Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that waiting time can be predicted. Queueing theory is considered a branch of operations research because the results are 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; the ideas have since seen applications including telecommunication, traffic engineering, computing and in industrial engineering, in the design of factories, shops and hospitals, as well as in project management. The spelling "queueing" over "queuing" is encountered in the academic research field. In fact, one of the flagship journals of the profession is named Queueing Systems. Simple description and analogyA queue, or "queueing node" can be thought of as nearly a black box. Jobs or "customers" arrive to the queue wait some time, take some time being processed, depart from the queue.
The queueing node is not quite a pure black box, 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. An analogy used is that of the cashier at a supermarket. There are other models, but this is one encountered in the literature. Customers arrive, are processed by the cashier, depart; each cashier processes one customer at a time, hence this is a queueing node with only one server. A setting, where a customer will leave when in arriving he finds the cashier busy, is called a queue with no buffer. A setting with a waiting zone for up to n customers is called a queue with a buffer of size n. Birth-death processThe behaviour / state of a single queue can be described by a Birth-death process, which describe the arrivals and departures from the queue, along with the number of jobs in the system.
An arrival increases the number of jobs by 1, a departure decreases k by 1. Kendall's notationSingle queueing nodes are 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. 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 and have exponentially distributed service times. In an M/G/1 queue, the G stands for general and indicates an arbitrary probability distribution for inter-arrival times. 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, 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. In Kendall's notation: M stands for Markov or memoryless and means arrivals occur according to a Poisson process.
If there are more jobs at the node than there are servers jobs will queue and wait for service The M/G/1 queue was solved by Felix Pollaczek in 1930, a solution recast in probabilistic terms by Aleksandr Khinchin and now known as the Pollaczek–Khinchine formula. After the 1940s queueing theory became an area of research interest to mathematicians. In 1953 David George Kendall solved the GI/M/k queue 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. John Kingman gave a formula for the mean waiting time in a G/G/1 queue: Kingman's formula. 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. Problems such as performance metrics for the M/G/k queue remain an open problem. Various scheduling policies can be used at queuing nodes: First in first out Also called first-come, first-served, this principle states that customers are served one at a time and that the customer, waiting the longest is served first. Last in first out This principle serves customers one at a time, but the customer with the shortest waiting time will be served first. Known as a stack. Processor sharing Service capacity is shared between customers. Priority Customers with high priority are served first. Priority queues can be of two types, preemptive. No work is lost in either model. Shortest job first The next job to be served is the one with the smallest sizePreemptive shortest job first The next job to be served is the one with the
David George Kendall
David George Kendall FRS was an English statistician and mathematician, known for his work on probability, statistical shape analysis, ley lines and queueing theory. He spent most of his academic life in the University of Cambridge, he worked with M. S. Bartlett during World War II, visited Princeton University after the war. David George Kendall was born on 15 January 1918 in Ripon, West Riding of Yorkshire, attended Ripon Grammar School before attending Queen's College, graduating in 1939, he worked on rocketry during the World War II, before moving to Magdalen College, Oxford, in 1946. In 1962 he was appointed the first Professor of Mathematical Statistics in the Statistical Laboratory, University of Cambridge, he was elected to a professorial fellowship at Churchill College, he was a founding trustee of the Rollo Davidson Trust. In 1986, he was awarded an Honorary Degree by the University of Bath. Kendall was an expert in probability and data analysis, pioneered statistical shape analysis including the study of ley lines.
