A Queueing System with Batch Renewal Input and Negative Arrivals

Abstract

This paper studies an infinite buffer single server queueing model with exponentially distributed service times and negative arrivals. The ordinary (positive) customers arrive in batches of random size according to renewal arrival process, and join the queue/server for service. The negative arrivals are characterized by two independent Poisson arrival processes, a negative customer which removes the positive customer undergoing service, if any, and a disaster which makes the system empty by simultaneously removing all the positive customers present in the system. Using the supplementary variable technique and difference equation method we obtain explicit formulae for the steady-state distribution of the number of positive customers in the system at pre-arrival and arbitrary epochs. Moreover, we discuss the results of some special models with or without negative arrivals along with their stability conditions. The results obtained throughout the analysis are computationally tractable as illustrated by few numerical examples. Furthermore, we discuss the impact of the negative arrivals on the performance of the system by means of some graphical representations.

Appendix

Theorem 1

The c.e. $A^*( \delta + (\mu +\eta ) (1-z)) \sum _{i=1}^{b} g_i z^{b-i} - z^b = 0$ have exactly b roots inside the unit circle |z| = 1 subject to the condition δ > 0.

Proof

Let us assume f ₁(z) = −z ^b and $f_2(z)=A^*( \delta + (\mu +\eta ) (1-z)) \sum _{i=1}^{b} g_i z^{b-i}= K(z)$. Since $A^*( \delta + (\mu +\eta ) (1-z)) \sum _{i=1}^{b} g_i z^{b-i}$ is an analytic function, it can be written in the form $K(z)= \sum _{i=1}^{\infty }k_iz^i$ such that k _i ≥ 0 for all i. Consider the circle |z| = 1 − 𝜖 where 𝜖 > 0 and is a sufficiently small quantity. Now

$$\displaystyle \begin{aligned} \begin{array}{rcl} |f_1(z)|&\displaystyle =&\displaystyle |z^b|=(1-\epsilon)^b=1-b\epsilon +o(\epsilon) \\ |f_2(z)|&\displaystyle =&\displaystyle |K(z)||\sum_{i=1}^{b} g_i z^{b-i}|\leq K(|z|)\sum_{i=1}^{b} g_i|z|{}^{b-i} =K(1-\epsilon)\sum_{i=1}^bg_i(1-\epsilon)^{b-i} \\ &\displaystyle =&\displaystyle A^*( \delta ) - \epsilon \{2A^*( \delta ) (b-\overline{g}) - (\mu+\eta) A^{*(1)}( \delta ) \} + o(\epsilon) \\ &\displaystyle &\displaystyle < 1- b\epsilon +o(\epsilon) \end{array} \end{aligned} $$

under the sufficient condition δ > 0. Thus from Rouch$\acute {e}$’s theorem we have exactly the same number of zeroes in f ₁(z) and f ₁(z) + f ₂(z) inside the unit circle, and hence the theorem. □