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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
125:. 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
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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
195:
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
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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
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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.
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2787:, 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
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1160:{\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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2013:. 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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1369:{\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}}}}
3160:(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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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
107:
The spelling "queueing" over "queuing" is typically encountered in the academic research field. In fact, one of the flagship journals of the field is
3172:, where a network with very general service time, regimes, and customer routing is shown to also exhibit a product–form stationary distribution. The
176:
75:, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. These ideas were seminal to the field of
634:
238:
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
161:
5456:
3268:. The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negative
64:. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of
4036:
2965:(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.
2718:
1836:: the reciprocal of the mean service time (the expected number of consecutive service completions per the same unit time, e.g. per 30 seconds)
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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
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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
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in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in the
410:
4224:
Dimitriou, I. (2019). "A Multiclass
Retrial System With Coupled Orbits And Service Interruptions: Verification of Stability Conditions".
157:, depending on the field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue.
4250:
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Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing.
3187:, where customers of different classes experience different priority levels at different service nodes. Another type of network are
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where (material) products have a spatiotemporal existence, in the sense that products have a certain volume and a certain duration.
3491:
Using queuing theory to analyse completion times in accident and emergency departments in the light of the
Government 4-hour target
1944:
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1531:{\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}
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A birth–death process. The values in the circles represent the state of the system, which evolves based on arrival rates
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234:, which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. If
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4601:
3578:"Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain"
3251:
1812:: the arrival rate (the reciprocal of the expected time between each customer arriving, e.g. 10 customers per second)
573:
4730:
Bobbio, A.; Gribaudo, M.; Telek, M. S. (2008). "Analysis of Large Scale
Interacting Systems by Mean Field Method".
68:
because the results are often used when making business decisions about the resources needed to provide a service.
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327:. For a queue, these rates are generally considered not to vary with the number of jobs in the queue, so a single
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3098:. When a customer is serviced at one node, it can join another node and queue for service, or leave the network.
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rate of arrivals/departures per unit time is assumed. Under this assumption, this process has an arrival rate of
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46:
31:
4771:
Chen, H.; Whitt, W. (1993). "Diffusion approximations for open queueing networks with service interruptions".
3053:
Several parallel servers (several queues): there are many counters and customers can decide for which to queue
2480:
1657:{\displaystyle P_{0}={\frac {1}{1+\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}}}}
231:
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5419:
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Ramaswami, V. (1988). "A stable recursion for the steady state vector in markov chains of m/g/1 type".
3195:
in 1993: these networks do not assume exponential time distributions like the classic
Jackson network.
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1734:
3009:(where a job in service can be interrupted by a higher-priority job). No work is lost in either model.
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D stands for "deterministic", and means jobs arriving at the queue require a fixed amount of service
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4810:"Diffusion Approximation for Open State-Dependent Queueing Networks in the Heavy Traffic Situation"
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When the system arrives at a steady state, the arrival rate should be equal to the departure rate.
72:
65:
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Morozov, E. (2017). "Stability analysis of a multiclass retrial system withcoupled orbit queues".
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After the 1940s, queueing theory became an area of research interest to mathematicians. In 1953,
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2016:
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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
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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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50:
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Several parallel servers (single queue): customers line up and there are several servers
160:
5520:
5077:
Quantitative System
Performance: Computer System Analysis Using Queueing Network Models
4498:
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This principle also serves customers one at a time, but the customer with the shortest
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310:
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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/
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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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4528:
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3925:
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The next job to serve is the one with the smallest remaining processing requirement.
3001:
Customers with high priority are served first. Priority queues can be of two types:
853:{\displaystyle \lambda _{n-1}P_{n-1}+\mu _{n+1}P_{n+1}=(\lambda _{n}+\mu _{n})P_{n}}
400:{\displaystyle \lambda ={\text{avg}}(\lambda _{1},\lambda _{2},\dots ,\lambda _{k})}
99:, 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.
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4003:"Simulation and queueing network modeling of single-product production campaigns"
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4547:"Computational algorithms for closed queueing networks with exponential servers"
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2792:
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5074:
Lazowska, Edward D.; John Zahorjan; G. Scott Graham; Kenneth C. Sevcik (1984).
4503:"Open, closed and mixed networks of queues with different classes of customers"
3804:
Pollaczek, F., Ueber eine Aufgabe der Wahrscheinlichkeitstheorie, Math. Z. 1930
1759:
is a simple model where a single server serves jobs that arrive according to a
37:
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3287:
known that a queueing network can be stable but have an unstable fluid limit.
3188:
3164:
and has been shown to also have a product–form stationary distribution by the
3086:. The average rate of dropouts is a significant parameter describing a queue.
17:
5352:
4694:
3563:, Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003
3078:
Reneging: customers leave the queue if they have waited too long for service
726:{\displaystyle \lambda _{0}P_{0}+\mu _{2}P_{2}=(\lambda _{1}+\mu _{1})P_{1}}
142:
88:
4341:
3644:
3635:
3618:
1747:
describes the distribution of durations between each arrival to the queue,
4565:
4519:
4502:
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4383:
4364:
3432:
4739:
4732:
2008 Fifth International Conference on Quantitative Evaluation of Systems
4304:
30:"First come, first served" redirects here. For the Kool Keith album, see
2759:{\displaystyle W={\frac {1}{\mu }}\mathrm {ln} {\frac {\lambda }{\mu }}}
121:
Queueing theory is one of the major areas of study in the discipline of
4876:
4835:
4786:
4671:
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the number of servers at the node. For an example of the notation, the
328:
4483:
4312:
2921:
Modern day application of queueing theory concerns among other things
508:
486:
61:
3886:; Atiyah (October 1961). "The single server queue in heavy traffic".
3094:
Queue networks are systems in which multiple queues are connected by
3027:
The next job to be served is the one with the smallest original size.
2654:
Assuming an exponential distribution for the rates, the waiting time
2408:{\displaystyle P_{n}={\frac {\lambda }{\mu }}P_{n-1},\ n=1,2,\ldots }
4663:
4617:
1842:: the parameter characterizing the number of customers in the system
1787:
Consider a queue with one server and the following characteristics:
1767:) and have exponentially distributed service times (the M denotes a
476:{\displaystyle \mu ={\text{avg}}(\mu _{1},\mu _{2},\dots ,\mu _{k})}
3970:. Lecture Notes in Computer Science. Vol. 17. pp. 85–98.
3237:(proportion of queues in different states) as the number of queues
5145:
4930:
3619:"An application of queuing theory to SIS and SEIS epidemic models"
2953:
2854:
queue and introduced the modern notation for queues, now known as
2828:
507:
485:
174:
159:
36:
4918:
Introduction to Queueing Theory and Stochastic Teletraffic Models
3072:
Balking: customers decide not to join the queue if it is too long
4604:(1975). "Networks of Queues with Customers of Different Types".
2319:{\displaystyle \lambda P_{n-1}+\mu P_{n+1}=(\lambda +\mu )P_{n}}
5149:
4326:
Jackson, James R. (Oct 1963). "Jobshop-like Queueing Systems".
4079:, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory.
3889:
Mathematical Proceedings of the Cambridge Philosophical Society
2915:
inter-arrival and service time distributions to be considered.
183:
is idle, and thus an arrival is given to it to process. Server
49:
of the network, although in many applications the study of the
3433:"Performance by Design: Computer Capacity Planning by Example"
3431:
Lawrence W. Dowdy, Virgilio A.F. Almeida, Daniel A. Menasce.
3047:
Single server: customers line up and there is only one server
2002:{\displaystyle \left\vert E_{n}-L_{n}\right\vert \in \{0,1\}}
5137:
4998:, (MIT, Cambridge, May 31, 1961) Proposal for a Ph.D. Thesis
3019:
The next job to be served is the one with the smallest size.
2232:{\displaystyle \lambda P_{0}+\mu P_{2}=(\lambda +\mu )P_{1}}
4650:(Sep 1993). "G-Networks with Triggered Customer Movement".
4460:(1967). "Closed Queuing Systems with Exponential Servers".
4365:"Mean-Value Analysis of Closed Multichain Queuing Networks"
4092:(2012). "Scheduling: Non-Preemptive, Size-Based Policies".
