems".
4088:, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory.
3898:
Mathematical Proceedings of the Cambridge Philosophical Society
2924:
inter-arrival and service time distributions to be considered.
192:
is idle, and thus an arrival is given to it to process. Server
57:
49:
of the network, although in many applications the study of the
4133:
3442:"Performance by Design: Computer Capacity Planning by Example"
3440:
Lawrence W. Dowdy, Virgilio A.F. Almeida, Daniel A. Menasce.
3056:
Single server: customers line up and there is only one server
2963:
2837:
2011:{\displaystyle \left\vert E_{n}-L_{n}\right\vert \in \{0,1\}}
5146:
5007:, (MIT, Cambridge, May 31, 1961) Proposal for a Ph.D. Thesis
3028:
The next job to be served is the one with the smallest size.
2241:{\displaystyle \lambda P_{0}+\mu P_{2}=(\lambda +\mu )P_{1}}
517:
495:
4659:(Sep 1993). "G-Networks with Triggered Customer Movement".
4469:(1967). "Closed Queuing Systems with Exponential Servers".
4374:"Mean-Value Analysis of Closed Multichain Queuing Networks"
4101:(2012). "Scheduling: Non-Preemptive, Size-Based Policies".
3231:, which affects the characteristics of the larger network.
3192:
Networks of customers have also been investigated, such as
2953:
Various scheduling policies can be used at queueing nodes:
2844:
in 1930, a solution later recast in probabilistic terms by
1734:, fully describes the required steady state probabilities.
932:{\displaystyle P_{1}={\frac {\lambda _{0}}{\mu _{1}}}P_{0}}
262:
by "births" and "deaths", which occur at the arrival rates
4505:
3468:"Hershey Medical Center to open redesigned emergency room"
3738:"The theory of probabilities and telephone conversations"
3254:
1784:, the G stands for "general" and indicates an arbitrary
541:
equations for the birth-and-death process, known as the
173:
A black box. Jobs arrive to, and depart from, the queue.
5147:
LINE: a general-purpose engine to solve queueing models
5103:
4164:(2012). "Scheduling: Preemptive, Size-Based Policies".
3668:"Agner Krarup Erlang (1878-1929) | plus.maths.org"
3496:
3153:
The simplest non-trivial networks of queues are called
3114:
nodes, the state of the system can be described by an
2825:
describes the number of servers at the queueing node (
1946:
represent the number of times the system leaves state
1915:
represent the number of times the system enters state
5019:
Communication Nets: Stochastic Message Flow and Delay
4860:"A stable queueing network with unstable fluid model"
3002:
Service capacity is shared equally between customers.
2730:
2711:{\displaystyle {\frac {\mu }{\lambda }}=e^{-W{\mu }}}
2676:
2624:
2550:
2492:
2433:
2347:
2255:
2180:
2133:
2068:
2028:
1956:
1925:
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1806:
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1708:
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1395:
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951:
881:
749:
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585:
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422:
346:
322:
295:
268:
4954:(revisited first ed.). Addison-Wesley. p.
2111:{\displaystyle \left\vert E_{n}-L_{n}\right\vert =1}
572:
denotes the steady state probability to be in state
5061:
Queueing Systems: Volume II – Computer Applications
4901:
4738:
4417:"On the arrival theorem for communication networks"
4202:
Performance Modeling and Design of Computer Systems
4166:
Performance Modeling and Design of Computer Systems
4103:
Performance Modeling and Design of Computer Systems
3014:(where a job in service cannot be interrupted) and
2576:{\displaystyle \rho ={\frac {\lambda }{\mu }}<1}
5132:Teknomo's Queueing theory tutorial and calculators
5029:
4973:
3157:. The first significant results in this area were
2767:
2710:
2652:{\displaystyle L={\frac {\lambda -\sigma }{\mu }}}
2651:
2575:
2533:
2471:
2416:
2327:
2240:
2165:
2110:
2054:
2010:
1938:
1907:
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408:
328:
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281:
217:customers is called a queue with a buffer of size
3777:(2009). "The first Erlang century—and the next".
3415:Sundarapandian, V. (2009). "7. Queueing Theory".
3150:represents the number of customers at each node.
2867:. In 1957, Pollaczek studied the GI/G/1 using an
5639:
4047:Business Process Modeling, Simulation and Design
3625:
2893:worked on the application of queueing theory to
2591:A common basic queueing system is attributed to
1760:the distribution of service times for jobs, and
4371:
3950:Communications in Statistics. Stochastic Models
632:{\displaystyle \mu _{1}P_{1}=\lambda _{0}P_{0}}
235:The behaviour of a single queue (also called a
80:Queueing theory has its origins in research by
4196:
4160:
4097:
3414:
2721:The second equation is commonly rewritten as:
27:Mathematical study of waiting lines, or queues
5170:
4009:
3891:
3691:
3539:
3537:
3535:
2937:Problems such as performance metrics for the
5012:Information Flow in Large Communication Nets
5005:Information Flow in Large Communication Nets
4464:
4080:
4078:
4076:
4074:
2586:
2005:
1993:
3823:
3821:
3819:
3497:Mayhew, Les; Smith, David (December 2006).
3417:Probability, Statistics and Queueing Theory
213:). A setting with a waiting zone for up to
5177:
5163:
5104:Jon Kleinberg; Éva Tardos (30 June 2013).
5014:(RLE Quarterly Progress Report, July 1961)
4977:Analysis and Synthesis of Computer Systems
4544:
3769:
3767:
3765:
3532:
3465:
5064:. New York: Wiley Interscience. pp.
5054:
5036:. New York: Wiley Interscience. pp.
5024:
4938:
4875:
4834:
4779:
4527:
4391:
4232:
4200:(2012). "Scheduling: SRPT and Fairness".
4134:Andrew S. Tanenbaum; Herbert Bos (2015).
4071:
4039:
4037:
3984:
3947:
3850:
3643:
3602:
2778:The two-stage one-box model is common in
258:The system transitions between values of
4923:
4414:
4050:. Pearson Education India. p. 178.
3816:
3299:
2962:
2534:{\displaystyle P_{n}=(1-\rho )\rho ^{n}}
516:
494:
183:
168:
137:
36:
4857:
4655:
4334:
4328:
4296:
3974:
3827:
3773:
3762:
3581:
3242:consider the limiting behaviour of the
2967:First in first out (FIFO) queue example
2166:{\displaystyle \mu P_{1}=\lambda P_{0}}
1884:customers in the system in steady state
188:A queueing node with 3 servers. Server
14:
5640:
4994:
4947:
4902:Gross, Donald; Carl M. Harris (1998).
4816:
4701:
4510:; Muntz, R.R.; Palacios, F.G. (1975).
4282:: CS1 maint: archived copy as title (
4043:
4034:
3833:"Applied Probability in Great Britain"
3732:
3543:
3255:Heavy traffic/diffusion approximations
2988:will be served first. Also known as a
2948:
2812:model in 1920. In Kendall's notation:
1676:which, together with the equation for
224:
41:
5158:
4609:
4550:
4372:Reiser, M.; Lavenberg, S. S. (1980).
