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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
129:. 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
199:
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.
735:
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2417:
2791:, 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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1164:{\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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2011:
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2017:. 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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1373:{\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}}}}
3164:(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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338:
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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
111:
The spelling "queueing" over "queuing" is typically encountered in the academic research field. In fact, one of the flagship journals of the field is
3176:, where a network with very general service time, regimes, and customer routing is shown to also exhibit a product–form stationary distribution. The
79:, who created models to describe the system of incoming calls at the Copenhagen Telephone Exchange Company. These ideas were seminal to the field of
638:
242:
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
3272:. The number of dimensions of the Brownian process is equal to the number of queueing nodes, with the diffusion restricted to the non-negative
68:. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of
4040:
2969:(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.
2722:
1840:: the reciprocal of the mean service time (the expected number of consecutive service completions per the same unit time, e.g. per 30 seconds)
200:
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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3732:
2339:
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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
2900:
in 1962, published in book form in 1964. His theoretical work published in the early 1970s underpinned the use of packet switching in the
414:
4228:
Dimitriou, I. (2019). "A Multiclass
Retrial System With Coupled Orbits And Service Interruptions: Verification of Stability Conditions".
161:, depending on the field) arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue.
4278:
4254:
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Systems with coupled orbits are an important part in queueing theory in the application to wireless networks and signal processing.
3191:, 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.
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Using queuing theory to analyse completion times in accident and emergency departments in the light of the
Government 4-hour target
1948:
2172:
1535:{\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}
873:
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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
2616:
238:, which describes the arrivals and departures from the queue, along with the number of jobs currently in the system. If
3036:
17:
3582:"Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain"
1816:: the arrival rate (the reciprocal of the expected time between each customer arriving, e.g. 10 customers per second)
577:
5377:
4734:
Bobbio, A.; Gribaudo, M.; Telek, M. S. (2008). "Analysis of Large Scale
Interacting Systems by Mean Field Method".
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72:
because the results are often used when making business decisions about the resources needed to provide a service.
331:. For a queue, these rates are generally considered not to vary with the number of jobs in the queue, so a single
5236:
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3102:. 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
3265:
4775:
Chen, H.; Whitt, W. (1993). "Diffusion approximations for open queueing networks with service interruptions".
3057:
Several parallel servers (several queues): there are many counters and customers can decide for which to queue
2484:
1661:{\displaystyle P_{0}={\frac {1}{1+\sum _{n=1}^{\infty }\prod _{i=0}^{n-1}{\frac {\lambda _{i}}{\mu _{i+1}}}}}}
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3943:
Ramaswami, V. (1988). "A stable recursion for the steady state vector in markov chains of m/g/1 type".
3199:
in 1993: these networks do not assume exponential time distributions like the classic
Jackson network.
3013:(where a job in service can be interrupted by a higher-priority job). No work is lost in either model.
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5334:
5301:
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3169:
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2814:
D stands for "deterministic", and means jobs arriving at the queue require a fixed amount of service
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5642:
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5157:
4814:"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.
1780:
46:
31:
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4944:
3970:
Morozov, E. (2017). "Stability analysis of a multiclass retrial system withcoupled orbit queues".
260:
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After the 1940s, queueing theory became an area of research interest to mathematicians. In 1953,
2020:
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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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76:
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Several parallel servers (single queue): customers line up and there are several servers
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5549:
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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
2840:
314:
126:
96:
84:
4920:
3877:
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/
225:
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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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3929:
3795:
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The next job to serve is the one with the smallest remaining processing requirement.
3005:
Customers with high priority are served first. Priority queues can be of two types:
857:{\displaystyle \lambda _{n-1}P_{n-1}+\mu _{n+1}P_{n+1}=(\lambda _{n}+\mu _{n})P_{n}}
404:{\displaystyle \lambda ={\text{avg}}(\lambda _{1},\lambda _{2},\dots ,\lambda _{k})}
103:, 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.
537:
113:
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4007:"Simulation and queueing network modeling of single-product production campaigns"
3320:
3310:
3153:
3149:
2836:
2792:
2591:
1764:
50:
5465:
4551:"Computational algorithms for closed queueing networks with exponential servers"
3980:
3846:
3827:
3513:
5524:
5078:
Lazowska, Edward D.; John Zahorjan; G. Scott Graham; Kenneth C. Sevcik (1984).
4507:"Open, closed and mixed networks of queues with different classes of customers"
4502:
3808:
Pollaczek, F., Ueber eine Aufgabe der Wahrscheinlichkeitstheorie, Math. Z. 1930
3360:
3340:
1772:
1763:
is a simple model where a single server serves jobs that arrive according to a
179:
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4830:
4813:
4710:
3956:
3913:
3787:
3706:
3598:
3581:
3291:
known that a queueing network can be stable but have an unstable fluid limit.
3168:
and has been shown to also have a product–form stationary distribution by the
3090:. The average rate of dropouts is a significant parameter describing a queue.
5636:
5577:
5562:
5539:
5351:
3188:
4698:
3567:, Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003
3082:
Reneging: customers leave the queue if they have waited too long for service
730:{\displaystyle \lambda _{0}P_{0}+\mu _{2}P_{2}=(\lambda _{1}+\mu _{1})P_{1}}
5534:
5361:
4651:
4345:
3887:
3769:
3648:
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3622:
3385:
3196:
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1751:
describes the distribution of durations between each arrival to the queue,
533:
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4736:
2008 Fifth International Conference on Quantitative Evaluation of Systems
4308:
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2933:
2875:
2800:
2796:
1776:
1760:
30:"First come, first served" redirects here. For the Kool Keith album, see
2763:{\displaystyle W={\frac {1}{\mu }}\mathrm {ln} {\frac {\lambda }{\mu }}}
125:
Queueing theory is one of the major areas of study in the discipline of
5280:
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the number of servers at the node. For an example of the notation, the
61:
4487:
4316:
2925:
Modern day application of queueing theory concerns among other things
65:
5356:
3890:; Atiyah (October 1961). "The single server queue in heavy traffic".
3098:
Queue networks are systems in which multiple queues are connected by
3031:
The next job to be served is the one with the smallest original size.
2658:
Assuming an exponential distribution for the rates, the waiting time
2412:{\displaystyle P_{n}={\frac {\lambda }{\mu }}P_{n-1},\ n=1,2,\ldots }
146:
92:
4667:
4621:
3434:
1846:: the parameter characterizing the number of customers in the system
1791:
Consider a queue with one server and the following characteristics:
1771:) and have exponentially distributed service times (the M denotes a
480:{\displaystyle \mu ={\text{avg}}(\mu _{1},\mu _{2},\dots ,\mu _{k})}
3974:. Lecture Notes in Computer Science. Vol. 17. pp. 85–98.
3241:(proportion of queues in different states) as the number of queues
37:
5149:
4934:
3623:"An application of queuing theory to SIS and SEIS epidemic models"
2858:
queue and introduced the modern notation for queues, now known as
4922:
Introduction to Queueing Theory and Stochastic Teletraffic Models
3273:
3076:
Balking: customers decide not to join the queue if it is too long
2901:
332:
4608:(1975). "Networks of Queues with Customers of Different Types".
2323:{\displaystyle \lambda P_{n-1}+\mu P_{n+1}=(\lambda +\mu )P_{n}}
4330:
Jackson, James R. (Oct 1963). "Jobshop-like Queueing Systems".
4083:, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory.
3893:
Mathematical Proceedings of the Cambridge Philosophical Society
2919:
inter-arrival and service time distributions to be considered.
187:
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
4128:
3437:"Performance by Design: Computer Capacity Planning by Example"
3435:
Lawrence W. Dowdy, Virgilio A.F. Almeida, Daniel A. Menasce.
3051:
Single server: customers line up and there is only one server
2958:
2832:
2006:{\displaystyle \left\vert E_{n}-L_{n}\right\vert \in \{0,1\}}
5141:
5002:, (MIT, Cambridge, May 31, 1961) Proposal for a Ph.D. Thesis
3023:
The next job to be served is the one with the smallest size.
2236:{\displaystyle \lambda P_{0}+\mu P_{2}=(\lambda +\mu )P_{1}}
512:
490:
4654:(Sep 1993). "G-Networks with Triggered Customer Movement".