3222:, which affects the characteristics of the larger network.
3183:
Networks of customers have also been investigated, such as
2944:
Various scheduling policies can be used at queueing nodes:
2835:
in 1930, a solution later recast in probabilistic terms by
1725:, fully describes the required steady state probabilities.
923:{\displaystyle P_{1}={\frac {\lambda _{0}}{\mu _{1}}}P_{0}}
253:
by "births" and "deaths", which occur at the arrival rates
3459:"Hershey Medical Center to open redesigned emergency room"
3729:"The theory of probabilities and telephone conversations"
1775:, the G stands for "general" and indicates an arbitrary
532:
equations for the birth-and-death process, known as the
164:
A black box. Jobs arrive to, and depart from, the queue.
5138:
LINE: a general-purpose engine to solve queueing models
4155:(2012). "Scheduling: Preemptive, Size-Based Policies".
3659:"Agner Krarup Erlang (1878-1929) | plus.maths.org"
3144:
The simplest non-trivial networks of queues are called
3105:
nodes, the state of the system can be described by an
2816:
describes the number of servers at the queueing node (
1937:
represent the number of times the system leaves state
1906:
represent the number of times the system enters state
5010:
Communication Nets: Stochastic Message Flow and Delay
4851:"A stable queueing network with unstable fluid model"
2993:
Service capacity is shared equally between customers.
2721:
2702:{\displaystyle {\frac {\mu }{\lambda }}=e^{-W{\mu }}}
2667:
2615:
2541:
2483:
2424:
2338:
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2124:
2059:
2019:
1947:
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4945:(revisited first ed.). Addison-Wesley. p.
2102:{\displaystyle \left\vert E_{n}-L_{n}\right\vert =1}
563:
denotes the steady state probability to be in state
5544:
5503:
5470:
5407:
5366:
5311:
5285:
5183:
5052:
Queueing Systems: Volume II – Computer Applications
4408:"On the arrival theorem for communication networks"
4193:
Performance Modeling and Design of Computer Systems
4157:
Performance Modeling and Design of Computer Systems
4094:
Performance Modeling and Design of Computer Systems
3005:(where a job in service cannot be interrupted) and
2567:{\displaystyle \rho ={\frac {\lambda }{\mu }}<1}
5123:Teknomo's Queueing theory tutorial and calculators
5020:
3148:. The first significant results in this area were
2758:
2701:
2643:{\displaystyle L={\frac {\lambda -\sigma }{\mu }}}
2642:
2566:
2524:
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2407:
2318:
2231:
2156:
2101:
2045:
2001:
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725:
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399:
319:
299:
272:
208:customers is called a queue with a buffer of size
3768:(2009). "The first Erlang century—and the next".
3406:Sundarapandian, V. (2009). "7. Queueing Theory".
3141:represents the number of customers at each node.
2858:. In 1957, Pollaczek studied the GI/G/1 using an
4038:Business Process Modeling, Simulation and Design
2884:worked on the application of queueing theory to
2582:A common basic queueing system is attributed to
1751:the distribution of service times for jobs, and
3941:Communications in Statistics. Stochastic Models
623:{\displaystyle \mu _{1}P_{1}=\lambda _{0}P_{0}}
226:The behaviour of a single queue (also called a
71:Queueing theory has its origins in research by
60:is the mathematical study of waiting lines, or
2712:The second equation is commonly rewritten as:
27:Mathematical study of waiting lines, or queues
5161:
3530:
3528:
3526:
2928:Problems such as performance metrics for the
8:
5003:Information Flow in Large Communication Nets
4996:Information Flow in Large Communication Nets
4071:
4069:
4067:
4065:
1996:
1984:
3814:
3812:
3810:
3488:Mayhew, Les; Smith, David (December 2006).
3408:Probability, Statistics and Queueing Theory
204:). A setting with a waiting zone for up to
5168:
5154:
5146:
5095:Jon Kleinberg; Éva Tardos (30 June 2013).
5005:(RLE Quarterly Progress Report, July 1961)
4968:Analysis and Synthesis of Computer Systems
3760:
3758:
3756:
5055:. New York: Wiley Interscience. pp.
5027:. New York: Wiley Interscience. pp.
4929:
4866:
4825:
4518:
4382:
4191:(2012). "Scheduling: SRPT and Fairness".
4125:Andrew S. Tanenbaum; Herbert Bos (2015).
4030:
4028:
3975:
3841:
3634:
3593:
2769:The two-stage one-box model is common in
2746:
2738:
2728:
2720:
2692:
2685:
2668:
2666:
2622:
2614:
2548:
2540:
2516:
2488:
2482:
2442:
2429:
2423:
2366:
2352:
2343:
2337:
2310:
2276:
2254:
2245:
2223:
2195:
2179:
2170:
2148:
2132:
2123:
2082:
2069:
2058:
2037:
2024:
2018:
1970:
1957:
1946:
1921:
1915:
1890:
1884:
1855:
1849:
1820:
1796:
1698:
1678:
1672:
1637:
1627:
1621:
1609:
1598:
1588:
1577:
1561:
1552:
1546:
1508:
1498:
1492:
1480:
1469:
1459:
1448:
1438:
1425:
1412:
1402:
1391:
1385:
1352:
1342:
1336:
1324:
1313:
1303:
1290:
1277:
1258:
1248:
1236:
1217:
1201:
1194:
1185:
1179:
1151:
1138:
1128:
1116:
1106:
1099:
1090:
1078:
1068:
1062:
1050:
1040:
1027:
1017:
1002:
993:
984:
972:
962:
956:
947:
941:
914:
902:
892:
886:
877:
871:
844:
831:
818:
796:
780:
761:
745:
739:
717:
704:
691:
675:
665:
652:
642:
636:
614:
604:
591:
581:
575:
547:
541:
464:
445:
432:
420:
412:
388:
369:
356:
344:
336:
312:
291:
285:
264:
258:
249:The system transitions between values of
4041:. Pearson Education India. p. 178.
2525:{\displaystyle P_{n}=(1-\rho )\rho ^{n}}
3393:
3233:consider the limiting behaviour of the
2958:First in first out (FIFO) queue example
2157:{\displaystyle \mu P_{1}=\lambda P_{0}}
1875:customers in the system in steady state
179:A queueing node with 3 servers. Server
4893:Gross, Donald; Carl M. Harris (1998).
4501:; Muntz, R.R.; Palacios, F.G. (1975).
4273:: CS1 maint: archived copy as title (
4266:
3824:"Applied Probability in Great Britain"
3246:Heavy traffic/diffusion approximations
2979:will be served first. Also known as a
2803:model in 1920. In Kendall's notation:
1667:which, together with the equation for
41:
4363:Reiser, M.; Lavenberg, S. S. (1980).
4291:(1957). "Networks of Waiting Lines".
3968:Proceedings of 14th European Workshop
3582:The Annals of Mathematical Statistics
3401:
3399:
3397:
2894:Massachusetts Institute of Technology
2463:{\displaystyle P_{0}+P_{1}+\cdots =1}
7:
4942:An introduction to operating systems
4007:Computers & Chemical Engineering
4001:Carlson, E.C.; Felder, R.M. (1992).
3539:(13th ed.). New York: Pearson.
3154:product-form stationary distribution
512:A queue with 1 server, arrival rate
79:and have since seen applications in
5133:Myron Hlynka's Queueing Theory Page
5023:Queueing Systems: Volume I – Theory
4965:Gelenbe, Erol; Isi Mitrani (2010).
1763:(where inter-arrival durations are
4986:Newell, Gordron F. (1 June 1971).
4412:Computer Networks and ISDN Systems
4077:Chapter 8 – Queueing Systems
3687:(2009). "Editorial introduction".
3537:Introduction to management science
3469:from the original on June 29, 2016
3457:Schlechter, Kira (March 2, 2009).
3168:. This result was extended to the
3033:Shortest remaining processing time
2742:
2739:
1783:Example analysis of an M/M/1 queue
1589:
1460:
1403:
25:
4855:The Annals of Applied Probability
4814:The Annals of Applied Probability
4695:"Applications of Queueing Theory"
3661:. Pass.maths.org.uk. 1997-04-30.
3617:Hernández-Suarez, Carlos (2010).
