Mathematical model for load balancing on digital platforms based on queueing theory and resource allocation optimization

Load balancing in digital platforms remains a fundamental problem in ensuring quality of service while minimizing operational expenditures. Conventional heuristic strategies — such as round-robin or fixed-capacity provisioning — often result in inefficient resource utilization and excessive user delays. To overcome these limitations, this paper presents a rigorous mathematical model that integrates queueing theory with discrete optimization to determine the optimal number of servers in an M/M/c system operating in a stationary regime Analytical expressions for key performance indicators — including traffic intensity, probability of system idle, and mean waiting time in the queue — are derived from classical results in stochastic modeling. The core contribution lies in the formulation of an economically consistent cost function that accounts for the total waiting burden across all concurrent requests, defined as This correction ensures that delay penalties scale with the actual number of waiting users, addressing a critical limitation of prior approaches. The optimal number of servers is obtained by solving a discrete minimization problem without simulation. Numerical experiments with realistic cloud-computing parameters ( requests/h, USD/h, USD/h) demonstrate that the model reduces total operational costs by over 98 % compared to minimally stable configurations and by 38 % compared to conservative overprovisioning. The framework is directly applicable to API gateways, web server pools, and homogeneous microservice frontends. Current limitations include the assumption of stationary Poisson arrivals and exponential service times. Future work will address non-stationary loads, server heterogeneity, migration overhead, and integration with real-time forecasting. The results contribute to the theory of optimal resource allocation and offer a practical, transparent tool for cost-aware infrastructure planning in modern digital platforms.