4300:(1957). "Networks of Waiting Lines".
3977:Proceedings of 14th European Workshop
3591:The Annals of Mathematical Statistics
3410:
3408:
3406:
3207:
2903:Massachusetts Institute of Technology
2472:{\displaystyle P_{0}+P_{1}+\cdots =1}
1737:
4951:An introduction to operating systems
4016:Computers & Chemical Engineering
4010:Carlson, E.C.; Felder, R.M. (1992).
3562:
3548:(13th ed.). New York: Pearson.
3234:
3163:product-form stationary distribution
3098:
532:
521:A queue with 1 server, arrival rate
88:and have since seen applications in
5184:
5142:Myron Hlynka's Queueing Theory Page
5032:Queueing Systems: Volume I – Theory
4974:Gelenbe, Erol; Isi Mitrani (2010).
1772:(where inter-arrival durations are
24:
4995:Newell, Gordron F. (1 June 1971).
4895:
4421:Computer Networks and ISDN Systems
4086:Chapter 8 – Queueing Systems
3696:(2009). "Editorial introduction".
3546:Introduction to management science
3478:from the original on June 29, 2016
3466:Schlechter, Kira (March 2, 2009).
3403:
3177:. This result was extended to the
3042:Shortest remaining processing time
2751:
2748:
1792:Example analysis of an M/M/1 queue
1598:
1469:
1412:
25:
5694:
5125:
4864:The Annals of Applied Probability
4823:The Annals of Applied Probability
4704:"Applications of Queueing Theory"
3670:. Pass.maths.org.uk. 1997-04-30.
3626:Hernández-Suarez, Carlos (2010).
1880:: the probability of there being
5624:
5623:
5137:Virtamo's Queueing Theory Course
4980:. World Scientific 2nd Edition.
2909:, a forerunner to the Internet.
4997:Applications of Queueing Theory
4904:Fundamentals of Queueing Theory
4851:
4810:
4773:
4732:
4695:
4649:
4603:
4592:from the original on 2016-05-13
4499:
4458:
4447:from the original on 2019-09-24
4408:
4365:
4290:
4265:from the original on 2017-03-29
4245:
4226:
4190:
4154:
4127:
4091:
4003:
3968:
3941:
3885:
3876:
3867:
3807:
3726:
3685:
3674:from the original on 2008-10-07
3448:from the original on 2016-05-06
3284:
251:by 1 and a departure decreases
4661:Journal of Applied Probability
4615:Journal of Applied Probability
3745:Nyt Tidsskrift for Matematik B
3660:
3619:
3575:
3570:Algorithmic Analysis of Queues
3490:
3459:
3433:
2518:
2506:
2312:
2300:
2225:
2213:
1721:
1709:
1065:
1019:
872:The first two equations imply
846:
820:
719:
693:
479:
434:
403:
358:
150:can be thought of as nearly a
125:
13:
1:
5454:Flow-equivalent server method
5021:(McGraw-Hill, New York, 1964)
3397:
3336:Project production management
3033:Preemptive shortest job first
65:is the mathematical study of
5535:Adversarial queueing network
5424:Continuous-time Markov chain
4433:10.1016/0169-7552(93)90073-D
4210:10.1017/CBO9781139226424.041
4174:10.1017/CBO9781139226424.040
4111:10.1017/CBO9781139226424.039
4028:10.1016/0098-1354(92)80018-5
3223:gives optimal throughput. A
282:{\displaystyle \lambda _{i}}
7:
5497:Heavy traffic approximation
5242:Pollaczek–Khinchine formula
4948:Deitel, Harvey M. (1984) .
3986:10.1007/978-3-319-66583-2_6
3852:10.1287/opre.50.1.227.17792
3544:Taylor, Bernard W. (2019).
3308:
3261:Heavy traffic approximation
3185:can be calculated with the
2850:Pollaczek–Khinchine formula
2124:Thus the balance equations
2055:{\displaystyle E_{n}=L_{n}}
1180:By mathematical induction,
111:
10:
5699:
3288:
3271:Ornstein–Uhlenbeck process
3258:
3219:visit more than one node,
3211:
2785:
1741:
228:
29:
5621:
5553:
5512:
5502:Reflected Brownian motion
5479:
5416:
5375:
5320:
5307:Markovian arrival process
5294:
5192:
4970:chap.15, pp. 380–412
4716:10.1007/978-94-009-5970-5
4563:Communications of the ACM
3962:10.1080/15326348808807077
3919:10.1017/S0305004100036094
3793:10.1007/s11134-009-9147-4
3712:10.1007/s11134-009-9151-8
3267:reflected Brownian motion
3161:, for which an efficient
3076:Customer waiting behavior
2920:have allowed queues with
2595:and is a modification of
2587:Simple two-equation queue
1774:exponentially distributed
1727:{\displaystyle (n\geq 1)}
5525:Layered queueing network
5312:Rational arrival process
4924:Zukerman, Moshe (2013).
4415:Van Dijk, N. M. (1993).
4137:Modern Operating Systems
3386:Traffic generation model
2972:first-come, first-served
2945:remain an open problem.
2599:. Given an arrival rate
1813:{\displaystyle \lambda }
1786:probability distribution
416:and a departure rate of
309:{\displaystyle \mu _{i}}
289:and the departure rates
239:) can be described by a
47:equilibrium distribution
32:First Come, First Served
5613:Teletraffic engineering
5408:Shortest remaining time
4877:10.1214/aoap/1029962815
4836:10.1214/aoap/1177004602
4235:Proceedings of FRUCT 24
4044:Manuel, Laguna (2011).
3751:: 33–39. Archived from
3604:10.1214/aoms/1177728975
3351:Queue management system
2918:matrix analytic methods
2914:matrix geometric method
2897:in the early 1960s and
2875:gave a formula for the
2607:, and a departure rate
545:, are as follows. Here
86:teletraffic engineering
5608:Scheduling (computing)
5247:Matrix analytic method
5089:. Prentice-Hall, Inc.
4702:Newell, G. F. (1982).
4351:10.1287/mnsc.1040.0268
3645:10.3934/mbe.2010.7.809
3376:Scheduling (computing)
3361:Random early detection
2968:
2922:phase-type distributed
2769:
2712:
2653:
2611:, length of the queue
2577:
2535:
2481:geometric distribution
2473:
2418:
2329:
2242:
2167:
2112:
2056:
2012:
1940:
1909:
1874:
1838:
1814:
1728:
1697:
1667:
1629:
1602:
1541:
1500:
1473:
1416:
1379:
1344:
1170:
933:
863:
736:
633:
566:
529:
514:
486:
410:
330:
310:
283:
201:
174:
106:industrial engineering
54:
5439:Product-form solution
5340:Gordon–Newell theorem
5302:Poisson point process
5193:Single queueing nodes
4575:10.1145/362342.362345
4529:10.1145/321879.321887
4485:10.1287/opre.15.2.254
4393:10.1145/322186.322195
3356:Queuing Rule of Thumb
3300:Queueing Applications
3214:Stochastic scheduling
3175:Gordon–Newell theorem
3118:–dimensional vector (
2966:
2848:and now known as the
2770:
2713:
2654:
2578:
2536:
2474:
2419:
2330:
2243:
2168:
2113:
2057:
2013:
1941:
1939:{\displaystyle L_{n}}
1910:
1908:{\displaystyle E_{n}}
1875:
1873:{\displaystyle P_{n}}
1839:
1815:
1729:
1698:
1696:{\displaystyle P_{n}}
1668:
1603:
1582:
1542:
1474:
1453:
1396:
1380:
1318:
1171:
934:
864:
737:
634:
567:
565:{\displaystyle P_{n}}
520:
498:
487:
411:
331:
311:
284:
187:
172:
138:Single queueing nodes
40:
5466:Decomposition method
4858:Bramson, M. (1999).