1871:: the probability of there being
5615:
5614:
5128:Virtamo's Queueing Theory Course
4971:. World Scientific 2nd Edition.
2900:, a forerunner to the Internet.
4988:Applications of Queueing Theory
4895:Fundamentals of Queueing Theory
4583:from the original on 2016-05-13
4438:from the original on 2019-09-24
4256:from the original on 2017-03-29
3665:from the original on 2008-10-07
3439:from the original on 2016-05-06
242:by 1 and a departure decreases
4652:Journal of Applied Probability
4606:Journal of Applied Probability
3736:Nyt Tidsskrift for Matematik B
3561:Algorithmic Analysis of Queues
2509:
2497:
2303:
2291:
2216:
2204:
1712:
1700:
1056:
1010:
863:The first two equations imply
837:
811:
710:
684:
470:
425:
394:
349:
141:can be thought of as nearly a
1:
5445:Flow-equivalent server method
5012:(McGraw-Hill, New York, 1964)
3327:Project production management
3024:Preemptive shortest job first
5526:Adversarial queueing network
5415:Continuous-time Markov chain
4424:10.1016/0169-7552(93)90073-D
4201:10.1017/CBO9781139226424.041
4165:10.1017/CBO9781139226424.040
4102:10.1017/CBO9781139226424.039
4019:10.1016/0098-1354(92)80018-5
3214:gives optimal throughput. A
273:{\displaystyle \lambda _{i}}
5488:Heavy traffic approximation
5233:Pollaczek–Khinchine formula
4939:Deitel, Harvey M. (1984) .
3977:10.1007/978-3-319-66583-2_6
3843:10.1287/opre.50.1.227.17792
3535:Taylor, Bernard W. (2019).
3252:Heavy traffic approximation
3176:can be calculated with the
2841:Pollaczek–Khinchine formula
2115:Thus the balance equations
2046:{\displaystyle E_{n}=L_{n}}
1171:By mathematical induction,
5690:
3279:
3262:Ornstein–Uhlenbeck process
3249:
3210:visit more than one node,
3202:
1732:
219:
29:
5612:
5493:Reflected Brownian motion
5298:Markovian arrival process
4961:chap.15, pp. 380–412
4707:10.1007/978-94-009-5970-5
4554:Communications of the ACM
3953:10.1080/15326348808807077
3910:10.1017/S0305004100036094
3784:10.1007/s11134-009-9147-4
3703:10.1007/s11134-009-9151-8
3258:reflected Brownian motion
3152:, for which an efficient
3067:Customer waiting behavior
2911:have allowed queues with
2586:and is a modification of
2578:Simple two-equation queue
1765:exponentially distributed
1718:{\displaystyle (n\geq 1)}
5516:Layered queueing network
5303:Rational arrival process
4915:Zukerman, Moshe (2013).
4406:Van Dijk, N. M. (1993).
4128:Modern Operating Systems
3377:Traffic generation model
2963:first-come, first-served
2936:remain an open problem.
2590:. Given an arrival rate
1804:{\displaystyle \lambda }
1777:probability distribution
1728:
407:and a departure rate of
300:{\displaystyle \mu _{i}}
280:and the departure rates
230:) can be described by a
47:equilibrium distribution
32:First Come, First Served
5604:Teletraffic engineering
5399:Shortest remaining time
4868:10.1214/aoap/1029962815
4827:10.1214/aoap/1177004602
4226:Proceedings of FRUCT 24
4035:Manuel, Laguna (2011).
3742:: 33–39. Archived from
3595:10.1214/aoms/1177728975
3342:Queue management system
2909:matrix analytic methods
2905:matrix geometric method
2888:in the early 1960s and
2866:gave a formula for the
2598:, and a departure rate
536:, are as follows. Here
77:teletraffic engineering
5599:Scheduling (computing)
5238:Matrix analytic method
5080:. Prentice-Hall, Inc.
4693:Newell, G. F. (1982).
4342:10.1287/mnsc.1040.0268
3636:10.3934/mbe.2010.7.809
3367:Scheduling (computing)
3352:Random early detection
2959:
2913:phase-type distributed
2760:
2703:
2644:
2602:, length of the queue
2568:
2526:
2472:geometric distribution
2464:
2409:
2320:
2233:
2158:
2103:
2047:
2003:
1931:
1900:
1865:
1829:
1805:
1719:
1688:
1658:
1620:
1593:
1532:
1491:
1464:
1407:
1370:
1335:
1161:
924:
854:
727:
624:
557:
520:
505:
477:
401:
321:
301:
274:
192:
165:
97:industrial engineering
54:
5430:Product-form solution
5331:Gordon–Newell theorem
5293:Poisson point process
5184:Single queueing nodes
4566:10.1145/362342.362345
4520:10.1145/321879.321887
4476:10.1287/opre.15.2.254
4384:10.1145/322186.322195
3347:Queuing Rule of Thumb
3291:Queueing Applications
3205:Stochastic scheduling
3166:Gordon–Newell theorem
3109:–dimensional vector (
2957:
2839:and now known as the
2761:
2704:
2645:
2569:
2527:
2465:
2410:
2321:
2234:
2159:
2104:
2048:
2004:
1932:
1930:{\displaystyle L_{n}}
1901:
1899:{\displaystyle E_{n}}
1866:
1864:{\displaystyle P_{n}}
1830:
1806:
1720:
1689:
1687:{\displaystyle P_{n}}
1659:
1594:
1573:
1533:
1465:
1444:
1387:
1371:
1309:
1162:
925:
855:
728:
625:
558:
556:{\displaystyle P_{n}}
511:
489:
478:
402:
322:
302:
275:
178:
163:
129:Single queueing nodes
40:
5457:Decomposition method
4849:Bramson, M. (1999).
4740:10.1109/QEST.2008.47
4305:10.1287/opre.5.4.518
4195:. pp. 518–530.
4159:. pp. 508–517.
4096:. pp. 499–507.
3725:Erlang, Agner Krarup
3512:on September 7, 2021
3496:Cass Business School
3212:backpressure routing
3191:, first proposed by
3180:, proposed in 1973.
3174:normalizing constant
2848:David George Kendall
2719:
2665:
2613:
2539:
2481:
2422:
2336:
2244:
2169:
2122:
2057:
2017:
1945:
1914:
1883:
1848:
1828:{\displaystyle \mu }
1819:
1795:
1697:
1671:
1545:
1384:
1178:
940:
870:
738:
635:
574:
540:
497:and departure rates
411:
335:
311:
284:
257:
5669:Network performance
5654:Operations research
5649:Customer experience
5644:Production planning
5589:Pipeline (software)
5569:Flow control (data)
5564:Erlang distribution
5546:Information systems
5336:Mean value analysis
5008:Leonard Kleinrock.
5001:Leonard Kleinrock.
4994:Leonard Kleinrock,
4990:. Chapman and Hall.
4808:Yamada, K. (1995).
4463:Operations Research
4293:Operations Research
3902:1961PCPS...57..902K
3829:Operations Research
3158:mean value analysis
2949:First in, first out
2940:Service disciplines
2923:product development
2785:Agner Krarup Erlang
1779:for service times.
516:and departure rate
232:birth–death process
216:Birth-death process
95:, and particularly
85:traffic engineering
73:Agner Krarup Erlang
66:operations research
5594:Quality of service
5579:Network congestion
5440:Quasireversibility
5420:Kendall's notation
5047:Kleinrock, Leonard
5019:(2 January 1975).
5017:Kleinrock, Leonard
4787:10.1007/BF01149260
4507:Journal of the ACM
4370:Journal of the ACM
4329:Management Science
4189:Harchol-Balter, M.
4153:Harchol-Balter, M.
4090:Harchol-Balter, M.
