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Queues of customers ‘arriving’ for service of one kind or another arise in many different fields of activity. Businesses of all types, government, industry, telephone exchanges, and airports, large and small all have queuing problems. Many of these congestion situations could benefit from OR analysis, which employs to this purpose a variety of queuing models, referred t as queuing systems or simply queues.

A queuing system involves a number of servers (or serving facilities) which we will also call service channels (in deference to the source field’ of the theory telephone communication system). The serving channels can be communications links; work stations, check-out counters, retailers, elevators, buses, to mention but a few. According to the number of servers, queuing systems can be of single and multi-channel type.

Customers arriving at a queuing system at random intervals of time are serviced generally for random times too. When a service is completed, the customer leaves the servicing facility rendering it empty and ready and get next arrival. The random nature of arrival and service times may be a cause of congestion at the input to the systemat some periods when the incoming customers either queue up for service or leave the system unserved; in other periods the system’ might not be completely busy because of the lack of customers, or even be idle altogether.

A queuing system operation is a random process with discrete states and continuous time. The state changes jump-wise at the instant some events occurs: an arrival occurs, a service is completed, or a customer unable to wait any longer leaves the queue.

The subject matter of queuing theory is to build mathematical models, which relate the specified operating conditions for the system (number of channels, their servicing mechanism, distribution of arrivals) to the concerned characteristics of value-measures of effectiveness describing the ability of the system to handle the incoming demands. Depending on the circumstances and the objective of the study, such measures may be : the expected (mean) number of arrivals served per unit time,. the expected number of busy channels, the expected number of customers in the queue and the mean waiting time for service, the probability that the Dumber in queue is above some limit, and so on. We do not single out purposely among intended for the given operating conditions, those intended for decision variables, since they may be either of these characteristics, for example, the number of channels, their capacity, service mechanism, etc. The most important part of a study is model establishment (primal problems) while its optimisation (inverse problem) is indeed depending on which parameters are selected to work with or to change. We are not going to consider optimization of .queuing models in this text with the exception made only for the simplest queuing situations.

The mathematical analysis of a queuing system simplifies considerably when the process concerned is Markovian. Markov process may be defined as, A random process is referred to as Markov, if for any moment of time, its probability characteristics in the future ‘depend only on its state at time and are to independent of when and how this state was acquired. ‘ As we already know a sufficient condition for this is that all the process changing system’s states (arrival intervals, service intervals) are Poisson. If this property does not hold, the mathematical description of the process complicates substantially and acquires an explicit analytical form only in seldom cases. However, the simplest mathematics of Markov queues may prove of value for approximate handling even of those queuing problems whose arrivals are distributed not in a Poisson process. In many situations a reasonable decision on queuing system organization suffices with an approximate model.

All of the queuing systems have certain common basic characteristics. They are (a) input process (arrival pattern) which may be specified by the source of arrivals, type of arrivals and the inter-arrival times, (by-service mechanism which is the duration and mode of service and may be characterized by the service-time distribution, capacity of the system, and service availability, and (c) queue discipline which includes all other factors regarding the rules of conduct of the queue.

We start illustrating the classification breakdown with a loss and delay system. In a purely loss system, customers arriving when all the servers are busy are denied service and are lost to the system. Examples of the loss system may be met in telephony: an incoming call arrived at an instant when all the channels are busy cannot be placed and leaves the exchange unserved. In a delay system an arrival incoming when all the channels are busy does not leave the system but joins the queue and waits (if there is enough waiting room) until a server is free. These latter situations more often occur in applications and are of great importance, which can be readily inferred from the name of the theory.

According to the type of the source-supplying customers to the system, the models are divided into those of a [mite population size, when the customers are only few, and the infinite-population systems. The length of the queue is subject to further limitations imposed by allowable waiting time or handling of impatient customers which are liable to be lost to the system.

The queue discipline, that is the rule followed by the server in taking the customers in service, may be according to such self-explanatory principles as “first-come, first-served”, “last-come, first-served”, or chain “random selection for serve”. In some situations priority disciplines need be introduced to allow for realistic queues with high priority arrivals. To illustrate, in extreme cases the server may stop the service of a customer of lower priority in order to deal with a customer of high priority ; this is called pre-emptive priority. For example a gantry crane working on a container ship may stop the unloading halfway and shift to another load to unload perishable goods of a later arrived ship. The situation when a service of a low priority customer stalled prior to the arrival of a high priority customer is completed and the high priority customer receives only a better position in the queue is called non-pre-emptive priority. This situation can be exemplified by an airplane which enters a queue ‘of a few other aircraft circling around an airport and asks a permission for an emergency landing; the ground control issues the permission on the condition that it lands next to the airplane being on a runway at the moment.

Turning over to the service mechanism, we may find systems whose servicing channels are placed in parallel or ill series. When in series, a customer leaving a previous server enters a queue for the next channel in the sequence. For example, a work piece being through the operation with one robot on a conveyor stacked to wait when the next robot in the process is free to handle it. These operation stages of a series-channel queuing system are called phases. The arrival pattern may and may not correlate with the other aspects of the system. Accordingly, the system can be loosely divided into “Open” and “Close”. In an open system, the distribution of arrivals does not depend on the status of the system say on how many channel are busy. To contrast, in a close system, in does For example, if a single operator tends a few Similar machines each of which has a chance of stopping i.e., arriving for serve, at random, then the arrival rate of stopping depends on how many mechanics have been already adjusted and put on the yet served.

An optimisation of a queuing system may be attempted from either of two stand points, the first in favour of “queuers” or owners of the queue” the second to favour the “queuers”, i.e., the customers. The first, stand makes a point of the efficiency of the system and would tend to load all the channels as high as possible, i.e., to cut down ideal times. The customers on the contrary would like to cut down waiting time in a queue. Therefore, any optimisation of congestion necessitates a “system approach” with the intrinsic complex evaluation and assessment of all consequences for each possible decision. The need for optimality over conflicting requirements may be illustrated with the viewpoint of the customer wishing to increase the number of channels, which, however, would increase the total servicing cost. The development of a reasonable model may help solving the optimisation problem by choosing the number of channels which account for all pros and cons. This is the reason why we do not suggest a single measure of effectiveness for all queuing problems, formulating them instead as multiple objective problems.

All the mentioned forms of queues (and many others for which we give no room here) are being studied queuing theory where there is a huge literature. The discussion, though, almost nowhere tunes an appropriate methodological level: the derivations are often too complicated, and deduced not in the very best way.