queuing theory

The study of how systems with limited resources distribute those resources to elements waiting in line, and how those elements respond. Examples include data traversing computer networks, phone calls traveling over voice networks and the distribution of cars on highways. Queuing theory is a complex area of engineering that is closely related to traffic engineering. See queuing and traffic engineering methods.

Queuing Theory

a mathematical discipline that studies systems intended for servicing a random flow of requests (the moments at which the requests appear as well as the time for servicing them may be random). A typical field of application of queuing theory is to automatic telephone exchanges to which requests for service—namely, the calls of the subscribers—arrive in a random fashion and where service consists in connecting subscribers to other subscribers and in maintaining the connection for the duration of the call. The purpose of methods developed in queuing theory is to organize service reasonably so that a given quality is ensured. Queuing theory from this standpoint can be considered as part of operations research.

Queuing theory widely uses the apparatus of probability theory and, to a lesser degree, mathematical statistics. Problems in queuing theory formulated mathematically usually reduce to the study of a special type of random process. Proceeding from given probabilistic characteristics of the incoming flow of calls and the service period and taking into account the organization of the system of service (presence of failures or of queues), queuing theory determines the corresponding characteristics of the quality of service (failure probability, mean time for service, mean down-time of communication lines). This determination is possible using analytic methods in a number of simpler cases, although more complex cases require the simulation of the corresponding random processes by means of a Monte Carlo method.

Example. Let us assume that an automatic communication system has n identical channels available to subscribers. Calls come in at random moments of time. If all the n channels of the communication system are busy when a new call arrives, that call will fail and be lost. If they are not all busy, a conversation will immediately begin on one of the free channels and in general will continue for a random period of time.

A measure of the operational efficiency of such a communication system is the fraction of calls that fail, that is, the limit p as T —. ∞ (if it exists) of the ratio v_T/N_T of the number V_T of calls lost over a period of time of length T to the total number N_T of incoming calls. This limit can be called the failure rate.

Another and no less essential measure of the operational efficiency of a communication system is the relative time during which it is busy, that is, the limit p* as T — ∞ (if it exists) of the ratio T_T/ T, where T_T is the total time during which all n channels of a communication line are simultaneously busy during a time period of length T. This limit may be called the busy rate. Let X(t) be the number of channels that are busy at time t. We can then show that (1) if the moments at which the calls come in form a time-homogeneous Poisson process of events and (2) if the lengths of conversations of successive subscribers are identically distributed, random variables, independent of one another and of the time at which the conversation begins, then the random process X(t), where t ≥ 0, possesses an ergodic distribution, that is, there exist [independently of the initial distribution X(0] the limits

and furthermore

where p is the product of the arrival rate of incoming calls and the mean length of a conversation of an individual subscriber. Moreover, in this case, p = p* and their common value is equal to p_n. The formulas (*) are used for calculating the minimum number of communication line channels that guarantee a given failure rate. These formulas are called Erlang formulas. It should be noted that if condition (1) is not satisfied, then the equality p = p* may not hold.

Queuing theory grew out of the interest the Danish engineer K. Erlang had in mathematical problems connected with organizing telephone networks. Erlang’s first published works appeared in the 1920’s. The theory was further developed in the 1950’s by C. Palm (Sweden), F. Pollaczek (France), and A. Ia. Khinchin (USSR). Their studies have been continued by the Soviet mathematician B. V. Gnedenko and others. The development of queuing theory has been significantly stimulated by the expansion of its range of application. Although formally part of the theory of random processes, queuing theory has become an independent field of research with its own set of problems and methods of solution and has in turn stimulated the development of the theory of random processes.

REFERENCES

Khinchin, A. Ia. Raboty po matematicheskoi teorii massovogo obsluzhivaniia. Moscow, 1963.
Rozenberg, V. Ia., and A. I. Prokhorov. Chto takoe teoriia massovogo obsluzhivaniia. Moscow, 1965.
Gnedenko, B. V., and I. N. Kovalenko. Vvedenie v teoriiu massovogo obsluzhivaniia. Moscow, 1966.
Saaty, T. L. Elementy teorii massovogo obsluzhivaniia i ee prilozheniia. Moscow, 1971. (Translated from English.)
Borovkov, A. A. Veroiatnostnye protsessy v teorii massovogo obsluzhivaniia. Moscow, 1972.

O. V. VISKOV

queuing theory

A theory by Danish mathematician AK Erlang (1878-1929), which he developed to determine the capacity of telephone networks. Queuing theory (QT) becomes immediately applicable to UK medical practice by substituting hospital beds for Erlang’s telephone lines, call arrival rate for admission rate into beds and average call duration for average length of stay. QT explains why average bed occupancy depends on the size and functional requirements of each specialty bed pool: whereas large hospitals in the USA and Europe have a 77–78% national average bed occupancy rate, the UK average is 87%, a rate which results in larger patient turnaway (e.g., due to queues for a bed, cancelled operations, medical patients in surgical beds), missed waiting time targets and organisational chaos manifested by increased use of antidepressants by overworked staff, increased hospital errors, cross-infection and patient deaths.
R Jones, author of a 2011 paper linking QT to the UK’s chronic bed shortage, argued that the Stafford Hospital scandal and the subsequent 400+ excess deaths were linked to the unrecognised role of excessive whole-hospital occupancy.