4464:(1967). "Closed Queuing Systems with Exponential Servers".
4369:"Mean-Value Analysis of Closed Multichain Queuing Networks"
4096:(2012). "Scheduling: Non-Preemptive, Size-Based Policies".
3226:, which affects the characteristics of the larger network.
3187:
Networks of customers have also been investigated, such as
2948:
Various scheduling policies can be used at queueing nodes:
2839:
in 1930, a solution later recast in probabilistic terms by
1729:, fully describes the required steady state probabilities.
927:{\displaystyle P_{1}={\frac {\lambda _{0}}{\mu _{1}}}P_{0}}
257:
by "births" and "deaths", which occur at the arrival rates
4500:
3463:"Hershey Medical Center to open redesigned emergency room"
3733:"The theory of probabilities and telephone conversations"
3249:
1779:, the G stands for "general" and indicates an arbitrary
536:
equations for the birth-and-death process, known as the
168:
A black box. Jobs arrive to, and depart from, the queue.
5142:
LINE: a general-purpose engine to solve queueing models
5098:
4159:(2012). "Scheduling: Preemptive, Size-Based Policies".
3663:"Agner Krarup Erlang (1878-1929) | plus.maths.org"
3491:
3148:
The simplest non-trivial networks of queues are called
3109:
nodes, the state of the system can be described by an
2820:
describes the number of servers at the queueing node (
1941:
represent the number of times the system leaves state
1910:
represent the number of times the system enters state
5014:
Communication Nets: Stochastic Message Flow and Delay
4855:"A stable queueing network with unstable fluid model"
2997:
Service capacity is shared equally between customers.
2725:
2706:{\displaystyle {\frac {\mu }{\lambda }}=e^{-W{\mu }}}
2671:
2619:
2545:
2487:
2428:
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1951:
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263:
4949:(revisited first ed.). Addison-Wesley. p.
2106:{\displaystyle \left\vert E_{n}-L_{n}\right\vert =1}
567:
denotes the steady state probability to be in state
5056:
Queueing Systems: Volume II – Computer Applications
4896:
4733:
4412:"On the arrival theorem for communication networks"
4197:
Performance Modeling and Design of Computer Systems
4161:
Performance Modeling and Design of Computer Systems
4098:
Performance Modeling and Design of Computer Systems
3009:(where a job in service cannot be interrupted) and
2571:{\displaystyle \rho ={\frac {\lambda }{\mu }}<1}
5127:Teknomo's Queueing theory tutorial and calculators
5024:
4968:
3152:. The first significant results in this area were
2762:
2705:
2647:{\displaystyle L={\frac {\lambda -\sigma }{\mu }}}
2646:
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2528:
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2235:
2160:
2105:
2049:
2005:
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403:
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276:
212:customers is called a queue with a buffer of size
3772:(2009). "The first Erlang century—and the next".
3410:Sundarapandian, V. (2009). "7. Queueing Theory".
3145:represents the number of customers at each node.
2862:. In 1957, Pollaczek studied the GI/G/1 using an
5634:
4042:Business Process Modeling, Simulation and Design
3620:
2888:worked on the application of queueing theory to
2586:A common basic queueing system is attributed to
1755:the distribution of service times for jobs, and
4366:
3945:Communications in Statistics. Stochastic Models
627:{\displaystyle \mu _{1}P_{1}=\lambda _{0}P_{0}}
230:The behaviour of a single queue (also called a
75:Queueing theory has its origins in research by
4191:
4155:
4092:
3409:
2716:The second equation is commonly rewritten as:
27:Mathematical study of waiting lines, or queues
5165:
4004:
3886:
3686:
3534:
3532:
3530:
2932:Problems such as performance metrics for the
5007:Information Flow in Large Communication Nets
5000:Information Flow in Large Communication Nets
4459:
4075:
4073:
4071:
4069:
2581:
2000:
1988:
3818:
3816:
3814:
3492:Mayhew, Les; Smith, David (December 2006).
3412:Probability, Statistics and Queueing Theory
208:). A setting with a waiting zone for up to
5172:
5158:
5099:Jon Kleinberg; Éva Tardos (30 June 2013).
5009:(RLE Quarterly Progress Report, July 1961)
4972:Analysis and Synthesis of Computer Systems
4539:
3764:
3762:
3760:
3527:
3460:
5059:. New York: Wiley Interscience. pp.
5049:
5031:. New York: Wiley Interscience. pp.
5019:
4933:
4870:
4829:
4774:
4522:
4386:
4227:
4195:(2012). "Scheduling: SRPT and Fairness".
4129:Andrew S. Tanenbaum; Herbert Bos (2015).
4066:
4034:
4032:
3979:
3942:
3845:
3638:
3597:
2773:The two-stage one-box model is common in
253:The system transitions between values of
4918:
4409:
4045:. Pearson Education India. p. 178.
3811:
3294:
2957:
2529:{\displaystyle P_{n}=(1-\rho )\rho ^{n}}
511:
489:
178:
163:
132:
36:
4852:
4650:
4329:
4323:
4291:
3969:
3822:
3768:
3757:
3576:
3237:consider the limiting behaviour of the
2962:First in first out (FIFO) queue example
2161:{\displaystyle \mu P_{1}=\lambda P_{0}}
1879:customers in the system in steady state
183:A queueing node with 3 servers. Server
14:
5635:
4989:
4942:
4897:Gross, Donald; Carl M. Harris (1998).
4811:
4696:
4505:; Muntz, R.R.; Palacios, F.G. (1975).
4277:: CS1 maint: archived copy as title (
4038:
4029:
3828:"Applied Probability in Great Britain"
3727:
3538:
3250:Heavy traffic/diffusion approximations
2983:will be served first. Also known as a
2943:
2807:model in 1920. In Kendall's notation:
1671:which, together with the equation for
219:
41:
5153:
4604:
4545:
4367:Reiser, M.; Lavenberg, S. S. (1980).
4295:(1957). "Networks of Waiting Lines".
3972:Proceedings of 14th European Workshop
3586:The Annals of Mathematical Statistics
3405:
3403:
3401:
3202:
2898:Massachusetts Institute of Technology
2467:{\displaystyle P_{0}+P_{1}+\cdots =1}
1732:
4946:An introduction to operating systems
4011:Computers & Chemical Engineering
4005:Carlson, E.C.; Felder, R.M. (1992).
3557:
3543:(13th ed.). New York: Pearson.
3229:
3158:product-form stationary distribution
3093:
527:
516:A queue with 1 server, arrival rate
83:and have since seen applications in
5179:
5137:Myron Hlynka's Queueing Theory Page
5027:Queueing Systems: Volume I – Theory
4969:Gelenbe, Erol; Isi Mitrani (2010).
1767:(where inter-arrival durations are
24:
4990:Newell, Gordron F. (1 June 1971).
4890:
4416:Computer Networks and ISDN Systems
4081:Chapter 8 – Queueing Systems
3691:(2009). "Editorial introduction".
3541:Introduction to management science
3473:from the original on June 29, 2016
3461:Schlechter, Kira (March 2, 2009).
3398:
3172:. This result was extended to the
3037:Shortest remaining processing time
2746:
2743:
1787:Example analysis of an M/M/1 queue
1593:
1464:
1407:
25:
5689:
5120:
4859:The Annals of Applied Probability
4818:The Annals of Applied Probability
4699:"Applications of Queueing Theory"
3665:. Pass.maths.org.uk. 1997-04-30.
3621:Hernández-Suarez, Carlos (2010).
1875:: the probability of there being
5619:
5618:
5132:Virtamo's Queueing Theory Course
4975:. World Scientific 2nd Edition.
2904:, a forerunner to the Internet.
