• Queuing Theory Tutorial Pdf
• Queuing Theory Formula Pdf

Queuing theory is the mathematical study of queuing, or waiting in lines. Queues contain customers (or “items”) such as people, objects, or information. Queues form when there are limited resources for providing a service. For example, if there are 5 cash registers in a grocery store, queues will form if more than 5 customers wish to pay for their items at the same time.

A basic queuing system consists of an arrival process (how customers arrive at the queue, how many customers are present in total), the queue itself, the service process for attending to those customers, and departures from the system.

Mathematical queuing models are often used in software and business to determine the best way of using limited resources. Queueing models can answer questions such as: What is the probability that a customer will wait 10 minutes in line? What is the average waiting time per customer?

The following situations are examples of how queueing theory can be applied:

In queueing theory these interarrival times are usually assumed to be independent and identicallydistributedrandomvariables.Theotherrandomvariableistheservicetime, sometimesitiscalledservicerequest,work.ItsdistributionfunctionisdenotedbyB(x), thatis B(x) = P( servicetime. Queuing theory is the study of queues and the random processes that characterize them. It deals with making mathematical sense of real-life scenarios. For example, a mob of people queuing up at a bank or the tasks queuing up on your computer’s back end.

• Waiting in line at a bank or a store
• Waiting for a customer service representative to answer a call after the call has been placed on hold
• Waiting for a train to come
• Waiting for a computer to perform a task or respond
• Waiting for an automated car wash to clean a line of cars

Characterizing a Queuing System

Queuing models analyze how customers (including people, objects, and information) receive a service. A queuing system contains:

• Arrival process. The arrival process is simply how customers arrive. They may come into a queue alone or in groups, and they may arrive at certain intervals or randomly.
• Behavior. How do customers behave when they are in line? Some might be willing to wait for their place in the queue; others may become impatient and leave. Yet others might decide to rejoin the queue later, such as when they are put on hold with customer service and decide to call back in hopes of receiving faster service.
• How customers are serviced. This includes the length of time a customer is serviced, the number of servers available to help the customers, whether customers are served one by one or in batches, and the order in which customers are serviced, also called service discipline.
• Service discipline refers to the rule by which the next customer is selected. Although many retail scenarios employ the “first come, first served” rule, other situations may call for other types of service. For example, customers may be served in order of priority, or based on the number of items they need serviced (such as in an express lane in a grocery store). Sometimes, the last customer to arrive will be served first (such s in the case in a stack of dirty dishes, where the one on top will be the first to be washed).

• Waiting room. The number of customers allowed to wait in the queue may be limited based on the space available.

Mathematics of Queuing Theory

Kendall’s notation is a shorthand notation that specifies the parameters of a basic queuing model. Kendall’s notation is written in the form A/S/c/B/N/D, where each of the letters stand for different parameters.

• The A term describes when customers arrive at the queue – in particular, the time between arrivals, or interarrival times. Mathematically, this parameter specifies the probability distribution that the interarrival times follow. One common probability distribution used for the A term is the Poisson distribution.
• The S term describes how long it takes for a customer to be serviced after it leaves the queue. Mathematically, this parameter specifies the probability distribution that these service times follow. The Poisson distribution is also commonly used for the S term.
• The c term specifies the number of servers in the queuing system. The model assumes that all servers in the system are identical, so they can all be described by the S term above.
• The B term specifies the total number of items that can be in the system, and includes items that are still in the queue and those that are being serviced. Though many systems in the real world have a limited capacity, the model is easier to analyze if this capacity is considered infinite. Consequently, if the capacity of a system is large enough, the system is commonly assumed to be infinite.

• The N term specifies the total number of potential customers – i.e., the number of customers that could ever enter the queueing system – which may be considered finite or infinite.
• The D term specifies the service discipline of the queuing system, such as first-come-first-served or last-in-first-out.

Little’s law, which was first proven by mathematician John Little, states that the average number of items in a queue can be calculated by multiplying the average rate at which the items arrive in the system by the average amount of time they spend in it.

• In mathematical notation, the Little's law is: L = λW
• L is the average number of items, λ is the average arrival rate of the items in the queuing system, and W is the average amount of time the items spend in the queuing system.
• Little’s law assumes that the system is in a “steady state” – the mathematical variables characterizing the system do not change over time.

