The method of calculating queuing system with impatient customers

The paper deals with queuing systems with impatient customers. The known methods for calculating these systems refers to either the purely Markov model MIMIn-M (in addition to the Kendall notation the hyphen indicates the type of a patience distribution law) or such methods have very restricted conditions, namely, the single-channel service and great loading, the exponential distribution service or patience. We propose a method for calculating the multi-channel QS with the second order hyperexponential distributions both service time and patience M^In-^. The method relies on the method of fictitious phases. The calculation consists of the following stages: • approximation by phase type distributions; • construction of the diagram of transitions; • forming the matrixes of transition intensities in accordance with the diagram; • deriving the balance equations and their solutions and calculating the QS stationary characteristics. The second order hyperexponential distribution (H2) and three initial moments of the origin distribution have been used. Then we have combined the proposed Roubos & Jouini diagram for MIMIn-H queue with the well known diagram of system without impatient customers MIH In. The balance equations were solved by the iterative Takahashi & Takami's method. The probability of impatience queue is defined as Pimp = ^ V V Cm + (k - j)Y2 К, j, k=1 j=0 where X is the intensity of the incoming flow; y! and y are the intensities for the exponential phase of the H -patience; %k j is the probability that the queue length is k, of which j are the first type of patience. It is shown the influence of the variation coefficients of service and patience. Also, the dependence of the probability of removing for impatience queue on the load coefficient is investigated. We concluded using Markov models for the calculation of systems, in which the service and patience distributions are markedly different from the exponential ones, gives incorrect results. The effect of fragmentation performance and its influence on the probability of impatience is made known. The results can be applied in the design of call centers, emergency services, systems of obsolescent information processing, etc.

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