4992:Applications of Queueing Theory
4899:Fundamentals of Queueing Theory
4846:
4805:
4768:
4727:
4690:
4644:
4598:
4587:from the original on 2016-05-13
4494:
4453:
4442:from the original on 2019-09-24
4403:
4360:
4285:
4260:from the original on 2017-03-29
4240:
4221:
4185:
4149:
4122:
4086:
3998:
3963:
3936:
3880:
3871:
3862:
3802:
3721:
3680:
3669:from the original on 2008-10-07
3443:from the original on 2016-05-06
3279:
246:by 1 and a departure decreases
4656:Journal of Applied Probability
4610:Journal of Applied Probability
3740:Nyt Tidsskrift for Matematik B
3655:
3614:
3570:
3565:Algorithmic Analysis of Queues
3485:
3454:
3428:
2513:
2501:
2307:
2295:
2220:
2208:
1716:
1704:
1060:
1014:
867:The first two equations imply
841:
815:
714:
688:
474:
429:
398:
353:
145:can be thought of as nearly a
120:
13:
1:
5449:Flow-equivalent server method
5016:(McGraw-Hill, New York, 1964)
3392:
3331:Project production management
3028:Preemptive shortest job first
60:is the mathematical study of
5530:Adversarial queueing network
5419:Continuous-time Markov chain
4428:10.1016/0169-7552(93)90073-D
4205:10.1017/CBO9781139226424.041
4169:10.1017/CBO9781139226424.040
4106:10.1017/CBO9781139226424.039
4023:10.1016/0098-1354(92)80018-5
3218:gives optimal throughput. A
277:{\displaystyle \lambda _{i}}
7:
5492:Heavy traffic approximation
5237:Pollaczek–Khinchine formula
4943:Deitel, Harvey M. (1984) .
3981:10.1007/978-3-319-66583-2_6
3847:10.1287/opre.50.1.227.17792
3539:Taylor, Bernard W. (2019).
3303:
3256:Heavy traffic approximation
3180:can be calculated with the
2845:Pollaczek–Khinchine formula
2119:Thus the balance equations
2050:{\displaystyle E_{n}=L_{n}}
1175:By mathematical induction,
106:
10:
5694:
3283:
3266:Ornstein–Uhlenbeck process
3253:
3214:visit more than one node,
3206:
2780:
1736:
223:
29:
5616:
5548:
5507:
5497:Reflected Brownian motion
5474:
5411:
5370:
5315:
5302:Markovian arrival process
5289:
5187:
4965:chap.15, pp. 380–412
4711:10.1007/978-94-009-5970-5
4558:Communications of the ACM
3957:10.1080/15326348808807077
3914:10.1017/S0305004100036094
3788:10.1007/s11134-009-9147-4
3707:10.1007/s11134-009-9151-8
3262:reflected Brownian motion
3156:, for which an efficient
3071:Customer waiting behavior
2915:have allowed queues with
2590:and is a modification of
2582:Simple two-equation queue
1769:exponentially distributed
1722:{\displaystyle (n\geq 1)}
5520:Layered queueing network
5307:Rational arrival process
4919:Zukerman, Moshe (2013).
4410:Van Dijk, N. M. (1993).
4132:Modern Operating Systems
3381:Traffic generation model
2967:first-come, first-served
2940:remain an open problem.
2594:. Given an arrival rate
1808:{\displaystyle \lambda }
1781:probability distribution
411:and a departure rate of
304:{\displaystyle \mu _{i}}
284:and the departure rates
234:) can be described by a
47:equilibrium distribution
32:First Come, First Served
5608:Teletraffic engineering
5403:Shortest remaining time
4872:10.1214/aoap/1029962815
4831:10.1214/aoap/1177004602
4230:Proceedings of FRUCT 24
4039:Manuel, Laguna (2011).
3746:: 33–39. Archived from
3599:10.1214/aoms/1177728975
3346:Queue management system
2913:matrix analytic methods
2909:matrix geometric method
2892:in the early 1960s and
2870:gave a formula for the
2602:, and a departure rate
540:, are as follows. Here
81:teletraffic engineering
5603:Scheduling (computing)
5242:Matrix analytic method
5084:. Prentice-Hall, Inc.
4697:Newell, G. F. (1982).
4346:10.1287/mnsc.1040.0268
3640:10.3934/mbe.2010.7.809
3371:Scheduling (computing)
3356:Random early detection
2963:
2917:phase-type distributed
2764:
2707:
2648:
2606:, length of the queue
2572:
2530:
2476:geometric distribution
2468:
2413:
2324:
2237:
2162:
2107:
2051:
2007:
1935:
1904:
1869:
1833:
1809:
1723:
1692:
1662:
1624:
1597:
1536:
1495:
1468:
1411:
1374:
1339:
1165:
928:
858:
731:
628:
561:
524:
509:
481:
405:
325:
305:
278:
196:
169:
101:industrial engineering
54:
5434:Product-form solution
5335:Gordon–Newell theorem
5297:Poisson point process
5188:Single queueing nodes
4570:10.1145/362342.362345
4524:10.1145/321879.321887
4480:10.1287/opre.15.2.254
4388:10.1145/322186.322195
3351:Queuing Rule of Thumb
3295:Queueing Applications
3209:Stochastic scheduling
3170:Gordon–Newell theorem
3113:–dimensional vector (
2961:
2843:and now known as the
2765:
2708:
2649:
2573:
2531:
2469:
2414:
2325:
2238:
2163:
2108:
2052:
2008:
1936:
1934:{\displaystyle L_{n}}
1905:
1903:{\displaystyle E_{n}}
1870:
1868:{\displaystyle P_{n}}
1834:
1810:
1724:
1693:
1691:{\displaystyle P_{n}}
1663:
1598:
1577:
1537:
1469:
1448:
1391:
1375:
1313:
1166:
929:
859:
732:
629:
562:
560:{\displaystyle P_{n}}
515:
493:
482:
406:
326:
306:
279:
182:
167:
133:Single queueing nodes
40:
5461:Decomposition method
4853:Bramson, M. (1999).
4744:10.1109/QEST.2008.47
4309:10.1287/opre.5.4.518
4199:. pp. 518–530.
4163:. pp. 508–517.
4100:. pp. 499–507.
3729:Erlang, Agner Krarup
3516:on September 7, 2021
3500:Cass Business School
3216:backpressure routing
3195:, first proposed by
3184:, proposed in 1973.
3178:normalizing constant
2852:David George Kendall
2723:
2669:
2617:
2543:
2485:
2426:
2340:
2248:
2173:
2126:
2061:
2021:
1949:
1918:
1887:
1852:
1832:{\displaystyle \mu }
1823:
1799:
1701:
1675:
1549:
1388:
1182:
944:
874:
742:
639:
578:
544:
501:and departure rates
415:
339:
315:
288:
261:
5673:Network performance
5658:Operations research
5653:Customer experience
5648:Production planning
5593:Pipeline (software)
5573:Flow control (data)
5568:Erlang distribution
5550:Information systems
5340:Mean value analysis
5012:Leonard Kleinrock.
5005:Leonard Kleinrock.
4998:Leonard Kleinrock,
4994:. Chapman and Hall.
4812:Yamada, K. (1995).
4467:Operations Research
4297:Operations Research
3906:1961PCPS...57..902K
3833:Operations Research
3162:mean value analysis
2953:First in, first out
2944:Service disciplines
2927:product development
2789:Agner Krarup Erlang
1783:for service times.
520:and departure rate
236:birth–death process
220:Birth-death process
99:, and particularly
89:traffic engineering
77:Agner Krarup Erlang
70:operations research
5598:Quality of service
5583:Network congestion
5444:Quasireversibility
5424:Kendall's notation
5051:Kleinrock, Leonard
5023:(2 January 1975).
5021:Kleinrock, Leonard
4791:10.1007/BF01149260
4511:Journal of the ACM
4374:Journal of the ACM
4333:Management Science
4193:Harchol-Balter, M.
4157:Harchol-Balter, M.
4094:Harchol-Balter, M.
3326:Network simulation
3268:, or more general
3224:queueing algorithm
3203:Routing algorithms
3019:Shortest job first
2975:Last in, first out
2964:
2860:Kendall's notation
2841:Aleksandr Khinchin
2760:
2703:
2644:
2568:
2526:
2464:
2409:
2320:
2233:
2158:
2103:
2047:
2003:
1931:
1900:
1865:
1829:
1805:
1739:Kendall's notation
1733:Kendall's notation
1719:
1688:
1658:
1532:
1370:
1161:
924:
854:
727:
624:
557:
525:
510:
477:
401:
321:
301:
274:
197:
170:
127:management science
97:project management
85:telecommunications
55:
18:Stochastic network
5630:
5629:
5588:Network scheduler
5487:Mean-field theory
5398:Shortest job next
5388:Processor sharing
5345:Buzen's algorithm
5328:Traffic equations
5316:Queueing networks
5290:Arrival processes
5264:Kingman's formula
5112:978-1-292-02394-6
5091:978-0-13-746975-8
5070:978-0-471-49111-8
5053:(22 April 1976).