He defined Kendall's notation for queueing theory. The Royal Statistical Society awarded him the Guy Medal in Silver in 1955, followed in 1981 by the Guy Medal in Gold. In 1980 the London Mathematical Society awarded Kendall their Senior Whitehead Prize, in 1989 their De Morgan Medal, he was elected a fellow of the Royal Society in 1964. He was married to Diana Fletcher from 1952 until his death, they had two sons and four daughters, including Wilfrid Kendall, professor in the Department of Statistics at the University of Warwick, journalist Bridget Kendall MBE, Felicity Kendall Hickman, Judy Kendall and poet at University of Salford, George Kendall, Harriet Strudwick, the Antipodean sibling. Kendall, David G. "Geometric ergodicity and the theory of queues", in Arrow, Kenneth J.. Janus: The Papers of Professor David Kendall Royal Society citation
Australian National University
The Australian National University is a national research university located in Canberra, the capital of Australia. Its main campus in Acton encompasses seven teaching and research colleges, in addition to several national academies and institutes. Founded in 1946, it is the only university to have been created by the Parliament of Australia. A postgraduate research university, ANU commenced undergraduate teaching in 1960 when it integrated the Canberra University College, established in 1929 as a campus of the University of Melbourne. ANU employs 3,753 staff; the university's endowment stood at A$1.13 billion in 2012. ANU is regarded as one of the world's leading research universities, it is ranked 1st in Australia and the whole of Oceania, 24th in the world by the 2019 QS World University Rankings, 49th in the world by the 2019 Times Higher Education. ANU was named the world's 7th most international university in a 2017 study by Times Higher Education. In the 2017 Times Higher Education Global Employability University Ranking, an annual ranking of university graduates' employability, ANU was ranked 21st in the world.
ANU is ranked 100th in the CWTS Leiden ranking. The university is well known for its programmes in the arts and social sciences, ranks among the best in the world for a number of disciplines including politics and international relations, social policy, geography. ANU counts six Nobel laureates and 49 Rhodes scholars among its faculty and alumni; the university has educated two prime ministers, 30 current Australian ambassadors and more than a dozen current heads of government departments of Australia. The latest releases of ANU's scholarly publications are held through ANU Press online. Calls for the establishment of a national university in Australia began as early as 1900. After the location of the nation's capital, was determined in 1908, land was set aside for the university at the foot of Black Mountain in the city designs by Walter Burley Griffin. Planning for the university was disrupted by World War II but resumed with the creation of the Department of Post-War Reconstruction in 1942 leading to the passage of the Australian National University Act 1946 by the Chifley Government on 1 August 1946.
A group of eminent Australian scholars returned from overseas to join the university, including Sir Howard Florey, Sir Mark Oliphant, Sir Keith Hancock and Sir Raymond Firth. Economist Sir Douglas Copland was appointed as ANU's first Vice-Chancellor and former Prime Minister Stanley Bruce served as the first Chancellor. ANU was organised into four centres—the Research Schools of Physical Sciences, Social Sciences and Pacific Studies and the John Curtin School of Medical Research; the first residents' hall, University House, was opened in 1954 for faculty members and postgraduate students. Mount Stromlo Observatory, established by the federal government in 1924, became part of ANU in 1957; the first locations of the ANU Library, the Menzies and Chifley buildings, opened in 1963. The Australian Forestry School, located in Canberra since 1927, was amalgamated by ANU in 1965. Canberra University College was the first institution of higher education in the national capital, having been established in 1929 and enrolling its first undergraduate pupils in 1930.
Its founding was led by Sir Robert Garran, one of the drafters of the Australian Constitution and the first Solicitor-General of Australia. CUC was affiliated with the University of Melbourne and its degrees were granted by that university. Academic leaders at CUC included historian Manning Clark, political scientist Finlay Crisp, poet A. D. Hope and economist Heinz Arndt. In 1960, CUC was integrated into ANU as the School of General Studies with faculties in arts, economics and science. Faculties in Oriental studies and engineering were introduced later. Bruce Hall, the first residential college for undergraduates, opened in 1961; the Canberra School of Music and the Canberra School of Art combined in 1988 to form the Canberra Institute of the Arts, amalgamated with the university as the ANU Institute of the Arts in 1992. ANU established its Medical School in 2002, after obtaining federal government approval in 2000. On 18 January 2003, the Canberra bushfires destroyed the Mount Stromlo Observatory.
ANU astronomers now conduct research from the Siding Spring Observatory, which contains 10 telescopes including the Anglo-Australian Telescope. In February 2013, financial entrepreneur and ANU graduate Graham Tuckwell made the largest university donation in Australian history by giving $50 million to fund an undergraduate scholarship program at ANU. ANU is well known for its history of student activism and, in recent years, its fossil fuel divestment campaign, one of the longest-running and most successful in the country; the decision of the ANU Council to divest from two fossil fuel companies in 2014 was criticised by ministers in the Abbott government, but defended by Vice Chancellor Ian Young, who noted:On divestment, it is clear we were in the right and played a national and international leadership role. E seem to have played a major role in a movement; as of 2014 ANU still had investments in major fossil fuel companies. A survey conducted by the Australian Human Rights Commission in 2017 found that the ANU had the second highest incidence of sexual assault and sexual harassment.
