Queue-length distribution of a batch service queue with random capacity and batch size dependent service: $M/{G^{Y}_{r}}/1$

Abstract

This paper considers a single-server batch-service queue with random service capacity of the server and service time depends on the size of the batch. Customers arrive according to Poisson process and service times of the batches are generally distributed. We obtain explicit closed-form expression for the steady-state queue-length distribution at departure epoch of a batch based on roots of the associated characteristic equation of the probability generating function. Moreover, we also discuss the case when the characteristic equation has non-zero multiple roots. The queue-length distribution at random epoch is obtained using the classical principle based on ‘rate in = rate out’ approach. Finally, variety of numerical results are presented for a number of service time distributions including gamma distribution.

Appendix When D(z)=0, the characteristic equation of P +(z), has repeated zeros inside the complex unit disk.

Let us assume that the characteristic equation D(z)=0 has some multiple roots inside the closed complex unit disk. Already, we know that D(z)=0 has total B roots inside the unit disk. Let us call the multiple roots as α ₁,α ₂,…,α _f with multiplicity r ₁,r ₂,…,r _f so that $m={\sum }_{i=1}^{f}r_{i}$. We call the other distinct roots as α _m+1,α _m+2,…,α _B in |z|≤1 with α _B =1. Analyticity of P ⁺(z) in $\{ z\in \mathbb {C}:~|z|\leq 1\}$ implies that

$$\begin{array}{@{}rcl@{}} U^{(i-1)}(\alpha_{1})&=&0, \quad i=1,2,\ldots,r_{1},\\ U^{(i-1)}(\alpha_{2})&=&0, \quad i=1,2,\ldots,r_{2},\\ \vdots\\ U^{(i-1)}(\alpha_{f})&=&0, \quad i=1,2,\ldots,r_{f},\\ U(\alpha_{i})&=&0, \quad i=m+1,m+2,\ldots,B-1,\\ U^{(1)}(1)&=&D^{(1)}(1), \end{array} $$

where f ⁽ⁱ⁾(ζ) is the i ^th derivative of f(z) at z=ζ. This gives total B linearly independent simultaneous equations in B unknowns. Solving these we obtain $p^{+}_{n}$’s (0≤n≤B−1).