5042:978-0-471-49110-1
4982:978-1-908978-42-4
4960:978-0-201-14502-1
4908:978-0-471-32812-4
4753:978-0-7695-3360-5
4720:978-94-009-5972-9
4422:(10): 1135–2013.
4214:978-1-139-22642-4
4178:978-1-139-22642-4
4142:978-0-13-359162-0
4115:978-1-139-22642-4
4052:978-81-317-6135-9
3991:978-3-319-66582-5
3888:Kingman, J. F. C.
3770:Kingman, J. F. C.
3687:Asmussen, S. R.;
3550:978-0-13-473066-0
3509:978-1-905752-06-5
3421:978-81-203-3844-9
3270:diffusion process
3239:empirical measure
3235:Mean-field models
3230:Mean-field limits
3220:network scheduler
3182:Buzen's algorithm
3094:Queueing networks
3062:Unreliable server
2993:Processor sharing
2890:message switching
2886:Leonard Kleinrock
2880:Kingman's formula
2872:mean waiting time
2864:integral equation
2758:
2740:
2680:
2642:
2598:, a dropout rate
2560:
2387:
2364:
1656:
1653:
1524:
1368:
1288:
1149:
1088:
1012:
982:
912:
538:balance equations
528:Balance equations
427:
351:
324:{\displaystyle i}
226:Survival analysis
16:(Redirected from
5685:
5622:
5621:
5439:Balance equation
5371:Service policies
5269:Lindley equation
5174:
5167:
5160:
5151:
5150:
5116:
5102:Algorithm Design
5095:
5074:
5046:
5030:
4995:
4986:
4964:
4939:
4937:
4927:
4912:
4885:
4884:
4874:
4850:
4844:
4843:
4833:
4809:
4803:
4802:
4778:Queueing Systems
4772:
4766:
4765:
4731:
4725:
4724:
4694:
4688:
4687:
4648:
4642:
4641:
4602:
4596:
4595:
4593:
4592:
4586:
4555:
4543:
4537:
4536:
4526:
4498:
4492:
4491:
4457:
4451:
4450:
4448:
4447:
4407:
4401:
4400:
4390:
4364:
4358:
4357:
4327:
4321:
4320:
4289:
4283:
4282:
4276:
4268:
4266:
4265:
4259:
4252:
4244:
4238:
4237:
4225:
4219:
4218:
4189:
4183:
4182:
4153:
4147:
4146:
4126:
4120:
4119:
4090:
4084:
4077:
4064:
4063:
4061:
4059:
4036:
4027:
4026:
4002:
3996:
3995:
3983:
3967:
3961:
3960:
3940:
3934:
3933:
3884:
3878:
3875:
3869:
3866:
3860:
3859:
3849:
3820:
3809:
3806:
3800:
3799:
3775:Queueing Systems
3766:
3755:
3754:
3752:
3737:
3725:
3719:
3718:
3694:Queueing Systems
3684:
3678:
3677:
3675:
3674:
3659:
3653:
3652:
3642:
3618:
3612:
3611:
3601:
3574:
3568:
3561:
3555:
3554:
3536:
3525:
3524:
3522:
3521:
3512:. Archived from
3489:
3483:
3482:
3480:
3478:
3467:The Patriot-News
3458:
3452:
3451:
3449:
3448:
3432:
3426:
3425:
3414:. PHI Learning.
3407:
3154:Jackson networks
3105:For networks of
3100:customer routing
3046:Service facility
2894:packet switching
2854:solved the GI/M/
2769:
2767:
2766:
2761:
2759:
2751:
2749:
2741:
2733:
2712:
2710:
2709:
2704:
2702:
2701:
2700:
2681:
2673:
2653:
2651:
2650:
2645:
2643:
2638:
2627:
2577:
2575:
2574:
2569:
2561:
2553:
2535:
2533:
2532:
2527:
2525:
2524:
2497:
2496:
2473:
2471:
2470:
2465:
2451:
2450:
2438:
2437:
2418:
2416:
2415:
2410:
2385:
2381:
2380:
2365:
2357:
2352:
2351:
2329:
2327:
2326:
2321:
2319:
2318:
2291:
2290:
2269:
2268:
2242:
2240:
2239:
2234:
2232:
2231:
2204:
2203:
2188:
2187:
2167:
2165:
2164:
2159:
2157:
2156:
2141:
2140:
2112:
2110:
2109:
2104:
2096:
2092:
2091:
2090:
2078:
2077:
2056:
2054:
2053:
2048:
2046:
2045:
2033:
2032:
2012:
2010:
2009:
2004:
1984:
1980:
1979:
1978:
1966:
1965:
1940:
1938:
1937:
1932:
1930:
1929:
1909:
1907:
1906:
1901:
1899:
1898:
1874:
1872:
1871:
1866:
1864:
1863:
1838:
1836:
1835:
1830:
1814:
1812:
1811:
1806:
1728:
1726:
1725:
1720:
1697:
1695:
1694:
1689:
1687:
1686:
1667:
1665:
1664:
1659:
1657:
1655:
1654:
1652:
1651:
1636:
1635:
1626:
1623:
1612:
1596:
1591:
1566:
1561:
1560:
1541:
1539:
1538:
1533:
1525:
1523:
1522:
1507:
1506:
1497:
1494:
1483:
1467:
1462:
1447:
1446:
1434:
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1421:
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1405:
1379:
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1371:
1369:
1367:
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1351:
1350:
1341:
1338:
1327:
1312:
1311:
1299:
1298:
1289:
1287:
1286:
1285:
1273:
1272:
1257:
1256:
1246:
1245:
1244:
1232:
1231:
1216:
1215:
1199:
1194:
1193:
1170:
1168:
1167:
1162:
1160:
1159:
1150:
1148:
1147:
1146:
1137:
1136:
1126:
1125:
1124:
1115:
1114:
1104:
1099:
1098:
1089:
1087:
1086:
1077:
1076:
1067:
1059:
1058:
1049:
1048:
1036:
1035:
1026:
1025:
1013:
1011:
1010:
998:
993:
992:
983:
981:
980:
971:
970:
961:
956:
955:
933:
931:
930:
925:
923:
922:
913:
911:
910:
901:
900:
891:
886:
885:
863:
861:
860:
855:
853:
852:
840:
839:
827:
826:
811:
810:
795:
794:
776:
775:
760:
759:
736:
734:
733:
728:
726:
725:
713:
712:
700:
699:
684:
683:
674:
673:
661:
660:
651:
650:
633:
631:
630:
625:
623:
622:
613:
612:
600:
599:
590:
589:
566:
564:
563:
558:
556:
555:
486:
484:
483:
478:
473:
472:
454:
453:
441:
440:
428:
425:
410:
408:
407:
402:
397:
396:
378:
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365:
364:
352:
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328:
327:
322:
310:
308:
307:
302:
300:
299:
283:
281:
280:
275:
273:
272:
114:Queueing Systems
21:
5693:
5692:
5688:
5687:
5686:
5684:
5683:
5682:
5663:Formal sciences
5643:Queueing theory
5633:
5632:
5631:
5626:
5612:
5544:
5503:
5470:
5456:Arrival theorem
5407:
5366:
5323:Jackson network
5311:
5285:
5276:Fork–join queue
5215:Burke's theorem
5183:
5181:Queueing theory
5178:
5147:
5123:
5113:
5092:
5071:
5043:
4983:
4961:
4925:
4909:
4893:
4891:Further reading
4888:
4851:
4847:
4810:
4806:
4773:
4769:
4754:
4738:. p. 215.