3322:Network simulation
3264:, or more general
3220:queueing algorithm
3199:Routing algorithms
3015:Shortest job first
2971:Last in, first out
2960:
2856:Kendall's notation
2837:Aleksandr Khinchin
2756:
2699:
2640:
2564:
2522:
2460:
2405:
2316:
2229:
2154:
2099:
2043:
1999:
1927:
1896:
1861:
1825:
1801:
1735:Kendall's notation
1729:Kendall's notation
1715:
1684:
1654:
1528:
1366:
1157:
920:
850:
723:
620:
553:
521:
506:
473:
397:
317:
297:
270:
193:
166:
123:management science
93:project management
81:telecommunications
55:
5626:
5625:
5584:Network scheduler
5483:Mean-field theory
5394:Shortest job next
5384:Processor sharing
5341:Buzen's algorithm
5324:Traffic equations
5312:Queueing networks
5286:Arrival processes
5260:Kingman's formula
5108:978-1-292-02394-6
5087:978-0-13-746975-8
5066:978-0-471-49111-8
5049:(22 April 1976).
5038:978-0-471-49110-1
4978:978-1-908978-42-4
4956:978-0-201-14502-1
4904:978-0-471-32812-4
4749:978-0-7695-3360-5
4716:978-94-009-5972-9
4418:(10): 1135–2013.
4210:978-1-139-22642-4
4174:978-1-139-22642-4
4138:978-0-13-359162-0
4111:978-1-139-22642-4
4048:978-81-317-6135-9
3987:978-3-319-66582-5
3884:Kingman, J. F. C.
3766:Kingman, J. F. C.
3683:Asmussen, S. R.;
3546:978-0-13-473066-0
3505:978-1-905752-06-5
3417:978-81-203-3844-9
3266:diffusion process
3235:empirical measure
3231:Mean-field models
3226:Mean-field limits
3216:network scheduler
3178:Buzen's algorithm
3090:Queueing networks
3058:Unreliable server
2989:Processor sharing
2886:message switching
2882:Leonard Kleinrock
2876:Kingman's formula
2868:mean waiting time
2860:integral equation
2754:
2736:
2676:
2638:
2594:, a dropout rate
2556:
2383:
2360:
1652:
1649:
1520:
1364:
1284:
1145:
1084:
1008:
978:
908:
534:balance equations
524:Balance equations
423:
347:
320:{\displaystyle i}
222:Survival analysis
16:(Redirected from
5681:
5618:
5617:
5435:Balance equation
5367:Service policies
5265:Lindley equation
5170:
5163:
5156:
5147:
5112:
5098:Algorithm Design
5091:
5070:
5042:
5026:
4991:
4982:
4960:
4935:
4933:
4923:
4908:
4881:
4880:
4870:
4846:
4840:
4839:
4829:
4805:
4799:
4798:
4774:Queueing Systems
4768:
4762:
4761:
4727:
4721:
4720:
4690:
4684:
4683:
4644:
4638:
4637:
4598:
4592:
4591:
4589:
4588:
4582:
4551:
4539:
4533:
4532:
4522:
4494:
4488:
4487:
4453:
4447:
4446:
4444:
4443:
4403:
4397:
4396:
4386:
4360:
4354:
4353:
4323:
4317:
4316:
4285:
4279:
4278:
4272:
4264:
4262:
4261:
4255:
4248:
4240:
4234:
4233:
4221:
4215:
4214:
4185:
4179:
4178:
4149:
4143:
4142:
4122:
4116:
4115:
4086:
4080:
4073:
4060:
4059:
4057:
4055:
4032:
4023:
4022:
3998:
3992:
3991:
3979:
3963:
3957:
3956:
3936:
3930:
3929:
3880:
3874:
3871:
3865:
3862:
3856:
3855:
3845:
3816:
3805:
3802:
3796:
3795:
3771:Queueing Systems
3762:
3751:
3750:
3748:
3733:
3721:
3715:
3714:
3690:Queueing Systems
3680:
3674:
3673:
3671:
3670:
3655:
3649:
3648:
3638:
3614:
3608:
3607:
3597:
3570:
3564:
3557:
3551:
3550:
3532:
3521:
3520:
3518:
3517:
3508:. Archived from
3485:
3479:
3478:
3476:
3474:
3463:The Patriot-News
3454:
3448:
3447:
3445:
3444:
3428:
3422:
3421:
3410:. PHI Learning.
3403:
3150:Jackson networks
3101:For networks of
3096:customer routing
3042:Service facility
2890:packet switching
2850:solved the GI/M/
2765:
2763:
2762:
2757:
2755:
2747:
2745:
2737:
2729:
2708:
2706:
2705:
2700:
2698:
2697:
2696:
2677:
2669:
2649:
2647:
2646:
2641:
2639:
2634:
2623:
2573:
2571:
2570:
2565:
2557:
2549:
2531:
2529:
2528:
2523:
2521:
2520:
2493:
2492:
2469:
2467:
2466:
2461:
2447:
2446:
2434:
2433:
2414:
2412:
2411:
2406:
2381:
2377:
2376:
2361:
2353:
2348:
2347:
2325:
2323:
2322:
2317:
2315:
2314:
2287:
2286:
2265:
2264:
2238:
2236:
2235:
2230:
2228:
2227:
2200:
2199:
2184:
2183:
2163:
2161:
2160:
2155:
2153:
2152:
2137:
2136:
2108:
2106:
2105:
2100:
2092:
2088:
2087:
2086:
2074:
2073:
2052:
2050:
2049:
2044:
2042:
2041:
2029:
2028:
2008:
2006:
2005:
2000:
1980:
1976:
1975:
1974:
1962:
1961:
1936:
1934:
1933:
1928:
1926:
1925:
1905:
1903:
1902:
1897:
1895:
1894:
1870:
1868:
1867:
1862:
1860:
1859:
1834:
1832:
1831:
1826:
1810:
1808:
1807:
1802:
1724:
1722:
1721:
1716:
1693:
1691:
1690:
1685:
1683:
1682:
1663:
1661:
1660:
1655:
1653:
1651:
1650:
1648:
1647:
1632:
1631:
1622:
1619:
1608:
1592:
1587:
1562:
1557:
1556:
1537:
1535:
1534:
1529:
1521:
1519:
1518:
1503:
1502:
1493:
1490:
1479:
1463:
1458:
1443:
1442:
1430:
1429:
1417:
1416:
1406:
1401:
1375:
1373:
1372:
1367:
1365:
1363:
1362:
1347:
1346:
1337:
1334:
1323:
1308:
1307:
1295:
1294:
1285:
1283:
1282:
1281:
1269:
1268:
1253:
1252:
1242:
1241:
1240:
1228:
1227:
1212:
1211:
1195:
1190:
1189:
1166:
1164:
1163:
1158:
1156:
1155:
1146:
1144:
1143:
1142:
1133:
1132:
1122:
1121:
1120:
1111:
1110:
1100:
1095:
1094:
1085:
1083:
1082:
1073:
1072:
1063:
1055:
1054:
1045:
1044:
1032:
1031:
1022:
1021:
1009:
1007:
1006:
994:
989:
988:
979:
977:
976:
967:
966:
957:
952:
951:
929:
927:
926:
921:
919:
918:
909:
907:
906:
897:
896:
887:
882:
881:
859:
857:
856:
851:
849:
848:
836:
835:
823:
822:
807:
806:
791:
790:
772:
771:
756:
755:
732:
730:
729:
724:
722:
721:
709:
708:
696:
695:
680:
679:
670:
669:
657:
656:
647:
646:
629:
627:
626:
621:
619:
618:
609:
608:
596:
595:
586:
585:
562:
560:
559:
554:
552:
551:
482:
480:
479:
474:
469:
468:
450:
449:
437:
436:
424:
421:
406:
404:
403:
398:
393:
392:
374:
373:
361:
360:
348:
345:
326:
324:
323:
318:
306:
304:
303:
298:
296:
295:
279:
277:
276:
271:
269:
268:
110:Queueing Systems
21:
5689:
5688:
5684:
5683:
5682:
5680:
5679:
5678:
5659:Formal sciences
5639:Queueing theory
5629:
5628:
5627:
5622:
5608:
5540:
5499:
5466:
5452:Arrival theorem
5403:
5362:
5319:Jackson network
5307:
5281:
5272:Fork–join queue
5211:Burke's theorem
5179:
5177:Queueing theory
5174:
5143:
5119:
5109:
5094:
5088:
5073:
5067:
5045:
5039:
5015:
4985:
4979:
4964:
4957:
4938:
4921:
4914:
4905:
4892:
4889:
4887:Further reading
4884:
4848:
4847:
4843:
4807:
4806:
4802:
4770:
4769:
4765:
4750:
4734:. p. 215.
