Analysis of Open Queueing Networks with Batch Services

On the Optimal Input Rate in Queues with Batch Service

“…The aim of this section is twofold. On one hand, it briefly recalls the main results for the M/M b /1 queue (see Section 2.1), focusing on the analysis method proposed in [26] and on the general expression of the sojourn time (more details can be found in [29] and the references therein). On the other hand, in Section 2.2, the characteristic equation is deeply investigated, highlighting the dependence of the asymptotic equivalent death rate M on the queue parameters; apart from the theoretical relevance of these results, some of the expressions will be used in Section 3 to derive the main contribution of the paper.…”

“…Product form state probabilities of the underlying Markov chain can be derived, as originally proposed by the authors (see [26] for details), by introducing an equivalent birth-death process, with constant birth rate λ and state-dependent death rates µ(n), determined by the following system of equations:…”

“…As it is shown in [26], for any consistent set of queue parameters, the characteristic equation has only one root satisfying the stability condition, which will be denoted by M in the following (in analogy with the notation already used in [29]). Some interesting mathematical properties of M will be further discussed in Section 2.2, where new material is presented, while this subsection just summarizes previous results and can be skipped by readers familiar with [29].…”

“…In [11], using the probability generating function method, the authors derived the closed-form expression for a steady-state queue length distribution, the expected queue length, and the expected waiting time of a customer. In [26], a multiplicative form of the stationary distribution for the M/M b /1 queue (i.e., with a fixed batch size b) was obtained on the basis of an equivalent birth-death process with (adequately chosen) state-dependent service rates, which can be derived by solving a polynomial equation of degree b + 1.…”

“…The rest of the paper is organized as follows. Section 2 deals with the analysis method based on the equivalent birth-death process we proposed in [26], which permitted us to find an explicit formula for the sojourn time. Then, the main contributions are reported in Section 3, which includes the general theoretical results on the existence of an optimal arrival rate λ opt as well as the first order approximation λ appr and the corresponding values of the minimum sojourn time as a function of the batch size.…”