4732:
4728:
4721:
4695:
4691:
4668:10.2307/3214781
4649:
4645:
4622:10.2307/3212869
4603:
4599:
4590:
4588:
4584:
4553:
4544:
4540:
4503:Chandy, K. Mani
4499:
4495:
4460:Gordon, W. J.;
4458:
4454:
4445:
4443:
4408:
4404:
4365:
4361:
4328:
4324:
4290:
4286:
4270:
4269:
4263:
4261:
4257:
4250:
4248:"Archived copy"
4246:
4245:
4241:
4226:
4222:
4215:
4190:
4186:
4179:
4154:
4150:
4143:
4127:
4123:
4116:
4091:
4087:
4078:
4067:
4057:
4055:
4053:
4037:
4030:
4003:
3999:
3992:
3968:
3964:
3941:
3937:
3885:
3881:
3876:
3872:
3867:
3863:
3821:
3812:
3807:
3803:
3767:
3758:
3750:
3735:
3726:
3722:
3685:
3681:
3672:
3670:
3661:
3660:
3656:
3619:
3615:
3575:
3571:
3562:
3558:
3551:
3537:
3528:
3519:
3517:
3510:
3490:
3486:
3476:
3474:
3459:
3455:
3446:
3444:
3433:
3429:
3422:
3408:
3399:
3395:
3390:
3321:Line management
3311:Ehrenfest model
3306:
3297:
3288:
3282:
3258:
3252:
3232:
3211:
3205:
3160:exists and the
3144:
3135:
3126:
3119:
3096:
2946:
2878:, now known as
2837:Felix Pollaczek
2824:= 1, 2, 3, ...)
2795:and solved the
2793:Poisson process
2783:
2750:
2742:
2732:
2724:
2721:
2720:
2696:
2689:
2685:
2672:
2670:
2667:
2666:
2628:
2626:
2618:
2615:
2614:
2610:is defined as:
2584:
2552:
2544:
2541:
2540:
2520:
2516:
2492:
2488:
2486:
2483:
2482:
2446:
2442:
2433:
2429:
2427:
2424:
2423:
2370:
2366:
2356:
2347:
2343:
2341:
2338:
2337:
2314:
2310:
2280:
2276:
2258:
2254:
2249:
2246:
2245:
2227:
2223:
2199:
2195:
2183:
2179:
2174:
2171:
2170:
2152:
2148:
2136:
2132:
2127:
2124:
2123:
2086:
2082:
2073:
2069:
2068:
2064:
2062:
2059:
2058:
2041:
2037:
2028:
2024:
2022:
2019:
2018:
1974:
1970:
1961:
1957:
1956:
1952:
1950:
1947:
1946:
1925:
1921:
1919:
1916:
1915:
1894:
1890:
1888:
1885:
1884:
1859:
1855:
1853:
1850:
1849:
1824:
1821:
1820:
1800:
1797:
1796:
1789:
1765:Poisson process
1741:
1735:
1702:
1699:
1698:
1682:
1678:
1676:
1673:
1672:
1641:
1637:
1631:
1627:
1625:
1613:
1602:
1592:
1581:
1570:
1565:
1556:
1552:
1550:
1547:
1546:
1512:
1508:
1502:
1498:
1496:
1484:
1473:
1463:
1452:
1442:
1438:
1429:
1425:
1416:
1412:
1406:
1395:
1389:
1386:
1385:
1356:
1352:
1346:
1342:
1340:
1328:
1317:
1307:
1303:
1294:
1290:
1281:
1277:
1262:
1258:
1252:
1248:
1247:
1240:
1236:
1221:
1217:
1205:
1201:
1200:
1198:
1189:
1185:
1183:
1180:
1179:
1155:
1151:
1142:
1138:
1132:
1128:
1127:
1120:
1116:
1110:
1106:
1105:
1103:
1094:
1090:
1082:
1078:
1072:
1068:
1066:
1054:
1050:
1044:
1040:
1031:
1027:
1021:
1017:
1006:
1002:
997:
988:
984:
976:
972:
966:
962:
960:
951:
947:
945:
942:
941:
918:
914:
906:
902:
896:
892:
890:
881:
877:
875:
872:
871:
848:
844:
835:
831:
822:
818:
800:
796:
784:
780:
765:
761:
749:
745:
743:
740:
739:
721:
717:
708:
704:
695:
691:
679:
675:
669:
665:
656:
652:
646:
642:
640:
637:
636:
618:
614:
608:
604:
595:
591:
585:
581:
579:
576:
575:
551:
547:
545:
542:
541:
530:
506:
499:
468:
464:
449:
445:
436:
432:
424:
416:
413:
412:
392:
388:
373:
369:
360:
356:
348:
340:
337:
336:
316:
313:
312:
295:
291:
289:
286:
285:
268:
264:
262:
259:
258:
228:
222:
135:
123:
109:
58:Queueing theory
53:is fundamental.
51:transient state
35:
28:
23:
22:
15:
12:
11:
5:
5691:
5681:
5680:
5675:
5670:
5665:
5660:
5655:
5650:
5645:
5628:
5627:
5617:
5614:
5613:
5611:
5610:
5605:
5600:
5595:
5590:
5585:
5580:
5575:
5570:
5565:
5560:
5554:
5552:
5546:
5545:
5543:
5542:
5537:
5532:
5527:
5525:Polling system
5522:
5517:
5511:
5509:
5505:
5504:
5502:
5501:
5500:
5499:
5489:
5484:
5478:
5476:
5475:Limit theorems
5472:
5471:
5469:
5468:
5463:
5458:
5453:
5452:
5451:
5446:
5441:
5431:
5426:
5421:
5415:
5413:
5409:
5408:
5406:
5405:
5400:
5395:
5390:
5385:
5380:
5374:
5372:
5368:
5367:
5365:
5364:
5359:
5354:
5349:
5348:
5347:
5342:
5332:
5331:
5330:
5319:
5317:
5313:
5312:
5310:
5309:
5304:
5299:
5293:
5291:
5287:
5286:
5284:
5283:
5278:
5273:
5272:
5271:
5266:
5256:
5251:
5246:
5245:
5244:
5239:
5229:
5224:
5219:
5218:
5217:
5207:
5202:
5197:
5191:
5189:
5185:
5184:
5177:
5176:
5169:
5162:
5154:
5145:
5144:
5139:
5134:
5129:
5122:
5121:External links
5119:
5118:
5117:
5111:
5096:
5090:
5075:
5069:
5047:
5041:
5017:
5010:
5003:
4996:
4987:
4981:
4966:
4959:
4940:
4916:
4907:
4892:
4889:
4887:
4886:
4865:(3): 818–853.
4845:
4824:(4): 958–982.
4804:
4767:
4752:
4726:
4719:
4689:
4662:(3): 742–748.
4643:
4616:(3): 542–554.
4597:
4564:(9): 527–531.
4538:
4517:(2): 248–260.
4493:
4452:
4402:
4359:
4340:(1): 131–142.
4322:
4303:(4): 518–521.
4293:Jackson, J. R.
4284:
4239:
4220:
4213:
4184:
4177:
4148:
4141:
4121:
4114:
4085:
4079:Penttinen A.,
4065:
4051:
4028:
4017:(7): 707–718.
3997:
3990:
3962:
3935:
3879:
3870:
3861:
3840:(1): 227–239.
3810:
3801:
3756:
3753:on 2011-10-01.
3720:
3679:
3654:
3633:(4): 809–823.
3613:
3592:(3): 338–354.
3578:Kendall, D. G.
