Busy Period Analysis of Multiserver Queueing Model with Simultaneous Service

This paper presents a mathematical analysis of a multiserver queueing model in which each arriving task requires the simultaneous use of multiple servers. The model reflects realistic scenarios in modern computing environments where parallel task execution is essential. Tasks arrive according to a Poisson process and require a randomly selected number of servers and an exponentially distributed service time. A key contribution of this study is the formulation of a recursive structure for analyzing the system’s busy period—the duration until the system becomes idle. Using combinatorial event conditions, the recursive equations are construct that describe the dynamics of task completion and waiting behavior. The mathematical expectation of the busy period is derived by incorporating transition probabilities and system parameters. This model extends existing results in multiserver queueing systems by explicitly accounting for simultaneous service requirements. The proposed analysis offers insights into system performance metrics and lays the groundwork for future computational and simulation-based studies.