Asymptotic analysis of retrial queue with N types of outgoing calls under low rate of retrials condition

In this paper, we consider Markovian retrial queue with two-way communication and multiple types of outgoing calls, which could be used as a mathematical model of a call center operator. Incoming calls arrive at system according to a Poisson process with rate λ. Service times of incoming calls follow the exponential distribution with rate μ1. Upon arrival, an incoming call either occupies the server if it is idle or joins an orbit if the server is busy. Incoming calls stay in orbit for exponentially distributed time with rate σ. From the orbit, an incoming call retries to occupy the server and behaves the same as a fresh incoming call. On the other hand, the server makes outgoing calls after some exponentially distributed idle time. We assume that there are N types of outgoing calls whole durations follow N distinct distributions. We consider a random process of the number of incoming calls at the system. The aim of the research is to derive an asymptotic stationary characteristic function of this process under the low rate of retrials condition and to find the parameters of the stationary distribution of this process. To use the asymptotic analysis method we have obtained the Kolmogorov equation system for probability distribution of a 2-dimentional random process of the number of incoming calls in the system and the state of the server. We have also converted the Kolmogorov equation system for probabilities to the Kolmogorov equation system for the partial characteristic functions. We derived the explicit expression for the characteristic function of the number of incoming calls in the system and discovered that it is difficult to apply this result. We then extend the study to use the asymptotic analysis method under the low rate of retrials limit condition to research the model. The first order asymptotic only defines the distribution of probabilities of the server state rk and the mean value κ1 of the random process of the number of incoming calls in the system. The second order asymptotic shows that the asymptotic probability distribution of the number of incoming calls in the system is Gaussian with the mean κ ∣∕σ and variance κ2∕σ. Based on the obtained asymptotic, we have built the Gaussian approximation of the probability distribution of the number of incoming calls in the system. Our numerical results have revealed that the accuracy of Gaussian approximation increases while decreasing σ.

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