4729:
4728:
4724:
4717:
4692:
4691:
4687:
4664:10.2307/3214781
4646:
4645:
4641:
4618:10.2307/3212869
4600:
4599:
4595:
4586:
4584:
4580:
4549:
4541:
4540:
4536:
4499:Chandy, K. Mani
4496:
4495:
4491:
4456:Gordon, W. J.;
4455:
4454:
4450:
4441:
4439:
4405:
4404:
4400:
4362:
4361:
4357:
4325:
4324:
4320:
4287:
4286:
4282:
4265:
4259:
4257:
4253:
4246:
4244:"Archived copy"
4242:
4241:
4237:
4223:
4222:
4218:
4211:
4187:
4186:
4182:
4175:
4151:
4150:
4146:
4139:
4124:
4123:
4119:
4112:
4088:
4087:
4083:
4074:
4063:
4053:
4051:
4049:
4034:
4033:
4026:
4000:
3999:
3995:
3988:
3965:
3964:
3960:
3938:
3937:
3933:
3882:
3881:
3877:
3872:
3868:
3863:
3859:
3818:
3817:
3808:
3803:
3799:
3764:
3763:
3754:
3746:
3731:
3723:
3722:
3718:
3682:
3681:
3677:
3668:
3666:
3657:
3656:
3652:
3616:
3615:
3611:
3572:
3571:
3567:
3558:
3554:
3547:
3534:
3533:
3524:
3515:
3513:
3506:
3487:
3486:
3482:
3472:
3470:
3456:
3455:
3451:
3442:
3440:
3430:
3429:
3425:
3418:
3405:
3404:
3395:
3391:
3386:
3317:Line management
3307:Ehrenfest model
3302:
3293:
3284:
3278:
3254:
3248:
3228:
3207:
3201:
3156:exists and the
3140:
3131:
3122:
3115:
3092:
2942:
2874:, now known as
2833:Felix Pollaczek
2820:= 1, 2, 3, ...)
2791:and solved the
2789:Poisson process
2779:
2717:
2716:
2681:
2663:
2662:
2624:
2611:
2610:
2606:is defined as:
2580:
2537:
2536:
2512:
2484:
2479:
2478:
2438:
2425:
2420:
2419:
2362:
2339:
2334:
2333:
2306:
2272:
2250:
2242:
2241:
2219:
2191:
2175:
2167:
2166:
2144:
2128:
2120:
2119:
2078:
2065:
2064:
2060:
2055:
2054:
2033:
2020:
2015:
2014:
1966:
1953:
1952:
1948:
1943:
1942:
1917:
1912:
1911:
1886:
1881:
1880:
1851:
1846:
1845:
1817:
1816:
1793:
1792:
1785:
1761:Poisson process
1737:
1731:
1695:
1694:
1674:
1669:
1668:
1633:
1623:
1566:
1548:
1543:
1542:
1504:
1494:
1434:
1421:
1408:
1382:
1381:
1348:
1338:
1299:
1286:
1273:
1254:
1244:
1243:
1232:
1213:
1197:
1196:
1181:
1176:
1175:
1147:
1134:
1124:
1123:
1112:
1102:
1101:
1086:
1074:
1064:
1046:
1036:
1023:
1013:
998:
980:
968:
958:
943:
938:
937:
910:
898:
888:
873:
868:
867:
840:
827:
814:
792:
776:
757:
741:
736:
735:
713:
700:
687:
671:
661:
648:
638:
633:
632:
610:
600:
587:
577:
572:
571:
543:
538:
537:
526:
502:
495:
460:
441:
428:
409:
408:
384:
365:
352:
333:
332:
309:
308:
287:
282:
281:
260:
255:
254:
224:
218:
131:
119:
105:
58:Queueing theory
53:is fundamental.
51:transient state
35:
28:
23:
22:
15:
12:
11:
5:
5687:
5685:
5677:
5676:
5671:
5666:
5661:
5656:
5651:
5646:
5641:
5631:
5630:
5624:
5623:
5613:
5610:
5609:
5607:
5606:
5601:
5596:
5591:
5586:
5581:
5576:
5571:
5566:
5561:
5556:
5550:
5548:
5542:
5541:
5539:
5538:
5533:
5528:
5523:
5521:Polling system
5518:
5513:
5507:
5505:
5501:
5500:
5498:
5497:
5496:
5495:
5485:
5480:
5474:
5472:
5471:Limit theorems
5468:
5467:
5465:
5464:
5459:
5454:
5449:
5448:
5447:
5442:
5437:
5427:
5422:
5417:
5411:
5409:
5405:
5404:
5402:
5401:
5396:
5391:
5386:
5381:
5376:
5370:
5368:
5364:
5363:
5361:
5360:
5355:
5350:
5345:
5344:
5343:
5338:
5328:
5327:
5326:
5315:
5313:
5309:
5308:
5306:
5305:
5300:
5295:
5289:
5287:
5283:
5282:
5280:
5279:
5274:
5269:
5268:
5267:
5262:
5252:
5247:
5242:
5241:
5240:
5235:
5225:
5220:
5215:
5214:
5213:
5203:
5198:
5193:
5187:
5185:
5181:
5180:
5175:
5173:
5172:
5165:
5158:
5150:
5141:
5140:
5135:
5130:
5125:
5118:
5117:External links
5115:
5114:
5113:
5107:
5092:
5086:
5071:
5065:
5043:
5037:
5013:
5006:
4999:
4992:
4983:
4977:
4962:
4955:
4936:
4912:
4903:
4888:
4885:
4883:
4882:
4861:(3): 818–853.
4841:
4820:(4): 958–982.
4800:
4763:
4748:
4722:
4715:
4685:
4658:(3): 742–748.
4639:
4612:(3): 542–554.
4593:
4560:(9): 527–531.
4534:
4513:(2): 248–260.
4489:
4448:
4398:
4355:
4336:(1): 131–142.
4318:
4299:(4): 518–521.
4289:Jackson, J. R.
4280:
4235:
4216:
4209:
4180:
4173:
4144:
4137:
4117:
4110:
4081:
4075:Penttinen A.,
4061:
4047:
4024:
4013:(7): 707–718.
3993:
3986:
3958:
3931:
3875:
3866:
3857:
3836:(1): 227–239.
3806:
3797:
3752:
3749:on 2011-10-01.
3716:
3675:
3650:
3629:(4): 809–823.
3609:
3588:(3): 338–354.
3574:Kendall, D. G.
