M/M/c Queueing Model with Variable Matching Rate

In recent years, with the development of the service industry, the queuing problem of improving the effectiveness of resource allocation to reduce the waiting time of customers has become more prominent. Existing researches mainly focus on the fixed matching rate model, and lack the in-depth exploration of the variable matching rate in real service scenarios. This paper constructs an M/M/c model of variable matching rate, considering the impact of various factors on the customer arrival rate and service rate in practical service contexts. Innovatively, we propose a variable matching rate formula applicable when the number of customers exceeds the number of service stations, addressing the gaps in the current research. By constructing a two-dimensional Markov quasi-birth-and-death process and generator matrix, it establishes an infinite-dimensional steady-state probability equation to calculate system performance metrics. Finally, a reasonable range of matching rates is determined by simulation, and system metrics are compared between fixed and variable matching rate M/M/c models.