单词	queuing theory
释义	queuing theory queuing theory n (Mathematics) a mathematical approach to the rate at which components queue to be processed by a machine, instructions are accessed by a computer, orders need to be serviced, etc, to achieve the optimum flow queuing theory queuing theory The study of how systems with limited resources distribute those resources to elements waiting in line, and how those elements respond. Examples include data traversing computer networks, phone calls traveling over voice networks and the distribution of cars on highways. Queuing theory is a complex area of engineering that is closely related to traffic engineering. See queuing and traffic engineering methods. Queuing Theory a mathematical discipline that studies systems intended for servicing a random flow of requests (the moments at which the requests appear as well as the time for servicing them may be random). A typical field of application of queuing theory is to automatic telephone exchanges to which requests for service—namely, the calls of the subscribers—arrive in a random fashion and where service consists in connecting subscribers to other subscribers and in maintaining the connection for the duration of the call. The purpose of methods developed in queuing theory is to organize service reasonably so that a given quality is ensured. Queuing theory from this standpoint can be considered as part of operations research. Queuing theory widely uses the apparatus of probability theory and, to a lesser degree, mathematical statistics. Problems in queuing theory formulated mathematically usually reduce to the study of a special type of random process. Proceeding from given probabilistic characteristics of the incoming flow of calls and the service period and taking into account the organization of the system of service (presence of failures or of queues), queuing theory determines the corresponding characteristics of the quality of service (failure probability, mean time for service, mean down-time of communication lines). This determination is possible using analytic methods in a number of simpler cases, although more complex cases require the simulation of the corresponding random processes by means of a Monte Carlo method. *Example. Let us assume that an automatic communication system has n identical channels available to subscribers. Calls come in at random moments of time. If all the n channels of the communication system are busy when a new call arrives, that call will fail and be lost. If they are not all busy, a conversation will immediately begin on one of the free channels and in general will continue for a random period of time. A measure of the operational efficiency of such a communication system is the fraction of calls that fail, that is, the limit p as T —. ∞ (if it exists) of the ratio v_T/N_T of the number V_T* of calls lost over a period of time of length T to the total number N_T of incoming calls. This limit can be called the failure rate. Another and no less essential measure of the operational efficiency of a communication system is the relative time during which it is busy, that is, the limit p* as T — ∞ (if it exists) of the ratio T_T/ T, where T_T is the total time during which all n channels of a communication line are simultaneously busy during a time period of length T. This limit may be called the busy rate. Let X(t) be the number of channels that are busy at time t. We can then show that (1) if the moments at which the calls come in form a time-homogeneous Poisson process of events and (2) if the lengths of conversations of successive subscribers are identically distributed, random variables, independent of one another and of the time at which the conversation begins, then the random process X(t), where t ≥ 0, possesses an ergodic distribution, that is, there exist [independently of the initial distribution X(0] the limits and furthermore where p is the product of the arrival rate of incoming calls and the mean length of a conversation of an individual subscriber. Moreover, in this case, p = p* and their common value is equal to p_n. The formulas () are used for calculating the minimum number of communication line channels that guarantee a given failure rate. These formulas are called Erlang formulas. It should be noted that if condition (1) is not satisfied, then the equality p = p may not hold. Queuing theory grew out of the interest the Danish engineer K. Erlang had in mathematical problems connected with organizing telephone networks. Erlang’s first published works appeared in the 1920’s. The theory was further developed in the 1950’s by C. Palm (Sweden), F. Pollaczek (France), and A. Ia. Khinchin (USSR). Their studies have been continued by the Soviet mathematician B. V. Gnedenko and others. The development of queuing theory has been significantly stimulated by the expansion of its range of application. Although formally part of the theory of random processes, queuing theory has become an independent field of research with its own set of problems and methods of solution and has in turn stimulated the development of the theory of random processes. REFERENCES Khinchin, A. Ia. Raboty po matematicheskoi teorii massovogo obsluzhivaniia. Moscow, 1963. Rozenberg, V. Ia., and A. I. Prokhorov. Chto takoe teoriia massovogo obsluzhivaniia. Moscow, 1965. Gnedenko, B. V., and I. N. Kovalenko. Vvedenie v teoriiu massovogo obsluzhivaniia. Moscow, 1966. Saaty, T. L. Elementy teorii massovogo obsluzhivaniia i ee prilozheniia. Moscow, 1971. (Translated from English.) Borovkov, A. A. Veroiatnostnye protsessy v teorii massovogo obsluzhivaniia. Moscow, 1972. O. V. VISKOV queuing theory queuing theory A theory by Danish mathematician AK Erlang (1878-1929), which he developed to determine the capacity of telephone networks. Queuing theory (QT) becomes immediately applicable to UK medical practice by substituting hospital beds for Erlang’s telephone lines, call arrival rate for admission rate into beds and average call duration for average length of stay. QT explains why average bed occupancy depends on the size and functional requirements of each specialty bed pool: whereas large hospitals in the USA and Europe have a 77–78% national average bed occupancy rate, the UK average is 87%, a rate which results in larger patient turnaway (e.g., due to queues for a bed, cancelled operations, medical patients in surgical beds), missed waiting time targets and organisational chaos manifested by increased use of antidepressants by overworked staff, increased hospital errors, cross-infection and patient deaths. R Jones, author of a 2011 paper linking QT to the UK’s chronic bed shortage, argued that the Stafford Hospital scandal and the subsequent 400+ excess deaths were linked to the unrecognised role of excessive whole-hospital occupancy. Queuing Theory Queuing Theory The use of quantitative measures to balance supply and demand for services in order to minimize wait times. Queuing theory is most beneficial in retail establishments; for example, one may use it to determine how many cashiers to have at the grocery store. It is also called waiting line theory. AcronymsSeecutie
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