3565:
3552:
3545:
3522:
3504:
3480:
3449:
3423:
3416:
3392:
3390:
3387:
3385:
3384:
3379:
3374:
3369:
3364:
3359:
3357:Renewal theory
3354:
3349:
3344:
3339:
3337:Queueing delay
3334:
3329:
3324:
3319:
3314:
3309:
3303:
3301:
3298:
3292:
3289:
3280:Main article:
3277:
3274:
3250:Main article:
3247:
3244:
3227:
3224:
3218:must choose a
3200:
3197:
3185:Kelly networks
3162:closed network
3136:
3127:
3120:
3113:
3091:
3088:
3080:
3079:
3076:
3073:
3069:
3068:
3060:
3059:
3055:
3054:
3051:
3048:
3044:
3043:
3039:
3038:
3035:
3029:
3028:
3025:
3021:
3020:
3017:
3011:
3010:
3003:non-preemptive
2999:
2995:
2994:
2991:
2985:
2984:
2973:
2967:
2966:
2951:
2941:
2938:
2831:was solved by
2822:
2821:
2811:
2808:
2778:
2775:
2767:
2766:
2753:
2750:
2744:
2741:
2735:
2732:
2727:
2724:
2710:
2709:
2695:
2691:
2688:
2684:
2680:
2675:
2672:
2652:
2651:
2637:
2633:
2630:
2627:
2621:
2618:
2579:
2576:
2563:
2560:
2555:
2552:
2547:
2544:
2533:
2532:
2519:
2515:
2511:
2508:
2505:
2502:
2499:
2496:
2491:
2487:
2459:
2456:
2453:
2450:
2445:
2441:
2437:
2432:
2428:
2418:The fact that
2416:
2415:
2404:
2401:
2398:
2395:
2392:
2389:
2386:
2380:
2375:
2372:
2369:
2365:
2359:
2356:
2351:
2346:
2342:
2327:
2326:
2313:
2309:
2305:
2302:
2299:
2296:
2293:
2290:
2285:
2282:
2279:
2275:
2271:
2268:
2263:
2260:
2257:
2253:
2249:
2239:
2226:
2222:
2218:
2215:
2212:
2209:
2206:
2203:
2198:
2194:
2190:
2187:
2182:
2178:
2174:
2164:
2151:
2147:
2143:
2140:
2135:
2131:
2127:
2098:
2095:
2091:
2085:
2081:
2077:
2072:
2068:
2063:
2040:
2036:
2032:
2027:
2023:
1998:
1995:
1992:
1989:
1986:
1983:
1979:
1973:
1969:
1965:
1960:
1956:
1951:
1924:
1920:
1893:
1889:
1877:
1876:
1858:
1854:
1843:
1837:
1824:
1813:
1800:
1784:
1781:
1769:Markov process
1733:Main article:
1730:
1727:
1714:
1711:
1708:
1705:
1702:
1681:
1677:
1665:
1664:
1646:
1643:
1640:
1636:
1630:
1626:
1618:
1615:
1612:
1607:
1604:
1601:
1597:
1591:
1586:
1583:
1580:
1576:
1572:
1569:
1565:
1560:
1555:
1551:
1527:
1524:
1517:
1514:
1511:
1507:
1501:
1497:
1489:
1486:
1483:
1478:
1475:
1472:
1468:
1462:
1457:
1454:
1451:
1447:
1441:
1437:
1433:
1428:
1424:
1420:
1415:
1411:
1405:
1400:
1397:
1394:
1390:
1380:The condition
1378:
1377:
1361:
1358:
1355:
1351:
1345:
1341:
1333:
1330:
1327:
1322:
1319:
1316:
1312:
1306:
1302:
1298:
1293:
1289:
1280:
1276:
1272:
1267:
1264:
1261:
1257:
1251:
1247:
1239:
1235:
1231:
1226:
1223:
1220:
1216:
1210:
1207:
1204:
1200:
1193:
1188:
1184:
1169:
1168:
1154:
1150:
1141:
1137:
1131:
1127:
1119:
1115:
1109:
1105:
1098:
1093:
1089:
1081:
1077:
1071:
1067:
1061:
1058:
1053:
1049:
1043:
1039:
1035:
1030:
1026:
1020:
1016:
1012:
1005:
1001:
997:
992:
987:
983:
975:
971:
965:
961:
955:
950:
946:
931:
930:
917:
913:
905:
901:
895:
891:
885:
880:
876:
861:
860:
847:
843:
839:
834:
830:
826:
821:
817:
813:
810:
805:
802:
799:
795:
789:
786:
783:
779:
775:
770:
767:
764:
760:
754:
751:
748:
744:
733:
720:
716:
712:
707:
703:
699:
694:
690:
686:
683:
678:
674:
668:
664:
660:
655:
651:
645:
641:
630:
617:
613:
607:
603:
599:
594:
590:
584:
580:
550:
546:
525:
522:
500:
493:
472:
467:
463:
459:
456:
453:
448:
444:
440:
435:
431:
427:
419:
416:
396:
391:
387:
383:
380:
377:
372:
368:
364:
359:
355:
351:
343:
340:
316:
294:
290:
267:
263:
217:
214:
130:
127:
118:
115:
104:
101:
42:Queue networks
26:
24:
18:Queueing model
14:
13:
10:
9:
6:
4:
3:
2:
5686:
5675:
5674:Markov models
5672:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5650:
5647:
5645:
5642:
5640:
5637:
5636:
5634:
5621:
5611:
5605:
5602:
5600:
5597:
5595:
5592:
5590:
5587:
5585:
5582:
5580:
5577:
5575:
5574:Message queue
5572:
5570:
5567:
5565:
5562:
5560:
5559:Erlang (unit)
5557:
5555:
5552:
5551:
5549:
5547:
5543:
5537:
5536:Retrial queue
5534:
5532:
5529:
5527:
5524:
5522:
5519:
5517:
5514:
5512:
5509:
5508:
5506:
5502:
5494:
5491:
5490:
5489:
5486:
5484:
5481:
5479:
5476:
5475:
5473:
5469:
5463:
5460:
5458:
5455:
5453:
5450:
5446:
5443:
5441:
5438:
5436:
5433:
5432:
5431:
5428:
5426:
5423:
5421:
5418:
5416:
5413:
5412:
5410:
5406:
5400:
5397:
5395:
5392:
5390:
5387:
5385:
5382:
5380:
5377:
5375:
5372:
5371:
5369:
5365:
5359:
5356:
5354:
5351:
5349:
5348:Kelly network
5346:
5342:
5339:
5337:
5334:
5333:
5332:
5329:
5325:
5322:
5321:
5320:
5317:
5316:
5314:
5310:
5304:
5301:
5299:
5296:
5294:
5291:
5290:
5288:
5284:
5278:
5275:
5273:
5270:
5266:
5263:
5261:
5258:
5257:
5256:
5253:
5251:
5248:
5246:
5243:
5239:
5236:
5234:
5231:
5230:
5229:
5226:
5224:
5221:
5219:
5216:
5212:
5209:
5208:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5189:
5188:
5186:
5182:
5178:
5171:
5166:
5164:
5159:
5157:
5152:
5151:
5148:
5144:
5139:
5136:
5134:
5131:
5129:
5126:
5124:
5121:
5120:
5116:
5110:
5104:
5100:
5099:
5093:
5089:
5083:
5079:
5078:
5072:
5068:
5062:
5058:
5054:
5053:
5048:
5044:
5040:
5034:
5030:
5025:
5024:
5018:
5014:
5011:
5007:
5004:
5000:
4997:
4993:
4989:
4984:
4980:
4974:
4970:
4969:
4963:
4958:
4952:
4948:
4944:
4943:
4937:
4932:
4927:
4920:
4919:
4913:
4911:
4906:
4900:
4896:
4891:
4890:
4886:
4878:
4874:
4869:
4864:
4860:
4856:
4852:
4845:
4842:
4837:
4833:
4828:
4823:
4819:
4815:
4811:
4804:
4801:
4796:
4792:
4788:
4784:
4780:
4776:
4775:
4767:
4764:
4759:
4755:
4751:
4745:
4741:
4737:
4733:
4726:
4723:
4718:
4712:
4708:
4704:
4700:
4696:
4689:
4686:
4681:
4677:
4673:
4669:
4665:
4661:
4657:
4653:
4649:
4648:Gelenbe, Erol
4643:
4640:
4635:
4631:
4627:
4623:
4619:
4615:
4611:
4607:
4603:
4597:
4594:
4579:
4575:
4571:
4567:
4563:
4559:
4555:
4548:
4544:
4538:
4535:
4530:
4526:
4521:
4516:
4512:
4508:
4504:
4500:
4497:Baskett, F.;
4493:
4490:
4485:
4481:
4477:
4473:
4469:
4465:
4464:
4459:
4458:Newell, G. F.