3569:
3556:
3549:
3526:
3508:
3484:
3453:
3427:
3420:
3396:
3394:
3391:
3389:
3388:
3383:
3378:
3373:
3368:
3363:
3361:Renewal theory
3358:
3353:
3348:
3343:
3341:Queueing delay
3338:
3333:
3328:
3323:
3318:
3313:
3307:
3305:
3302:
3296:
3293:
3284:Main article:
3281:
3278:
3254:Main article:
3251:
3248:
3231:
3228:
3222:must choose a
3204:
3201:
3189:Kelly networks
3166:closed network
3140:
3131:
3124:
3117:
3095:
3092:
3084:
3083:
3080:
3077:
3073:
3072:
3064:
3063:
3059:
3058:
3055:
3052:
3048:
3047:
3043:
3042:
3039:
3033:
3032:
3029:
3025:
3024:
3021:
3015:
3014:
3007:non-preemptive
3003:
2999:
2998:
2995:
2989:
2988:
2977:
2971:
2970:
2955:
2945:
2942:
2835:was solved by
2826:
2825:
2815:
2812:
2782:
2779:
2771:
2770:
2757:
2754:
2748:
2745:
2739:
2736:
2731:
2728:
2714:
2713:
2699:
2695:
2692:
2688:
2684:
2679:
2676:
2656:
2655:
2641:
2637:
2634:
2631:
2625:
2622:
2583:
2580:
2567:
2564:
2559:
2556:
2551:
2548:
2537:
2536:
2523:
2519:
2515:
2512:
2509:
2506:
2503:
2500:
2495:
2491:
2463:
2460:
2457:
2454:
2449:
2445:
2441:
2436:
2432:
2422:The fact that
2420:
2419:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2384:
2379:
2376:
2373:
2369:
2363:
2360:
2355:
2350:
2346:
2331:
2330:
2317:
2313:
2309:
2306:
2303:
2300:
2297:
2294:
2289:
2286:
2283:
2279:
2275:
2272:
2267:
2264:
2261:
2257:
2253:
2243:
2230:
2226:
2222:
2219:
2216:
2213:
2210:
2207:
2202:
2198:
2194:
2191:
2186:
2182:
2178:
2168:
2155:
2151:
2147:
2144:
2139:
2135:
2131:
2102:
2099:
2095:
2089:
2085:
2081:
2076:
2072:
2067:
2044:
2040:
2036:
2031:
2027:
2002:
1999:
1996:
1993:
1990:
1987:
1983:
1977:
1973:
1969:
1964:
1960:
1955:
1928:
1924:
1897:
1893:
1881:
1880:
1862:
1858:
1847:
1841:
1828:
1817:
1804:
1788:
1785:
1773:Markov process
1737:Main article:
1734:
1731:
1718:
1715:
1712:
1709:
1706:
1685:
1681:
1669:
1668:
1650:
1647:
1644:
1640:
1634:
1630:
1622:
1619:
1616:
1611:
1608:
1605:
1601:
1595:
1590:
1587:
1584:
1580:
1576:
1573:
1569:
1564:
1559:
1555:
1531:
1528:
1521:
1518:
1515:
1511:
1505:
1501:
1493:
1490:
1487:
1482:
1479:
1476:
1472:
1466:
1461:
1458:
1455:
1451:
1445:
1441:
1437:
1432:
1428:
1424:
1419:
1415:
1409:
1404:
1401:
1398:
1394:
1384:The condition
1382:
1381:
1365:
1362:
1359:
1355:
1349:
1345:
1337:
1334:
1331:
1326:
1323:
1320:
1316:
1310:
1306:
1302:
1297:
1293:
1284:
1280:
1276:
1271:
1268:
1265:
1261:
1255:
1251:
1243:
1239:
1235:
1230:
1227:
1224:
1220:
1214:
1211:
1208:
1204:
1197:
1192:
1188:
1173:
1172:
1158:
1154:
1145:
1141:
1135:
1131:
1123:
1119:
1113:
1109:
1102:
1097:
1093:
1085:
1081:
1075:
1071:
1065:
1062:
1057:
1053:
1047:
1043:
1039:
1034:
1030:
1024:
1020:
1016:
1009:
1005:
1001:
996:
991:
987:
979:
975:
969:
965:
959:
954:
950:
935:
934:
921:
917:
909:
905:
899:
895:
889:
884:
880:
865:
864:
851:
847:
843:
838:
834:
830:
825:
821:
817:
814:
809:
806:
803:
799:
793:
790:
787:
783:
779:
774:
771:
768:
764:
758:
755:
752:
748:
737:
724:
720:
716:
711:
707:
703:
698:
694:
690:
687:
682:
678:
672:
668:
664:
659:
655:
649:
645:
634:
621:
617:
611:
607:
603:
598:
594:
588:
584:
554:
550:
529:
526:
504:
497:
476:
471:
467:
463:
460:
457:
452:
448:
444:
439:
435:
431:
423:
420:
400:
395:
391:
387:
384:
381:
376:
372:
368:
363:
359:
355:
347:
344:
320:
298:
294:
271:
267:
221:
218:
134:
131:
122:
119:
108:
105:
42:Queue networks
26:
9:
6:
4:
3:
2:
5690:
5679:
5678:Markov models
5676:
5674:
5671:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5649:
5646:
5644:
5641:
5640:
5638:
5625:
5615:
5609:
5606:
5604:
5601:
5599:
5596:
5594:
5591:
5589:
5586:
5584:
5581:
5579:
5578:Message queue
5576:
5574:
5571:
5569:
5566:
5564:
5563:Erlang (unit)
5561:
5559:
5556:
5555:
5553:
5551:
5547:
5541:
5540:Retrial queue
5538:
5536:
5533:
5531:
5528:
5526:
5523:
5521:
5518:
5516:
5513:
5512:
5510:
5506:
5498:
5495:
5494:
5493:
5490:
5488:
5485:
5483:
5480:
5479:
5477:
5473:
5467:
5464:
5462:
5459:
5457:
5454:
5450:
5447:
5445:
5442:
5440:
5437:
5436:
5435:
5432:
5430:
5427:
5425:
5422:
5420:
5417:
5416:
5414:
5410:
5404:
5401:
5399:
5396:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5375:
5373:
5369:
5363:
5360:
5358:
5355:
5353:
5352:Kelly network
5350:
5346:
5343:
5341:
5338:
5337:
5336:
5333:
5329:
5326:
5325:
5324:
5321:
5320:
5318:
5314:
5308:
5305:
5303:
5300:
5298:
5295:
5294:
5292:
5288:
5282:
5279:
5277:
5274:
5270:
5267:
5265:
5262:
5261:
5260:
5257:
5255:
5252:
5250:
5247:
5243:
5240:
5238:
5235:
5234:
5233:
5230:
5228:
5225:
5223:
5220:
5216:
5213:
5212:
5211:
5208:
5206:
5203:
5201:
5198:
5196:
5193:
5192:
5190:
5186:
5182:
5175:
5170:
5168:
5163:
5161:
5156:
5155:
5152:
5148:
5143:
5140:
5138:
5135:
5133:
5130:
5128:
5125:
5124:
5114:
5108:
5104:
5103:
5097:
5093:
5087:
5083:
5082:
5076:
5072:
5066:
5062:
5058:
5057:
5052:
5048:
5044:
5038:
5034:
5029:
5028:
5022:
5018:
5015:
5011:
5008:
5004:
5001:
4997:
4993:
4988:
4984:
4978:
4974:
4973:
4967:
4962:
4956:
4952:
4948:
4947:
4941:
4936:
4931:
4924:
4923:
4917:
4915:
4910:
4904:
4900:
4895:
4894:
4882:
4878:
4873:
4868:
4864:
4860:
4856:
4849:
4841:
4837:
4832:
4827:
4823:
4819:
4815:
4808:
4800:
4796:
4792:
4788:
4784:
4780:
4779:
4771:
4763:
4759:
4755:
4749:
4745:
4741:
4737:
4730:
4722:
4716:
4712:
4708:
4704:
4700:
4693:
4685:
4681:
4677:
4673:
4669:
4665:
4661:
4657:
4653:
4652:Gelenbe, Erol
4647:
4639:
4635:
4631:
4627:
4623:
4619:
4615:
4611:
4607:
4601:
4583:
4579:
4575:
4571:
4567:
4563:
4559:
4552:
4548:
4542:
4534:
4530:
4525:
4520:
4516:
4512:
4508:
4504:
4501:Baskett, F.;
4497:
4489:
4485:
4481:
4477:
4473:
4469:
4468:
4463:
4462:Newell, G. F.
