| Queuing Theory |
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HISTORY AND NOTATION Agner Krarup Erlang , a Danish engineer who worked for the Copenhagen Telephone Exchange, published the first paper on queueing theory in 1909 . David G. Kendall introduced an ''A/B/C'' queueing notation in 1953 . Kendall's notation for describing queues and their characteristics can be found in Tijms, H.C, ''Algorithmic Analysis of Queues", Chapter 9 in A First Course in Stochastic Models, Wiley, Chichester, 2003.. It has since been extended to '''1/2/3/(4/5/6)''' where the numbers are replaced with: #A code describing the arrival process. The codes used are:
#A similar code representing the service process. The same symbols are used. #The Number of service channels (or servers). #The capacity of the system, or the maximum number of customers allowed in the system including those in service. When the number is at this maximum, further arrivals are turned away. #The Priority order that jobs in the line are served:
#The size of calling source. The size of the population from which the customers come. This limits the Arrival Rate . As more Job s queue up there are fewer available to arrive into the system. Queueing theory is directly applicable to Intelligent Transportation System s, Call Center s, PABX s, Networks , Telecommunication s, Server queueing, Mainframe Computer queueing of telecommunications terminals, advanced telecommunications systems, and traffic flow. APPLICATION OF QUEUEING THEORY TO TELEPHONY The Public Switched Telephone Networks ( to divert calls via different paths -- even these systems have a finite or maximum traffic carrying capacity . However, the use of queueing in PSTNs allows the systems to queue their customer's requests until free resources become available. This means that if traffic intensity levels exceed available capacity, customer’s calls are here no longer lost; they instead wait until they can be served Bose S.J., ''Chapter 1 - An Introduction to Queueing Systems'', Kluwer/Plenum Publishers, 2002.. This method is used in queueing customers for the next available operator. A queueing discipline determines the manner in which the exchange handles calls from customers . It defines the way they will be served, the order in which they are served, and the way in which resources are divided between the customers ,Penttinen A., ''Chapter 8 – Queueing Systems'', Lecture Notes: S-38.145 - Introduction to Teletraffic Theory, .. Here are details of three queueing disciplines:
Queueing is handled by control processes within exchanges, which can be modelled using state equations ,. Queueing systems use a particular form of State Equation s known as Markov Chain s which model the system in each state {Link without Title} . Incoming traffic to these systems is modelled via a Poisson Distribution and is subject to Erlang’s queueing theory assumptions viz. :
Classic queueing theory involves complex calculations to determine call waiting time, service time, server utilisation and many other metrics which are used to measure queueing performance ,. QUEUEING NETWORKS Queues can be chained to form queueing networks where the departures from one queue enter the next queue. Queueing networks can be classified into two categories: open queueing networks and closed queueing networks. Open queueing networks have an external input and an external final destination. Closed queueing networks are completely contained and the customers circulate continually never leaving the network. THE ROLE OF THE POISSON AND EXPONENTIAL DISTRIBUTIONS To derive a queueing model that represents a real-life system, it is necessary to use a form that is both simple and sufficiently realistic. For queueing theory, it is most convenient to work with probability distributions which exhibit the Memoryless property, as it vastly simplifies the mathematics involved. As a result, queuing models are frequently modeled as Poisson Processes through the use of the Exponential Distribution . LIMITATIONS OF THE MATHEMATICAL APPROACH Classic queueing is too mathematically restrictive to be able to model all real-world situations. This restriction arises because the underlying assumptions of the theory do not always hold in the real world. Alternative means of analysis have been devised in order to provide some insight into problems which do not fall under the scope of queueing theory, though they are often scenario-specific since they generally consist of computer simulations and/or of analysis of experimental data. See Network Traffic Simulation . SEE ALSO REFERENCES FURTHER READING EXTERNAL LINKS
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