4749:10.1109/QEST.2008.47
4314:10.1287/opre.5.4.518
4204:. pp. 518–530.
4168:. pp. 508–517.
4105:. pp. 499–507.
3734:Erlang, Agner Krarup
3521:on September 7, 2021
3505:Cass Business School
3221:backpressure routing
3200:, first proposed by
3189:, proposed in 1973.
3183:normalizing constant
2857:David George Kendall
2728:
2674:
2622:
2548:
2490:
2431:
2345:
2253:
2178:
2131:
2066:
2026:
1954:
1923:
1892:
1857:
1837:{\displaystyle \mu }
1828:
1804:
1706:
1680:
1554:
1393:
1187:
949:
879:
747:
644:
583:
549:
506:and departure rates
420:
344:
320:
293:
266:
5678:Network performance
5663:Operations research
5658:Customer experience
5653:Production planning
5598:Pipeline (software)
5578:Flow control (data)
5573:Erlang distribution
5555:Information systems
5345:Mean value analysis
5017:Leonard Kleinrock.
5010:Leonard Kleinrock.
5003:Leonard Kleinrock,
4999:. Chapman and Hall.
4817:Yamada, K. (1995).
4472:Operations Research
4302:Operations Research
3911:1961PCPS...57..902K
3838:Operations Research
3167:mean value analysis
2958:First in, first out
2949:Service disciplines
2932:product development
2794:Agner Krarup Erlang
1788:for service times.
525:and departure rate
241:birth–death process
225:Birth-death process
104:, and particularly
94:traffic engineering
82:Agner Krarup Erlang
75:operations research
5603:Quality of service
5588:Network congestion
5449:Quasireversibility
5429:Kendall's notation
5056:Kleinrock, Leonard
5028:(2 January 1975).
5026:Kleinrock, Leonard
4796:10.1007/BF01149260
4516:Journal of the ACM
4379:Journal of the ACM
4338:Management Science
4198:Harchol-Balter, M.
4162:Harchol-Balter, M.
4099:Harchol-Balter, M.
3331:Network simulation
3273:, or more general
3229:queueing algorithm
3208:Routing algorithms
3024:Shortest job first
2980:Last in, first out
2969:
2865:Kendall's notation
2846:Aleksandr Khinchin
2765:
2708:
2649:
2573:
2531:
2469:
2414:
2325:
2238:
2163:
2108:
2052:
2008:
1936:
1905:
1870:
1834:
1810:
1744:Kendall's notation
1738:Kendall's notation
1724:
1693:
1663:
1537:
1375:
1166:
929:
859:
732:
629:
562:
530:
515:
482:
406:
326:
306:
279:
202:
175:
132:management science
102:project management
90:telecommunications
55:
5635:
5634:
5593:Network scheduler
5492:Mean-field theory
5403:Shortest job next
5393:Processor sharing
5350:Buzen's algorithm
5333:Traffic equations
5321:Queueing networks
5295:Arrival processes
5269:Kingman's formula
5117:978-1-292-02394-6
5096:978-0-13-746975-8
5075:978-0-471-49111-8
5058:(22 April 1976).
5047:978-0-471-49110-1
4987:978-1-908978-42-4
4965:978-0-201-14502-1
4913:978-0-471-32812-4
4758:978-0-7695-3360-5
4725:978-94-009-5972-9
4427:(10): 1135–2013.
4219:978-1-139-22642-4
4183:978-1-139-22642-4
4147:978-0-13-359162-0
4120:978-1-139-22642-4
4057:978-81-317-6135-9
3996:978-3-319-66582-5
3893:Kingman, J. F. C.
3775:Kingman, J. F. C.
3692:Asmussen, S. R.;
3555:978-0-13-473066-0
3514:978-1-905752-06-5
3426:978-81-203-3844-9
3275:diffusion process
3244:empirical measure
3240:Mean-field models
3235:Mean-field limits
3225:network scheduler
3187:Buzen's algorithm
3099:Queueing networks
3067:Unreliable server
2998:Processor sharing
2895:message switching
2891:Leonard Kleinrock
2885:Kingman's formula
2877:mean waiting time
2869:integral equation
2763:
2745:
2685:
2647:
2603:, a dropout rate
2565:
2392:
2369:
1661:
1658:
1529:
1373:
1293:
1154:
1093:
1017:
987:
917:
543:balance equations
533:Balance equations
432:
356:
329:{\displaystyle i}
231:Survival analysis
16:(Redirected from
5690:
5627:
5626:
5444:Balance equation
5376:Service policies
5274:Lindley equation
5179:
5172:
5165:
5156:
5155:
5121:
5107:Algorithm Design
5100:
5079:
5051:
5035:
5000:
4991:
4969:
4944:
4942:
4932:
4917:
4890:
4889:
4879:
4855:
4849:
4848:
4838:
4814:
4808:
4807:
4783:Queueing Systems
4777:
4771:
4770:
4736:
4730:
4729:
4699:
4693:
4692:
4653:
4647:
4646:
4607:
4601:
4600:
4598:
4597:
4591:
4560:
4548:
4542:
4541:
4531:
4503:
4497:
4496:
4462:
4456:
4455:
4453:
4452:
4412:
4406:
4405:
4395:
4369:
4363:
4362:
4332:
4326:
4325:
4294:
4288:
4287:
4281:
4273:
4271:
4270:
4264:
4257:
4249:
4243:
4242:
4230:
4224:
4223:
4194:
4188:
4187:
4158:
4152:
4151:
4131:
4125:
4124:
4095:
4089:
4082:
4069:
4068:
4066:
4064:
4041:
4032:
4031:
4007:
4001:
4000:
3988:
3972:
3966:
3965:
3945:
3939:
3938:
3889:
3883:
3880:
3874:
3871:
3865:
3864:
3854:
3825:
3814:
3811:
3805:
3804:
3780:Queueing Systems
3771:
3760:
3759:
3757:
3742:
3730:
3724:
3723:
3699:Queueing Systems
3689:
3683:
3682:
3680:
3679:
3664:
3658:
3657:
3647:
3623:
3617:
3616:
3606:
3579:
3573:
3566:
3560:
3559:
3541:
3530:
3529:
3527:
3526:
3517:. Archived from
3494:
3488:
3487:
3485:
3483:
3472:The Patriot-News
3463:
3457:
3456:
3454:
3453:
3437:
3431:
3430:
3419:. PHI Learning.