Although Little’s law only needs three inputs, it is quite general and can be applied to many queuing systems, regardless of the types of items in the queue or the way items are processed in the queue. Little’s law can be useful in analyzing how a queue has performed over some time, or to quickly gauge how a queue is currently performing.

For example: a shoebox company wants to figure out the average number of shoeboxes that are stored in a warehouse. The company knows that the average arrival rate of the boxes into the warehouse is 1,000 shoeboxes/year, and that the average time they spend in the warehouse is about 3 months, or ¼ of a year. Thus, the average number of shoeboxes in the warehouse is given by (1000 shoeboxes/year) x (¼ year), or 250 shoeboxes.

Key Takeaways

• Queuing theory is the mathematical study of queuing, or waiting in lines.
• Queues contain “customers” such as people, objects, or information. Queues form when there are limited resources for providing a service.
• Queuing theory can be applied to situations ranging from waiting in line at the grocery store to waiting for a computer to perform a task. It is often used in software and business applications to determine the best way of using limited resources.
• Kendall’s notation can be used to specify the parameters of a queuing system.
• Little’s law is a simple but general expression that can provide a quick estimate of the average number of items in a queue.

Sources

• Beasley, J. E. “Queuing theory.”
• Boxma, O. J. “Stochastic performance modelling.” 2008.
• Lilja, D. Measuring Computer Performance: A Practitioner’s Guide, 2005.
• Little, J., and Graves, S. “Chapter 5: Little’s law.” In Building Intuition: Insights from Basic Operations Management Models and Principles. Springer Science+Business Media, 2008.
• Mulholland, B. “Little’s law: How to analyze your processes (with stealth bombers).”Process.st, 2017.

Queueing theory is the mathematical study of waiting lines, or queues.^[1] A queueing model is constructed so that queue lengths and waiting time can be predicted.^[1] Queueing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service.

Queueing theory has its origins in research by Agner Krarup Erlang when he created models to describe the Copenhagen telephone exchange.^[1] The ideas have since seen applications including telecommunication, traffic engineering, computing^[2]and, particularly in industrial engineering, in the design of factories, shops, offices and hospitals, as well as in project management.^[3][4]

• 6Queueing networks

Spelling[edit]

The spelling 'queueing' over 'queuing' is typically encountered in the academic research field. In fact, one of the flagship journals of the profession is named Queueing Systems.

'Queueing' is, incidentally, the only English word having more than four consecutive vowels.

Single queueing nodes[edit]

Simple description and analogy

A queue, or 'queueing node' can be thought of as nearly a black box. Jobs or 'customers' arrive to the queue, possibly wait some time, take some time being processed, and then depart from the queue (see Fig. 1).

The queueing node is not quite a pure black box, however, since there is some information we need to specify about the inside of the queuing node. The queue has one or more 'servers' which can each be paired with an arriving job until it departs, after which that server will be free to be paired with another arriving job (see Fig. 2).

An analogy often used is that of the cashier at a supermarket. There are other models, but this is one commonly encountered in the literature. Customers arrive, are processed by the cashier, and depart. Each cashier processes one customer at a time, and hence this is a queueing node with only one server. A setting, where a customer will leave immediately, when in arriving he finds the cashier busy, is called a queue with no buffer (or no 'waiting area', or similar terms). A setting with a waiting zone for up to n customers is called a queue with a buffer of size n.

Birth-death process

The behaviour / state of a single queue (also called a 'queueing node') can be described by a Birth-death process, which describe the arrivals and departures from the queue, along with the number of jobs (also called 'customers' or 'requests', or any number of other things, depending on the field) currently in the system. An arrival increases the number of jobs by 1, and a departure (a job completing its service) decreases k by 1 (see Fig. 3).

Kendall's notation

Single queueing nodes are usually described using Kendall's notation in the form A/S/c where A describes the distribution of durations between each arrival to the queue, S the distribution of service times for jobs and c the number of servers at the node.^[5][6] For an example of the notation, the M/M/1 queue is a simple model where a single server serves jobs that arrive according to a Poisson process (inter-arrival durations are exponentially distributed) and have exponentially distributed service times. In an M/G/1 queue, the G stands for general and indicates an arbitrary probability distribution for service times.