3.5 per cent of respondents from the ANU re
University of Bristol
The University of Bristol is a red brick research university located in Bristol, United Kingdom. It received its royal charter in 1909, although like the University of the West of England and the University of Bath, it can trace its roots to the Merchant Venturers' Technical College, founded as a school in 1595 by the Society of Merchant Venturers, its key predecessor institution, University College, had been in existence since 1876. Bristol is organised into six academic faculties composed of multiple schools and departments running over 200 undergraduate courses situated in the Tyndalls Park area of the city; the university had a total income of £642.7 million in 2017/18, of which £164.0 million was from research grants and contracts. It is the largest independent employer in Bristol; the University of Bristol is ranked 44th by the QS World University Rankings 2018, is ranked amongst the top 10 of UK universities by QS, THE, ARWU. A selective institution, it has an average of 6.4 to 13.1 applicants for each undergraduate place.
It was ranked 9th in the UK amongst multi-faculty institutions for the quality of its research and for its Research Power in the 2014 Research Excellence Framework. Current academics include 21 fellows of the Academy of Medical Sciences, 13 fellows of the British Academy, 13 fellows of the Royal Academy of Engineering and 44 fellows of the Royal Society; the university has been associated with 13 Nobel laureates throughout its history, including Paul Dirac, Sir William Ramsay, Cecil Frank Powell, Sir Winston Churchill, Dorothy Hodgkin, Hans Albrecht Bethe, Max Delbrück, Gerhard Herzberg, Sir Nevill Francis Mott, Sir Paul Nurse, Harold Pinter, Jean-Marie Gustave Le Clézio and most 2015 Economics Nobel Prize winner Angus Deaton. Bristol is a member of the Russell Group of research-intensive British universities, the European-wide Coimbra Group and the Worldwide Universities Network, of which the university's previous vice-chancellor, Eric Thomas, was chairman from 2005 to 2007. In addition, the university holds an Erasmus Charter, sending more than 500 students per year to partner institutions in Europe.
The earliest antecedent of the university was the engineering department of the Merchant Venturers' Technical College which became the engineering faculty of Bristol University. The university was preceded by Bristol Medical School and University College, founded in 1876, where its first lecture was attended by only 99 students; the university was able to apply for a royal charter due to the financial support of the Wills and Fry families, who made their fortunes in tobacco plantations and chocolate, respectively. The Wills Family made a vast fortune from the tobacco industry and gave generously to the city and university; the royal charter was gained in May 1909, with 288 undergraduates and 400 other students entering the university in October 1909. Henry Overton Wills III became its first chancellor; the University College was the first such institution in the country to admit women on the same basis as men. However, women were forbidden to take examinations in medicine until 1906. Since the founding of the university itself in 1909, it has grown and is now one of the largest employers in the local area, although it is smaller by student numbers than the nearby University of the West of England.
Bristol is spread over a considerable geographic area. Most of its activities, are concentrated in the area of the city centre, referred to as the "University Precinct", it is a member of the Russell Group of research-led UK universities, the Coimbra Group of leading European universities and the Worldwide Universities Network. After the founding of the University College in 1876, Government support began in 1889. After mergers with the Bristol Medical School in 1893 and the Merchant Venturers' Technical College in 1909, this funding allowed the opening of a new medical school and an engineering school—two subjects that remain among the university's greatest strengths. In 1908, gifts from the Fry and Wills families £100,000 from Henry Overton Wills III, were provided to endow a University for Bristol and the West of England, provided that a royal charter could be obtained within two years. In December 1909, the King erected the University of Bristol. Henry Wills became Conwy Lloyd Morgan the first vice-chancellor.