3412:
3159:Jackson networks
3110:For networks of
3105:customer routing
3051:Service facility
2899:packet switching
2859:solved the GI/M/
2774:
2772:
2771:
2766:
2764:
2756:
2754:
2746:
2738:
2717:
2715:
2714:
2709:
2707:
2706:
2705:
2686:
2678:
2658:
2656:
2655:
2650:
2648:
2643:
2632:
2582:
2580:
2579:
2574:
2566:
2558:
2540:
2538:
2537:
2532:
2530:
2529:
2502:
2501:
2478:
2476:
2475:
2470:
2456:
2455:
2443:
2442:
2423:
2421:
2420:
2415:
2390:
2386:
2385:
2370:
2362:
2357:
2356:
2334:
2332:
2331:
2326:
2324:
2323:
2296:
2295:
2274:
2273:
2247:
2245:
2244:
2239:
2237:
2236:
2209:
2208:
2193:
2192:
2172:
2170:
2169:
2164:
2162:
2161:
2146:
2145:
2117:
2115:
2114:
2109:
2101:
2097:
2096:
2095:
2083:
2082:
2061:
2059:
2058:
2053:
2051:
2050:
2038:
2037:
2017:
2015:
2014:
2009:
1989:
1985:
1984:
1983:
1971:
1970:
1945:
1943:
1942:
1937:
1935:
1934:
1914:
1912:
1911:
1906:
1904:
1903:
1879:
1877:
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988:
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491:
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277:
119:Queueing Systems
21:
5698:
5697:
5693:
5692:
5691:
5689:
5688:
5687:
5668:Formal sciences
5648:Queueing theory
5638:
5637:
5636:
5631:
5617:
5549:
5508:
5475:
5461:Arrival theorem
5412:
5371:
5328:Jackson network
5316:
5290:
5281:Fork–join queue
5220:Burke's theorem
5188:
5186:Queueing theory
5183:
5152:
5128:
5118:
5097:
5076:
5048:
4988:
4966:
4930:
4914:
4898:
4896:Further reading
4893:
4856:
4852:
4815:
4811:
4778:
4774:
4759:
4743:. p. 215.
4737:
4733:
4726:
4700:
4696:
4673:10.2307/3214781
4654:
4650:
4627:10.2307/3212869
4608:
4604:
4595:
4593:
4589:
4558:
4549:
4545:
4508:Chandy, K. Mani
4504:
4500:
4465:Gordon, W. J.;
4463:
4459:
4450:
4448:
4413:
4409:
4370:
4366:
4333:
4329:
4295:
4291:
4275:
4274:
4268:
4266:
4262:
4255:
4253:"Archived copy"
4251:
4250:
4246:
4231:
4227:
4220:
4195:
4191:
4184:
4159:
4155:
4148:
4132:
4128:
4121:
4096:
4092:
4083:
4072:
4062:
4060:
4058:
4042:
4035:
4008:
4004:
3997:
3973:
3969:
3946:
3942:
3890:
3886:
3881:
3877:
3872:
3868:
3826:
3817:
3812:
3808:
3772:
3763:
3755:
3740:
3731:
3727:
3690:
3686:
3677:
3675:
3666:
3665:
3661:
3624:
3620:
3580:
3576:
3567:
3563:
3556:
3542:
3533:
3524:
3522:
3515:
3495:
3491:
3481:
3479:
3464:
3460:
3451:
3449:
3438:
3434:
3427:
3413:
3404:
3400:
3395:
3326:Line management
3316:Ehrenfest model
3311:
3302:
3293:
3287:
3263:
3257:
3237:
3216:
3210:
3165:exists and the
3149:
3140:
3131:
3124:
3101:
2951:
2883:, now known as
2842:Felix Pollaczek
2829:= 1, 2, 3, ...)
2800:and solved the
2798:Poisson process
2788:
2755:
2747:
2737:
2729:
2726:
2725:
2701:
2694:
2690:
2677:
2675:
2672:
2671:
2633:
2631:
2623:
2620:
2619:
2615:is defined as:
2589:
2557:
2549:
2546:
2545:
2525:
2521:
2497:
2493:
2491:
2488:
2487:
2451:
2447:
2438:
2434:
2432:
2429:
2428:
2375:
2371:
2361:
2352:
2348:
2346:
2343:
2342:
2319:
2315:
2285:
2281:
2263:
2259:
2254:
2251:
2250:
2232:
2228:
2204:
2200:
2188:
2184:
2179:
2176:
2175:
2157:
2153:
2141:
2137:
2132:
2129:
2128:
2091:
2087:
2078:
2074:
2073:
2069:
2067:
2064:
2063:
2046:
2042:
2033:
2029:
2027:
2024:
2023:
1979:
1975:
1966:
1962:
1961:
1957:
1955:
1952:
1951:
1930:
1926:
1924:
1921:
1920:
1899:
1895:
1893:
1890:
1889:
1864:
1860:
1858:
1855:
1854:
1829:
1826:
1825:
1805:
1802:
1801:
1794:
1770:Poisson process
1746:
1740:
1707:
1704:
1703:
1687:
1683:
1681:
1678:
1677:
1646:
1642:
1636:
1632:
1630:
1618:
1607:
1597:
1586:
1575:
1570:
1561:
1557:
1555:
1552:
1551:
1517:
1513:
1507:
1503:
1501:
1489:
1478:
1468:
1457:
1447:
1443:
1434:
1430:
1421:
1417:
1411:
1400:
1394:
1391:
1390:
1361:
1357:
1351:
1347:
1345:
1333:
1322:
1312:
1308:
1299:
1295:
1286:
1282:
1267:
1263:
1257:
1253:
1252:
1245:
1241:
1226:
1222:
1210:
1206:
1205:
1203:
1194:
1190:
1188:
1185:
1184:
1160:
1156:
1147:
1143:
1137:
1133:
1132:
1125:
1121:
1115:
1111:
1110:
1108:
1099:
1095:
1087:
1083:
1077:
1073:
1071:
1059:
1055:
1049:
1045:
1036:
1032:
1026:
1022:
1011:
1007:
1002:
993:
989:
981:
977:
971:
967:
965:
956:
952:
950:
947:
946:
923:
919:
911:
907:
901:
897:
895:
886:
882:
880:
877:
876:
853:
849:
840:
836:
827:
823:
805:
801:
789:
785:
770:
766:
754:
750:
748:
745:
744:
726:
722:
713:
709:
700:
696:
684:
680:
674:
670:
661:
657:
651:
647:
645:
642:
641:
623:
619:
613:
609:
600:
596:
590:
586:
584:
581:
580:
556:
552:
550:
547:
546:
535:
511:
504:
473:
469:
454:
450:
441:
437:
429:
421:
418:
417:
397:
393:
378:
374:
365:
361:
353:
345:
342:
341:
321:
318:
317:
300:
296:
294:
291:
290:
273:
269:
267:
264:
263:
233:
227:
140:
128:
114:
63:Queueing theory
53:is fundamental.