4456:
4441:
4437:
4433:
4429:
4425:
4421:
4417:
4413:
4406:
4398:
4394:
4389:
4384:
4380:
4376:
4375:
4370:
4363:
4355:
4351:
4347:
4343:
4339:
4335:
4334:
4326:
4318:
4314:
4310:
4306:
4302:
4298:
4294:
4288:
4280:
4274:
4256:
4249:
4243:
4235:
4231:
4224:
4216:
4210:
4206:
4202:
4198:
4194:
4188:
4180:
4174:
4170:
4166:
4162:
4158:
4152:
4144:
4138:
4134:
4133:
4125:
4117:
4111:
4107:
4103:
4099:
4095:
4089:
4082:
4076:
4074:
4072:
4070:
4054:
4048:
4044:
4043:
4035:
4033:
4024:
4020:
4016:
4012:
4008:
4001:
3993:
3987:
3982:
3977:
3973:
3966:
3958:
3954:
3950:
3946:
3939:
3931:
3927:
3923:
3919:
3915:
3911:
3907:
3903:
3899:
3895:
3894:
3889:
3883:
3874:
3865:
3857:
3853:
3848:
3843:
3839:
3835:
3834:
3829:
3825:
3819:
3817:
3815:
3805:
3797:
3793:
3789:
3785:
3781:
3777:
3776:
3771:
3765:
3763:
3761:
3749:
3745:
3741:
3734:
3730:
3724:
3716:
3712:
3708:
3704:
3700:
3696:
3695:
3690:
3683:
3668:
3664:
3658:
3650:
3646:
3641:
3636:
3632:
3628:
3624:
3617:
3609:
3605:
3600:
3595:
3591:
3587:
3583:
3579:
3573:
3566:
3560:
3552:
3546:
3542:
3535:
3533:
3531:
3515:
3511:
3505:
3501:
3497:
3496:
3488:
3472:
3468:
3464:
3457:
3442:
3438:
3431:
3423:
3417:
3413:
3406:
3404:
3402:
3397:
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:
3312:
3309:
3308:
3301:
3292:
3287:
3277:
3275:
3271:
3267:
3263:
3257:
3247:
3244:
3240:
3236:
3227:
3225:
3221:
3217:
3210:
3200:
3198:
3194:
3190:
3185:
3183:
3179:
3175:
3171:
3167:
3163:
3159:
3155:
3151:
3150:tandem queues
3146:
3143:
3139:
3134:
3130:
3123:
3116:
3112:
3108:
3103:
3101:
3091:
3089:
3081:
3078:
3075:
3074:
3070:
3069:
3068:
3061:
3060:
3056:
3053:
3050:
3049:
3045:
3044:
3040:
3038:
3035:
3034:
3030:
3027:
3026:
3022:
3020:
3017:
3016:
3012:
3008:
3004:
3001:
3000:
2996:
2994:
2991:
2990:
2986:
2982:
2978:
2976:
2973:
2972:
2968:
2960:
2956:
2954:
2951:
2950:
2949:
2941:
2939:
2937:
2930:
2928:
2923:
2920:
2918:
2914:
2910:
2905:
2903:
2899:
2895:
2891:
2887:
2883:
2881:
2877:
2873:
2869:
2865:
2861:
2857:
2853:
2848:
2846:
2842:
2838:
2834:
2829:
2823:
2819:
2816:
2813:
2810:
2809:
2808:
2806:
2804:
2798:
2794:
2790:
2785:
2778:
2776:
2755:
2752:
2737:
2734:
2729:
2726:
2719:
2718:
2717:
2697:
2693:
2690:
2686:
2682:
2677:
2674:
2665:
2664:
2663:
2661:
2639:
2635:
2632:
2629:
2623:
2620:
2613:
2612:
2611:
2609:
2605:
2601:
2597:
2593:
2589:
2579:
2565:
2562:
2557:
2554:
2549:
2546:
2521:
2517:
2510:
2507:
2504:
2498:
2493:
2489:
2481:
2480:
2479:
2477:
2474:leads to the
2461:
2458:
2455:
2452:
2447:
2443:
2439:
2434:
2430:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2382:
2377:
2374:
2371:
2367:
2361:
2358:
2353:
2348:
2344:
2336:
2335:
2334:
2315:
2311:
2304:
2301:
2298:
2292:
2287:
2284:
2281:
2277:
2273:
2270:
2265:
2262:
2259:
2255:
2251:
2244:
2228:
2224:
2217:
2214:
2211:
2205:
2200:
2196:
2192:
2189:
2184:
2180:
2176:
2169:
2153:
2149:
2145:
2142:
2137:
2133:
2129:
2122:
2121:
2120:
2117:
2114:
2100:
2097:
2093:
2087:
2083:
2079:
2074:
2070:
2065:
2042:
2038:
2034:
2029:
2025:
2016:
1997:
1994:
1991:
1985:
1981:
1975:
1971:
1967:
1962:
1958:
1953:
1944:
1926:
1922:
1913:
1895:
1891:
1883:Further, let
1878:
1860:
1856:
1848:
1845:
1842:
1839:
1826:
1818:
1815:
1802:
1794:
1793:
1792:
1784:
1782:
1778:
1774:
1770:
1766:
1762:
1758:
1754:
1750:
1746:
1740:
1730:
1713:
1710:
1707:
1683:
1679:
1648:
1645:
1642:
1638:
1632:
1628:
1620:
1617:
1614:
1609:
1606:
1603:
1599:
1588:
1585:
1582:
1578:
1574:
1571:
1567:
1562:
1557:
1553:
1545:
1544:
1543:
1529:
1526:
1519:
1516:
1513:
1509:
1503:
1499:
1491:
1488:
1485:
1480:
1477:
1474:
1470:
1459:
1456:
1453:
1449:
1443:
1439:
1435:
1430:
1426:
1422:
1417:
1413:
1402:
1399:
1396:
1392:
1363:
1360:
1357:
1353:
1347:
1343:
1335:
1332:
1329:
1324:
1321:
1318:
1314:
1308:
1304:
1300:
1295:
1291:
1282:
1278:
1274:
1269:
1266:
1263:
1259:
1253:
1249:
1241:
1237:
1233:
1228:
1225:
1222:
1218:
1212:
1209:
1206:
1202:
1195:
1190:
1186:
1178:
1177:
1176:
1156:
1152:
1143:
1139:
1133:
1129:
1121:
1117:
1111:
1107:
1100:
1095:
1091:
1083:
1079:
1073:
1069:
1063:
1055:
1051:
1045:
1041:
1037:
1032:
1028:
1022:
1018:
1007:
1003:
999:
994:
989:
985:
977:
973:
967:
963:
957:
952:
948:
940:
939:
938:
919:
915:
907:
903:
897:
893:
887:
882:
878:
870:
869:
868:
849:
845:
836:
832:
828:
823:
819:
812:
807:
804:
801:
797:
791:
788:
785:
781:
777:
772:
769:
766:
762:
756:
753:
750:
746:
738:
722:
718:
709:
705:
701:
696:
692:
685:
680:
676:
670:
666:
662:
657:
653:
647:
643:
635:
619:
615:
609:
605:
601:
596:
592:
586:
582:
574:
573:
572:
570:
552:
548:
539:
535:
523:
519:
514:
507:
500:
492:
488:
469:
465:
461:
458:
455:
450:
446:
442:
437:
433:
421:
418:
393:
389:
385:
382:
379:
374:
370:
366:
361:
357:
345:
342:
334:
318:
311:for each job
296:
292:
269:
265:
256:
251:
249:
245:
241:
237:
233:
232:queueing node
227:
217:
215:
211:
207:
203:
194:
190:
186:
181:
177:
175:
166:
162:
160:
156:
153:(also called
152:
148:
144:
143:queueing node
140:
130:
128:
118:
116:
115:
104:
102:
98:
94:
90:
86:
82:
78:
73:
71:
67:
63:
62:waiting lines
59:
52:
48:
43:
39:
33:
19:
5535:Loss network
5466:Beneš method
5429:Little's law
5412:Key concepts
5362:BCMP network
5180:
5146:
5101:
5080:
5055:
5026:
5013:
5006:
4991:
4971:
4945:
4921:
4898:
4862:
4858:
4848:
4821:
4817:
4807:
4782:
4776:
4770:
4735:
4729:
4703:SpringerLink
4702:
4692:
4659:
4655:
4646:
4613:
4609:
4606:Kelly, F. P.
4600:
4589:. Retrieved
4561:
4557:
4547:Buzen, J. P.