Wills died in 1911 and in tribute his sons George and Harry built the Wills Memorial Building, starting in 1913 and finishing in 1925. Today, it houses parts of the academic provision for earth sciences and law, graduation ceremonies are held in its Great Hall; the Wills Memorial Building is a Grade II* listed building. In 1920, George Wills bought the Victoria Rooms and endowed them to the university as a Students' Union; the building now is a Grade II * listed building. At the point of foundation, the university was required to provide for the local community; this mission was behind the creation of the Department of Extra-Mural Adult Education in 1924 to provide courses to the local community. This mission continues today. Among the famous names associated with Bristol in this early period is Paul Dirac, who graduated in 1921 with a degree in engineering, before obtaining a second degree in mathematics in 1923 from Cambridge. For his subsequent pioneering work on quantum mechanics, he was awarded the 1933 Nobel Prize for Physics.
Royal Statistical Society
The Royal Statistical Society is one of the world's most distinguished and renowned statistical societies. It has three main goals; the RSS is a British learned society for statistics, a professional body for statisticians, a charity which promotes statistics for the public good. The society was founded in 1834 as the Statistical Society of London, though a unrelated London Statistical Society was in existence at least as early as 1824. At that time there were many provincial statistics societies throughout Britain, but most have not survived; the Manchester Statistical Society is a notable exception. The associations were formed with the object of gathering information about society; the idea of statistics referred more to political knowledge rather than a series of methods. The members called themselves "statists" and the original aim was "...procuring and publishing facts to illustrate the condition and prospects of society" and the idea of interpreting data, or having opinions, was explicitly excluded.
The original seal had the motto Aliis Exterendum but this separation was found to be a hindrance and the motto was dropped in logos. It was many decades. Instrumental in founding the LSS were Richard Jones, Charles Babbage, Adolphe Quetelet, William Whewell, Thomas Malthus. Among its famous members was Florence Nightingale, the society's first female member in 1858. Stella Cunliffe was the first female president. Other notable RSS presidents have included William Beveridge, Ronald Fisher, Harold Wilson, David Cox. Honorary Secretaries include Gerald Goodhardt; the LSS became the RSS by Royal Charter in 1887, merged with the Institute of Statisticians in 1993. The merger enabled the society to take on the role of a professional body as well as that of a learned society; as of 2017 the society has more than 8.000 members around the world, of whom some 1,500 are professionally qualified, with the status of Chartered Statistician. In January 2009, the RSS received Licensed Body status from the UK Science Council to award Chartered Scientist status.
Since February 2009 the society has awarded Chartered Scientist status to suitably qualified members. Unusually among professional societies, all members of the RSS are known as "Fellows". Fellowship is nowadays not used by post-merger members as a post-nominal mark of distinction. However, before the 1993 merger with the Institute of Statisticians, Fellows did use the post-nominal letters FSS. Before the merger, Fellows were required to have a statistical qualification; the alternative route was to be proposed by two Fellows. The nomination paper had to be approved by the Council. After the merger these requirements were dropped and all the previous members of the Institute of Statisticians became Fellows as well. Since use by new members of their unearned post-nominal FSS qualification was viewed as inappropriate and discouraged, it became less common; the RSS has premises in Errol Street, EC1, in the London Borough of Islington close to the boundary with the City of London, between Old Street and Barbican stations.
The society has various local groups in the UK, together with a wide range of topic-related sections and study groups. Each of these sections and groups organises seminars on statistical topics; the society was engaged with the passage of the Statistics and Registration Service Act 2007, having long argued for legislation on statistics. The RSS organises an annual conference. Among the society's awards are the Guy Medals in gold and bronze, in honour of William Guy; the RSS team reached the finals of University Challenge: The Professionals 2006, where they were beaten 230 to 125 by a team from the Bodleian Library, Oxford. The society publishes the Journal of the Royal Statistical Society, which consists of three separate series of journals whose contents include papers presented at ordinary meetings of the society, namely Series A, Series B and Series C, as well as a general audience magazine called Significance published in conjunction with the American Statistical Association. In September 2013, the society established StatsLife, an online magazine website that features news and opinion from the world of statistics and data.
