Calculation Of Characteristics Of Queuing Systems Using The Erlang Method And Conservation Laws

Abstract

Any human activity (economic, entrepreneurial, commercial) is associated with the performance of many operations at the stages of movement of raw materials, goods, products from the supply sector to the sphere of production and consumption. Such operations are transportation, storage, processing, sales, etc. These types of activities are characterized by the massive receipt of products, goods, money, and clients at random times; their sequential servicing is carried out by corresponding operations, the execution time of which, as a rule, is also random. All this leads to inequalities in work, creates idle time, underload and overload in operations. Therefore, the tasks arise of analyzing existing options for performing a certain set of operations, identifying bottlenecks and reserves for developing and making management decisions to improve the operating efficiency of any organization. Such problems are successfully solved using queuing theory. The basis for making management decisions are parameters that characterize different aspects of the operation of both the system as a whole and its individual elements. To describe individual parts of the system, the distribution of random variables and their numerical properties are traditionally used. The main characteristics of the action and state of the QS are the average number of requests in the queue or system, the average waiting time for service and others, as well as the value of some probabilities (the probability of service denial, the probability that there are at least a certain number of requirements in the system, the probability that the system is free from maintenance, etc.). The article describes some methods and techniques for studying Markov systems (the processes of which have no history), which turn out to be applicable to more general systems. A Markov queuing system with a restriction on the largest number of requirements in the system is considered. The probabilities of state transitions for closed and open-loop systems are described and calculated using Erlang methods and using the laws of conservation of system probabilities and the Laplace transform. Calculations of queue safety under stationary operating modes of queuing systems are demonstrated.