Discrete-time queue with batch renewal input and random serving capacity rule: $GI^X/ Geo^Y/1$

Abstract

In this paper, we provide a complete analysis of a discrete-time infinite buffer queue in which customers arrive in batches of random size such that the inter-arrival times are arbitrarily distributed. The customers are served in batches by a single server according to the random serving capacity rule, and the service times are geometrically distributed. We model the system via the supplementary variable technique and further use the displacement operator method to solve the non-homogeneous difference equation. The analysis done using these methods results in an explicit expression for the steady-state queue-length distribution at pre-arrival and arbitrary epochs simultaneously, in terms of roots of the underlying characteristic equation. Our approach enables one to estimate the asymptotic distribution at a pre-arrival epoch by a unique largest root of the characteristic equation lying inside the unit circle. With the help of few numerical results, we demonstrate that the methodology developed throughout the work is computationally tractable and is suitable for light-tailed inter-arrival distributions and can also be extended to heavy-tailed inter-arrival distributions. The model considered in this paper generalizes the previous work done in the literature in many ways.

Appendix

Theorem 2

The equation $A(\overline{\mu }+ \mu Y(s))\sum _{i=1}^{b} g_i s^{b-i}-s^b =0$ has exactly b roots inside the unit circle $|s|=1$.

Proof

Let us suppose that $g(s)= A(\overline{\mu }+ \mu Y(s)) \sum _{i=1}^{b} g_i s^{b-i}$ and $f(s)=-s^b$. Let $K(s)=A(\overline{\mu }+ \mu Y(s))$, which can be expressed in the form $K(s)=\sum _{i=0}^{\infty }k_is^i,$ where the coefficients are such that $k_i \ge 0~~ \forall ~ i \ge 0$. Consider the circle $|s|=1- \delta$ where $\delta >0$ and is sufficiently small.

$$\begin{aligned} \text {Here}~~|f(s)|= & {} (1- \delta )^b= 1- b \delta + o(\delta )~~\text {and}\\ |g(s)|= & {} \big |K(s)\big |\big |\sum \limits _{i=1}^{b} g_i s^{b-i}\big | \le K(|s|)\sum _{i=1}^bg_i|s|^{b-i} \\= & {} K(1-\delta )\sum _{i=1}^bg_i(1-\delta )^{b-i}\\= & {} 1-b\delta +(\bar{g}-a \mu \bar{y})\delta + o(\delta ). \end{aligned}$$

Since $\rho <1$ and $\delta$ is a very small quantity, we have $|g(s)|<|f(s)|$ on the circle $|s|=1- \delta$. Thus, from Rouché’s theorem, we can say that f(s) and $f(s)+g(s)$ have exactly b zeroes inside the unit circle. From the proof, one can also observe that the condition $\bar{g} <a \mu \bar{y}$ is necessary as well as sufficient for the steady-state solution of the model (see Abolnikov and Dukhovny [1]). $\square$

Theorem 3

The roots of the equation $1-Y(s)=0$ lie outside or on the unit circle $|s|=1$.

Proof

As before, we assume $f(s)=1$, $g(s)= -Y(s)$ and the circle $|s|=1-\delta$ where $\delta >0$ and is sufficiently small. Now $|f(s)|=1$ and $|g(s)| \le Y(|s|)= 1- \overline{y}\delta +o(\delta )< |f(s)|$. Since f(s) has no zeroes inside $|s|=1$, using Rouché’s theorem we can say that $f(s)+g(s)$ has no zeroes inside the unit circle. Or in other words, all the zeroes of $f(s)+g(s)$ lie outside or on the unit circle. Hence, the theorem is proved. $\square$