Council for the Mathematical Sciences Royal Statistical Society of Belgium Royal Statistical Society official website RSS StatsLife official website Official page on Twitter MacTutor: The Royal Statistical Society Scholarly Societies Project: RSS RoyalStatSoc's channel on YouTube
University of Cambridge
The University of Cambridge is a collegiate public research university in Cambridge, United Kingdom. Founded in 1209 and granted a Royal Charter by King Henry III in 1231, Cambridge is the second-oldest university in the English-speaking world and the world's fourth-oldest surviving university; the university grew out of an association of scholars who left the University of Oxford after a dispute with the townspeople. The two'ancient universities' share many common features and are referred to jointly as'Oxbridge'; the history and influence of the University of Cambridge has made it one of the most prestigious universities in the world. Cambridge is formed from a variety of institutions which include 31 constituent Colleges and over 100 academic departments organised into six schools. Cambridge University Press, a department of the university, is the world's oldest publishing house and the second-largest university press in the world; the university operates eight cultural and scientific museums, including the Fitzwilliam Museum, as well as a botanic garden.
Cambridge's libraries hold a total of around 15 million books, eight million of which are in Cambridge University Library, a legal deposit library. In the fiscal year ending 31 July 2018, the university had a total income of £1.965 billion, of which £515.5 million was from research grants and contracts. In the financial year ending 2017, the central university and colleges had combined net assets of around £11.8 billion, the largest of any university in the country. However, the true extent of Cambridge's wealth is much higher as many colleges hold their historic main sites, which date as far back as the 13th century, at depreceated valuations. Furthermore, many of the wealthiest colleges do not account for “heritage assets” such as works of art, libraries or artefacts, whose value many college accounts describe as “immaterial”; the university is linked with the development of the high-tech business cluster known as'Silicon Fen'. It is a member of numerous associations and forms part of the'golden triangle' of English universities and Cambridge University Health Partners, an academic health science centre.
As of 2018, Cambridge is the top-ranked university in the United Kingdom according to all major league tables. As of September 2017, Cambridge is ranked the world's second best university by the Times Higher Education World University Rankings, is ranked 3rd worldwide by Academic Ranking of World Universities, 6th by QS, 7th by US News. According to the Times Higher Education ranking, no other institution in the world ranks in the top 10 for as many subjects; the university has educated many notable alumni, including eminent mathematicians, politicians, philosophers, writers and foreign Heads of State. As of March 2019, 118 Nobel Laureates, 11 Fields Medalists, 7 Turing Award winners and 15 British Prime Ministers have been affiliated with Cambridge as students, faculty or research staff. By the late 12th century, the Cambridge area had a scholarly and ecclesiastical reputation, due to monks from the nearby bishopric church of Ely. However, it was an incident at Oxford, most to have led to the establishment of the university: two Oxford scholars were hanged by the town authorities for the death of a woman, without consulting the ecclesiastical authorities, who would take precedence in such a case, but were at that time in conflict with King John.
The University of Oxford went into suspension in protest, most scholars moved to cities such as Paris and Cambridge. After the University of Oxford reformed several years enough scholars remained in Cambridge to form the nucleus of the new university. In order to claim precedence, it is common for Cambridge to trace its founding to the 1231 charter from King Henry III granting it the right to discipline its own members and an exemption from some taxes. A bull in 1233 from Pope Gregory IX gave graduates from Cambridge the right to teach "everywhere in Christendom". After Cambridge was described as a studium generale in a letter from Pope Nicholas IV in 1290, confirmed as such in a bull by Pope John XXII in 1318, it became common for researchers from other European medieval universities to visit Cambridge to study or to give lecture courses; the colleges at the University of Cambridge were an incidental feature of the system. No college is as old as the university itself; the colleges were endowed fellowships of scholars.
There were institutions without endowments, called hostels. The hostels were absorbed by the colleges over the centuries, but they have left some traces, such as the name of Garret Hostel Lane. Hugh Balsham, Bishop of Ely, founded Peterhouse, Cambridge's first college, in 1284. Many colleges were founded during the 14th and 15th centuries, but colleges continued to be established until modern times, although there was a gap of 204 years between the founding of Sidney Sussex in 1596 and that of Downing in 1800; the most established college is Robinson, built in the late 1970s. However, Homerton College only achieved full university college status in March 2010, making it the newest full college. In medieval times, many colleges were founded so that their members would pray for the souls of the founders, were associated with chapels or abbeys; the colleges' focus changed in 1536 with the Dissolution of the Monasteries. King Henry VIII ordered the university to disband its Faculty of Canon Law and to stop teaching "scholastic philosophy".
In response, colleges changed