Cost Optimization in a GI / M /2/ N Queue with Heterogeneous Servers, Working Vacations, and Impatient Customers via the Bat Algorithm

Abdelhak Guendouzi and Salim Bouzebda ()
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Abdelhak Guendouzi: Institute of Science, University Center of Aflou, B. P. 603, Aflou 03001, Algeria
Salim Bouzebda: Université de Technologie de Compiègne, LMAC (Laboratory of Applied Mathematics of Compiègne), CS 60 319, 60 203 Compiègne, France

Mathematics, 2025, vol. 13, issue 21, 1-34

Abstract: This paper analyzes a finite-capacity G I / M / 2 / N queue with two heterogeneous servers operating under a multiple working-vacation policy, Bernoulli feedback, and customer impatience. Using the supplementary-variable technique in tandem with a tailored recursive scheme, we derive the stationary distributions of the system size as observed at pre-arrival instants and at arbitrary epochs. From these, we obtain explicit expressions for key performance metrics, including blocking probability, average reneging rate, mean queue length, mean sojourn time, throughput, and server utilizations. We then embed these metrics in an economic cost function and determine service-rate settings that minimize the total expected cost via the Bat Algorithm. Numerical experiments implemented in R validate the analysis and quantify the managerial impact of the vacation, feedback, and impatience parameters through sensitivity studies. The framework accommodates general renewal arrivals ( G I ), thereby extending classical ( M / M / 2 / N ) results to more realistic input processes while preserving computational tractability. Beyond methodological interest, the results yield actionable design guidance: (i) they separate Palm and time-stationary viewpoints cleanly under non-Poisson input, (ii) they retain heterogeneity throughout all formulas, and (iii) they provide a cost–optimization pipeline that can be deployed with routine numerical effort. Methodologically, we (i) characterize the generator of the augmented piecewise–deterministic Markov process and prove the existence/uniqueness of the stationary law on the finite state space, (ii) derive an explicit Palm–time conversion formula valid for non-Poisson input, (iii) show that the boundary-value recursion for the Laplace–Stieltjes transforms runs in linear time O ( N ) and is numerically stable, and (iv) provide influence-function (IPA) sensitivities of performance metrics with respect to ( μ 1 , μ 2 , ν , α , ϕ , β ) .

Keywords: GI / M /2/ N queue; heterogeneous servers; multiple working vacations; impatient customers; Bernoulli feedback; Bat algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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