4541:
4514:
4510:
4496:
4471:
4465:
4455:
4444:. Retrieved
4419:
4415:
4405:
4378:
4372:
4362:
4337:
4331:
4325:
4300:
4296:
4287:
4262:. Retrieved
4242:
4233:
4229:
4223:
4196:
4187:
4160:
4151:
4131:
4124:
4097:
4088:
4080:
4056:. Retrieved
4041:
4014:
4010:
4000:
3971:
3965:
3948:
3944:
3938:
3897:
3891:
3882:
3873:
3864:
3837:
3831:
3804:
3782:(1–4): 3–4.
3779:
3773:
3748:the original
3743:
3739:
3723:
3701:(1–4): 1–2.
3698:
3692:
3689:Boxma, O. J.
3682:
3671:. Retrieved
3657:
3630:
3627:Math. Biosci
3626:
3616:
3589:
3585:
3572:
3564:
3563:Tijms, H.C,
3559:
3540:
3518:. Retrieved
3514:the original
3494:
3487:
3475:. Retrieved
3466:
3456:
3445:. Retrieved
3430:
3411:
3386:Flow network
3298:
3289:
3280:Fluid limits
3259:
3242:
3233:
3212:
3197:Erol Gelenbe
3186:
3174:BCMP network
3165:
3147:
3141:
3137:
3132:
3128:
3121:
3114:
3110:
3106:
3104:
3099:
3097:
3087:
3085:
3065:
3010:
3006:
2981:waiting time
2966:
2965:Also called
2947:
2935:
2931:
2924:
2921:
2906:
2884:
2868:John Kingman
2855:
2849:
2830:
2827:
2821:
2817:
2802:
2799:in 1917 and
2786:
2784:
2775:epidemiology
2772:
2715:
2659:
2657:
2607:
2603:
2599:
2595:
2592:Little's Law
2585:
2538:
2421:
2332:
2118:
2115:
2014:
1942:
1911:
1882:
1876:
1843:
1819:
1795:
1790:
1756:
1752:
1748:
1744:
1742:
1670:
1383:
1174:
936:
866:
568:
534:steady state
531:
521:
517:
502:
495:
254:
252:
247:
243:
239:
231:
229:
213:
209:
206:waiting area
205:
201:
198:
192:
188:
184:
173:
171:
158:
154:
150:
142:
138:
136:
124:
112:
110:
74:
57:
56:
5558:Data buffer
5515:Fluid queue
5482:Fluid limit
5393:Round-robin
5259:G/G/1 queue
5254:G/M/1 queue
5249:M/G/k queue
5232:M/G/1 queue
5227:M/M/∞ queue
5222:M/M/c queue
5210:M/M/1 queue
5205:M/D/c queue
5200:M/D/1 queue
5195:D/M/1 queue
5105:. Pearson.
4135:. Pearson.
3951:: 183–188.
3824:Whittle, P.
3376:Traffic jam
3316:Erlang unit
3286:Fluid limit
2876:G/G/1 queue
2833:M/G/1 queue
2797:M/D/1 queue
1777:M/G/1 queue
1761:M/M/1 queue
121:Description
5637:Categories
5508:Extensions
5281:Bulk queue
4785:(4): 335.
4591:2015-09-01
4474:(2): 254.
4446:2019-09-24
4381:(2): 313.
4264:2018-08-02
3900:(4): 902.
3673:2013-04-22
3520:2008-05-20
3447:2009-07-08
3393:References
3366:Throughput
3336:Queue area
3207:See also:
3193:G-networks
3011:preemptive
2057:) or not (
224:See also:
5668:Rationing
5357:G-network
4935:1307.2968
4901:. Wiley.
4684:121673725
4058:6 October
3477:March 12,
2787:In 1909,
2756:μ
2753:λ
2738:μ
2698:μ
2691:−
2678:λ
2675:μ
2640:μ
2636:σ
2633:−
2630:λ
2558:μ
2555:λ
2547:ρ
2518:ρ
2511:ρ
2508:−
2456:⋯
2407:…
2375:−
2362:μ
2359:λ
2305:μ
2299:λ
2274:μ
2263:−
2252:λ
2218:μ
2212:λ
2193:μ
2177:λ
2146:λ
2130:μ
2080:−
1986:∈
1968:−
1827:μ
1803:λ
1775:). In an
1711:≥
1639:μ
1629:λ
1618:−
1600:∏
1594:∞
1579:∑
1542:leads to
1510:μ
1500:λ
1489:−
1471:∏
1465:∞
1450:∑
1408:∞
1393:∑
1354:μ
1344:λ
1333:−
1315:∏
1279:μ
1275:⋯
1267:−
1260:μ
1250:μ
1238:λ
1234:⋯
1226:−
1219:λ
1210:−
1203:λ
1140:μ
1130:μ
1118:λ
1108:λ
1080:μ
1070:λ
1042:λ
1038:−
1019:μ
1004:μ
974:μ
964:λ
904:μ
894:λ
833:μ
820:λ
782:μ
770:−
754:−
747:λ
706:μ
693:λ
667:μ
644:λ
606:λ
583:μ
466:μ
459:…
447:μ
434:μ
419:μ
390:λ
383:…
371:λ
358:λ
343:λ
293:μ
266:λ
155:customers
147:black box
93:computing
5624:Category
4638:51917794
4582:Archived
4549:(1973).
4533:15204199
4440:Archived
4436:45218280
4273:cite web
4255:Archived
4236:: 75–82.
3930:62590290
3826:(2002).
3796:38588726
3731:(1909).
3715:45664707
3667:Archived
3649:21077709
3580:(1953).
3471:Archived
3441:Archived
3304:See also
3136:) where
3088:dropouts
3002:Priority
2805:queueing
2478:formula
2013:for all
159:requests
107:Spelling
4881:2667284
4840:2245101
4799:1180930
4762:2714909
4676:3214781
4630:3212869
4397:8694947
4354:2627213
3922:2984229
3902:Bibcode
3856:3088474
3608:2236285
3274:orthant
3127:, ...,
2902:ARPANET
2781:History
1945:. Then
333:average
204:(or no
174:servers
5109:
5088:
5067:
5039:
4979:
4957:
4914:Online
4905:
4879:
4838:
4797:
4760:
4750:
4717:
4682:
4674:
4636:
4628:
4576:
4531:
4488:168557
4486:
4434:
4395:
4352:
4317:167249
4315:
4211:
4175:
4139:
4112:
4049:
3988:
3928:
3920:
3854:
3794:
3713:
3647:
3606:
3547:
3506:
3418:
2588:Erlang
2539:where
2386:
2333:imply
1914:, and
1747:where
250:by 1.
202:buffer
66:queues
4930:arXiv
4926:(PDF)
4877:JSTOR
4836:JSTOR
4795:S2CID
4758:S2CID
4680:S2CID
4672:JSTOR
4634:S2CID
4626:JSTOR
4585:(PDF)
4578:10702
4574:S2CID
4554:(PDF)
4529:S2CID
4484:JSTOR
4432:S2CID
4393:S2CID
4350:JSTOR
4313:JSTOR
4258:(PDF)
4251:(PDF)
3926:S2CID
3918:JSTOR
3852:JSTOR
3792:S2CID
3751:(PDF)
3736:(PDF)
3711:S2CID
3604:JSTOR
2985:stack
2938:queue
2874:in a
139:queue
64:, or
5383:LIFO
5378:FIFO
5107:ISBN
5086:ISBN
5065:ISBN
5037:ISBN
4977:ISBN
4955:ISBN
4903:ISBN
4748:ISBN
4715:ISBN
4279:link
4209:ISBN
4173:ISBN
4137:ISBN
4110:ISBN
4060:2017
4047:ISBN
3986:ISBN
3645:PMID
3545:ISBN
3504:ISBN
3479:2009
3416:ISBN
2934:M/G/
2911:and
2907:The
2831:The
2801:M/D/
2563:<
937:and
532:The
151:Jobs
5061:576
5033:417
4951:673
4867:doi
4826:doi
4787:doi
4740:doi
4707:doi
4664:doi
4618:doi
4566:doi
4519:doi
4476:doi
4424:doi
4383:doi
4342:doi
4305:doi
4201:doi
4165:doi
4102:doi
4019:doi
3976:doi
3953:doi
3910:doi
3842:doi
3784:doi
3703:doi
3635:doi
3594:doi
2113:).
426:avg
350:avg
157:or
141:or
5639::
5063:.
5035:.
4953:.
4928:.
4875:.
4861:.
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