51:transient state
35:
28:
23:
22:
18:Queueing models
15:
12:
11:
5:
5696:
5686:
5685:
5680:
5675:
5670:
5665:
5660:
5655:
5650:
5633:
5632:
5622:
5619:
5618:
5616:
5615:
5610:
5605:
5600:
5595:
5590:
5585:
5580:
5575:
5570:
5565:
5559:
5557:
5551:
5550:
5548:
5547:
5542:
5537:
5532:
5530:Polling system
5527:
5522:
5516:
5514:
5510:
5509:
5507:
5506:
5505:
5504:
5494:
5489:
5483:
5481:
5480:Limit theorems
5477:
5476:
5474:
5473:
5468:
5463:
5458:
5457:
5456:
5451:
5446:
5436:
5431:
5426:
5420:
5418:
5414:
5413:
5411:
5410:
5405:
5400:
5395:
5390:
5385:
5379:
5377:
5373:
5372:
5370:
5369:
5364:
5359:
5354:
5353:
5352:
5347:
5337:
5336:
5335:
5324:
5322:
5318:
5317:
5315:
5314:
5309:
5304:
5298:
5296:
5292:
5291:
5289:
5288:
5283:
5278:
5277:
5276:
5271:
5261:
5256:
5251:
5250:
5249:
5244:
5234:
5229:
5224:
5223:
5222:
5212:
5207:
5202:
5196:
5194:
5190:
5189:
5182:
5181:
5174:
5167:
5159:
5150:
5149:
5144:
5139:
5134:
5127:
5126:External links
5124:
5123:
5122:
5116:
5101:
5095:
5080:
5074:
5052:
5046:
5022:
5015:
5008:
5001:
4992:
4986:
4971:
4964:
4945:
4921:
4912:
4897:
4894:
4892:
4891:
4870:(3): 818–853.
4850:
4829:(4): 958–982.
4809:
4772:
4757:
4731:
4724:
4694:
4667:(3): 742–748.
4648:
4621:(3): 542–554.
4602:
4569:(9): 527–531.
4543:
4522:(2): 248–260.
4498:
4457:
4407:
4364:
4345:(1): 131–142.
4327:
4308:(4): 518–521.
4298:Jackson, J. R.
4289:
4244:
4225:
4218:
4189:
4182:
4153:
4146:
4126:
4119:
4090:
4084:Penttinen A.,
4070:
4056:
4033:
4022:(7): 707–718.
4002:
3995:
3967:
3940:
3884:
3875:
3866:
3845:(1): 227–239.
3815:
3806:
3761:
3758:on 2011-10-01.
3725:
3684:
3659:
3638:(4): 809–823.
3618:
3597:(3): 338–354.
3583:Kendall, D. G.
3574:
3561:
3554:
3531:
3513:
3489:
3458:
3432:
3425:
3401:
3399:
3396:
3394:
3393:
3388:
3383:
3378:
3373:
3368:
3366:Renewal theory
3363:
3358:
3353:
3348:
3346:Queueing delay
3343:
3338:
3333:
3328:
3323:
3318:
3312:
3310:
3307:
3301:
3298:
3289:Main article:
3286:
3283:
3259:Main article:
3256:
3253:
3236:
3233:
3227:must choose a
3209:
3206:
3194:Kelly networks
3171:closed network
3145:
3136:
3129:
3122:
3100:
3097:
3089:
3088:
3085:
3082:
3078:
3077:
3069:
3068:
3064:
3063:
3060:
3057:
3053:
3052:
3048:
3047:
3044:
3038:
3037:
3034:
3030:
3029:
3026:
3020:
3019:
3012:non-preemptive
3008:
3004:
3003:
3000:
2994:
2993:
2982:
2976:
2975:
2960:
2950:
2947:
2840:was solved by
2831:
2830:
2820:
2817:
2787:
2784:
2776:
2775:
2762:
2759:
2753:
2750:
2744:
2741:
2736:
2733:
2719:
2718:
2704:
2700:
2697:
2693:
2689:
2684:
2681:
2661:
2660:
2646:
2642:
2639:
2636:
2630:
2627:
2588:
2585:
2572:
2569:
2564:
2561:
2556:
2553:
2542:
2541:
2528:
2524:
2520:
2517:
2514:
2511:
2508:
2505:
2500:
2496:
2468:
2465:
2462:
2459:
2454:
2450:
2446:
2441:
2437:
2427:The fact that
2425:
2424:
2413:
2410:
2407:
2404:
2401:
2398:
2395:
2389:
2384:
2381:
2378:
2374:
2368:
2365:
2360:
2355:
2351:
2336:
2335:
2322:
2318:
2314:
2311:
2308:
2305:
2302:
2299:
2294:
2291:
2288:
2284:
2280:
2277:
2272:
2269:
2266:
2262:
2258:
2248:
2235:
2231:
2227:
2224:
2221:
2218:
2215:
2212:
2207:
2203:
2199:
2196:
2191:
2187:
2183:
2173:
2160:
2156:
2152:
2149:
2144:
2140:
2136:
2107:
2104:
2100:
2094:
2090:
2086:
2081:
2077:
2072:
2049:
2045:
2041:
2036:
2032:
2007:
2004:
2001:
1998:
1995:
1992:
1988:
1982:
1978:
1974:
1969:
1965:
1960:
1933:
1929:
1902:
1898:
1886:
1885:
1867:
1863:
1852:
1846:
1833:
1822:
1809:
1793:
1790:
1778:Markov process
1742:Main article:
1739:
1736:
1723:
1720:
1717:
1714:
1711:
1690:
1686:
1674:
1673:
1655:
1652:
1649:
1645:
1639:
1635:
1627:
1624:
1621:
1616:
1613:
1610:
1606:
1600:
1595:
1592:
1589:
1585:
1581:
1578:
1574:
1569:
1564:
1560:
1536:
1533:
1526:
1523:
1520:
1516:
1510:
1506:
1498:
1495:
1492:
1487:
1484:
1481:
1477:
1471:
1466:
1463:
1460:
1456:
1450:
1446:
1442:
1437:
1433:
1429:
1424:
1420:
1414:
1409:
1406:
1403:
1399:
1389:The condition
1387:
1386:
1370:
1367:
1364:
1360:
1354:
1350:
1342:
1339:
1336:
1331:
1328:
1325:
1321:
1315:
1311:
1307:
1302:
1298:
1289:
1285:
1281:
1276:
1273:
1270:
1266:
1260:
1256:
1248:
1244:
1240:
1235:
1232:
1229:
1225:
1219:
1216:
1213:
1209:
1202:
1197:
1193:
1178:
1177:
1163:
1159:
1150:
1146:
1140:
1136:
1128:
1124:
1118:
1114:
1107:
1102:
1098:
1090:
1086:
1080:
1076:
1070:
1067:
1062:
1058:
1052:
1048:
1044:
1039:
1035:
1029:
1025:
1021:
1014:
1010:
1006:
1001:
996:
992:
984:
980:
974:
970:
964:
959:
955:
940:
939:
926:
922:
914:
910:
904:
900:
894:
889:
885:
870:
869:
856:
852:
848:
843:
839:
835:
830:
826:
822:
819:
814:
811:
808:
804:
798:
795:
792:
788:
784:
779:
776:
773:
769:
763:
760:
757:
753:
742:
729:
725:
721:
716:
712:
708:
703:
699:
695:
692:
687:
683:
677:
673:
669:
664:
660:
654:
650:
639:
626:
622:
616:
612:
608:
603:
599:
593:
589:
559:
555:
534:
531:
509:
502:
481:
476:
472:
468:
465:
462:
457:
453:
449:
444:
440:
436:
428:
425:
405:
400:
396:
392:
389:
386:
381:
377:
373:
368:
364:
360:
352:
349:
325:
303:
299:
276:
272:
226:
223:
139:
136:
127:
124:
113:
110:
42:Queue networks
26:
9:
6:
4:
3:
2:
5695:
5684:
5683:Markov models
5681:
5679:
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5645:
5643:
5630:
5620:
5614:
5611:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5583:Message queue
5581:
5579:
5576:
5574:
5571:
5569:
5568:Erlang (unit)
5566:
5564:
5561:
5560:
5558:
5556:
5552:
5546:
5545:Retrial queue
5543:
5541:
5538:
5536:
5533:
5531:
5528:
5526:
5523:
5521:
5518:
5517:
5515:
5511:
5503:
5500:
5499:
5498:
5495:
5493:
5490:
5488:
5485:
5484:
5482:
5478:
5472:
5469:
5467:
5464:
5462:
5459:
5455:
5452:
5450:
5447:
5445:
5442:
5441:
5440:
5437:
5435:
5432:
5430:
5427:
5425:
5422:
5421:
5419:
5415:
5409:
5406:
5404:
5401:
5399:
5396:
5394:
5391:
5389:
5386:
5384:
5381:
5380:
5378:
5374:
5368:
5365:
5363:
5360:
5358:
5357:Kelly network
5355:
5351:
5348:
5346:
5343:
5342:
5341:
5338:
5334:
5331:
5330:
5329:
5326:
5325:
5323:
5319:
5313:
5310:
5308:
5305:
5303:
5300:
5299:
5297:
5293:
5287:
5284:
5282:
5279:
5275:
5272:
5270:
5267:
5266:
5265:
5262:
5260:
5257:
5255:
5252:
5248:
5245:
5243:
5240:
5239:
5238:
5235:
5233:
5230:
5228:
5225:
5221:
5218:
5217:
5216:
5213:
5211:
5208:
5206:
5203:
5201:
5198:
5197:
5195:
5191:
5187:
5180:
5175:
5173:
5168:
5166:
5161:
5160:
5157:
5153:
5148:
5145:
5143:
5140:
5138:
5135:
5133:
5130:
5129:
5119:
5113:
5109:
5108:
5102:
5098:
5092:
5088:
5087:
5081:
5077:
5071:
5067:
5063:
5062:
5057:
5053:
5049:
5043:
5039:
5034:
5033:
5027:
5023:
5020:
5016:
5013:
5009:
5006:
5002:
4998:
4993:
4989:
4983:
4979:
4978:
4972:
4967:
4961:
4957:
4953:
4952:
4946:
4941:
4936:
4929:
4928:
4922:
4920:
4915:
4909:
4905:
4900:
4899:
4887:
4883:
4878:
4873:
4869:
4865:
4861:
4854:
4846:
4842:
4837:
4832:
4828:
4824:
4820:
4813:
4805:
4801:
4797:
4793:
4789:
4785:
4784:
4776:
4768:
4764:
4760:
4754:
4750:
4746:
4742:
4735:
4727:
4721:
4717:
4713:
4709:
4705:
4698:
4690:
4686:
4682:
4678:
4674:
4670:
4666:
4662:
4658:
4657:Gelenbe, Erol
4652:
4644:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4612:
4606:
4588:
4584:
4580:
4576:
4572:
4568:
4564:
4557:
4553:
4547:
4539:
4535:
4530:
4525:
4521:
4517:
4513:
4509:
4506:Baskett, F.;
4502:
4494:
4490:
4486:
4482:
4478:
4474:
4473:
4468:
4467:Newell, G. F.
4461:
4446:
4442:
4438:
4434:
4430:
4426:
4422:
4418:
4411:
4403:
4399:
4394:
4389:
4385:
4381:
4380:
4375:
4368:
4360:
4356:
4352:
4348:
4344:
4340:
4339:
4331:
4323:
4319:
4315:
4311:
4307:
4303:
4299:
4293:
4285:
4279:
4261:
4254:
4248:
4240:
4236:
4229:
4221:
4215:
4211:
4207:
4203:
4199:
4193:
4185:
4179:
4175:
4171:
4167:
4163:
4157:
4149:
4143:
4139:
4138:
4130:
4122:
4116:
4112:
4108:
4104:
4100:
4094:
4087:
4081:
4079:
4077:
4075:
4059:
4053:
4049:
4048:
4040:
4038:
4029:
4025:
4021:
4017:
4013:
4006:
3998:
3992:
3987:
3982:
3978:
3971:
3963:
3959:
3955:
3951:
3944:
3936:
3932:
3928:
3924:
3920:
3916:
3912:
3908:
3904:
3900:
3899:
3894:
3888:
3879:
3870:
3862:
3858:
3853:
3848:
3844:
3840:
3839:
3834:
3830:
3824:
3822:
3820:
3810:
3802:
3798:
3794:
3790:
3786:
3782:
3781:
3776:
3770:
3768:
3766:
3754:
3750:
3746:
3739:
3735:
3729:
3721:
3717:
3713:
3709:
3705:
3701:
3700:
3695:
3688:
3673:
3669:
3663:
3655:
3651:
3646:
3641:
3637:
3633:
3629:
3622:
3614:
3610:
3605:
3600:
3596:
3592:
3588:
3584:
3578:
3571:
3565:
3557:
3551:
3547:
3540:
3538:
3536:
3520:
3516:
3510:
3506:
3502:
3501:
3493:
3477:
3473:
3469:
3462:
3447:
3443:
3436:
3428:
3422:
3418:
3411:
3409:
3407:
3402:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3314:
3313:
3306:
3297:
3292:
3282:
3280:
3276:
3272:
3268:
3262:
3252:
3249:
3245:
3241:
3232:
3230:
3226:
3222:
3215:
3205:
3203:
3199:
3195:
3190:
3188:
3184:
3180:
3176:
3172:
3168:
3164:
3160:
3156:
3155:tandem queues
3151:
3148:
3144:
3139:
3135:
3128:
3121:
3117:
3113:
3108:
3106:
3096:
3094:
3086:
3083:
3080:
3079:
3075:
3074:
3073:
3066:
3065:
3061:
3058:
3055:
3054:
3050:
3049:
3045:
3043:
3040:
3039:
3035:
3032:
3031:
3027:
3025:
3022:
3021:
3017:
3013:
3009:
3006:
3005:
3001:
2999:
2996:
2995:
2991:
2987:
2983:
2981:
2978:
2977:
2973:
2965:
2961:
2959:
2956:
2955:
2954:
2946:
2944:
2942:
2935:
2933:
2928:
2925:
2923:
2919:
2915:
2910:
2908:
2904:
2900:
2896:
2892:
2888:
2886:
2882:
2878:
2874:
2870:
2866:
2862:
2858:
2853:
2851:
2847:
2843:
2839:
2834:
2828:
2824:
2821:
2818:
2815:
2814:
2813:
2811:
2809:
2803:
2799:
2795:
2790:
2783:
2781:
2760:
2757:
2742:
2739:
2734:
2731:
2724:
2723:
2722:
2702:
2698:
2695:
2691:
2687:
2682:
2679:
2670:
2669:
2668:
2666:
2644:
2640:
2637:
2634:
2628:
2625:
2618:
2617:
2616:
2614:
2610:
2606:
2602:
2598:
2594:
2584:
2570:
2567:
2562:
2559:
2554:
2551:
2526:
2522:
2515:
2512:
2509:
2503:
2498:
2494:
2486:
2485:
2484:
2482:
2479:leads to the
2466:
2463:
2460:
2457:
2452:
2448:
2444:
2439:
2435:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2387:
2382:
2379:
2376:
2372:
2366:
2363:
2358:
2353:
2349:
2341:
2340:
2339:
2320:
2316:
2309:
2306:
2303:
2297:
2292:
2289:
2286:
2282:
2278:
2275:
2270:
2267:
2264:
2260:
2256:
2249:
2233:
2229:
2222:
2219:
2216:
2210:
2205:
2201:
2197:
2194:
2189:
2185:
2181:
2174:
2158:
2154:
2150:
2147:
2142:
2138:
2134:
2127:
2126:
2125:
2122:
2119:
2105:
2102:
2098:
2092:
2088:
2084:
2079:
2075:
2070:
2047:
2043:
2039:
2034:
2030:
2021:
2002:
1999:
1996:
1990:
1986:
1980:
1976:
1972:
1967:
1963:
1958:
1949:
1931:
1927:
1918:
1900:
1896:
1888:Further, let
1883:
1865:
1861:
1853:
1850:
1847:
1844:
1831:
1823:
1820:
1807:
1799:
1798:
1797:
1789:
1787:
1783:
1779:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1745:
1735:
1718:
1715:
1712:
1688:
1684:
1653:
1650:
1647:
1643:
1637:
1633:
1625:
1622:
1619:
1614:
1611:
1608:
1604:
1593:
1590:
1587:
1583:
1579:
1576:
1572:
1567:
1562:
1558:
1550:
1549:
1548:
1534:
1531:
1524:
1521:
1518:
1514:
1508:
1504:
1496:
1493:
1490:
1485:
1482:
1479:
1475:
1464:
1461:
1458:
1454:
1448:
1444:
1440:
1435:
1431:
1427:
1422:
1418:
1407:
1404:
1401:
1397:
1368:
1365:
1362:
1358:
1352:
1348:
1340:
1337:
1334:
1329:
1326:
1323:
1319:
1313:
1309:
1305:
1300:
1296:
1287:
1283:
1279:
1274:
1271:
1268:
1264:
1258:
1254:
1246:
1242:
1238:
1233:
1230:
1227:
1223:
1217:
1214:
1211:
1207:
1200:
1195:
1191:
1183:
1182:
1181:
1161:
1157:
1148:
1144:
1138:
1134:
1126:
1122:
1116:
1112:
1105:
1100:
1096:
1088:
1084:
1078:
1074:
1068:
1060:
1056:
1050:
1046:
1042:
1037:
1033:
1027:
1023:
1012:
1008:
1004:
999:
994:
990:
982:
978:
972:
968:
962:
957:
953:
945:
944:
943:
924:
920:
912:
908:
902:
898:
892:
887:
883:
875:
874:
873:
854:
850:
841:
837:
833:
828:
824:
817:
812:
809:
806:
802:
796:
793:
790:
786:
782:
777:
774:
771:
767:
761:
758:
755:
751:
743:
727:
723:
714:
710:
706:
701:
697:
690:
685:
681:
675:
671:
667:
662:
658:
652:
648:
640:
624:
620:
614:
610:
606:
601:
597:
591:
587:
579:
578:
577:
575:
557:
553:
544:
540:
528:
524:
519:
512:
505:
497:
493:
474:
470:
466:
463:
460:
455:
451:
447:
442:
438:
426:
423:
398:
394:
390:
387:
384:
379:
375:
371:
366:
362:
350:
347:
339:
323:
316:for each job
301:
297:
274:
270:
261:
256:
254:
250:
246:
242:
238:
237:queueing node
232:
222:
220:
216:
212:
208:
199:
195:
191:
186:
182:
180:
171:
167:
165:
161:
158:(also called
157:
153:
149:
148:queueing node
145:
135:
133:
123:
121:
120:
109:
107:
103:
99:
95:
91:
87:
83:
78:
76:
72:
68:
67:waiting lines
64:
60:
59:
52:
48:
43:
39:
33:
19:
5540:Loss network
5471:Beneš method
5434:Little's law
5417:Key concepts
5367:BCMP network
5185:
5151:
5106:
5085:
5060:
5031:
5018:
5011:
4996:
4976:
4950:
4926:
4903:
4867:
4863:
4853:
4826:
4822:
4812:
4787:
4781:
4775:
4740:
4734:
4708:SpringerLink
4707:
4697:
4664:
4660:
4651:
4618:
4614:
4611:Kelly, F. P.
4605:
4594:. Retrieved
4566:
4562:
4552:Buzen, J. P.
4546:
4519:
4515:
4501:
4476:
4470:
4460:
4449:. Retrieved
4424:
4420:
4410:
4383:
4377:
4367:
4342:
4336:
4330:
4305:
4301:
4292:
4267:. Retrieved
4247:
4238:
4234:
4228:
4201:
4192:
4165:
4156:
4136:
4129:
4102:
4093:
4085:
4061:. Retrieved
4046:
4019:
4015:
4005:
3976:
3970:
3953:
3949:
3943:
3902:
3896:
3887:
3878:
3869:
3842:
3836:
3809:
3787:(1–4): 3–4.
3784:
3778:
3753:the original
3748:
3744:
3728:
3706:(1–4): 1–2.
3703:
3697:
3694:Boxma, O. J.