Overview of the development of the theory[edit]

In 1909, Agner Krarup Erlang, a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on what would now be called queueing theory.^[7][8][9] He modeled the number of telephone calls arriving at an exchange by a Poisson process and solved the M/D/1 queue in 1917 and M/D/k queueing model in 1920.^[10] In Kendall's notation:

• M stands for Markov or memoryless and means arrivals occur according to a Poisson process;
• D stands for deterministic and means jobs arriving at the queue which require a fixed amount of service;
• k describes the number of servers at the queueing node (k = 1, 2, ..).

If there are more jobs at the node than there are servers, then jobs will queue and wait for service

The M/G/1 queue was solved by Felix Pollaczek in 1930,^[11] a solution later recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek–Khinchine formula.^[10][12]

After the 1940s queueing theory became an area of research interest to mathematicians.^[12] In 1953 David George Kendall solved the GI/M/k queue^[13] and introduced the modern notation for queues, now known as Kendall's notation. In 1957 Pollaczek studied the GI/G/1 using an integral equation.^[14]John Kingman gave a formula for the mean waiting time in a G/G/1 queue: Kingman's formula.^[15]

Leonard Kleinrock worked on the application of queueing theory to message switching and packet switching. His initial contribution to this field was his doctoral thesis at the Massachusetts Institute of Technology in 1962, published in book form in 1964 in the field of digital message switching. His theoretical work after 1967 underpinned the use of packing switching in the ARPANET, a forerunner to the Internet.

The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival and service time distributions to be considered.^[16]

Problems such as performance metrics for the M/G/k queue remain an open problem.^[10][12]

Service disciplines[edit]

Various scheduling policies can be used at queuing nodes:

First in first out

Also called first-come, first-served (FCFS),^[17] this principle states that customers are served one at a time and that the customer that has been waiting the longest is served first.^[18]

Last in first out

This principle also serves customers one at a time, but the customer with the shortest waiting time will be served first.^[18] Also known as a stack.

Processor sharing

Service capacity is shared equally between customers.^[18]

Priority

Customers with high priority are served first.^[18] Priority queues can be of two types, non-preemptive (where a job in service cannot be interrupted) and preemptive (where a job in service can be interrupted by a higher-priority job). No work is lost in either model.^[19]

Shortest job first

The next job to be served is the one with the smallest size

Preemptive shortest job first

The next job to be served is the one with the original smallest size^[20]

Shortest remaining processing time

The next job to serve is the one with the smallest remaining processing requirement.^[21]

Service facility

• Single server: customers line up and there is only one server
• Several parallel servers–Single queue: customers line up and there are several servers
• Several servers–Several queues: there are many counters and customers can decide going where to queue

Customer's behavior of waiting

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

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

Simple two-equation queue[edit]

A common basic queuing system is attributed to Erlang, and is a modification of Little's Law. Given an arrival rate λ, a dropout rate σ, and a departure rate μ, length of the queue L is defined as:

${\displaystyle L=\frac{\lambda -\sigma }{\mu }.}$

Assuming an exponential distribution for the rates, the waiting time W can be defined as the proportion of arrivals that are served. This is equal to the exponential survival rate of those who do not drop out over the waiting period, giving:

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${\displaystyle \frac{\mu }{\lambda }=e^{-W\mu }}$

Vampire diaries full episodes download. The second equation is commonly rewritten as:

${\displaystyle W=\frac{1}{\mu }\mathrm{l}\mathrm{n}\frac{\lambda }{\mu }}$

The two-stage one-box model is common in epidemiology.^[22]

Queueing networks[edit]

Networks of queues are systems in which a number of queues are connected by what's known as customer routing. When a customer is serviced at one node it can join another node and queue for service, or leave the network.

For networks of m nodes, the state of the system can be described by an m–dimensional vector (x₁, x₂, .., x_m) where x_i represents the number of customers at each node.

The simplest non-trivial network of queues is called tandem queues.^[23] The first significant results in this area were Jackson networks,^[24][25] for which an efficient product-form stationary distribution exists and the mean value analysis^[26] which allows average metrics such as throughput and sojourn times to be computed.^[27] If the total number of customers in the network remains constant the network is called a closed network and has also been shown to have a product–form stationary distribution in the Gordon–Newell theorem.^[28] This result was extended to the BCMP network^[29] where a network with very general service time, regimes and customer routing is shown to also exhibit a product-form stationary distribution. The normalizing constant can be calculated with the Buzen's algorithm, proposed in 1973.^[30]

Networks of customers have also been investigated, Kelly networks where customers of different classes experience different priority levels at different service nodes.^[31] Another type of network are G-networks first proposed by Erol Gelenbe in 1993:^[32] these networks do not assume exponential time distributions like the classic Jackson Network.