4452:
4449:
4437:
4433:
4429:
4425:
4421:
4417:
4413:
4409:
4402:
4399:
4394:
4390:
4385:
4380:
4376:
4372:
4371:
4366:
4359:
4356:
4351:
4347:
4343:
4339:
4335:
4331:
4330:
4322:
4319:
4314:
4310:
4306:
4302:
4298:
4294:
4290:
4284:
4281:
4276:
4270:
4252:
4245:
4239:
4236:
4231:
4227:
4220:
4217:
4212:
4206:
4202:
4198:
4194:
4190:
4184:
4181:
4176:
4170:
4166:
4162:
4158:
4154:
4148:
4145:
4140:
4134:
4130:
4129:
4121:
4118:
4113:
4107:
4103:
4099:
4095:
4091:
4085:
4082:
4078:
4072:
4070:
4068:
4066:
4062:
4050:
4044:
4040:
4039:
4031:
4029:
4025:
4020:
4016:
4012:
4008:
4004:
3997:
3994:
3989:
3983:
3978:
3973:
3969:
3962:
3959:
3954:
3950:
3946:
3942:
3935:
3932:
3927:
3923:
3919:
3915:
3911:
3907:
3903:
3899:
3895:
3891:
3890:
3885:
3879:
3876:
3870:
3867:
3861:
3858:
3853:
3849:
3844:
3839:
3835:
3831:
3830:
3825:
3821:
3815:
3813:
3811:
3807:
3801:
3798:
3793:
3789:
3785:
3781:
3777:
3773:
3772:
3767:
3761:
3759:
3757:
3753:
3745:
3741:
3737:
3730:
3726:
3720:
3717:
3712:
3708:
3704:
3700:
3696:
3692:
3691:
3686:
3679:
3676:
3664:
3660:
3654:
3651:
3646:
3642:
3637:
3632:
3628:
3624:
3620:
3613:
3610:
3605:
3601:
3596:
3591:
3587:
3583:
3579:
3575:
3569:
3566:
3562:
3556:
3553:
3548:
3542:
3538:
3531:
3529:
3527:
3523:
3511:
3507:
3501:
3497:
3493:
3492:
3484:
3481:
3468:
3464:
3460:
3453:
3450:
3438:
3434:
3427:
3424:
3419:
3413:
3409:
3402:
3400:
3398:
3394:
3388:
3383:
3380:
3378:
3375:
3373:
3370:
3368:
3365:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3304:
3299:
3297:
3290:
3288:
3283:
3275:
3273:
3271:
3267:
3263:
3259:
3253:
3245:
3243:
3240:
3236:
3232:
3225:
3223:
3221:
3217:
3213:
3206:
3198:
3196:
3194:
3190:
3186:
3181:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3151:
3147:
3146:tandem queues
3142:
3139:
3135:
3130:
3126:
3119:
3112:
3108:
3104:
3099:
3097:
3089:
3087:
3085:
3077:
3074:
3071:
3070:
3066:
3065:
3064:
3057:
3056:
3052:
3049:
3046:
3045:
3041:
3040:
3036:
3034:
3031:
3030:
3026:
3023:
3022:
3018:
3016:
3013:
3012:
3008:
3004:
3000:
2997:
2996:
2992:
2990:
2987:
2986:
2982:
2978:
2974:
2972:
2969:
2968:
2964:
2956:
2952:
2950:
2947:
2946:
2945:
2939:
2937:
2935:
2933:
2926:
2924:
2919:
2916:
2914:
2910:
2906:
2901:
2899:
2895:
2891:
2887:
2883:
2879:
2877:
2873:
2869:
2865:
2861:
2857:
2853:
2849:
2844:
2842:
2838:
2834:
2830:
2825:
2819:
2815:
2812:
2809:
2806:
2805:
2804:
2802:
2800:
2794:
2790:
2786:
2781:
2776:
2774:
2772:
2751:
2748:
2733:
2730:
2725:
2722:
2715:
2714:
2713:
2693:
2689:
2686:
2682:
2678:
2673:
2670:
2661:
2660:
2659:
2657:
2635:
2631:
2628:
2625:
2619:
2616:
2609:
2608:
2607:
2605:
2601:
2597:
2593:
2589:
2585:
2577:
2575:
2561:
2558:
2553:
2550:
2545:
2542:
2517:
2513:
2506:
2503:
2500:
2494:
2489:
2485:
2477:
2476:
2475:
2473:
2470:leads to the
2457:
2454:
2451:
2448:
2443:
2439:
2435:
2430:
2426:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2378:
2373:
2370:
2367:
2363:
2357:
2354:
2349:
2344:
2340:
2332:
2331:
2330:
2311:
2307:
2300:
2297:
2294:
2288:
2283:
2280:
2277:
2273:
2269:
2266:
2261:
2258:
2255:
2251:
2247:
2240:
2224:
2220:
2213:
2210:
2207:
2201:
2196:
2192:
2188:
2185:
2180:
2176:
2172:
2165:
2149:
2145:
2141:
2138:
2133:
2129:
2125:
2118:
2117:
2116:
2113:
2110:
2096:
2093:
2089:
2083:
2079:
2075:
2070:
2066:
2061:
2038:
2034:
2030:
2025:
2021:
2012:
1993:
1990:
1987:
1981:
1977:
1971:
1967:
1963:
1958:
1954:
1949:
1940:
1922:
1918:
1909:
1891:
1887:
1879:Further, let
1874:
1856:
1852:
1844:
1841:
1838:
1835:
1822:
1814:
1811:
1798:
1790:
1789:
1788:
1782:
1780:
1778:
1774:
1770:
1766:
1762:
1758:
1754:
1750:
1746:
1742:
1736:
1726:
1709:
1706:
1703:
1679:
1675:
1644:
1641:
1638:
1634:
1628:
1624:
1616:
1613:
1610:
1605:
1602:
1599:
1595:
1584:
1581:
1578:
1574:
1570:
1567:
1563:
1558:
1553:
1549:
1541:
1540:
1539:
1525:
1522:
1515:
1512:
1509:
1505:
1499:
1495:
1487:
1484:
1481:
1476:
1473:
1470:
1466:
1455:
1452:
1449:
1445:
1439:
1435:
1431:
1426:
1422:
1418:
1413:
1409:
1398:
1395:
1392:
1388:
1359:
1356:
1353:
1349:
1343:
1339:
1331:
1328:
1325:
1320:
1317:
1314:
1310:
1304:
1300:
1296:
1291:
1287:
1278:
1274:
1270:
1265:
1262:
1259:
1255:
1249:
1245:
1237:
1233:
1229:
1224:
1221:
1218:
1214:
1208:
1205:
1202:
1198:
1191:
1186:
1182:
1174:
1173:
1172:
1152:
1148:
1139:
1135:
1129:
1125:
1117:
1113:
1107:
1103:
1096:
1091:
1087:
1079:
1075:
1069:
1065:
1059:
1051:
1047:
1041:
1037:
1033:
1028:
1024:
1018:
1014:
1003:
999:
995:
990:
985:
981:
973:
969:
963:
959:
953:
948:
944:
936:
935:
934:
915:
911:
903:
899:
893:
889:
883:
878:
874:
866:
865:
864:
845:
841:
832:
828:
824:
819:
815:
808:
803:
800:
797:
793:
787:
784:
781:
777:
773:
768:
765:
762:
758:
752:
749:
746:
742:
734:
718:
714:
705:
701:
697:
692:
688:
681:
676:
672:
666:
662:
658:
653:
649:
643:
639:
631:
615:
611:
605:
601:
597:
592:
588:
582:
578:
570:
569:
568:
566:
548:
544:
535:
531:
523:
519:
515:
510:
503:
496:
488:
484:
465:
461:
457:
454:
451:
446:
442:
438:
433:
429:
417:
414:
389:
385:
381:
378:
375:
370:
366:
362:
357:
353:
341:
338:
330:
314:
307:for each job
292:
288:
265:
261:
252:
247:
245:
241:
237:
233:
229:
228:queueing node
223:
215:
213:
211:
207:
203:
199:
190:
186:
182:
177:
173:
171:
162:
158:
156:
152:
149:(also called
148:
144:
140:
139:queueing node
136:
128:
126:
124:
116:
114:
112:
111:
102:
100:
98:
94:
90:
86:
82:
78:
74:
69:
67:
63:
59:
52:
48:
43:
39:
33:
19:
5531:Loss network
5462:Beneš method
5425:Little's law
5408:Key concepts
5358:BCMP network
5176:
5142:
5097:
5076:
5051:
5022:
5009:
5002:
4987:
4967:
4941:
4917:
4894:
4858:
4854:
4844:
4817:
4813:
4803:
4778:
4772:
4766:
4731:
4725:
4699:SpringerLink
4698:
4688:
4655:
4651:
4642:
4609:
4605:
4602:Kelly, F. P.
4596:
4585:. Retrieved
4557:
4553:
4543:Buzen, J. P.
4537:
4510:
4506:
4492:
4467:
4461:
4451:
4440:. Retrieved
4415:
4411:
4401:
4374:
4368:
4358:
4333:
4327:
4321:
4296:
4292:
4283:
4258:. Retrieved
4238:
4229:
4225:
4219:
4192:
4183:
4156:
4147:
4127:
4120:
4093:
4084:
4076:
4052:. Retrieved
4037:
4010:
4006:
3996:
3967:
3961:
3944:
3940:
3934:
3893:
3887:
3878:
3869:
3860:
3833:
3827:
3800:
3778:(1–4): 3–4.
