Abstract
The model of a fully available group of servers with a Poisson flow of primary calls and the possibility of losses before and after occupying a free server is considered. Additionally, a call can leave the system because of the aging of transmitted information. After each loss, there is some probability that a customer repeats the call. Such models are seen in the modeling of various telecommunication systems such as emergency information services, call and contact centers, access nodes, etc., functioning in overloading situations. The stationary behavior of the system is described by the infinite-state Markov process. It is shown that stationary characteristics of the model can be calculated with the help of an auxiliary model of the same class but without call repetitions due to losses occurring before and after the occupation of a free server and the aging of transmitted information. The performance measurements of the auxiliary model are calculated by solving a system of state equations using a recursive algorithm based on the concept of the truncation of the used state space. This approach allows significant savings of computer resources to be made by ignoring highly unlikely states in the process of calculation. The error caused by truncation is estimated. The presented numerical examples illustrate the use of the model for the elimination of the negative effects of emergency information service overload based on the filtering of the input flow of calls.
Highlights
Queueing models that take into account customer behavior after being refused service provide a powerful tool for performance evaluation and the planning of many resourcesharing systems functioning in overload conditions
The value of I is the mean number of busy servers, the value of J is the mean number of repeating customers, the value of πc is the ratio of lost calls, the value of τ is the ratio of repeated attempts in the total flow of incoming calls, and the value of M is the mean number of retrials per one primary call can be found from expressions v
The first approach is time-consuming; the last is numerically unstable for real values of input parameters, because of the necessity to convert a large matrix. We overcome these difficulties by showing that values of P(i) and J(i) can be calculated with the help of the corresponding characteristics of the basic model after a suitable choice of values of its input parameters. We prove it by constructing the number of auxiliary models using another probabilistic interpretation of input parameters of the generalized model or by algebraic transformations of the system of state equations
Summary
Queueing models that take into account customer behavior after being refused service provide a powerful tool for performance evaluation and the planning of many resourcesharing systems functioning in overload conditions. By taking into account customer behavior, we can model the input flow in a way that is close to the real processes happening in telecommunication systems that is considered in the overload situation This aspect creates problems with the theoretical analysis of such models, because time intervals between successive call arrivals are dependent random variables. The first approach is time-consuming, and the last is numerically unstable for real values of input parameters because of the necessity to convert a large matrix Such a situation seriously complicates the use of models with retrials as a component of planning tools, because it requires first numerical stability of the calculation algorithm, and second the possibility to estimate performance measures for any choice of input parameters for a reasonable time. This method of performance measurement estimation allows significant savings of computer resources to be made by ignoring highly unlikely states in the process of calculation
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