A revisiting to queueing theory for mobile instant messaging with keep-alive mechanism in cellular networks

When we talk about the queueing theory in cellular networks, M/M/1 always comes first, which supposes the inter-arrival time and service time are both exponentially distributed. However, with the prosperity of Mobile Instant Messaging (MIM), many researchers found that the inter-arrival times of users' MIM follow lognormal distribution and there are also many keep-alive(KA) messages produced by the keep-alive mechanism accompanied with the users' messages. Based on this discovery, we rethink the queueing model that people might apply to analyze the MIM traffic. Firstly, we focus on a queueing system with lognormally distributed inter-arrival time and exponentially distributed service time, and find that the time a customer spends for waiting in queue and the length of the queue are shorter and smaller than the situation that the inter-arrival time is exponential one as proposed by 3GPP. Secondly, we study the influence of the keep-alive mechanism on queueing process. After that, we carry out simulations for this model, and the numerical results show that there exists an optimal KA duration, which determines the minimal average waiting time.