We investigate an M/G/1 queueing-inventory (QI) system where service operations consume inventory units, arrivals follow a Poisson process with replenishment which are both random in size and subject to exponentially distributed lead times. Unlike the conventional QI models, customer demands are not assumed to be a fixed quantity. Instead, they are modeled using a geometric batch-size distribution, which accounts for the variability in the number of units or services requested by each customer. Customers may deplete multiple inventory units per service, and replenishment is triggered once inventory falls to or below a predefined reorder level \(r~(>0)\) . When inventory level becomes \(\le r\) after serving the random customer demand an variable ordering quantity \(o_j~(1\le j \le Q=M-r)\) , where M is the maximum inventory level the server may hold at time. To analyze the system, we construct an embedded Markov chain at post-departure epochs and apply the roots’ method to obtain the joint stationary distribution of number of customers in the systems and the corresponding inventory levels. Explicit vector-generating functions are obtained from which joint stationary probabilities, random epoch probabilities, and performance measures are calculated. A reward-based optimization model is incorporated to study the effect of pricing and demand sensitivity on the expected profit. Numerical results under light and heavy-tailed service-time distributions validate the approach and highlight the influence of service-time variability and replenishment uncertainty on profitability. The study extends existing classical QI models and provides insights for supply chain, retail, and service operations where inventory-dependent service is essential.