Example analysis of an M/M/1 queue[edit]

Consider a queue with 1 server and the following variables:

Queuing Theory Tutorial Pdf

• λ: the average arrival rate (customers arriving to the system per unit time, e.g. per 30 seconds);
• μ: the average departure rate (customers leaving the system—completing service—per the same unit time, e.g. per 30 seconds);
• n: the number of customers in the system at the given time;
• P_n: the probability of there being n customers in the system.

Further, let E_n represent the number of times the system enters state n, and L_n represent the number of times the system leaves state n. For all n, we have E_n − L_n ∈ {0, 1}, that is, the number of times the system leaves a state differs by at most 1 from the number of times it enters that state, since it will either return into that state at some time in the future (E_n = L_n) or not ( E_n − L_n = 1).

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

Balance equation

situation 0: ${\displaystyle {\mu }_1{P}_1={\lambda }_0{P}_0}$

situation 1: ${\displaystyle {\lambda }_0{P}_0+{\mu }_2{P}_2=({\lambda }_1+{\mu }_1)P_1}$

situation n: ${\displaystyle {\lambda }_{n-1}{P}_{n-1}+{\mu }_{n+1}{P}_{n+1}=({\lambda }_n+{\mu }_n)P_n}$

By balance equation, ${\displaystyle P_1=\frac{{\lambda }_0}{{\mu }_1}{P}_0{P}_2=\frac{{\lambda }_1}{{\mu }_2}{P}_1+\frac{1}{{\mu }_2}({\mu }_1{P}_1-{\lambda }_0{P}_0)=\frac{{\lambda }_1}{{\mu }_2}{P}_1=\frac{{\lambda }_1{\lambda }_0}{{\mu }_2{\mu }_1}{P}_0}$

By mathematical induction, ${\displaystyle P_n=\frac{{\lambda }_{n-1}{\lambda }_{n-2}\cdots {\lambda }_0}{{\mu }_n{\mu }_{n-1}\cdots {\mu }_1}{P}_0=P_0\prod _{i=0}^{n-1}\frac{{\lambda }_i}{{\mu }_{i+1}}}$

Because ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{P}_n=P_0+P_0\sum _{n=1}^{\mathrm{\infty }}\prod _{i=0}^{n-1}\frac{{\lambda }_i}{{\mu }_{i+1}}=1}$

We get ${\displaystyle P_0=\frac{1}{1+\sum _{n=1}^{\mathrm{\infty }}\prod _{i=0}^{n-1}\frac{{\lambda }_i}{{\mu }_{i+1}}}}$

Routing algorithms[edit]

In discrete time networks where there is a constraint on which service nodes can be active at any time, the max-weight scheduling algorithm chooses a service policy to give optimal throughput in the case that each job visits only a single person ^[17] service node. In the more general case where jobs can visit more than one node, backpressure routing gives optimal throughput. A network scheduler must choose a queuing algorithm, which affects the characteristics of the larger network^{[citation needed]}. See also Stochastic scheduling for more about scheduling of queueing systems.

Mean field limits[edit]

Mean field models consider the limiting behaviour of the empirical measure (proportion of queues in different states) as the number of queues (m above) goes to infinity. The impact of other queues on any given queue in the network is approximated by a differential equation. The deterministic model converges to the same stationary distribution as the original model.^[33]

Fluid limits[edit]

Fluid models are continuous deterministic analogs of queueing networks obtained by taking the limit when the process is scaled in time and space, allowing heterogeneous objects. This scaled trajectory converges to a deterministic equation which allows the stability of the system to be proven. It is known that a queueing network can be stable, but have an unstable fluid limit.^[34]

Heavy traffic/diffusion approximations[edit]

In a system with high occupancy rates (utilisation near 1) a heavy traffic approximation can be used to approximate the queueing length process by a reflected Brownian motion,^[35]Ornstein–Uhlenbeck process or more general diffusion process.^[36] The number of dimensions of the RBM is equal to the number of queueing nodes and the diffusion is restricted to the non-negative orthant.

