The Time-Dependent Mean and Variance of the Non-Stationary Markovian Infinite Server System

Problem statement: In many queuing situations the average arrival and service rates vary over time. In those situations a transient solution for the state probabilities and mean and variance must be obtained. Approach: The mean and the variance of a particular infinite server model will be obtained using the state differential-difference equations and the factorial moment generating function. The average arrival and service rates will be taken to be dependent on time. The individual customer interarrival times and service time are assumed to be exponentially distributed. This is known as the Markovian system. Results: The mean and variance of the system will be established as solutions to two sequential linear ordinary differential equations. A comparison is also made to a previously known result for the corresponding system with a finite number of servers. Conclusion: Simple closed-form equations for the mean and variance of the system are presented.