3775:
3769:
3744:the original
3739:
3735:
3719:
3697:(1–4): 1–2.
3694:
3688:
3685:Boxma, O. J.
3678:
3667:. Retrieved
3653:
3626:
3623:Math. Biosci
3622:
3612:
3585:
3581:
3568:
3560:
3559:Tijms, H.C,
3555:
3536:
3514:. Retrieved
3510:the original
3490:
3483:
3471:. Retrieved
3462:
3452:
3441:. Retrieved
3426:
3407:
3382:Flow network
3294:
3285:
3276:Fluid limits
3255:
3238:
3229:
3208:
3193:Erol Gelenbe
3182:
3170:BCMP network
3161:
3143:
3137:
3133:
3128:
3124:
3117:
3110:
3106:
3102:
3100:
3095:
3093:
3083:
3081:
3061:
3006:
3002:
2977:waiting time
2962:
2961:Also called
2943:
2931:
2927:
2920:
2917:
2902:
2880:
2864:John Kingman
2851:
2845:
2826:
2823:
2817:
2813:
2798:
2795:in 1917 and
2782:
2780:
2771:epidemiology
2768:
2711:
2655:
2653:
2603:
2599:
2595:
2591:
2588:Little's Law
2581:
2534:
2417:
2328:
2114:
2111:
2010:
1938:
1907:
1878:
1872:
1839:
1815:
1791:
1786:
1752:
1748:
1744:
1740:
1738:
1666:
1379:
1170:
932:
862:
564:
530:steady state
527:
517:
513:
498:
491:
250:
248:
243:
239:
235:
227:
225:
209:
205:
202:waiting area
201:
197:
194:
188:
184:
180:
169:
167:
154:
150:
146:
138:
134:
132:
120:
108:
106:
70:
57:
56:
5554:Data buffer
5511:Fluid queue
5478:Fluid limit
5389:Round-robin
5255:G/G/1 queue
5250:G/M/1 queue
5245:M/G/k queue
5228:M/G/1 queue
5223:M/M/∞ queue
5218:M/M/c queue
5206:M/M/1 queue
5201:M/D/c queue
5196:M/D/1 queue
5191:D/M/1 queue
5101:. Pearson.
4131:. Pearson.
3947:: 183–188.
3820:Whittle, P.
3372:Traffic jam
3312:Erlang unit
3282:Fluid limit
2872:G/G/1 queue
2829:M/G/1 queue
2793:M/D/1 queue
1773:M/G/1 queue
1757:M/M/1 queue
117:Description
5633:Categories
5504:Extensions
5277:Bulk queue
4781:(4): 335.
4587:2015-09-01
4470:(2): 254.
4442:2019-09-24
4377:(2): 313.
4260:2018-08-02
3896:(4): 902.
3669:2013-04-22
3516:2008-05-20
3443:2009-07-08
3389:References
3362:Throughput
3332:Queue area
3203:See also:
3189:G-networks
3007:preemptive
2053:) or not (
220:See also:
5664:Rationing
5353:G-network
4931:1307.2968
4897:. Wiley.
4680:121673725
4054:6 October
3473:March 12,
2783:In 1909,
2752:μ
2749:λ
2734:μ
2694:μ
2687:−
2674:λ
2671:μ
2636:μ
2632:σ
2629:−
2626:λ
2554:μ
2551:λ
2543:ρ
2514:ρ
2507:ρ
2504:−
2452:⋯
2403:…
2371:−
2358:μ
2355:λ
2301:μ
2295:λ
2270:μ
2259:−
2248:λ
2214:μ
2208:λ
2189:μ
2173:λ
2142:λ
2126:μ
2076:−
1982:∈
1964:−
1823:μ
1799:λ
1771:). In an
1707:≥
1635:μ
1625:λ
1614:−
1596:∏
1590:∞
1575:∑
1538:leads to
1506:μ
1496:λ
1485:−
1467:∏
1461:∞
1446:∑
1404:∞
1389:∑
1350:μ
1340:λ
1329:−
1311:∏
1275:μ
1271:⋯
1263:−
1256:μ
1246:μ
1234:λ
1230:⋯
1222:−
1215:λ
1206:−
1199:λ
1136:μ
1126:μ
1114:λ
1104:λ
1076:μ
1066:λ
1038:λ
1034:−
1015:μ
1000:μ
970:μ
960:λ
900:μ
890:λ
829:μ
816:λ
778:μ
766:−
750:−
743:λ
702:μ
689:λ
663:μ
640:λ
602:λ
579:μ
462:μ
455:…
443:μ
430:μ
415:μ
386:λ
379:…
367:λ
354:λ
339:λ
289:μ
262:λ
151:customers
143:black box
89:computing
5620:Category
4634:51917794
4578:Archived
4545:(1973).
4529:15204199
4436:Archived
4432:45218280
4269:cite web
4251:Archived
4232:: 75–82.
3926:62590290
3822:(2002).
3792:38588726
3727:(1909).
3711:45664707
3663:Archived
3645:21077709
3576:(1953).
3467:Archived
3437:Archived
3300:See also
3132:) where
3084:dropouts
2998:Priority
2801:queueing
2474:formula
2009:for all
155:requests
103:Spelling
4877:2667284
4836:2245101
4795:1180930
4758:2714909
4672:3214781
4626:3212869
4393:8694947
4350:2627213
3918:2984229
3898:Bibcode
3852:3088474
3604:2236285
3270:orthant
3123:, ...,
2898:ARPANET
2777:History
1941:. Then
329:average
200:(or no
170:servers
5105:
5084:
5063:
5035:
4975:
4953:
4910:Online
4901:
4875:
4834:
4793:
4756:
4746:
4713:
4678:
4670:
4632:
4624:
4572:
4527:
4484:168557
4482:
4430:
4391:
4348:
4313:167249
4311:
4207:
4171:
4135:
4108:
4045:
3984:
3924:
3916:
3850:
3790:
3709:
3643:
3602:
3543:
3502:
3414:
2584:Erlang
2535:where
2382:
2329:imply
1910:, and
1743:where
246:by 1.
198:buffer
62:queues
4926:arXiv
4922:(PDF)
4873:JSTOR
4832:JSTOR
4791:S2CID
4754:S2CID
4676:S2CID
4668:JSTOR
4630:S2CID
4622:JSTOR
4581:(PDF)
4574:10702
4570:S2CID
4550:(PDF)
4525:S2CID
4480:JSTOR
4428:S2CID
4389:S2CID
4346:JSTOR
4309:JSTOR
4254:(PDF)
4247:(PDF)
3922:S2CID
3914:JSTOR
3848:JSTOR
3788:S2CID
3747:(PDF)
3732:(PDF)
3707:S2CID
3600:JSTOR
2981:stack
2934:queue
2870:in a
135:queue
5379:LIFO
5374:FIFO
5103:ISBN
5082:ISBN
5061:ISBN
5033:ISBN
4973:ISBN
4951:ISBN
4899:ISBN
4744:ISBN
4711:ISBN
4275:link
4205:ISBN
4169:ISBN
4133:ISBN
4106:ISBN
4056:2017
4043:ISBN
3982:ISBN
3641:PMID
3541:ISBN
3500:ISBN
3475:2009
3412:ISBN
2930:M/G/
2907:and
2903:The
2827:The
2797:M/D/
2559:<
933:and
528:The
147:Jobs
5057:576
5029:417
4947:673
4863:doi
4822:doi
4783:doi
4736:doi
4703:doi
4660:doi
4614:doi
4562:doi
4515:doi
4472:doi
4420:doi
4379:doi
4338:doi
4301:doi
4197:doi
4161:doi
4098:doi
4015:doi
3972:doi
3949:doi
3906:doi
3838:doi
3780:doi
3699:doi
3631:doi
3590:doi
2109:).
422:avg
346:avg
153:or
137:or
5635::
5059:.
5031:.
4949:.
4924:.
4871:.
4857:.
4853:.
4830:.
4816:.
4812:.
4789:.
4779:13
4777:.
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