A Fuzzy M/M/1 Queueing Model for Single-Charger Electric Vehicle Charging Stations: An α-Cut Approach

Abstract

The growing deployment of electric vehicle (EV) charging infrastructure has renewed interest in queueing-theoretic models of charging-station congestion, yet the arrival and service rates that drive these models are rarely known with certainty in practice. This paper develops a fuzzy M/M/1 queueing model for a single-charger EV station in which the arrival rate and the charging (service) rate are represented as triangular fuzzy numbers rather than fixed constants. The α-cut method is used as the principal computational tool: each classical M/M/1 performance measure is shown to be monotone in the arrival and service rates, so that its fuzzy counterpart’s α-cut is obtained in closed form by evaluating the classical expression at the corresponding extreme combination of the arrival- and service-rate α-cuts. Using this approach, we derive the α-cut representations of the fuzzy traffic intensity, the fuzzy expected number of vehicles in the system and in the queue, and the fuzzy waiting times in the system and in the queue, and we prove a fuzzy stability condition and a fuzzy form of Little’s Law that holds as an exact interval identity at every membership level, without additional independence assumptions. Each theorem is shown to reduce to the corresponding classical M/M/1 formula when the fuzzy parameters collapse to crisp numbers. A numerical illustration with λ = (3, 4, 5) vehicles/hour and µ = (6, 8, 10) vehicles/hour is computed at eleven α-levels and cross-validated against the crisp limit, together with a sensitivity analysis across four illustrative demand and service scenarios. The results show that the width of the fuzzy performance measures narrows monotonically as α → 1 and widens sharply as the traffic intensity approaches saturation, indicating that a single-charger station is particularly exposed to worst-case parameter combinations under the assumed uncertainty. The model and its validated α-cut formulas provide station operators with an interval-based, rather than single-point, description of expected congestion, which is directly useful for capacity and reliability planning at single-charger EV facilities.