Abstract: The paper considers the model of a heterogeneous queuing system $$HK(t)/M/\infty$$ and limit evolutions in the averaging and diffusion approximation scheme. The input process of the system under study is modelled as a mixture of independent heterogeneous Poisson processes, which reflects the behaviour of modern cloud serverless platforms, where tasks arrive simultaneously from different event sources (HTTP gateways, object stores, message queues, event buses). Based on the apparatus of semi-Markovian random evolutions, the weak convergence of the queue length process $$L(t)$$ in two asymptotic schemes: averaging (law of large numbers) and diffusion approximation (central limit theorem), with the Ornstein–Uhlenbeck limit process, is proved. As a result, simple analytical estimates of the queue length $$L(t)$$ and the probability of exceeding the maximum length $$P\left(\sup_{t\in[0,T]}L(t)<L_{\mathrm{MAX}}\right)$$ are constructed. Also, a necessary and sufficient condition for the queue's finiteness is expressed in terms of the parameters of the input and task-processing processes. The results obtained enable us to develop analytical methods for parameter estimation and predictive scaling for complex serverless systems in cloud environments, without the need to train models or neural networks on large amounts of data.

Keywords: queue, heterogeneous queuing system, Poisson process, semi-Markov process, queue length estimation, model, modelling, cloud computing, serverless

Pages: 476-488