See also[edit]

• Queueing Systems – a journal of queueing theory

References[edit]

Queuing Theory Formula Pdf

1. ^ ^abcSundarapandian, V. (2009). '7. Queueing Theory'. Probability, Statistics and Queueing Theory. PHI Learning. ISBN978-8120338449.
2. ^Lawrence W. Dowdy, Virgilio A.F. Almeida, Daniel A. Menasce. 'Performance by Design: Computer Capacity Planning by Example'.
3. ^Schlechter, Kira (March 2, 2009). 'Hershey Medical Center to open redesigned emergency room'. The Patriot-News.
4. ^Mayhew, Les; Smith, David (December 2006). Using queuing theory to analyse completion times in accident and emergency departments in the light of the Government 4-hour target. Cass Business School. ISBN978-1-905752-06-5. Retrieved 2008-05-20.^{[permanent dead link]}
5. ^Tijms, H.C, Algorithmic Analysis of Queues', Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003
6. ^Kendall, D. G. (1953). 'Stochastic Processes Occurring in the Theory of Queues and their Analysis by the Method of the Imbedded Markov Chain'. The Annals of Mathematical Statistics. 24 (3): 338–354. doi:10.1214/aoms/1177728975. JSTOR2236285.
7. ^'Agner Krarup Erlang (1878-1929) plus.maths.org'. Pass.maths.org.uk. 1997-04-30. Retrieved 2013-04-22.
8. ^Asmussen, S. R.; Boxma, O. J. (2009). 'Editorial introduction'. Queueing Systems. 63 (1–4): 1–2. doi:10.1007/s11134-009-9151-8.
9. ^Erlang, Agner Krarup (1909). 'The theory of probabilities and telephone conversations'(PDF). NYT Tidsskrift for Matematik B. 20: 33–39. Archived from the original(PDF) on 2011-10-01.
10. ^ ^abcKingman, J. F. C. (2009). 'The first Erlang century—and the next'. Queueing Systems. 63 (1–4): 3–4. doi:10.1007/s11134-009-9147-4.
11. ^Pollaczek, F., Ueber eine Aufgabe der Wahrscheinlichkeitstheorie, Math. Z. 1930
12. ^ ^abcWhittle, P. (2002). 'Applied Probability in Great Britain'. Operations Research. 50 (1): 227–239. doi:10.1287/opre.50.1.227.17792. JSTOR3088474.
13. ^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
14. ^Pollaczek, F., Problèmes Stochastiques posés par le phénomène de formation d'une queue
15. ^Kingman, J. F. C.; Atiyah (October 1961). 'The single server queue in heavy traffic'. Mathematical Proceedings of the Cambridge Philosophical Society. 57 (4): 902. doi:10.1017/S0305004100036094. JSTOR2984229.
16. ^Ramaswami, V. (1988). 'A stable recursion for the steady state vector in markov chains of m/g/1 type'. Communications in Statistics. Stochastic Models. 4: 183–188. doi:10.1080/15326348808807077.
17. ^ ^abManuel, Laguna (2011). Business Process Modeling, Simulation and Design. Pearson Education India. p. 178. ISBN9788131761359. Retrieved 6 October 2017.
18. ^ ^abcdPenttinen A., Chapter 8 – Queueing Systems, Lecture Notes: S-38.145 - Introduction to Teletraffic Theory.
19. ^Harchol-Balter, M. (2012). 'Scheduling: Non-Preemptive, Size-Based Policies'. Performance Modeling and Design of Computer Systems. pp. 499–507. doi:10.1017/CBO9781139226424.039. ISBN9781139226424.
20. ^Harchol-Balter, M. (2012). 'Scheduling: Preemptive, Size-Based Policies'. Performance Modeling and Design of Computer Systems. pp. 508–517. doi:10.1017/CBO9781139226424.040. ISBN9781139226424.
21. ^Harchol-Balter, M. (2012). 'Scheduling: SRPT and Fairness'. Performance Modeling and Design of Computer Systems. pp. 518–530. doi:10.1017/CBO9781139226424.041. ISBN9781139226424.
22. ^Hernández-Suarez, Carlos (2010). 'An application of queuing theory to SIS and SEIS epidemic models'. Math. Biosci. 7 (4): 809–823. doi:10.3934/mbe.2010.7.809. PMID21077709.
23. ^http://www.stats.ox.ac.uk/~winkel/bs3a07l13-14.pdf#page=4
24. ^Jackson, J. R. (1957). 'Networks of Waiting Lines'. Operations Research. 5 (4): 518–521. doi:10.1287/opre.5.4.518. JSTOR167249.
25. ^Jackson, James R. (Oct 1963). 'Jobshop-like Queueing Systems'. Management Science. 10 (1): 131–142. doi:10.1287/mnsc.1040.0268. JSTOR2627213.
26. ^Reiser, M.; Lavenberg, S. S. (1980). 'Mean-Value Analysis of Closed Multichain Queuing Networks'. Journal of the ACM. 27 (2): 313. doi:10.1145/322186.322195.
27. ^Van Dijk, N. M. (1993). 'On the arrival theorem for communication networks'. Computer Networks and ISDN Systems. 25 (10): 1135–2013. doi:10.1016/0169-7552(93)90073-D.
28. ^Gordon, W. J.; Newell, G. F. (1967). 'Closed Queuing Systems with Exponential Servers'. Operations Research. 15 (2): 254. doi:10.1287/opre.15.2.254. JSTOR168557.
29. ^Baskett, F.; Chandy, K. Mani; Muntz, R.R.; Palacios, F.G. (1975). 'Open, closed and mixed networks of queues with different classes of customers'. Journal of the ACM. 22 (2): 248–260. doi:10.1145/321879.321887.
30. ^Buzen, J. P. (1973). 'Computational algorithms for closed queueing networks with exponential servers'(PDF). Communications of the ACM. 16 (9): 527–531. doi:10.1145/362342.362345.
31. ^Kelly, F. P. (1975). 'Networks of Queues with Customers of Different Types'. Journal of Applied Probability. 12 (3): 542–554. doi:10.2307/3212869. JSTOR3212869.
32. ^Gelenbe, Erol (Sep 1993). 'G-Networks with Triggered Customer Movement'. Journal of Applied Probability. 30 (3): 742–748. doi:10.2307/3214781. JSTOR3214781.
33. ^Bobbio, A.; Gribaudo, M.; Telek, M. S. (2008). 'Analysis of Large Scale Interacting Systems by Mean Field Method'. 2008 Fifth International Conference on Quantitative Evaluation of Systems. p. 215. doi:10.1109/QEST.2008.47. ISBN978-0-7695-3360-5.
34. ^Bramson, M. (1999). 'A stable queueing network with unstable fluid model'. The Annals of Applied Probability. 9 (3): 818–853. doi:10.1214/aoap/1029962815. JSTOR2667284.
35. ^Chen, H.; Whitt, W. (1993). 'Diffusion approximations for open queueing networks with service interruptions'. Queueing Systems. 13 (4): 335. doi:10.1007/BF01149260.
36. ^Yamada, K. (1995). 'Diffusion Approximation for Open State-Dependent Queueing Networks in the Heavy Traffic Situation'. The Annals of Applied Probability. 5 (4): 958–982. doi:10.1214/aoap/1177004602. JSTOR2245101.