3687:
3676:. Retrieved
3662:
3635:
3632:Math. Biosci
3631:
3621:
3594:
3590:
3577:
3569:
3568:Tijms, H.C,
3564:
3545:
3523:. Retrieved
3519:the original
3499:
3492:
3480:. Retrieved
3471:
3461:
3450:. Retrieved
3435:
3416:
3391:Flow network
3303:
3294:
3285:Fluid limits
3264:
3247:
3238:
3217:
3202:Erol Gelenbe
3191:
3179:BCMP network
3170:
3152:
3146:
3142:
3137:
3133:
3126:
3119:
3115:
3111:
3109:
3104:
3102:
3092:
3090:
3070:
3015:
3011:
2986:waiting time
2971:
2970:Also called
2952:
2940:
2936:
2929:
2926:
2911:
2889:
2873:John Kingman
2860:
2854:
2835:
2832:
2826:
2822:
2807:
2804:in 1917 and
2791:
2789:
2780:epidemiology
2777:
2720:
2664:
2662:
2612:
2608:
2604:
2600:
2597:Little's Law
2590:
2543:
2426:
2337:
2123:
2120:
2019:
1947:
1916:
1887:
1881:
1848:
1824:
1800:
1795:
1761:
1757:
1753:
1749:
1747:
1675:
1388:
1179:
941:
871:
573:
539:steady state
536:
526:
522:
507:
500:
259:
257:
252:
248:
244:
236:
234:
218:
214:
211:waiting area
210:
206:
203:
197:
193:
189:
178:
176:
163:
159:
155:
147:
143:
141:
129:
117:
115:
79:
62:
61:
56:
5563:Data buffer
5520:Fluid queue
5487:Fluid limit
5398:Round-robin
5264:G/G/1 queue
5259:G/M/1 queue
5254:M/G/k queue
5237:M/G/1 queue
5232:M/M/∞ queue
5227:M/M/c queue
5215:M/M/1 queue
5210:M/D/c queue
5205:M/D/1 queue
5200:D/M/1 queue
5110:. Pearson.
4140:. Pearson.
3956:: 183–188.
3829:Whittle, P.
3381:Traffic jam
3321:Erlang unit
3291:Fluid limit
2881:G/G/1 queue
2838:M/G/1 queue
2802:M/D/1 queue
1782:M/G/1 queue
1766:M/M/1 queue
126:Description
5642:Categories
5513:Extensions
5286:Bulk queue
4790:(4): 335.
4596:2015-09-01
4479:(2): 254.
4451:2019-09-24
4386:(2): 313.
4269:2018-08-02
3905:(4): 902.
3678:2013-04-22
3525:2008-05-20
3452:2009-07-08
3398:References
3371:Throughput
3341:Queue area
3212:See also:
3198:G-networks
3016:preemptive
2062:) or not (
229:See also:
5673:Rationing
5362:G-network
4940:1307.2968
4906:. Wiley.
4689:121673725
4063:6 October
3482:March 12,
2792:In 1909,
2761:μ
2758:λ
2743:μ
2703:μ
2696:−
2683:λ
2680:μ
2645:μ
2641:σ
2638:−
2635:λ
2563:μ
2560:λ
2552:ρ
2523:ρ
2516:ρ
2513:−
2461:⋯
2412:…
2380:−
2367:μ
2364:λ
2310:μ
2304:λ
2279:μ
2268:−
2257:λ
2223:μ
2217:λ
2198:μ
2182:λ
2151:λ
2135:μ
2085:−
1991:∈
1973:−
1832:μ
1808:λ
1780:). In an
1716:≥
1644:μ
1634:λ
1623:−
1605:∏
1599:∞
1584:∑
1547:leads to
1515:μ
1505:λ
1494:−
1476:∏
1470:∞
1455:∑
1413:∞
1398:∑
1359:μ
1349:λ
1338:−
1320:∏
1284:μ
1280:⋯
1272:−
1265:μ
1255:μ
1243:λ
1239:⋯
1231:−
1224:λ
1215:−
1208:λ
1145:μ
1135:μ
1123:λ
1113:λ
1085:μ
1075:λ
1047:λ
1043:−
1024:μ
1009:μ
979:μ
969:λ
909:μ
899:λ
838:μ
825:λ
787:μ
775:−
759:−
752:λ
711:μ
698:λ
672:μ
649:λ
611:λ
588:μ
471:μ
464:…
452:μ
439:μ
424:μ
395:λ
388:…
376:λ
363:λ
348:λ
298:μ
271:λ
160:customers
152:black box
98:computing
5629:Category
4643:51917794
4587:Archived
4554:(1973).
4538:15204199
4445:Archived
4441:45218280
4278:cite web
4260:Archived
4241:: 75–82.
3935:62590290
3831:(2002).
3801:38588726
3736:(1909).
3720:45664707
3672:Archived
3654:21077709
3585:(1953).
3476:Archived
3446:Archived
3309:See also
3141:) where
3093:dropouts
3007:Priority
2810:queueing
2483:formula
2018:for all
164:requests
112:Spelling
4886:2667284
4845:2245101
4804:1180930
4767:2714909
4681:3214781
4635:3212869
4402:8694947
4359:2627213
3927:2984229
3907:Bibcode
3861:3088474
3613:2236285
3279:orthant
3132:, ...,
2907:ARPANET
2786:History
1950:. Then
338:average
209:(or no
179:servers
5114:
5093:
5072:
5044:
4984:
4962:
4919:Online
4910:
4884:
4843:
4802:
4765:
4755:
4722:
4687:
4679:
4641:
4633:
4581:
4536:
4493:168557
4491:
4439:
4400:
4357:
4322:167249
4320:
4216:
4180:
4144:
4117:
4054:
3993:
3933:
3925:
3859:
3799:
3718:
3652:
3611:
3552:
3511:
3423:
2593:Erlang
2544:where
2391:
2338:imply
1919:, and
1752:where
255:by 1.
207:buffer
71:queues
58:queues
4935:arXiv
4931:(PDF)
4882:JSTOR
4841:JSTOR
4800:S2CID
4763:S2CID
4685:S2CID
4677:JSTOR
4639:S2CID
4631:JSTOR
4590:(PDF)
4583:10702
4579:S2CID
4559:(PDF)
4534:S2CID
4489:JSTOR
4437:S2CID
4398:S2CID
4355:JSTOR
4318:JSTOR
4263:(PDF)
4256:(PDF)
3931:S2CID
3923:JSTOR
3857:JSTOR
3797:S2CID
3756:(PDF)
3741:(PDF)
3716:S2CID
3609:JSTOR
2990:stack
2943:queue
2879:in a
144:queue
69:, or
5388:LIFO
5383:FIFO
5112:ISBN
5091:ISBN
5070:ISBN
5042:ISBN
4982:ISBN
4960:ISBN
4908:ISBN
4753:ISBN
4720:ISBN
4284:link
4214:ISBN
4178:ISBN
4142:ISBN
4115:ISBN
4065:2017
4052:ISBN
3991:ISBN
3650:PMID
3550:ISBN
3509:ISBN
3484:2009
3421:ISBN
2939:M/G/
2916:and
2912:The
2836:The
2806:M/D/
2568:<
942:and
537:The
156:Jobs
5066:576
5038:417
4956:673
4872:doi
4831:doi
4792:doi
4745:doi
4712:doi
4669:doi
4623:doi
4571:doi
4524:doi
4481:doi
4429:doi
4388:doi
4347:doi
4310:doi
4206:doi
4170:doi
4107:doi
4024:doi
3981:doi
3958:doi
3915:doi
3847:doi
3789:doi
3708:doi
3640:doi
3599:doi
2118:).
431:avg
355:avg
162:or
146:or
5644::
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