Further reading[edit]

• Gross, Donald; Carl M. Harris (1998). Fundamentals of Queueing Theory. Wiley. ISBN978-0-471-32812-4.Online
• Deitel, Harvey M. (1984) [1982]. An introduction to operating systems (revisited first ed.). Addison-Wesley. p. 673. ISBN978-0-201-14502-1. chap.15, pp. 380–412
• Kleinrock, Leonard (2 January 1975). Queueing Systems: Volume I – Theory. New York: Wiley Interscience. p. 417. ISBN978-0471491101.
• Kleinrock, Leonard (22 April 1976). Queueing Systems: Volume II – Computer Applications. New York: Wiley Interscience. p. 576. ISBN978-0471491118.
• Lazowska, Edward D.; John Zahorjan; G. Scott Graham; Kenneth C. Sevcik (1984). Quantitative System Performance: Computer System Analysis Using Queueing Network Models. Prentice-Hall, Inc. ISBN978-0-13-746975-8.
• Zukerman, Moshe. Introduction to Queueing Theory and Stochastic Teletraffic Models(PDF).

External links[edit]

• Office Fire Emergency Evacuation Simulation on YouTube
• What You Hate Most About Waiting in Line: (It’s not the length of the wait.), by Seth Stevenson, Slate, 2012 – popular introduction